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+{"text":"\\section{introduction}\n\nExcitations in quantum spin liquids can be viewed as strongly\ninteracting bosonic quasiparticles. This circumstance enables\nexperimental studies of the physics of Bose liquids in prototypical\nquantum magnetic\nmaterials\\cite{GiamarchiPRB99,Nikuni00,GiamarchiNaturePhys}. Such\nexperiments are often possible under conditions that can not be\nrealized in more conventional models, such as\n$^4$He\\cite{London,Reppy} and ultracold trapped\nions\\cite{MHanderson,Wynar}. One recent topic of interest is teh\nbehavior of bosonic quasiparticles in the presence of disorder.\nExotic new phases such as the Random Singlet state\\cite{Ma79}, Bose\nand Mott glasses\\cite{Fisher89} have been predicted for systems with\nquenched disorder. In real prototype materials one usually tries to\ncreate such disorder by chemical doping\\cite{ManakaPRL,OosawaPRB}.\nIn the present work we demonstrate an alternative approach: a random\nmagnetic field created by disordered (paramagnetic) ions. We show\nthat such a random field acting on a simple dimer-based quantum spin\nliquid dramatically alters the excitation spectrum.\n\nLet us consider the effect of different types of magnetic fields on\nan isolated $S=1\/2$ dimer, as shown in Fig.~\\ref{fig5}. In a uniform\nfield the excited triplet is split into three levels. Eventually, at\nhigh field, $\\lvert S=1, S^z=0\\rangle $ will cross the singlet\nground state. In the presence of inter-dimer interactions, BEC of\nmagnon will occur. If local fields applied to each dimer spin are\nantiparallel to each other (referred to as ``staggered field''\nhereafter) the triplet is split into a singlet and a doublet.  The\nsinglet ground state becomes mixed with $\\lvert S=1, S^z=0\\rangle $\nand the total spin is no longer a good quantum number. Even in an\ninfinitesimal staggered field the ground state becomes ploarized.\nNow, if the field direction is spatially randomized, each dimer will\nexperience both a staggered and uniform component. The corresponding\nenergy levels can be calculated numerically. The resulting density\nof state (DOS) for excitations in a set of $N$ dimers is plotted in\nthe right panel in Fig.~\\ref{fig5}. The DOS  lower and higher\nboundaries of the DOS spread coincide with the levels of $\\lvert\nS_{z}=1\\rangle $ and $\\lvert S_{z}=-1\\rangle $ in the uniform field.\n\nThe quantum ferrimagnet \\CuFeGeO\\ \\cite{Masuda03} is a rare\npotential realization  of this random field effect. The compound\nincludes $S$ = 1\/2 Cu$^{2+}$ dimers coupled to classical Fe$^{3+}$\nchains \\cite{Masuda05}. At low temperature the cooperative ordered\nstate with classical spin and quantum spin is stabilized by a weak\ninter-subsystem coupling. In the adiabatic approximation, the\nquantum spins are effectively under the internal field from the much\nslower fluctuating classical spins. In this compound, the staggered\nnature of the exchange field is due to the magnetic structure. The\nstaggered magnetization curves of dimers in \\CuFeGeO\n~\\cite{Masuda04a} were experimentally obtained by measuring the\ntemperature dependence of sublattice moments in neutron diffraction.\nAt high temperature, in the paramagnetic phase, the classical spins\nare thermally disordered and the effective field on the quantum\nspins is randomly oriented. Then the system can be considered as the\nensemble of $N$ dimers in a random quasi-static field. As shown in\nFig.~\\ref{fig5} the effect of this random field is to broaden the\ndimer excitations at $T > T_{\\rm N}$.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8.5cm]{fig1.eps}\n\\end{center}\n\\caption{ Schematic diagrams of triplet excitations in $S=1\/2$\ndimers in different types of locally applied magnetic field. }\n\\label{fig5}\n\\end{figure}\n\nIn the previous inelastic neutron scattering study it was shown that\nthe energy scales of excitations in the Fe chains and Cu dimers are\nwell separated~\\cite{Masuda05,Masuda07a}. The lower energy\nexcitations up to 10 meV are Fe-based spin waves. Preliminary powder\nexperiments~\\cite{Masuda05} and comparative studies in an\nisostructural compound \\CuScGeO ~\\cite{Masuda06} associated the\ndispersionless excitations at 24 meV  with Cu-dimers. However the\neffect of a staggered and\/or fluctuating field could not been\nidentified in powders samples. In the present paper we study the\ndimer excitations by single crystal inelastic neutron scattering. By\nadopting a high resolution setup, we identify the split peaks due to\nthe staggered exchange field. Furthermore, we observe a drastic\nbroadening of the peak profile at $T > T_{\\rm N}$ that can be\nascribed to randomly oriented field from thermally fluctuated Fe\nmoments.\n\nHigh quality single crystals were grown by floating zone method. The\ncrystal (monoclinic $P2_1\/m$) were found to be twinned, so that both\nmicroscopic domains share $a^*$ - $b^*$ plane. To avoid\ncomplications due to twinning, we restrict the measurements to the\n$a^*$ - $b^*$ plane. In the setups Ia and Ib PG (002) were used for\nboth monochromator and analyzer. The Soller collimations were 48' -\n60' - 60' - 120' and open - 80' - 80' - open for Ia and Ib,\nrespectively. In setup II, to achieve high energy resolution, PG\n(004) for monochromator and PG (002) for analyzer with 30' - 20' -\n40' - 120' were used. The setups Ia and II were performed on HB1\nspectrometer in HFIR, ORNL. The setup Ib was performed on TAS1\nspectrometer in JRR-3M, JAEA. In all setups final energy of the\nneutron was fixed at $E_f$ = 14.7meV and PG filter was installed\nafter the sample to eliminate higher order contamination. A closed\ncycle He refrigerator was used to achieve low temperatures.\n\nIn a series of energy scans in a wide range of $(h~k~0)$ space shown\nin Fig.\\ref{fig1}(a) two dispersionless peaks are readily\nidentified: a pronounced one at $\\hbar \\omega \\sim 24$ meV and a\nweaker feature at $\\hbar \\omega \\sim 31$ meV. The experiments were\nperformed in setups Ia and Ib. The former is consistent with the\nCu-centered magnetic excitation in previous\nstudies~\\cite{Masuda05,Masuda06}. Constant energy scan at $\\hbar\n\\omega = 24$ meV and its temperature dependence are shown in\nFig.\\ref{fig1}(b). The observed sinusoidal intensity modulation is\ncharacteristics of dimer excitations and is observed in a wide\ntemperature range. In fig.~\\ref{fig1}(c) the temperature dependence\nof the peak intensity is shown. The intensity at ${\\bm q} =\n(0~2.5~0)$ was measured at each temperature and then subtracted as\nbackground. The decrease of the intensity at high temperature is\ncommon behavior for magnetic excitations in local spin clusters. The\nsmaller peak at $\\hbar \\omega \\sim 31$ meV was identified as a\nFe-centered excitation, as will be discussed below.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8.7cm]{fig2.eps}\n\\end{center}\n\\caption{ Inelastic neutron scattering using experimental setup Ia\nand Ib. (a) Typical energy scans at $(h~k~0)$. Dispersionless\nexcitations are observed at $\\hbar \\omega = 24$ and 31 meV. Two\npeaks are separately fit by Gaussians (dotted curves). (b) $h$ scans\nat $\\hbar \\omega =$ 24 meV at various temperatures. Sinusoidal\nintensity modulations are fitted to the dimer structure factor\ncalculated for zero field, plus a constant background (solid\ncurves). (c) Temperature dependence of the peak intensity at ${\\bm\nq} = (h~2.5~0)$ and $\\hbar \\omega =$ 24 meV. } \\label{fig1}\n\\end{figure}\n\nTo obtain a more detailed profile, we performed an energy scan using\nsetup II at $T$ = 2.0 K. As shown in Fig.~\\ref{fig3} it is revealed\nthat the primary peak at $\\hbar \\omega = 24$ meV actually has a\nshoulder structure. The main peak is located at 23.5 meV, and a\nsmaller bump is centered around 25.0 meV. This splitting is\nattributed to the staggered exchange field from the adjacent Fe\nmoments. The main peak corresponds to the excitation doublet and the\nsmall one to the singlet.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=7.5cm]{fig3.eps}\n\\end{center}\n\\caption{ Energy scan collected using the high resolution setup II\nat $T$ = 2.0 K. The shoulder structure is reproduced by the doublet\n(shaded with white background) and singlet (shaded with gray\nbackground) dimer excitations, split by a staggered exchange field.\n} \\label{fig3}\n\\end{figure}\n\nEnergy scans collected at several temperatures are shown\nFig.~\\ref{fig4}(a). The small peak at $\\hbar \\omega \\sim 31$ meV in\nFig.~\\ref{fig1} is temperature independent and has been  subtracted\nfrom the data. Well-defined peaks are observed at all temperatures.\nWhile at low temperature the peak profile is sharp and the width is\nwithin resolution limit, at $T \\gtrsim T_N$ the peak becomes\ndrastically broadened. This qualitative behavior is consistent with\nthe effect a random exchange field should have on the dimer\nexcitation triplet. The data were analyzed using Gaussian fits. The\nestimated peak positions, widths, and the integrated intensities are\nplotted as functions of temperature in Figs.~\\ref{fig4}(b)-(d). With\nincreasing temperature the peak energy decreases at $T \\sim T_{\\rm\nN}$,  and stays constant beyond. The peak width drastically\nincreases at $T \\sim T_{\\rm N}$, but also remains constant at higher\ntemperature. The integrated intensity decreases by 10 $\\sim $ 20\\%.\nIt is noted that in the previous powder experiment the peak cannot\nbe distinguished at $T \\ge 41$ K \\cite{Masuda05}. This is because\nthe powder integration in wide ${\\bm q}$ space collects phonon\nexcitations and accidental suprious peaks, masking magnetic\nexcitations at higher temperatures.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8.7cm]{fig4.eps}\n\\end{center}\n\\caption{ (a) Temperature dependence of the peak profile at ${\\bm q}\n= (2~3~0)$. Experimental resolution is indicated by the gray area.\nSmall peaks due to Fe-centered excitation at 31 meV are separately\nfitted and subtracted. Profiles at $T \\ge 54$ K are reproduced by\ndimers in a randomly oriented field model (solid curves). The\ntemperature dependence of peak positions (b), widths (c) and\nintegrated intensities (d), as estimated from Gaussin fits. }\n\\label{fig4}\n\\end{figure}\n\n\n\nFor $T < T_{\\rm N}$ we will consider the following effective\nHamiltonian:\n\\begin{equation}\nH = J_{\\rm Cu}{\\bm S_1}\\cdot {\\bm S_2} +g\\mu _{\\rm B}{\\bm S_1}\\cdot {\\bm h_1} + g\\mu _{\\rm B}{\\bm S_2}\\cdot {\\bm h_2},\n\\label{dimerHamiltonian}\n\\end{equation}\nwhere ${\\bm h_1}=(0~0~h)$ and ${\\bm h_2}=(0~0~-h)$. Here  the\n$z$-axis is chosen along the ordered Cu moment. The ground state\nenergy $E_{\\rm G}$ decreases with the field, $E_{\\rm G} =-J_{\\rm\nCu}\/4-\\sqrt {(2g\\mu _{\\rm B} h)^2+J_{\\rm Cu}^2}\/2$, and the\nexcitation triplet splits into a singlet and doublet. The\ncorresponding energy levels are giben by\n\\begin{eqnarray}\n\\Delta_{\\rm s}&=&\\sqrt {(2g\\mu _{\\rm B} h)^2+J_{\\rm Cu}^2}\n\\label{singlet} \\\\\n\\Delta_{\\rm d}&=&J_{\\rm Cu}\/2+\\sqrt{(2g\\mu _{\\rm B} h)^2+\nJ_{\\rm Cu}^2}\/2, \\label{doublet}\n\\end{eqnarray}\nand plotted vs. $h$ in Fig.~\\ref{fig5}. For $g\\mu _{\\rm B}  h\\ll\n\\Delta$,  the neutron cross section is approximately given by:\n\\begin{eqnarray}\n\\frac{d^2\\sigma}{d\\Omega dE}\\sim N(\\gamma r_0)^2\\frac{k'}{k}\n\\sin ^2(\\bm {q.d})\\left( f(q)\\right) ^2P(T) \\nonumber \\\\\n\\times \\{A(h)(1+\\cos ^2 \\theta)\\delta (\\hbar \\omega-\\Delta_{\\rm d})+\nB(h)\\sin ^2 \\theta\n\\delta(\\hbar \\omega -\\Delta _{\\rm s})\\}. \\label{crosssection}\n\\end{eqnarray}\nThe doublet and singlet terms correspond to transverse and\nlongitudinal spin fluctuation, respectively. $A(h)$ and $B(h)$ are\n$h$ dependent parameters with $A(h) \\le 1, B(h) \\le 1,$ and $A(0) =\nB(0) = 1$. Since the staggered field stabilizes the polarized spin\nconfiguration and suppresses longitudinal fluctuation, $B(h)$\ndecrease with $h$. Meanwhile $A(h)$ is almost constant in the low\nfield. $P(T)$ is a temperature factor, $P(T)=1\/\\{ 1+2\\exp (-\\beta\n\\Delta _{\\rm d}) + \\exp (-\\beta \\Delta _{\\rm s}\\}$. $\\bm q$ is the\nscattering vector, $\\bm d$ is the spin separation in each dimer, and\n$\\theta$ is the angle between $\\bm q$ and the moment of Cu. We used\n${\\bm m}_{\\rm Cu}$ = (-0.227, 0.035, -0.301) $\\mu _{\\rm\nB}$\\cite{Masuda04a} to calculate $\\theta$. Two types of domains,\nnamely antiferromagntic and crystallographic ones due to twinning,\nare considered.\n\nThe peak profile in Fig.~\\ref{fig3} is reasonably well reproduced by\nthe cross section convoluted by experimental resolution function\nwith $\\Delta _{\\rm s}$ = 25.0 meV and $\\Delta _{\\rm d}$ = 23.5 meV.\nFrom eqs.(\\ref{singlet}) and (\\ref{doublet}), $J_{\\rm Cu}=22.0$ meV\nand $h$ = 51 T are obtained. Let us check the consistency of $h$\nwith the previous study~\\cite{Masuda04a}. From the staggered\nmagnetization curve by neutron diffraction $J_{\\rm Cu-Fe}\/J_{\\rm\nCu}=0.105$ was obtained. Here $J_{\\rm Cu-Fe}$ is the interaction\nbetween Cu and Fe spins. Using the molecular field relation\n$h=m_{\\rm Fe}J_{\\rm Cu-Fe}\/(g\\mu _{\\rm B})^2$ and previously\nobtained parameters, $h \\sim 40$ T is estimated. Thus statically\nestimated value is consistent with that obtained in the present\ndynamic measurement.\n\nThe energy splitting between the singlet and doublet states is about\n1.5 meV. This value is small compared energy resolution in the\ntypical experimental setup. In setup Ia and Ib at $T < T_{\\rm N}$,\ntherefore, staggered field effect is smeared and two terms in\neq.(\\ref{crosssection}) are integrated. Then the cross section is\napproximately equivalent to that at $h$ = 0. Indeed, the constant\nenergy scan at $T$ = 3.3 K in Fig.~\\ref{fig1}(b) is reasonably\nfitted by dimers cross section in zero field shown by the thick\ncurve.\n\n\nAt $T>T_{\\rm N}$ effective field on the Cu dimers is randomly\noriented. We consider an ensemble of $N$ dimers in random field. The\nrandomly oriented field ${\\bm h}$ is assumed to have a constant\nmagnitude in eq.(\\ref{dimerHamiltonian}). The resulting DOS of the\nexcited states is then calculated numerically. The neutron cross\nsection is assumed to be approximately proportional to the DOS,\n\\begin{equation}\n\\frac{d^2\\sigma}{d\\Omega dE}=(\\gamma r_0)^2\\frac{k'}{k}\\sin ^2(\\bm {q.d})f(q)^2P_{\\rm rand}(T) D(\\hbar \\omega)  \\label{crosssection2}\n\\end{equation}\nwith $\\int D(\\epsilon )d\\epsilon = 3N$ and $P_{\\rm rand}(T) =\nN\/(N+\\int D(\\epsilon )e^{-\\frac{\\epsilon }{k_{\\rm B}T}}d\\epsilon )$.\nThe data collected at $T \\ge 54$ K are well reproduced by this cross\nsection convoluted by experimental resolution function, as indicated\nby solid curves in Fig.~\\ref{fig4}(a). The obtained fit parameters\nare $J_{\\rm Cu}$ = 22.3(4) meV and $h$ = 41.(8) T. The values are\nreasonably consistent with those obtained at $T \\le T_{\\rm N}$. The\n${\\bm q}$ dependence of the cross section is the same as for zero\nfield, and is given by dimer structure factor $\\sin ^2 (\\bm {q.d})$.\nIndeed, the ${\\bm q}$ scans at 80 K and 300 K in Fig.~\\ref{fig1}(b)\nare reproduced by this model. The temperature dependence in\nFig.~\\ref{fig1}(c) is well accounted for by the temperature factor\n$P_{\\rm rand}(T)$.\n\nWe shall now discuss the small decrease of the intensity at $T <\nT_{\\rm N}$ in Fig.~\\ref{fig4}(d). At $T > T_{\\rm N}$ dimer spins are\nfluctuated equally in all directions and the dynamical spin\ncorrelation is fully detected by neutron. In the ordered state the\npolarized magnetic ordering suppresses the longitudinal fluctuation\nof Cu spins. To estimate the reduction of the longitudinal\nexcitation we calculate $B(h = 51 {\\rm T})=0.77$. The reduction of\n$B(h)$ is about 20\\% that is consistent with the experiment. This\nmeans that 51 T is rather modest compared with the intradimer\ninteraction $J_{\\rm Cu}$ = 22 meV. If the effective field was large\nand the moment were fully polarized, the suppression would be more\ndrastic. Such a situation is in fact realized in Haldane spin chains\ncoupled to rare earth moment in Pr$_2$BaNiO$_5$ with fully saturated\nNi$^{2+}$ moment at $T < T_{\\rm N}$~\\cite{Zheludev96c}. The\nHaldane-gap mode lost half of its intensity at $T < T_{\\rm N}$ and\nit was ascribed to the total suppression of longitudinal mode.\n\nFinally we will mention the temperature independent small peak at\n$\\hbar \\omega \\sim$ 31 meV in Fig.~\\ref{fig1}(a). If the Fe $S=5\/2$\nchains were perfectly isolated from the Cu subsystem, the Fe\nexcitation spectrum  would be dominated by one-magnon excitation at\n$\\hbar \\omega \\le 5J_{\\rm Fe}$. However, a recent theory predicts\nthat the introduction of Cu dimer enhances the multi-magnon\nexcitation of Fe spins at $\\hbar \\omega$ = $10J_{\\rm Fe}$, $15J_{\\rm\nFe}$, $20J_{\\rm Fe}$ and $25J_{\\rm Fe}$. According to the Bond\noperator method~\\cite{Matsumoto04,Sachdev90} the excitation at\n$\\hbar \\omega$ = $20J_{\\rm Fe}$ is the particularly\nenhanced~\\cite{Matsumoto09}. Since $J_{\\rm Fe}=1.6$\nmeV~\\cite{Masuda07a}, the observed small peak at $\\hbar \\omega$ = 31\nmeV could be ascribed to the Fe centered longitudinal excitation.\nFurther details will be published somewhere else.\n\nTo conclude, we have experimentally investigated the dynamics of $S\n= 1\/2$ dimers in staggered and random fields in \\CuFeGeO . The\nstaggered field is realized at $T < T_{\\rm N}$ and produces a\nsplitting of the excitation triplet.  At $T > T_{\\rm N}$ a random\nexchange field produces a drastic broadening of these modes. In teh\nfuture, polarized neutron experiments may be useful to separate the\nlongitudinal and transverse excitations. Recently\nCu$_2$CdB$_2$O$_6$~\\cite{Hase05} and\nCu$_3$Mo$_2$O$_9$~\\cite{Hamasaki08} identified as new realizationsof\nthe coupled dimers and chains models. Particularly in the latter\ncompound, the dimer energy is close to that of the  chains, and more\ncomplex physics is expected.\n\nProf. M. Matsumoto is greatly appreciated for fruitful discussion.\nThis work was partly supported by Yamada Science Foundation, Asahi\nglass foundation, and Grant-in-Aid for Scientific Research (No.s\n19740215 and 19052004) of Ministry of Education, Culture, Sports,\nScience and Technology of Japan.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $G$ be a finite group of automorphisms of an abelian variety $A\/\\bbc$.  It is a classical result \\cite[II. \\S1]{La} that $A$ itself cannot contain a rational curve.\nFor $|G| > 1$, there may or may not be rational curves on $A\/G$. For general abelian varieties, $\\Aut(A)= \\pm1$, and Pirola proved \\cite{P} that for $A$ sufficiently general and of dimension at least three, $A\/\\pm1$ has no rational curves. At the other extreme, regarding $A=E^n$ as the set of $n+1$-tuples of points on the elliptic curve $E$ which sum to $0$, $A\/S_{n+1}$ can be interpreted as the set of effective divisors linearly equivalent to $(n+1)[0]$ and, as such, is just $\\bbp^n$. More generally, Looijenga has shown  \\cite{L} that the quotient of $E^n$ by the Weyl group of a root system of rank $n$ is a weighted projective space.\n\nRational curves on $A\/G$ over a field $K$ are potentially a source of rational points over $G$-extensions of $K$.  For instance, the method \\cite{Larsen} for finding pairs $a,b\\in \\bbq^\\times$ such that the quadratic twists\n$E_a$, $E_b$, and $E_{ab}$ all have positive rank amounts to finding a rational curve on $E^3\/(\\bbz\/2\\bbz)^2$.   Likewise, the theorem of Looijenga cited above gives for each elliptic curve $E$ over a number field $K$ and for each Weyl group $W$, a source of $W$-extensions $L_i$ of $K$ such that the representation of $W$ on each $E(L_i)\\otimes\\bbq$ contains the reflection representation.  On the other hand, the result of Pirola cited above dims the hope of using geometric methods to show that every abelian variety over a number field $K$ gains rank over infinitely many quadratic extensions of $K$.  Thus, it is desirable from the viewpoint of arithmetic to understand when $A\/G$ can be expected to have a rational curve over a given field $K$, and to begin with, one would like to know when $A\/G$ has a rational curve over $\\bbc$.\n\nAny automorphism $g$ of an abelian variety $A$ defines an invertible linear transformation (also denoted $g$) on $\\Lie(A)$.\nIf $g$ is of finite order, there exists a unique sequence of rationals $0\\le x_1\\le x_2\\le \\cdots\\le x_n < 1$ such that\nthe eigenvalues of $g$ are $e(x_1),\\ldots,e(x_n)$, where $e(x) := e^{2\\pi i x}$.  We say $g$ is of \\emph{type} $(x_1,\\ldots,x_n)$. Following\nKoll\\'ar and Larsen \\cite{KL}, we write $\\age(g) = x_1+\\cdots+x_n$.\nFor instance, $\\age(g) = 1\/2$ for every reflection $g$.  The main result of \\cite{KL} asserts that\n$A\/G$ is uniruled if and only if $0<\\age(g)<1$ for some $g\\in G$.  In this paper, we prove that to find a single rational curve in $A\/G$, it sufices that $\\age(g) \\le 1$.\n\nSince we need only consider the case $\\age(g) = 1$, we first classify all types of weight $1$.  This requires a combinatorial analysis, which we carried out using a computer algebra system to minimize the risk of an oversight.\nThere are thirty-five cases (see Table 2 below), and our strategy for finding rational curves depends on case analysis.  Abelian surfaces play a special role, since here we can use known results on K3 surfaces.\nThe other key  idea is to find a non-singular projective curve $X$ on which $G$ acts with quotient\n$\\bbp^1$ and a $G$-equivariant map from $X$ to $A$, or, equivalently, a $G$-homomorphism from the Jacobian variety of $X$ to $A$.\n\nWe would like to thank Yuri Tschinkel and Alessio Corti for helpful comments on earlier versions of this paper.\n\n\\section{Classifying types}\n\nIf $A = V\/\\Lambda$, then the Hodge decomposition $\\Lambda\\otimes\\bbc \\cong V\\oplus \\bar V$ respects the action of $\\Aut(A)$.\nTherefore, if $g$ is of finite order with eigenvalues $e(x_1),\\ldots,e(x_n)$, then the multiset\n\\begin{equation}\n\\tag{$\\ast$}\\{e(x_1),\\ldots,e(x_n),e(-x_1),\\ldots,e(-x_n)\\}\n\\end{equation}\nis $\\Aut(\\bbc)$-stable.\nBy a \\emph{type}, we mean a multiset $\\{x_1,\\ldots,x_n\\}$ with $x_i\\in [0,1)$ such that the multiset ($\\ast$) is $\\Aut(\\bbc)$-stable.\nEquivalently, a type can be identified with a finitely supported function $f\\colon \\bbq\/\\bbz\\to \\bbn$ such that $f(x)+f(-x)$ depends only on the order of $x$ in $\\bbq\/\\bbz$.\nBy the \\emph{weight} of $\\{x_1,\\ldots,x_n\\}$, we mean the sum $x_1+\\cdots+x_n$, so that $\\age(g)$ is the weight of the type of $g$.\n\nA type is \\emph{reduced} if $0$ does not appear, and the reduced type of a given type is obtained by discarding all copies of $0$.\nThe \\emph{sum} of types is the union in the sense of multisets; at the level of associated functions on $\\bbq\/\\bbz$ it is the usual sum.\nA type which is not the sum of non-zero types is \\emph{primitive}.\nAll the elements of a primitive type appear with multiplicity one, and they all have the same denominator.\nEvery type can be realized (not necessarily uniquely) as a sum of primitive types; if the weight of the type is $1$, each of the primitive types has weight $\\le 1$,\nso our first task is to classify primitive types with weight $\\le 1$.\n\nA primitive type $X$ of denominator $n\\ge 2$ consists of fractions $a_i\/n$ where $0<a_i<n$, and $(a_i,n)=1$.\nMoveover, if $n\\ge 3$, for each positive integer $a<n$ prime to $n$, exactly one of $a\/n$ and $1-a\/n$ belongs to $X$.  If $a<b<n\/2$ and $1-a\/n\\in X$,\nthen the weight of $X$ exceeds $1$ since either $b\/n$ or $1-b\/n$ belongs to $X$.\nThus, if $0<a<n\/2$ and $1-a\/n\\in X$, then $a$ must be the largest integer in $(0,n\/2)$ prime to $n$.\n\n\\begin{lem}\\label{primineq} If $\\phi(n) > 24$, then  $$\\sum_{x\\in S_n} \\min(x,n-x) > 2n,$$ where $S_n$ is the set of  positive integers $< n$ and prime to $n$. Moreover, the largest integer $n$ such that $\\phi(n)\\le 24$ is $90$.\n\\end{lem}\n\n\\begin{proof} Note that\n$$\\min(x,n-x) > \\frac{x(n-x)}{n} = \\frac{n^2-x^2-(n-x)^2}{2n}.$$\nIn order to prove the first statement, we want to prove if $\\phi(n)>24$, then\n$$\\sum_{x\\in S_n} \\Big(n^2-x^2-(n-x)^2\\Big) > 4n^2,$$ or equivalently,\n$$\\phi(n)n^2  - 2\\sum_{x\\in S_n} x^2-4n^2>0.$$\nBy  M\\\"{o}bius inversion, one can prove that\n$$\\sum_{x\\in S_n} x^2 = \\frac{\\phi(n) n^2}{3} +(-1)^{d_n}\\frac{\\phi(f(n)) n}6,$$\nwhere $f(n)$ denotes the larges squarefree divisor of $n$ and $d_n $ is the number of distinct prime divisors of $n$.  Thus, if $\\phi(n) > 24$, then $\\phi(n)>24\\geq 12 \\frac{n}{n-1}$, so $(n-1)\\phi(n)-12n >0$ and since $\\phi(f(n))\\leq \\phi(n)$,\n\\begin{align*}\n\\phi(n)n^2 - 2\\sum_{x\\in S_n} x^2-4n^2 &\\geq \\Big(\\frac{\\phi(n)}{3}-4\\Big)n^2-\\frac{\\phi(n)n}{3} \\\\\n&=\\frac{n((n-1)\\phi(n)-12n)}{3}>0,\n\\end{align*}\nwhich is the desired inequality.\n\nFor the second statement, if $\\phi(n)\\le 24$ and $p$ is a prime factor of $n$, then $\\phi(p)=p-1\\leq\\phi(n)\\leq 24$. Hence $p\\le 23$.  Writing\n$$n=2^{n_2}3^{n_3}5^{n_5}7^{n_7}11^{n_{11}}13^{n_{13}}17^{n_{17}}19^{n_{19}}23^{n_{23}},$$\nwe have \n$$0\\leq n_2\\leq 5,0\\leq n_3\\leq 3,0\\leq n_5\\leq 2,$$\nand $0\\leq n_i\\leq 1$ for $7\\le i\\le 23$.\nCase analysis now shows $n\\leq 90$.\n\n\\end{proof}\n\n\\begin{prop}\\label{primlist} There are $28$  primitive types with weight $ \\leq 1$:\n\\end{prop}\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|}\n\\hline\n\\#& n & \\text{primitive types} &  \\text{weight}\\\\\\hline\n1&  2 & 1\/2 & 1\/2\\\\\\hline\n2&  3 & 1\/3 & 1\/3\\\\\n3&    & 2\/3 & 2\/3\\\\\\hline\n4&  4 & 1\/4 & 1\/4 \\\\\n5&    & 3\/4 & 3\/4\\\\\\hline\n6&  5 & 1\/5, 2\/5 & 3\/5 \\\\\n7&    & 1\/5, 3\/5 & 4\/5 \\\\\\hline\n8&  6 & 1\/6 &1\/6\\\\\n9&    & 5\/6 & 5\/6\\\\\\hline\n10&  7 & 1\/7, 2\/7, 3\/7 & 6\/7\\\\\n11&    & 1\/7, 2\/7, 4\/7 & 1\\\\\\hline\n12&  8 & 1\/8, 3\/8 & 1\/2 \\\\\n13&    & 1\/8, 5\/8 & 3\/4 \\\\\\hline\n14&  9 & 1\/9, 2\/9, 4\/9 & 7\/9 \\\\\n15&    & 1\/9, 2\/9, 5\/9 & 8\/9\\\\\\hline\n16&  10 & 1\/10, 3\/10 & 2\/5 \\\\\n17&    & 1\/10, 7\/10 & 4\/5 \\\\\\hline\n18&  12 & 1\/12, 5\/12 & 1\/2\\\\\n19&    & 1\/12, 7\/12 & 2\/3\\\\\\hline\n20&  14 & 1\/14, 3\/14, 5\/14 & 9\/14\\\\\n21&    & 1\/14, 3\/14, 9\/14 & 13\/14\\\\\\hline\n22&  15 & 1\/15, 2\/15, 4\/15, 7\/15 & 14\/15\\\\\n23&    &1\/15,2\/15, 4\/15, 8\/15 & 1\\\\\\hline\n24&  16 & 1\/16, 3\/16, 5\/16, 7\/16 & 1\\\\\\hline\n25&  18 & 1\/18, 5\/18, 7\/18 & 13\/18\\\\\n26&    & 1\/18, 5\/18, 11\/18 & 17\/18 \\\\\\hline\n27&  20 & 1\/20, 3\/20, 7\/20, 9\/20 & 1\\\\\\hline\n28&  24 & 1\/24, 5\/24, 7\/24, 11\/24 & 1\\\\\n  \\hline\n\\end{tabular}\n\\vskip 4pt\nTable 1\n\\end{center}\n\\begin{proof} For $n\\ge 3$, the weight of a primitive type of denominator $n$ is at least\n$$\\sum_{\\{x\\in S_n\\mid x<n\/2\\}}\\frac xn \\ge \\frac 1{2n}\\sum_{x\\in S_n} \\min(x,n-x).$$\nBy Lemma~\\ref{primineq}, it suffices to carry out an exhaustive search  up to $n = 90$.\n\\end{proof}\n\\begin{lem} There are $35$ types with $\\age ~~1$ of automorphisms given in Table 2 below.\n\\end{lem}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\#&  n & \\text{types} &  \\text{notes}\\\\\\hline\n1&  2 & 1\/2, 1\/2 & Prop.~\\ref{kummer} \\\\\\hline\n2&  3 & 1\/3, 1\/3, 1\/3 &  Th.~\\ref{small-factor}\\\\\n3&   & 1\/3, 2\/3 &  Prop.~\\ref{cube}\\\\\\hline\n4&  4 & 1\/4, 1\/4, 1\/4, 1\/4 &    Th.~\\ref{small-factor}\\\\\n5&   & 1\/4, 1\/4, 2\/4 & $g^2\\to\\#1$ \\\\\n6&   & 1\/4, 3\/4 & $g^2\\to\\#1$  \\\\\\hline\n7&  6 & 1\/6, 1\/6, 1\/6, 1\/6, 1\/6, 1\/6 &    Th.~\\ref{small-factor}\\\\\n8&  & 1\/6, 1\/6, 1\/6, 1\/6, 2\/6  & Th.~\\ref{small-factor} \\\\\n9&   & 1\/6, 1\/6, 1\/6, 3\/6  &  $g^2\\to\\#2$ \\\\\n10&  & 1\/6, 1\/6,  4\/6  &  $g^3\\to\\#1$ \\\\\n11&    & 1\/6, 5\/6  &  $g^3\\to\\#1$ \\\\\n12&   & 1\/6, 2\/6, 3\/6  &  $g^3\\to\\#1$  \\\\\n13&    & 1\/6, 1\/6,  2\/6, 2\/6  &  $g^3\\to\\#1$  \\\\\\hline\n14&   7 & 1\/7, 2\/7, 4\/7 & Cor.~\\ref{seven}\\\\\\hline\n15&   8 & 1\/8, 2\/8, 5\/8  &   $g^4\\to\\#1$\\\\\n16&   &  1\/8, 3\/8, 4\/8 & $g^4\\to\\#1$ \\\\\n17&   & 1\/8,  1\/8, 3\/8, 3\/8 & Th.~\\ref{small-factor} \\\\\n18&    & 1\/8, 2\/8, 2\/8, 3\/8 & $g^4\\to\\#1$  \\\\\\hline\n19&   10 &  1\/10, 2\/10, 3\/10, 4\/10 & $g^5\\to\\#1$ \\\\\\hline\n20&   12 & 4\/12, 2\/12, 1\/12, 5\/12& $g^6\\to\\#1$\\\\\n21&    & 4\/12, 3\/12, 3\/12, 2\/12 &$g^6\\to\\#1$\\\\\n22&    & 6\/12, 1\/12, 5\/12 &$g^6\\to\\#1$\\\\\n23&    & 3\/12, 3\/12, 1\/12, 5\/12 &$g^4\\to \\#3$\\\\\n24&    & 2\/12, 2\/12, 2\/12, 1\/12, 5\/12 & $g^6\\to\\#1$\\\\\n25&    & 1\/12, 1\/12, 5\/12, 5\/12 & Th.~\\ref{small-factor} \\\\\n26&    & 4\/12, 1\/12, 7\/12 & $g^6\\to\\#1$ \\\\\n27&    & 2\/12, 2\/12, 1\/12, 7\/12 & $g^6\\to\\#1$ \\\\\n28&    & 3\/12, 3\/12, 2\/12, 2\/12, 2\/12 & $g^6\\to\\#1$\\\\\\hline\n29&   15 & 1\/15, 2\/15, 4\/15, 8\/15 & Cor.~\\ref{fifteen}\\\\\\hline\n30&   16 & 1\/16, 3\/16, 5\/16, 7\/16 & Cor.~\\ref{mult-one}\\\\ \\hline\n31&   20 & 1\/20, 3\/20, 7\/20, 9\/20 & Cor.~\\ref{mult-one}\\\\\\hline\n32&   24 & 1\/24, 5\/24, 7\/24, 11\/24 & Cor.~\\ref{mult-one}\\\\\n33&    & 8\/24, 4\/24, 3\/24, 9\/24 & $g^{12}\\to\\#1$\\\\\n34&    & 3\/24, 9\/24, 2\/24, 10\/24 & $g^{12}\\to\\#1$\\\\\n35&    & 4\/24, 4\/24, 4\/24, 3\/24, 9\/24 & $g^{12}\\to\\#1$ \\\\\n  \\hline\n\\end{tabular}\n\\vskip 4pt\nTable 2\n\\end{center}\n\n\\begin{proof} Let $[a_i]$ be a formal variable representing the $i$th primitive type in Table $1$,  and let  $w_i$ denote the weight of the $i$th type.\nA monomial $\\prod a_i^{m_i}$ stands for a sum of primitive types in which the $i$th type appears $m_i$ times.\nThe g.c.d. of the denominators of the $w_i$  is $5040$. Let $y=x^{1\/5040}$, so $(1-[a_i]x^{w_i})^{-1}$ is a power series in $y$ for every $i$.\nBy \\textsc{Maple 13},   the coefficient of $y^{5040}$ in the product\n$$\\prod_{i=1}^{28} (1-[a_i]x^{w_i})^{-1}=\\prod_{i=1}^{28} (1-[a_i]y^{w_i\\cdot 5040})^{-1},$$\nis\n${{\\it a1}}^{2}+{{\\it a2}}^{3}+{\\it a2}\\,{\\it a3}+{\\it a4}\\,{\\it a5}+{\n\\it a8}\\,{\\it a9}+{\\it a4}\\,{\\it a13}+{\\it a6}\\,{\\it a16}+{{\\it a4}}^{\n2}{\\it a1}+{\\it a28}+{\\it a27}+{\\it a23}+{\\it a24}+{\\it a11}+{\\it a19}\n\\,{\\it a2}+{\\it a19}\\,{{\\it a8}}^{2}+{\\it a18}\\,{\\it a1}+{\\it a18}\\,{{\n\\it a4}}^{2}+{\\it a18}\\,{{\\it a8}}^{3}+{\\it a18}\\,{\\it a12}+{{\\it a18}\n}^{2}+{\\it a12}\\,{\\it a1}+{\\it a12}\\,{{\\it a4}}^{2}+{\\it a12}\\,{{\\it\na8}}^{3}+{{\\it a12}}^{2}+{{\\it a8}}^{2}{{\\it a2}}^{2}+{{\\it a8}}^{2}{\n\\it a3}+{{\\it a8}}^{3}{\\it a1}+{{\\it a8}}^{3}{{\\it a4}}^{2}+{{\\it a8}}\n^{4}{\\it a2}+{{\\it a8}}^{6}+{{\\it a4}}^{4}+{\\it a18}\\,{\\it a8}\\,{\\it\na2}+{\\it a12}\\,{\\it a8}\\,{\\it a2}+{\\it a8}\\,{\\it a1}\\,{\\it a2}+{\\it a8\n}\\,{\\it a2}\\,{{\\it a4}}^{2}$.\n\nEach monomial in this sum corresponds to an entry in Table 2.\n\\end{proof}\n\n\\section{Rational curves in $A\/\\langle g\\rangle$}\n\nIn this section we explain how to find rational curves on $A\/\\langle g\\rangle$ in each case in Table 2.\n\n\\begin{lem}\nIf $A\/\\langle g^n\\rangle$ has a rational curve for some positive integer $n$, then $A\/\\langle g\\rangle$ has a rational curve.\n\\end{lem}\n\n\\begin{proof}\nThe morphism $A\/\\langle g^n\\rangle\\to A\/\\langle g\\rangle$ is finite, so the image of a rational curve is again a rational curve.\n\\end{proof}\n\n\\begin{prop}\n\\label{reduce}\nLet $A$ be an abelian variety and $g$ an automorphism of finite order.  Suppose that for every abelian variety $B$ and finite-order automorphism $h\\in \\Aut(B)$\nwhose type is the reduced type of $g$, $B\/\\langle h\\rangle$ has a rational curve.  Then $A\/\\langle g\\rangle$ has a rational curve.\n\\end{prop}\n\n\\begin{proof}\nLet $B$ denote the image of $1-g$ acting on $A$.  Then $B$ is an abelian subvariety of $A$, and $g$ restricts to an automorphism $h$ of $B$\nwhose type is the reduced type of $g$.  As $B\/\\langle h\\rangle \\subset A\/\\langle g\\rangle$ has a rational curve, the same is true of $A\/\\langle g\\rangle$.\n\\end{proof}\n\n\nThe following proposition is well known.\n\n\\begin{prop}\\label{kummer}\nIf $A$ is an abelian surface, then $A\/\\pm 1$ has a rational curve.\n\\end{prop}\n\n\\begin{proof}\nResolving the $16$ singularities of $A\/\\pm 1$, we obtain a K3 surface with Picard number $\\ge 16\\ge 5$.  By work of Bogomolov and Tschinkel \\cite{BT},  any such surface is either elliptic or has infinite automorphism\ngroup and in either case has infinitely many rational curves, all but finitely many of which lie on $A\/\\pm 1$.\n\\end{proof}\n\nNote that Proposition~\\ref{kummer} covers not only case \\#1 in Table 2 but twenty other cases as well, namely those (indicated in the ``notes'' column) for which the reduced type of some power of $g$ is $(1\/2,1\/2)$.\n\n\\begin{prop}\\label{cube} Let $A\\cong V\/\\Lambda$ be an abelian surface with an automorphism $g$ of type $(1\/3,2\/3)$.  Then $A\/\\langle g\\rangle$ contains a rational curve.\n\\end{prop}\n\n\\begin{proof}\nLet $G=\\langle g\\rangle$ and $X=A\/G$. Regarding  $1-g$  as an isogeny of $A$, the number of fixed points of $g$ is\n$$\\deg(1-g)=\\#\\ker(1-g)=\\det(1-g|\\Lambda )=3^2=9.$$\nThese are singularities of  type $A_2$, since under $(x,y)\\mapsto(\\omega x, \\omega^2 y)$ where $\\omega$ is a cube root of unity, the invariants are generated by $X=x^3, Y=y^3, Z=xy$, and so\n    $$\\bbc[[x,y]]^G = \\bbc[[X,Y,Z]]\/(XY-Z^3).$$\nThis is isomorphic to $\\bbc[[x, y, z]]\/(x^2 + y^2 + z^3)$,\n    which has a Du Val singularity of type $A_2$ (see \\cite[Ch.4,~4.2]{R}).\n\nConsider the  minimal resolution $f: Y\\rightarrow X$, for which the $9$ exceptional divisors $Y_i$ each consists of two projective lines $D_{i,1}$ and $D_{i,2}$ intersecting at one point.\nThe canonical divisor of $Y$ is $K_Y=f^*K_X=0$.  Hence $Y$ is a K$3$-surface of Picard number $\\ge 18$, and again by \\cite{BT}, we deduce that $X$ has infinitely many rational curves..\n\\end{proof}\n\n\\begin{thm}\n\\label{disjoint}\nLet $B$ be an abelian variety and $h\\in \\Aut(B)$ an automorphism of finite order such that $h$ and $h^{-1}$ have disjoint types.\nLet $A$ be an abelian variety with a finite order automorphism $g$ whose type is contained in that of $h$.\nIf there is a rational curve on $B\/\\langle h\\rangle$ then there is a rational curve on $A\/\\langle g\\rangle$.\n\\end{thm}\n\n\\begin{proof}\nIt suffices to prove that there exists a surjective homomorphism $p\\colon B\\to A$ such that the diagram\n$$\\xymatrix{B\\ar[r]^p\\ar[d]_h&A\\ar[d]^g \\\\ B\\ar[r]^p&A}$$\ncommutes.  Writing $B = \\Lie(B)\/\\Lambda_B$ and $A = \\Lie(A)\/\\Lambda_A$, the goal is to find a surjective $\\bbc[t]$-linear map $\\phi\\colon \\Lie(B)\\to \\Lie(A)$\n(where $t$ acts as $g$ on $\\Lie(A)$ and as $h$ on $\\Lie(B)$)\nsuch that $\\phi(\\Lambda_B)\\subset \\Lambda_A$.  If $\\psi$ is a surjective $\\bbc[ t]$-linear map $\\Lie(B)\\to \\Lie(A)$\nsuch that $\\psi(\\Lambda_B\\otimes\\bbq) = \\Lambda_A\\otimes  \\bbq$, then we can define $\\phi := n\\psi$ for $n$ a sufficiently divisible positive integer.  It suffices, therefore, to find $\\psi$\nwith the desired properties.\n\nAs the type of $A$ is a subset of the type of $B$, there exists a surjective $\\bbq[t]$-linear map $T\\colon\\Lambda_B\\otimes\\bbq\\to \\Lambda_A\\otimes  \\bbq$.\nExtending scalars to $\\bbc$, $T\\otimes 1$ maps $\\Lambda_B\\otimes\\bbc = \\Lie(B) \\oplus \\overline{\\Lie(B)}$ to $\\Lambda_A\\otimes \\bbc = \\Lie(A) \\oplus \\overline{\\Lie(A)}$.\nThe type of $g$ acting on $\\overline{\\Lie(A)}$ is the same as the type of $g^{-1}$ acting on  $\\Lie(A)$ and therefore disjoint from the type of $g$ acting on $\\Lie(A)$, and the same is true for the type of $h$ acting on $\\Lie(B)$.\nAs $\\Lie(B)$ and $\\Lie(A)$ are direct sums of certain $t$-eigenspaces of $\\Lambda_B\\otimes\\bbc$ and $\\Lambda_A\\otimes\\bbc$ respectively and\nas the spectrum of $t$ acting on $\\Lie(A)$ is the intersection of the spectra of $t$ acting on $\\Lie(B)$ and on $\\Lambda_A\\otimes\\bbc$, it follows that\n$T\\otimes 1$ maps $\\Lie(B)$ to $\\Lie(A)$.  The restriction of $\\psi$ to $\\Lie(B)$ is therefore the desired map.\n\n\\end{proof}\n\n\\begin{cor}\n\\label{mult-one}\nLet $n\\ge 3$ be a positive integer, and let $m = \\lceil n\/2\\rceil - 1$.  If $A$ is an abelian variety and $g\\in \\Aut(A)$ is an automorphism of order $n$\nwhose type is contained in $\\{1\/n,2\/n,\\ldots,m\/n\\}$, then $A\/\\langle g\\rangle$ has a rational curve.\n\\end{cor}\n\n\\begin{proof}\nLet $X$ denote the non-singular projective hyperelliptic curve of genus $m$ which contains the affine curve $y^2 = x^n-1$.\nThe order-$n$ automorphism $h(x,y) = (e(1\/n)x,y)$ extends to an automorphism of $X$ and therefore defines an automorphism of $B := \\Jac(X)$.\nThe Lie algebra $\\Lie(B)$ can be identified with the space $H^0(X,\\Omega_X)$ of holomorphic differential forms on $X$, which has a basis\n$$\\Bigl\\{\\frac {dx}y, \\frac{x\\,dx}y,\\ldots,\\frac{x^{m-1}\\,dx}y \\Bigr\\}.$$\nTherefore, the type of $h$ acting on $B$ is $\\{1\/n,2\/n,\\ldots,m\/n\\}$, which is disjoint from the type of $h^{-1}$.\nOn the other hand, $B\/\\langle h\\rangle$ contains the rational curve $X\/\\langle h\\rangle$.\nThus, Theorem~\\ref{disjoint} applies.\n\\end{proof}\n\n\\begin{cor}\n\\label{seven}\nIf $A$ is an abelian $3$-fold and $g$ is an automorphism of $A$ of type $(1\/7,2\/7,4\/7)$, then $A\/\\langle g\\rangle$ has a rational curve.\n\\end{cor}\n\n\\begin{proof}\n\nLet $X$ denote the Klein quartic:\n$$X : x^3y+y^3z+z^3x=0$$\nand $B$ the Jacobian of $X$.\nThe self-map $h(x,y,z)=(\\zeta_7x, \\zeta_7^4y, \\zeta_7^2z)$ of $X$ belongs to the automorphism group\n$\\PSL_2(\\bbf_7)$ of $X$ which acts non-trivially on the Jacobian variety $B$ and therefore on $\\Lie(B) = H^0(X,\\Omega_X)$.\nConjugating $h$ by the cyclic permutations of $(x,y,z)$, we see that $h$ is conjugate to $h^2$ and $h^4$\nin $\\Aut(X)$, and therefore the type of $h$ is invariant under multiplication by $2$ (mod $1$).\nIt is therefore $(1\/7,2\/7,4\/7)$ or $(3\/7,5\/7,6\/7)$, and\nreplacing $h$ by $h^{-1}$ if necessary, we may assume that it is the former.\n\\end{proof}\n\nWe remark that $B\/\\langle h\\rangle$ in the proof of Corollary~\\ref{seven} has appeared\nin the literature; it is known to have  a Calabi-Yau resolution  \\cite[Example~6.3]{BT2}.\n\n\n\n\\begin{cor}\\label{fifteen}\nIf $A$ is an abelian variety and $g$ an automorphism such that the type of $A$ is contained in $(1\/15,2\/15,3\/15,4\/15,8\/15,9\/15)$,\nthen $A\/\\langle g\\rangle$ has a rational curve.\n\\end{cor}\n\n\\begin{proof}\nLet $X$ be the non-singular projective curve which has a (singular, affine) model $X'\\colon y^{15} = x^2(x-1)$.  This is singular only at $(0,0)$, and the inverse image of this singularity\nunder the normalization map $X\\setminus \\{P_\\infty\\}\\to X'$ is a single point $P_0\\in X$.\nThe automorphism $h\\colon (x,y)\\mapsto (x,e(1\/15)y)$ of the affine curve induces an automorphism of $X$ of order $15$.\nAs $15y^{14}dy = (3x^2-2x)dx$, any differential form $\\frac {x^m y^n  dy}{3x^2-2x}$, $m,n\\ge 0$, is holomorphic except possibly at $P_0$ and $P_\\infty$.\nOne checks that\n$$\\frac{dy}{3x-2}, \\frac{ydy}{3x-2}, \\frac{y^2dy}{3x-2}, \\frac{y^3dy}{3x-2},  \\frac{y^7dy}{3x^2-2x},   \\frac{y^8dy}{3x^2-2x}$$\nare all holomorphic, and their eigenvalues under $h$ are $e(1\/15)$, $e(2\/15)$, $e(3\/15)$, $e(4\/15)$, $e(8\/15)$, $e(9\/15)$ respectively.\nApplying the Riemann-Hurwitz theorem to the map $X\\to \\bbp^1$ given by $y$, we see that $X$ is of genus $6$, and therefore, that these differential\nforms form a basis of $\\Lie(\\Jac(B)) = H^0(X,\\Omega_X)$.\n\\end{proof}\n\n\\begin{thm}\n\\label{small-factor}\nIf $A$ is an abelian variety and $g$ is an automorphism of finite order such that $g$ and $g^{-1}$ have disjoint types and the type of $g$ is a sum\nof primitive types at least one of which has $weight$ less than $1$, then $A\/\\langle g\\rangle$ has a rational curve.\n\\end{thm}\n\n\\begin{proof}\nFor every primitive type, there exists an abelian variety $B_i$ with complex multiplication and an automorphism $h_i$ of $B_i$ with the given type.\nIndeed, the primitive types of denominator $n$ are in natural correspondence with CM-types on $\\bbq(\\zeta_n) = \\bbq(e(1\/n))$.  Any CM-type $\\Phi$ on $\\bbq(\\zeta_n)$, $n\\ge 3$\ndefines an embedding $\\bbq(\\zeta_n) \\to \\bbc^{\\phi(n)\/2}$.  The image of $\\bbz[\\zeta_n]$ defines a lattice $\\Lambda\\subset \\bbc^{\\phi(n)\/2}$, and the quotient $\\bbc^{\\phi(n)\/2}\/\\Lambda$\nis a complex torus with a natural action of $\\bbz\/n\\bbz$ of the type associated with $\\Phi$.  The quotient $\\bbc^{\\phi(n)\/2}\/\\Lambda$ admits a polarization \\cite[II~6~Theorem 4]{S}, so there\nexists a pair $(B_i,h_i)$ as claimed.  If $\\age(h_1) < 1$, then by \\cite{KL}, $B_1\/\\langle g_1\\rangle$ has a rational curve.\nIf $A = A_1\\times\\cdots\\times A_m$, and $g = (g_1,\\ldots,g_m)$ is a finite order automorphism of $A$ which stabilizes each factor, then $A_1\/\\langle g_1\\rangle \\subset A\/\\langle g\\rangle$,\nso $A\/\\langle g\\rangle$ has a rational curve.  The theorem now follows from Theorem~\\ref{disjoint}.\n\\end{proof}\n\n\\\n\nTo summarize, we have the following theorem:\n\n\\begin{thm}\\label{main2}\nLet $A$ be an abelian variety with a nontrivial automorphism $g$ of finite order such that $\\age(g)\\le 1$.  Then $A\/\\langle g\\rangle$ contains a rational curve.\n\\end{thm}\n\n\n\\begin{cor}\\label{hyper} Let $A$ be an abelian variety of dimension $n$ with a nontrivial automorphism $g$ of finite order. If $\\dim(\\ker(1-g))\\geq n-2$ (i.e. the codimension of the fixed subspace of $A$ under $g$ is less than or equal to $2$), then the quotient $A\/\\langle g\\rangle$ contains a uniruled hypersurface.\n\\end{cor}\n\n\\begin{proof} Since $\\dim(\\ker(1-g))\\geq n-2$, $B:=\\mathrm{im}(1-g)$ is an abelian variety of dimension $n-\\dim (\\ker (1-g))\\le 2$.\nLet $h$ denote the restriction of $g$ to $B$.  As $\\age(h)+\\age(h^{-1}) \\ge 2$, we may assume without loss of generality that $\\age(h)\\le 1$,\nso $B\/\\langle h\\rangle$ has a rational curve $Z$ by Theorem~\\ref{main2}.\n  Let $C$ denote the identity component of $\\ker (1-g)$, which is an abelian subvariety of dimension $\\dim \\ker(1-g)$ on which $g$ acts trivially.  The addition morphism $B\\times C\\to A$ is an isogeny and respects the action of  $\\langle g\\rangle$.  We therefore obtain a finite morphism from\n$$Z\\times C \\subset (B\/\\langle g\\rangle)\\times C \\cong (B\\times C)\/\\langle g\\rangle$$\nto $A\/\\langle g\\rangle$.  The image of an $n-1$-dimensional ruled variety under a finite morphism is a uniruled hypersurface.\n\\end{proof}\n\n\\begin{cor} Let $E$ be an elliptic curve. If $W$ is a Weyl group of simple roots of rank $n\\geq 3$ acting on $E^n$ and $W^+$ is an index $2$-subgroup of $W$, then the quotient $E^n\/W^+$ contains a rational curve.\n\\end{cor}\n\\begin{proof} $W$ is generated by reflections $s_j$ of simple roots. Since $W^+$ is an index $2$-subgroup of $W$, there exist  two reflections $s_1$ and $s_2$ such that $s_1s_2\\in W^+$. Then for each $i$, $\\ker(1-s_i)$ has codimension $1$ and their intersection has codimension $\\leq 2$. Since $\\ker(1-s_1s_2)$ contains the intersection of $\\ker(1-s_1)$ and $\\ker(1-s_2)$,  this follows from Corollary~\\ref{hyper}.\n\\end{proof}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe main aim of this work is to present a combined theory of the term structure of interest rates and of price impact, with applications to optimal execution. This objective entails the inclusion of a specific type of cross price impact that is specific to fixed income. \n\nTerm structure modeling with a view to derivatives valuation and hedging has been developed over several decades. For the purposes of this work and what we might call the classic theory we refer to the monographs by Bjork \\cite{bjork1997interest}, Brigo and Mercurio \\cite{brigo2007interest} and Filipovic \\cite{filipovic2009term}. After the crisis that started in 2007, the gap between two of the rates that were used as benchmarks for risk free rates, namely interbank and overnight rates, widened considerably, peaking in October 2008, following the defaults of several financial institutions in the space of one month \\cite{brigopallatorre}. This highlighted the fact that interbank rates could no longer be used as benchmarks for risk free rates and neither could they be used to derive zero-coupon curves that were not contaminated by credit and liquidity risk. This led to the necessity to model multiple interest rate curves, treating interbank rates as risky rates affected by credit and liquidity risk and adopting overnight based rates as new risk free rates. \n\nThis multiple interest rate curve academic literature was initiated by practitioners, see in particular the monograph by Henrard \\cite{henrard2014interest}. Substantial contributions were made later by numerous academics, where we refer to a monograph by Grbac and Runggaldier \\cite{grbac2015interest}, Crepey et al. \\cite{crepey2015levy}, Grbac et al. \\cite{grbac2015affine}, Cuchiero et al. \\cite{cuchiero2019affine,cuchiero2016general}, Nguyen and Seifried \\cite{nguyen2015multi} and finally to Bormetti et al. \\cite{bormetti2018}, for multiple curves in conjunction with valuation adjustments and credit risk.\n\nFurther recent developments include the presence of negative interest rates in many currencies, see for example the BIS report  \\cite{negativeratesbis}, and the ongoing project of eliminating current interbank rates like the London Interbank Offer Rate (LIBOR), replacing them with new types of risk free rates inspired by overnight rates. This puts the multiple-curve area in a state of uncertainty, while negative rates prompted the mainstream resurgence of Gaussian models that were previously justified only in very special economies exhibiting negative rates, such as for example Switzerland in the seventies. \n\nGiven the state of market uncertainty on benchmark interest rates, products and markets, we will not consider these recent developments in this work, except for allowing for negative rates in our formulation. We are interested in developing a combination of term structure modeling and price impact in the classic theory of interest rates. We are confident that multiple curves and further discussion of negative rates, if still present in the market after reforms and updated central bank policies, can be incorporated in further work after the classic theory has been developed.\n\nDespite the fact that the bond market size is considerably larger than the equity market size, relevance of liquidity risk in the context of bonds was pointed out in several papers. A report from the Federal Reserve Bank of New York from 2003 \\cite{chordia2005empirical}, for instance, showed that bond and stock markets have common factors driving their liquidity. A strong relationship between liquidity in the Treasury bond market and in the stock market was later highlighted also by Goyenko and Ukhov \\cite{goyenko2009stock}. Another significant contribution appears in a recent paper by Schneider and Lillo \\cite{schneider2019cross}, which studied cross price impact among sovereign bonds.\n\nThe financial crisis of 2008-2009 have also raised concerns about the inventories kept by intermediaries.\nRegulators and policy makers took advantage of two main regulatory changes (Reg NMS in the US and MiFID in Europe) and enforced more transparency on the transactions and hence on market participants positions, which pushed the trading processes toward electronic platforms \\cite{citeulike:12047995}.\nSimultaneously, consumers and producers of financial products asked for less complexity and more transparency.\n\nThis tremendous pressure on the business habits of the financial system, shifted it from a customized and high margins industry, in which intermediaries could keep large (and potentially risky) inventories, to a mass market industry where logistics have a central role. As a result, investment banks nowadays unwind their risks as fast as possible.\n In the context of small margins and high velocity of position changes, trading costs  are of paramount importance. A major factor of the trading costs is the price impact: the faster the trading rate, the more the buying or selling pressure will move the price in a detrimental way. \n \nAcademic efforts to quantify and reduce the transaction costs of large trades trace back to the seminal papers of Almgren and Chriss \\cite{OPTEXECAC00} and Bertsimas and Lo \\cite{BLA98}. In both models one large market participant (for instance an asset manager or a bank) would like to buy or sell a large amount of shares or contracts during a specified duration.\nThe cost minimization problem turned out to be quite involved, due to multiple constraints on the trading strategies.\nOn one hand, the price impact demands to trade slowly, or at least at a pace which takes into account the available liquidity (see \\cite{citeulike:13497373} and references therein).\nOn the other hand, traders have an incentive to trade rapidly, because they do not want to carry the risk of an adverse price move far away from their decision price. The importance of optimal trading in the industry generated a lot of variations for the initial mean-variance minimization of the trading costs (see \\cite{citeulike:12047995,cartea15book,olivier16book} for details). These type of problems are usually formulated as optimisation problems in the context of stochastic control where the agent tries to minimize the transaction costs which result by the price impact and to reduce the risk associated with holding the assets for too long (see e.g. \\cite{citeulike:5282787,GLFT, citeulike:13675369}). \n\nIn this spirit, we will follow the term structure theory as developed by Bjork \\cite{bjork1997interest}, modifying it to allow for the inclusion of price impact.  We will start from the simplest possible dynamics, namely one factor short rate models, and extend it later to instantaneous forward rate models. We will introduce price impact formulated on zero-coupon bonds, since they are a possible choice of building blocks for the term structure. We will later connect this with impact on coupon bearing bonds that are more commonly traded. \n\nWe assume that an agent who is executing a large order of bonds is creating two types of price impact which are extensively used in the literature. The first one is an instantaneous (or temporary) price impact, which affects the asset  price only while trading, and fades away immediately after. This type of price impact occurs due to the fact that a large buy (sell) trade consumes the liquidity which is available in the market by ``walking through'' the first few price levels of the limit order book (see e.g. \\cite{OPTEXECAC00}  and Chapter 6.3 of \\cite{cartea15book}). However, empirical studies have shown that price impact also has a \\emph{transient} effect. A short of liquidity due to a large trade creates an imbalance between supply and demand, which in turn pushes the price in a detrimental direction. This effect decays within a short time period after each trade (see \\cite{citeulike:13497373} and \\cite{Ob-Wan2005}). Execution in presence of transient price impact was studied extensively in the context of optimal control problems (see e.g.  \\cite{GSS, GSS2, AFS2,neuman2020optimal,stat-adp18}).  In this work we incorporate these two types of price impact models into a bonds trading framework, as was done in \\cite{neuman2020optimal} for equities. \n\nApplying these price impact models to the term structure will be challenging. At every point in time the term structure of interest rates is a high dimensional object, or even an infinite dimensional one when considering all possible maturities for interest rates or zero-coupon bonds at a given time. This is a unique feature of term structure modeling, where differently from FX or equity modeling for example, we model a whole curve dynamics rather than a point dynamics. We can expect that trading a bond with a specific maturity may impact the price of bonds with different maturities {\\emph{on the same currency curve}}. In this sense, the cross price impact is endogenous to the same underlying asset, differently from what happens in other markets.  \nWe will investigate how price impact interacts with no-arbitrage dynamics, and we will encapsulate the effect of price impact in the definition of a new no-arbitrage pricing measure embedding impact itself.  This will be done by extending the market price of risk to an impacted version embedding the bond price impact speed. The impacted zero-coupon bond dynamics will then be written as the unimpacted bond dynamics but under a different measure. We will also introduce an impacted physical measure that could be useful for risk management and risk analysis. Finally, we will illustrate our theory by proposing an application to optimal execution. \n\nWe will not limit ourselves to short rate models. We will also see how in the Heath-Jarrow-Morton model the no-arbitrage drift condition for instantaneous forward rate dynamics can be maintained under price impact by resorting to the modified pricing measure.\n\n\nThe paper is structured as follow. \nIn section \\ref{sec:road} we propose a roadmap for readers who are not familiar with at least one of the two areas of the paper, namely term structure modeling and market impact.  \nIn Section \\ref{sec:main} we introduce the short rate models setup and the main theoretical results. In particular, we introduce price impact for zero-coupon bonds, and we look at the impacted market price of risk, absence of arbitrage and the impacted risk neutral measure. We define the impacted yield curve and extend impact to coupon bearing bonds. We further show how to use the impact setup in a HJM framework.  \nSection \\ref{sec:examples} features a few examples including valuation of impacted Eurodollar futures with the Hull and White model.\nSection \\ref{sec:numerical_results}\t\npresents some numerical results illustrating how the yield curve behaves under impact.\nFinally, we introduce a result on optimal execution with impacted bonds in Section \\ref{sec:optexec}.\nSections \\ref{sec:proofs} and \\ref{sec:proof_linear_feedback} include proofs that have not been included in the main text. \n\n\n\n\\section{Roadmap: how to read this paper}\\label{sec:road}\n\nThis paper combines areas that are typically disjointed in the literature, such as term structure modeling and price impact\/optimal execution. It is possible that most readers will be familiar with one area but not the other one. We expect to find few readers who may read the whole paper without effort. We therefore provide a roadmap for the reader who wishes to have a quick understanding of what the paper deals with and how it is organized, with a particular focus on where finding what and what needs to be read more carefully depending on the reader's background. \n\n{\\bf Main section of the paper.} \nSection \\ref{sec:main} is the section where both the term structure model is introduced and  the impact model is postulated and developed in detail, linking the developments to no-arbitrage pricing. The reader should start ideally from here. \n\n\\medskip\n\n{\\bf Interest rate and bond dynamics under different measures.} In Section \n\\ref{subsec:def} we introduce our initial assumption on the interest rate dynamics, assuming it is a one-dimensional short rate model dynamics. This will be relaxed later with a Heath-Jarrow-Morton setting in Section \\ref{subsec:HJM} but the paper arguments and innovations are best appreciated in a short-rate setting. We start postulating a diffusion dynamics for the short rate $r$ under the physical measure, giving then also the dynamics of the bond price $P$ under the same measure. We introduce the market price of risk leading to the risk neutral measure dynamics for the zero-coupon bond and show that, as classically done in Bjork with no-arbitrage assumptions related to self-financing and locally risk free portfolios, it does not depend on the bond maturity. With this market price of risk we then derive the dynamics of the short rate $r$ under the risk neutral measure. All this is standard theory that the term structure expert will find immediate and may want to read quickly. For readers who are more into price impact but relatively unfamiliar with term structure modeling, we recommend a careful reading of this introductory part. \n\n\\medskip\n\n{\\bf Inventory and impact.} We define the trader's inventory $X$, the execution speed $v$ and the instantaneous and transient impacts, leading to the impacted bond price $\\tilde{P}$ in Section \\ref{subsec:def}. This part will be easier to the market microstructure expert but, given that applications of price impact to full term structure modeling are not known, we recommend to read carefully this part  given the nuances involved in developing an impact theory under term structure.\nWe then show how it is possible to incorporate the effect of price impact in a new definition of impacted market price of risk, leading to an impacted risk neutral measure.  \n \n{\\bf Impacted risk neutral and physical probability measures.} \nThis measure is introduced in Section \\ref{subsec:rnm}. Under this measure, the impacted zero-coupon bond dynamics is the same as the risk neutral dynamics of the unimpacted bond provided one replaces the Brownian motion under the original risk neutral measure with a Brownian motion under the new impacted measure. The impacted risk neutral measure allows us to define an impacted physical measure as well, and again the bond price dynamics under the impacted physical measure is the same as the bond price dynamics under the original physical measure, provided that the Brownian motion is replaced with a Brownian motion under the new measure. In this setting, we can show that the impacted zero-coupon bond is the impacted risk neutral expectation of the stochastic discount factor. In a way, the impacted risk neutral measure behaves indeed as much as possible as a risk neutral measure in presence of impact. \n\nOne of the main purposes of introducing the risk neutral measure in classic derivatives pricing is the idea that the existence of this measure implies absence of arbitrage. The reader might then wonder whether we can say the same of the impacted risk measure. This is dealt with in the Section \\ref{subsec:pricing}.\n\n\\medskip\n\n{\\bf Derivatives pricing and absence of arbitrage.} First, we show that the existence of an impacted equivalent pricing measure implies that the impacted model is arbitrage free. It then follows that we can extend our result for zero-coupon bonds, namely we can price interest-rate derivatives under impact by taking the impacted risk neutral expectation of their discounted cash flows. We present an example involving Eurodollar futures. \n\n\\medskip\n\nAn important part of the theory of price impact for interest rates is what we might call {\\bf endogenous cross-impact}. With this we point to the impact of a zero-coupon bond with a given maturity on the price of a zero-coupon bond with a different maturity {\\emph{on the same currency curve}}. Exogenous cross impact would be the impact of a bond in one curve on the bond on another curve, but this is not what we deal with here. The theory of endogenous cross impact is developed in Section \\ref{subsec:yield}. This has to be read very carefully by all readers, as it represents a very peculiar aspect of any theory of price impact applied to a term structure. In term structure modeling the underlying of derivatives is a whole curve, so price impact has to be postulated also for one part of the curve over other parts. This is not a problem in markets where assets are points rather than curves, like stocks or FX rates. \n\n\\medskip\n\n{\\bf Coupon bonds.} The theory developed up to this point of the paper deals with zero-coupon bonds. However, most traded bonds are coupon bonds, so it is necessary to develop a theory of price impact for coupon bearing bonds. This is done in Section \\ref{subsec:coupon}. First, by looking at prices as expectations of discounted cash flows, we deduce that the price of impacted coupon bearing bond is simply obtained by the unimpacted one, by replacing all unimpacted zero-coupon bonds in its expression with impacted ones. \nWe find a way to connect the impact on the coupon bearing bond to the impact of single zero-coupon bonds, and find assumptions under which the impacted dynamics may even be the same.\n\n\\medskip\n\nFinally, as explained earlier, Section \\ref{subsec:HJM} goes into quite some detail to show that the theory developed so far can be equally formulated in the {\\bf Heath-Jarrow-Morton (HJM) framework}, using instantaneous forward rates to model the term structure. It is well known to interest rate experts that the no-arbitrage condition in the HJM framework is expressed by requiring that the drift in the forward rate dynamics is a specific transformation of the dynamics volatility. We show that the same relationship between drift and volatility holds for impacted instantaneous forward rates under the impact-adjusted risk neutral measure. The classic HJM theory then carries over to the impacted case. This part may be helpful to readers who wish to see our framework work outside the short rate models setting, but if the reader is not too interested in the specific interest rate model used, she can skip this part and still get all the ideas on impacted term structure modeling. \n\n\n\\bigskip\n\n{\\bf Pricing examples.} These are presented in Section \\ref{sec:examples}, including pricing impacted Eurodollar Futures with Gaussian short rate models. This section is interesting for readers who wish to see how the model can be effectively applied to price interest rate derivatives under price impact. \n\n\n\\bigskip\n\n{\\bf Numerical results.} Section \\ref{sec:numerical_results} presents numerical examples of impacted yield curves, showing how the term structure changes once price impact is included. We analyze the interplay between the bond pull-to-par and cross-price impact parameters. This can be a good section for readers who wish to develop some intuition on how price impact may change the shape of the term structure.  \n\n\\bigskip\n\n\n{\\bf Optimal execution with impacted bonds.} For the reader wondering how this theory of price impact can be included in classic problems such as optimal execution, we solve the optimal execution problem in a specific setting. This is presented in Section \\ref{sec:optexec}. The example is rather technical and the solution involves a system of forward-backward SDEs, showing that optimal execution with term structure models can be technically demanding, and this is recommended only for readers who are interested in optimal execution. \n\n\\bigskip\n\nConcluding, the best way to read the paper is sequentially, skipping the parts that are not of interest and looking more carefully at areas outside the specific expertise of the reader. We tried to provide a guide to what can be skipped and on the different emphasis for different areas of expertise above. \n\n\n\\section{Model setup and main results}\\label{sec:main}\n\n\\subsection{Impacted market price of risk, impacted risk neutral measure and absence of arbitrage} \\label{subsec:def}\nWe introduce our initial assumption on the interest rate dynamics, assuming it is a one-dimensional short rate model dynamics. This will be relaxed later with a Heath-Jarrow-Morton setting in Section \\ref{subsec:HJM}.\n\nLet us fix a maturity $T>0$ and let $(\\Omega, \\mathcal{F}, (\\mathcal{F}_t)_{t \\geq 0}, \\mathbb{P})$ be a filtered probability space satisfying the usual conditions, on which there is a standard $(\\mathcal{F}_t)_{t \\in [0,T]}$-Brownian motion $W^\\mathbb{P}$. We consider the following dynamics of the short rate under the real world measure $\\mathbb{P}$\n\\begin{equation} \\label{r-sde} \nd r(t) = \\mu(t,r(t)) dt + \\sigma(t,r(t)) d W^\\mathbb{P}(t),\n\\end{equation}\nwhere $\\mu(t,r), \\sigma(t,r)$ are given real valued functions, assumed to be regular enough to ensure the SDE has a unique strong solution. For example one can assume that both $\\mu$ and $\n\\sigma$ are Lipschitz continuous in the $r$ coordinate, and has at most linear growth in $r$ uniformly in $t\\in [0,T]$. \nWe moreover assume that $\\sigma(t,r(t))$ is $\\mathbb{P}$-a.s.  strictly positive for any $t>0$.  \n\nAssume that the dynamics of a zero-coupon bond with maturity at time $T$, under the real world measure, is given by\n\\begin{equation} \\label{class-b}\nd P(t,T) = \\mu_T(t,r(t)) dt + \\sigma_T(t,r(t)) d W^\\mathbb{P}(t),\n\\end{equation}\nwith $\\mu_T,\\sigma_T$ depending on the maturity $T$ and regular enough as in \\eqref{r-sde}. Typically, one might assume the price process of the $T$-bond to be of the form $P(t,T) = F(t,r(t);T)$ for some function $F$ smooth in three variables. Then, under suitable assumptions (see Assumption 3.2 in Chapter 3.2 of \\cite{bjork1997interest}) one can define for any finite maturity $T>0$ the stochastic process\n\\begin{equation}\n\\lambda(t) = \\frac{\\mu_T(t,r(t)) - r(t) P(t,T)}{\\sigma_T(t,r(t))},\n\\label{market_price_of_risk}\n\\end{equation}\nand show that $\\lambda$ may depend on $r$ but it does not actually depend on $T$. Such process is called \\emph{market price of risk}. Provided the Novikov condition holds, this process can be used to define a change of measure from the real world measure $\\mathbb{P}$ to the risk neutral measure $\\mathbb{Q}$:\n\\begin{equation}\n\\frac{d \\mathbb{Q}}{d \\mathbb{P}} = \\exp \\left( \\int_0^t \\lambda(s) d W^\\mathbb{P}(t)- \\frac{1}{2} \\int_0^t \\lambda^2(s) ds\\right).\n\\label{risk_neutral_measure_Q}\n\\end{equation}\nThe dynamics of the short rate under $\\mathbb{Q}$ becomes\n\\begin{equation*}\nd r(t) = [\\mu(t,r(t)) - \\lambda(t) \\sigma(t,r(t))] dt + \\sigma(t,r(t)) d W^\\mathbb{Q}(t).\n\\end{equation*}\nThe model will be fully specified once the stochastic process $\\lambda$ is defined.\nOur strategy for establishing a mathematical framework that encompasses both risk neutral pricing and price impact in the context of interest rates derivatives consists, first of all, in specifying the dynamics for an impacted bond with maturity $T$ under the real world measure $\\mathbb{P}$. \n\nWe consider a trader with an initial position of $x_T>0$ zero-coupon bonds with maturity $T$. Let $0< \\tau \\leq T$ denote some finite deterministic time horizon. In an optimal execution problem, the objective of the trader would be to complete her transaction by time $\\tau$, starting from the $x_T$ position at time $0$. In this sense, we should avoid confusion between $T$, which is the traded bond maturity, and $\\tau$, which is the trading horizon of the $T$-maturity bond. The number of bonds the trader holds at time $t \\in [0,\\tau]$ is given by\n\\begin{equation}  \\label{inv} \nX_T(t) = x_T- \\int_{0}^t v_T(s) ds.\n\\end{equation} \nwhere the function $v_T$ denotes the trader's selling rate, which takes negative values in case of a buy strategy. In what follows we assume that $v_T= \\{v_T(t)\\}_{0\\leq t\\leq \\tau}$ is progressively measurable and has a $\\mathbb P$-a.s. bounded derivative (in the $t$-variable), that is, there exists $M>0$ such that \n \\begin{equation} \\label{v-bnd} \n \\sup_{0\\leq t\\leq \\tau^{+}} |\\partial_{t}  v_{T}(t)|  < M, \\quad \\mathbb P-\\rm{a.s.}, \n\\end{equation} \nwhere $0\\leq t\\leq \\tau$. After the trading stops, we assume that $v_{T}(t)= 0$.\nWe denote the class of such trading speeds as $\\mathcal{A}_{T}$.  \n \nThe main idea behind the assumption of the differentiability of $v_{T}$ is that the overall impact we add to the zero-coupon bond should affect the drift only (see \\eqref{impacted_bond_differential_form}). Moreover due to price impact effect, we allow  bond price which are large than $1$ for some time intervals but we do need to control their upper bound. \n\nWe consider a price impact model with both transient and instantaneous impact, which is a slight generalization of the model which was considered in \\cite{neuman2020optimal}. The impacted bond price is therefore given by \n\\begin{equation} \\label{impacted_bond}\n\\tilde{P}(t,T) = P(t,T) - l(t,T)  v_T(t) - K(t,T) \\Upsilon_T^v(t). \n\\end{equation}\nHere $\\Upsilon_T^v$ represents the transient impact effect and it has the form \n\\begin{equation}\\label{def:transient_impact}\n\\Upsilon_T^v(t):= y e^{-\\rho t} + \\gamma \\int_0^t e^{-\\rho(t-s)} v_T(s) ds,\n\\end{equation}\nwhere $y,\\rho$ and $\\gamma$ are positive constants. The term $v_T(t)$ in \\eqref{impacted_bond} represents the instantaneous price impact, where we absorb in the function $l$ any constants that should factor it. \nLastly, $l$ and $K$ are differentiable functions with respect to both variables $(t,T)$ which take positive values on $0\\leq t <T$ and for any $0\\leq \\tau <T$ we have\n\\begin{equation}  \\label{l-pos}\n\\inf_{0\\leq t \\leq \\tau} l(t,T) >0.\n\\end{equation} \nMoreover we assume that   \n\\begin{equation}  \\label{k-assump} \n\\begin{aligned} \n&\\sup_{0\\leq t\\leq \\tau} |\\partial_{t}l(t,T)| <\\infty, \\quad \\lim_{t \\rightarrow T} l(t,T) =0,  \\\\\n&    \\sup_{0\\leq t\\leq \\tau} |\\partial_{t}K(t,T)| <\\infty, \\quad \\lim_{t \\rightarrow T} K(t,T) =0.\n\\end{aligned} \n\\end{equation}  \nWhile the assumption on boundedness of the derivatives of functions $K$ and $l$ arise from technical reasons which has similar motivation as the reason for \\eqref{v-bnd}, the assumptions on the behaviour at expiration is meant to enforce the boundary condition on the price of the impacted bond at expiration, which is $\\tilde{P}(T,T) =1$. Note that $K$ and $l$ are time-dependent versions of the parameters $\\lambda,k$ in \\cite{neuman2020optimal}. A prominent example of such functions is \n\\[\nl(t,T) = \\kappa \\left(1-\\frac{t}{T} \\right)^\\alpha, \\quad K(t,T) = \\left(1-\\frac{t}{T} \\right)^\\beta, \n\\]\nfor some constants $\\alpha , \\beta \\geq1$ and $\\kappa>0$. \n\nWe define for convenience the overall price impact:\n\\begin{equation}\\label{def:overall_price_impact}\nI_T(t) := l(t,T) v_T(t) + K(t,T) \\Upsilon_T^v(t).\n\\end{equation}\nThen, since $v_{T}$ is in $\\mathcal A_{T}$ we can rewrite \\eqref{impacted_bond} as follows: \n\\begin{equation} \\label{impacted_bond_differential_form}\nd \\tilde{P}(t,T) = d P(t,T) - J_T(t) dt, \\quad \\tilde P(T,T) = 1,\n\\end{equation}\nwith\n\\begin{equation} \\label{impact_density}\n\\begin{aligned}\nJ_T(t) & := \\partial_t I_T(t) \\\\\n& = \\partial_t l(t,T) v_T(t) + l(t,T) \\partial_t v_T(t) + \\partial_t K(t,T) \\Upsilon_T^v(t) + K(t,T) [-\\rho \\Upsilon_T^v(t) + v_T(t)].\n\\end{aligned}\n\\end{equation} \n\nOur model so far describes how trading a $T$-bond affects its price. Next, we show the existence of an \\emph{impacted market price of risk process} which will be a generalization of \\eqref{market_price_of_risk}. Using this process we will define an equivalent martingale measure, under which bonds and derivatives prices can be computed. Such a measure will be called an \\emph{impacted risk neutral measure}. It is important to remark that, as in the classic case, this change of measure will be unique for all bond maturities. \n\nBefore stating the main theorem of this section, let us first introduce a few important definitions.\n\n\\begin{definition}[Impacted portfolio]\\label{def:impacted_portfolio}\nLet $\\hat{T}<+\\infty$ be some finite time horizon. An \\emph{impacted portfolio} is a $(n+1)-$dimensional, bounded progressively measurable process $\\tilde{h} = (\\tilde{h}_t)_{t \\in [0, \\hat T]}$ with $\\tilde{h}_t = (\\tilde{h}_t^0,\\tilde{h}_t^1,\\dots,\\tilde{h}_t^n)$, where $\\tilde{h}_t^i$ represents the number of shares in the impacted bond $\\tilde{P}(t,T_i)$ held in the portfolio at time $t$. The value at time $t$ of such a portfolio $\\tilde{h}$ is defined as\n\\begin{equation*}\n\\tilde{V}(t) \\equiv \\tilde{V}(t,\\tilde{h}) := \\sum_{i=0}^n \\tilde{h}^i(t) \\tilde{P}(t,T_i).\n\\end{equation*}\n\\end{definition}\n\n\\begin{definition}[Self-financing]\\label{def:self_financing}\nLet $\\hat{T}<+\\infty$ be some finite time horizon. and let $\\tilde{h}$ be an impacted portfolio as in Definition \\ref{def:impacted_portfolio}. We say that $\\tilde{h}$ is \\emph{self-financing} if its value $\\tilde{V}$ is such that\n\\begin{equation} \\label{imp-port} \nd \\tilde{V}(t,\\tilde{h}) = \\sum_{i=0}^n \\tilde{h}^i(t) d  \\tilde P(t,T_i), \\quad \\textrm{for all } 0\\leq t \\leq \\hat T. \n\\end{equation}\n\\end{definition}\n\n\\begin{definition}[Locally risk free]\\label{def:locally_risk_free}\nLet $\\tilde{h}$ be an impacted portfolio as in Definition \\ref{def:impacted_portfolio} and let $\\tilde{V}$ be its value. Let also $\\alpha=(\\alpha_t)_{t \\in [0,\\hat T]}$ be an adapted process. We say that $\\tilde{h}$ is \\emph{locally risk-free} if, for almost all $t$,\n\\begin{equation*}\nd \\tilde{V}(t) = \\alpha(t) \\tilde{V}(t) \\implies \\alpha(t) = r(t),\n\\end{equation*}\nwhere $r(t)$ is the risk-free interest rate introduced in \\eqref{r-sde}.\n\\end{definition}\n\n\nHere is the main result of this section.\n\n\\begin{theorem}[Impacted market price of risk]\\label{thm_impacted_market_price}\nLet $\\hat{T}<+\\infty$ be some finite time horizon and let $\\mathbb T:=(0,\\hat{T}]$.\nLet $J_T$ be the impact density defined in \\eqref{impact_density}. Given an impacted portfolio $\\tilde{h}$ as in \\eqref{imp-port}, we assume that it is self-financing and locally risk-free, as in Definitions \\eqref{def:self_financing} and \\eqref{def:locally_risk_free}, respectively. Then, there exists a progressively measurable stochastic process $\\tilde{\\lambda}(t)$ such that \n\\begin{equation} \\label{def:lambda_tilde}\n\\tilde{\\lambda}(t) = \\frac{\\mu_{T_i}(t,r(t)) - r(t) \\tilde{P}(t,T_i) - J_{T_i}(t)}{\\sigma_{T_i}(t,r(t))}, \\quad t\\geq 0, \n\\end{equation}\nfor each maturity $T_i $, $i=1,..,n$, with $\\tilde{\\lambda}$  depending on the short rate $r$ but not on $T_i$.\n \\end{theorem}\n The proof of Theorem \\ref{thm_impacted_market_price} is given in Section \\ref{sec:proofs}.\n\n \n\n \\begin{remark}[Self-financing in presence of price impact]\nIn presence of price impact it is of course not obvious that the self-financing condition should hold. Adjusted self-financing conditions have been proposed, for instance, by Carmona and Webster \\cite{carmona2013self}. We notice that, in their work, the adjustment consists of two parts: the covariation between the inventory and the price process, and the bid-ask spread. In our work we will assume the inventory is a finite variation process and that the bid ask spread is negligible, thereby obtaining the classic self-financing condition.\n\\end{remark}\n\n\\begin{remark} [Intrinsic price impact] \nFrom Theorem \\ref{thm_impacted_market_price} it follows that \\emph{endogenous cross price impact} naturally emerges in our framework. Indeed, once an agent trades a bond with maturity $T_1$, the process $\\tilde{\\lambda}$ is uniquely determined. Note that $\\tilde \\lambda$ does not depend on the maturity. For any bond with maturity $T_2 \\in \\mathbb{T}$, which is not traded, we have $J_{T_2}\\equiv 0$ but by \\eqref{def:lambda_tilde}, the price $\\tilde P(t,T_2)$ will be affected by the trade on the bond with maturity $T_1$. We remark that, by \\emph{endogenous}, we mean that the bonds with different maturities $T_1$ and $T_2$ are thought of as belonging to the \\emph{same currency curve}. If we were to discuss multiple interest rate curves, then exogenous cross price impact should be taken into account as well.\n\\end{remark} \n\n\\subsection{Impacted risk-neutral measure}\\label{subsec:rnm}\nWe previously introduced two measures: the real world measure $\\mathbb{P}$ and the classic risk neutral measure $\\mathbb{Q}$, as defined in \\eqref{risk_neutral_measure_Q}. Now we use the result of Theorem \\ref{thm_impacted_market_price} to define a third measure, which we call \\emph{impacted risk neutral measure} and denote by $\\tilde{\\mathbb{Q}}$. This is defined as follows:\n\\begin{equation}   \\label{radon_nikodym_der_Q_tilde}\n\\frac{d \\tilde{\\mathbb{Q}}}{d \\mathbb{P}} = \\exp \\left\\{ \\int_0^t \\tilde{\\lambda}(s) d W^\\mathbb{P}(s) - \\frac{1}{2} \\int_0^t \\tilde{\\lambda}^2(s) ds \\right\\}.\n\\end{equation}\nThe well posedness of $\\tilde{\\mathbb{Q}}$ can be checked via the Novikov condition. It might be useful to recall that the usual approach does not consist in determining the conditions on $\\mu_T,\\sigma_T$ under which the Novikov condition is fulfilled. Rather, one chooses a specific short rate model to begin with. Then, one can specify the market price of risk process, exploiting the fact that it depends on $t$ and $r$, but not on $T$. For example, in the case of Vasicek model, the market price of risk is assumed to be $\\lambda(t) = \\lambda r(t)$, for some constant $\\lambda$. At this point, Novikov condition can be checked much more easily. Since we proved that $\\tilde{\\lambda}$ depends on $t$ and $r$ only, we can assume the two processes to have the same structure and follow the same idea. In the case of Vasicek model, for example, we can assume $\\tilde{\\lambda}(t) = \\tilde{\\lambda} r(t)$, for some constant $\\tilde{\\lambda}$ incorporating the impact. Consequently, determining the existence and well-posedness of $\\tilde{\\mathbb{Q}}$ is fundamentally equivalent to determining the existence and well-posedness of $\\mathbb{Q}$.\n\nThe Girsanov change of measure from the classic risk neutral measure to the impacted one is given by\n\\begin{equation*}\n\\frac{d \\tilde{\\mathbb{Q}}}{d \\mathbb{Q}} = \\frac{d \\tilde{\\mathbb{Q}}}{d \\mathbb{P}} \\frac{d \\mathbb{P}}{d \\mathbb{Q}}.\n\\end{equation*}\nwith\n\\begin{equation*}\n\\frac{d \\mathbb{P}}{d \\mathbb{Q}} = \\exp \\left\\{- \\int_0^t \\lambda(s) d W^\\mathbb{Q}(s) - \\frac{1}{2} \\int_0^t \\lambda^2(s) ds \\right\\}.\n\\end{equation*}\nwhere $\\lambda$ was defined in \\eqref{market_price_of_risk}.\nHence,\n\\begin{equation*}\n\\frac{d \\tilde{\\mathbb{Q}}}{d \\mathbb{Q}} = \\exp \\left\\{ \\int_0^t \\tilde{\\lambda}(s) d W^\\mathbb{P}(s) - \\frac{1}{2} \\int_0^t \\tilde{\\lambda}^2(s)ds + \\int_0^t \\lambda(s) d W^\\mathbb{Q}(s) - \\frac{1}{2} \\int_0^t \\lambda^2(s)ds \\right\\}.\n\\end{equation*}\nSince $W^\\mathbb{P}(t) := W^\\mathbb{Q}(t) + \\int_0^t \\lambda(s) ds$ is a Brownian motion under the measure $\\mathbb{P}$, we have\n\\begin{equation*}\n\\frac{d \\tilde{\\mathbb{Q}}}{d \\mathbb{Q}} = \\exp \\left\\{ \\int_0^t (\\tilde{\\lambda}(s) - \\lambda(s)) d W^\\mathbb{Q}(s) - \\frac{1}{2} \\int_0^t \\left(\\lambda^2(s) + \\tilde{\\lambda}^2(s) - 2 \\lambda(s) \\tilde{\\lambda}(s) \\right) ds \\right\\}.\n\\end{equation*}\nIn other words,\n\\begin{equation*}\nW^{\\tilde{\\mathbb{Q}}}(t) := W^\\mathbb{Q}(t) - \\int_0^t (\\tilde{\\lambda}(s) - \\lambda(s)) ds,\n\\end{equation*}\nis a Brownian motion under the measure $\\tilde{\\mathbb{Q}}$. It is then straightforward to notice that the impacted zero-coupon bond under the impacted measure $\\tilde{\\mathbb{Q}}$ will be described by the dynamics\n\\begin{equation} \\label{qt-p}\nd \\tilde{P}(t,T) = r(t) \\tilde{P}(t,T) dt + \\sigma_T(t,r(t)) dW^{\\tilde{\\mathbb{Q}}}(t).\n\\end{equation}\nWe further remark that, in principle, we could start by defining a new measure $\\tilde{\\mathbb{P}}$ to get rid of the additional drift due to impact. Just rewrite the dynamics of the impacted zero-coupon bond as\n\\begin{equation*}\nd \\tilde{P}(t,T) = \\mu_T(t,r(t)) dt + \\sigma_T(t,r(t)) \\left( \\frac{J_T(t)}{\\sigma_T(t,r(t))} dt  + d W^{\\mathbb{P}}(t) \\right).\n\\end{equation*}\nThis suggests to define\n\\begin{equation*}\n\\frac{d \\tilde{\\mathbb{P}}}{d \\mathbb{P}} = \\exp \\left\\{ \\int_0^t \\frac{J_T(s)}{\\sigma_T(s,r(s))} d W^\\mathbb{P}(s) - \\frac{1}{2} \\int_0^t \\left(\\frac{J_T(s)}{\\sigma_T(s,r(s))} \\right)^2 ds \\right\\}.\n\\end{equation*}\nThe impacted bond under this measure would follow the dynamics\n\\begin{equation}\nd \\tilde{P}(t,T) = \\mu_T(t,r(t)) dt + \\sigma_T(t,r(t)) dW^{\\tilde{\\mathbb{P}}}(t).\n\\label{impacted_bond_under_impacted_P}\n\\end{equation}\nAt this point, $\\tilde{\\mathbb{Q}}$ can be defined from $\\tilde{\\mathbb{P}}$ by using the classic market price of risk $\\lambda(t)$. In other words,\n\\begin{equation*}\n\\frac{d \\tilde{\\mathbb{Q}}}{d \\tilde{\\mathbb{P}}} = \\frac{d \\tilde{\\mathbb{Q}}}{d \\mathbb{Q}} \\frac{d \\mathbb{Q}}{d \\tilde{\\mathbb{P}}} = \\frac{d \\tilde{\\mathbb{P}}}{d \\mathbb{P}} \\frac{d \\mathbb{Q}}{d \\tilde{\\mathbb{P}}} = \\frac{d \\mathbb{Q}}{d \\mathbb{P}}.\n\\end{equation*}\nPutting everything together, we have the following commuting diagram\n\\[\n\\begin{tikzcd}\n\\mathbb{P} \\arrow{r}{\\lambda}  \\arrow[swap]{dr}{\\tilde{\\lambda}} \\arrow[swap]{d}{\\tilde{\\lambda}-\\lambda} & \\mathbb{Q} \\arrow{d}{\\tilde{\\lambda} - \\lambda} \\\\\n\\tilde{\\mathbb{P}} \\arrow{r}{\\lambda} & \\tilde{\\mathbb{Q}}\n\\end{tikzcd}\n\\]\nBy \\eqref{qt-p} and usual arguments it follows that discounted impacted traded prices, that is $\\{\\tilde P(\\cdot,T)\/B(t)\\}_{t\\geq 0}$, are martingales for any $0\\leq T \\leq \\hat T$ under $\\tilde{\\mathbb{Q}}$. Here $B$ is the usual money market account at time $t$ given by \n\\begin{equation}\\label{def:bank_account}\nB(t) = e^{\\int_0^t r(s) ds}.\n\\end{equation}\nWe therefore have \n\\begin{equation} \\label{discounted_impacted_ZC_bond}\n\\frac{\\tilde{P}(t,T)}{B(t)} = \\mathbb{E}^{\\tilde{\\mathbb{Q}}} \\left[ \\frac{\\tilde{P}(T,T)}{B(T)} \\Bigg| \\mathcal{F}_t \\right].\n\\end{equation}\nMultiplying both sides by $B(t)$ and exploiting the boundary condition $\\tilde{P}(T,T)=1$, we obtain the fundamental equation\n\\begin{equation}\n\\tilde{P}(t,T) = \\mathbb{E}^{\\tilde{\\mathbb{Q}}} \\left[ e^{-\\int_t^T r(s) ds} \\big | \\mathcal{F}_t \\right].\n\\label{impacted_ZCB_expectation}\n\\end{equation}\n\n\\begin{remark}[Interpretation impacted real world measure]\nFrom \\eqref{impacted_bond_under_impacted_P} we observe that, financially speaking, under the impacted real world measure $\\tilde{\\mathbb{P}}$, impacted bond dynamics $\\tilde P(\\cdot,T)$ has the same dynamics as the classic bond (without price impact modeling) in \\eqref{class-b} under $\\mathbb{P}$. In particular we have that $\\tilde P(T,T)=1$ for all maturities $T$.\n\\end{remark} \n\n\\subsection{Applications to pricing of interest rate derivatives} \\label{subsec:pricing}\n\nWe start this section by remarking that the notion of arbitrage we use in our work is the classic one (see e.g. Harrison and Kreps \\cite{harrison1979martingales} or Harrison and Pliska \\cite{harrison1981martingales}), adjusted with impacted portfolios.\n\\begin{definition}[Arbitrage portfolio]\nAn \\emph{arbitrage portfolio} is an impacted self-financing portfolio $\\tilde{h}$ such that its corresponding value process $\\tilde{V}$ satisfies\n\\begin{enumerate}\n\\item $\\tilde{V}(0) = 0$ . \n\\item $\\tilde{V}(T)  \\geq 0$ $\\mathbb{P}$-a.s.\n\\item $\\mathbb{P}(\\tilde{V}(T)>0)>0$\n\\end{enumerate}\n\\end{definition}\nUsing this definition of arbitrage we show that the first fundamental theorem of asset pricing holds in our setting.\n\n\\begin{theorem}[Absence of arbitrage]\\label{thm_absence_arbitrage}\nAssume that there exists an impacted equivalent martingale measure $\\tilde{\\mathbb{Q}}$ as in \\eqref{radon_nikodym_der_Q_tilde}. Then, our impacted model is arbitrage  free.\n\\end{theorem}\nThe proof of Theorem \\ref{thm_absence_arbitrage} is given in Section \\ref{sec:proofs}.\n\nA key consequence Theorem \\ref{thm_absence_arbitrage} is that our term structure model with price impact is indeed free of arbitrage. This allows to price interest rate derivatives by taking the expectation of discounted payoffs under the impacted risk neutral measure $\\tilde{\\mathbb{Q}}$. As a benchmark example, we consider the price of an impacted Eurodollar future. In the classic context, a Eurodollar-futures contract provides its owner with the payoff (see Chapter 13.12 of \\cite{brigo2007interest})\n\\begin{equation*}\nN (1-L(S,T)),\n\\end{equation*}\nwhere $N$ denotes the notional and $L(S,T)$ is the LIBOR rate, defined as (see Chapter 1 of \\cite{brigo2007interest}, Definition 1.2.4) \n\\begin{equation}\nL(S,T) := \\frac{1 - P(S,T)}{\\tau(S,T) P(S,T)},\n\\label{libor_rate}\n\\end{equation}\nwith $\\tau(S,T)$ denoting the year fraction between $S$ and $T$. Motivated by this, we introduce the impacted counterpart of the LIBOR rate in \\eqref{libor_rate}, i.e.\n\\begin{equation} \\label{l-t}\n\\tilde{L}(S,T) := \\frac{1 - \\tilde{P}(S,T)}{\\tau(S,T) \\tilde{P}(S,T)},\n\\end{equation}\nwith $\\tau$ defined as above and the impacted zero-coupon bond in place of the classic one. This new rate $\\tilde{L}$ is interpreted as the simply-compounded rate which is consistent with the impacted bond. This corresponds to the classic LIBOR rate which is the constant rate at which one needs to invest $P(t,T)$ units of currency at time $t$ in order to get an amount of one unit of currency at maturity $T$. Then, the fair price of an impacted Eurodollar future at time $t$ is (see \\cite{brigo2007interest}, Chapter 13, eq. (13.19))\n\\begin{equation}\\label{fair_price_impacted_Eurodollar}\n\\begin{split}\n\\tilde{C}_t & = \\mathbb{E}_t^{\\tilde{\\mathbb{Q}}} [N (1-\\tilde{L}(S,T))], \\\\\n& = N \\left(1 + \\frac{1}{\\tau(S,T)} - \\frac{1}{\\tau(S,T)} \\mathbb{E}_t^{\\tilde{\\mathbb{Q}}} \\left[\\frac{1}{\\tilde{P}(S,T)} \\right] \\right),\n\\end{split}\n\\end{equation}\nwhere the discounting was left out due to continuous rebalancing (see again Chapter 13.12 of \\cite{brigo2007interest}). We will demonstrate in Section \\ref{sec:examples} how such expectation can be computed analytically provided the short rate model is simple enough as in Vasicek and Hull-White models. \n\n\\begin{remark}[Linear and nonlinear pricing equations]\nOur success in retaining analytical tractability and linearity in the pricing equation may look surprising at first. In the context of equities, pricing derivatives in presence of price impact typically leads to nonlinear PDEs. This, in turn, motivated the study of super-replicating strategies and the so-called gamma constrained strategies. Several works provide also necessary and sufficient conditions ensuring the parabolicity of the pricing equation, hence the existence and uniqueness of a self-financing, perfectly replicating strategy. We refer, for example, to Abergel and Loeper \\cite{abergel2013pricing}, Bourchard, Loeper et al. \\cite{bouchard2016almost,bouchard2017hedging} and Loeper \\cite{loeper2018option}. The point we would like to stress here is that the nonlinearity of the pricing equation is a consequence of the trading strategy having a diffusion term, or a consequence of the presence of transaction costs. In other words, under the assumption that trading strategies have bounded variation and no transaction costs are present, the pricing PDE becomes linear again. Hence, our work is actually in agreement to what can be found in the context of equities.\n\\end{remark}\n\n\\subsection{Cross price impact and impacted yield curve} \\label{subsec:yield}\nIn this section we discuss how trading a bond $P(t,T)$ impacts the yield curve. For the sake of analytical tractability, we will consider affine short-rate models, that is, those models where bond prices are of the form\n\\begin{equation}\nP(t,T) = A(t,T) e^{-B(t,T) r(t)}, \\quad 0\\leq t \\leq T, \n\\label{affine:bond_price}\n\\end{equation}\nfor some deterministic, smooth functions $A$ and $B$ and $r$ is given by \\eqref{r-sde}. The remarkable property of these models is that they can be completely characterized as in the following theorem (see, e.g., Filipovic \\cite{filipovic2009term}, Section 5.3, Brigo and Mercurio \\cite{brigo2007interest}, Section 3.2.4, Bjork \\cite{bjork1997interest} Section 3.4 and references therein). \n\n\\begin{lemma}[Characterization affine short-rate models]\nThe short rate model \\eqref{r-sde} is affine if and only if there exist deterministic, continuous functions $a,\\alpha,b,\\beta$ such that the diffusion and the drift terms in \\eqref{r-sde} are of the form\n\\begin{align*}\n\\begin{split}\n\\sigma^2(t,r) & = a(t) + \\alpha(t) r, \\\\\n\\mu(t,r) & = b(t) + \\beta(t) r,\n\\end{split}\n\\end{align*}\nand the functions $A,B$ satisfy the following system of ODEs\n\\begin{align*}\n\\begin{split}\n- \\frac{\\partial}{\\partial t}  \\ln A(t,T) & = \\frac{1}{2} a(t) B^2(t,T) - b(t) B(t,T), \\ \\ \\ A(T,T) = 1, \\\\\n\\frac{\\partial}{\\partial t} B(t,T) & = \\frac{1}{2} \\alpha(t) B^2(t,T) - \\beta(t) B(t,T) - 1, \\ \\ \\ B(T,T) = 0, \\\\\n\\end{split}\n\\end{align*}\nfor all $t \\leq T$.\n\\end{lemma}\nAs explained in \\cite{filipovic2009term}, the functions $a,\\alpha,b,\\beta$ can be further specified by observing that any non-degenerate short rate affine model, that is a model with $\\sigma(t,r) \\ne 0$ for all $t>0$, can be transformed, by means of an affine transformation, in two cases only, depending on whether the state space of the short rate $r$ is the whole real line $\\mathbb{R}$ or only the positive part $\\mathbb{R}_+$. In the first case, it must hold $\\alpha(t)=0$ and $a(t) \\geq 0$, with $b,\\beta$ arbitrary. In the second case, it must hold $a(t)=0, \\alpha(t), b(t) \\geq 0$ and $\\beta$ arbitrary.\n\nLet $\\hat{T}<+\\infty$ be some finite time horizon. The yield curve at a pre-trading time $t_0$ (i.e. before price impact effects kick in) according to classic theory of interest rates is defined by \n\\begin{equation}\nY(t,T) := P(t,T)^{-1\/T} - 1, \\quad 0\\leq t \\leq t_0,\n\\label{def:classic_yield}\n\\end{equation}\nfor all maturities $0 \\leq T \\leq \\hat T$. Next, we consider the impacted bond dynamics \n\\begin{equation*}\nd \\tilde{P}(t,T) = d P(t,T) - J_T(t) dt, \n\\end{equation*}\nwhere $J_T$ was defined in \\eqref{impact_density}. Recall that the dynamics of $r(t)$ is given in \\eqref{r-sde}. Applying Ito's formula on $P(t,T)$ in  \\eqref{affine:bond_price} we get\n\\begin{multline*}\nd P(t,T) = e^{-B(t,T) r(t)} \\bigg[\\frac{\\partial A}{\\partial t} - A(t,T) \\frac{\\partial B}{\\partial t} r(t) + \\frac{1}{2} A(t,T) B^2(t,T) \\sigma^2(t,r(t)) + \\\\\n- A(t,T) B(t,T) \\mu(t,r(t)) \\bigg] dt - \\sigma(t,r(t)) B(t,T) A(t,T) e^{-B(t,T) r(t)} d W^{\\mathbb{P}}(t). \\\\\n\\end{multline*}\nFrom this equation, we readily extract the drift and the diffusion of the zero-coupon bond with maturity $T$: \n\\begin{align} \\label{affine:drift_vol_zero_coupon_bond}\n\\begin{split}\n\\mu_T(t,r(t)) &:= e^{-B(t,T) r(t)} \\bigg[\\frac{\\partial A}{\\partial t} - A(t,T) \\frac{\\partial B}{\\partial t} r(t) + \\frac{1}{2} A(t,T) B^2(t,T) \\sigma^2(t,r(t))  \\\\\n& \\quad - A(t,T) B(t,T) \\mu(t,r(t)) \\bigg], \\\\\n\\sigma_T(t,r(t)) & := - \\sigma(t,r) A(t,T) B(t,T) e^{-B(t,T) r(t)}.\n\\end{split}\n\\end{align}\nNext, we consider the effect of an agent trading on the bond with maturity $T$ on a bond which is not traded by the agent with maturity $S$. We call this effect the \\emph{endogenous cross-impact} on the bond with maturity $S$. Recall that in this case the dynamics of the $S$-bond is given by\n\\begin{equation} \\label{sde:impacted_bond_S}\nd \\tilde{P}(t,S) = \\mu_S(t,r(t)) dt + \\sigma_S(t,r(t)) d W^{\\mathbb{P}}(t), \n\\end{equation}\nwhere the coefficients $\\mu_S$ and $\\sigma_S$ are given by analogous formulas to \\eqref{affine:drift_vol_zero_coupon_bond}. Since we are trading the $T$-bond only, $J_S$ in \\eqref{impact_density} will be identically equal to zero. Hence, the definition of the impacted market price of risk \\eqref{def:lambda_tilde} implies the following relationship\n\\begin{equation*}\n\\frac{\\mu_T(t,r(t)) - r(t) \\tilde{P}(t,T) - J_T(t)}{\\sigma_T(t,r(t))} = \\frac{\\mu_S(t,r(t)) - r(t) \\tilde{P}(t,S)}{\\sigma_S(t,r(t))}.\n\\end{equation*}\nThis equation tells us how the drift the $S$-bond has to change in order to avoid arbitrage. That is, this equation describes the \\emph{cross-price impact}. Specifically we have\n\\begin{equation*}\n\\mu_S(t,r(t)) = \\frac{\\sigma_S(t,r(t))}{\\sigma_T(t,r(t))} \\left[\\mu_T(t,r(t)) - r(t) \\tilde{P}(t,T) - J_T(t)\\right] + r(t) \\tilde{P}(t,S).\n\\end{equation*}\nSubstituting this drift in \\eqref{sde:impacted_bond_S} we get \n\\begin{align}\\label{sde:cross_impacted_bond_S}\n\\begin{split}\nd \\tilde{P}(t,S) &= r(t) \\tilde{P}(t,S) dt + \\frac{\\sigma_S(t,r(t))}{\\sigma_T(t,r(t))} \\left[\\mu_T(t,r(t)) - r(t) \\tilde{P}(t,T) - J_T(t) \\right] dt \\\\\n& \\quad + \\sigma_S(t,r(t)) d W^{\\mathbb{P}}(t). \\\\\n\\end{split}\n\\end{align}\nFinally, we define the impacted yield curve for all $t_0 \\leq T \\leq \\hat T$ as follows: \n\\begin{equation} \\label{def:impacted_yield}\n\\tilde{Y}(t,T) := \\tilde{P}(t, T)^{-1\/T} - 1.\n\\end{equation}\n\n\\begin{remark}[Cross impacted bonds at maturity]\\label{rem:cross_impacted_bonds_at_maturity}\nWe have shown in \\eqref{impacted_bond_differential_form} that according to our model $\\tilde P(T,T) =1$. However, we should also ensure that all cross-impacted bonds with maturity $S \\not = T$ reach value $1$ at their  maturities. This of course, would make the model much more involved and we may lose tractability. \n\\end{remark}\n\n\\subsection{Coupon bonds}\\label{subsec:coupon}\nIt is worth recalling that the zero-coupon bond $P(t,T)$ is rarely traded. In practice, its price is derived using some bootstrapping procedure applied, for instance, to coupon bonds. In the classic theory, coupon bonds are defined as\n\\begin{equation*}\nB(t,T) = \\sum_{i=1}^n c_i P(t,T_i) + N P(t,T_n),\n\\end{equation*}\nwhere $N$ denotes the reimbursement notional, $(c_i,T_i)_{i=1}^n$ are the coupons and the maturities at which the coupons are paid, respectively. In order to determine an expression for the impacted coupon bond, we start from its cash flow\n\\begin{equation*}\nC(t) := \\sum_{i=1}^n c_i D(t,T_i) + N D(t,T_n),\n\\end{equation*}\nwhere $D(t,T)$ is the stochastic discount factor defined by\n\\begin{equation*}\nD(t,T) := e^{- \\int_t^T r(s) ds},\n\\end{equation*}\nwhere $r$ is given by \\eqref{r-sde}.\nThen, we define the impacted coupon bond as the expectation of this cash flow under the impacted risk neutral measure $\\tilde{\\mathbb{Q}}$ (see \\eqref{radon_nikodym_der_Q_tilde}):\n\\begin{equation*}\n\\tilde{B}(t,T) := \\mathbb{E}^{\\tilde{\\mathbb{Q}}} \\left[C(t) \\right].\n\\end{equation*}\nSubstituting the expression of $C$ immediately yields\n\\begin{align}\\label{impacted_coupon_bond_linear_combination}\n\\begin{split}\n\\tilde{B}(t,T) & = \\sum_{i=1}^n c_i \\mathbb{E}^{\\tilde{\\mathbb{Q}}} \\left[ D(t,T_i) \\right] + N \\mathbb{E}^{\\tilde{\\mathbb{Q}}} \\left[ D(t,T_n) \\right] \\\\\n& = \\sum_{i=1}^n c_i \\tilde{P}(t,T_i) + N \\tilde{P}(t,T_n),\n\\end{split}\n\\end{align}\nwhere $\\tilde{P}(\\cdot, T_i)$ is the (directly) impacted price of a zero-coupon bond as defined in \\eqref{impacted_bond}. Note that \\eqref{impacted_coupon_bond_linear_combination} gives the price of the impacted coupon bond in terms of impacted zero-coupon bonds. Since zero-coupon bonds are not always traded, we would like to get a direct pricing formula for impacted coupon bonds. Let $\\{v_{T_i}\\}_{i=1}^n$ be admissible trading speeds on zero-coupon bonds with maturities $\\{T_i\\}_{i=1}^n$ as defined in Section \\ref{sec:main}, that is $v_{T_i} \\in \\mathcal A_{T_i}$ for any $i=1,...n$.\nFrom \\eqref{impacted_bond} and \\eqref{impacted_coupon_bond_linear_combination} we get\n\\begin{subequations}\n\\begin{align*}\n\\tilde{B}(t,T) & = \\sum_{i=1}^n c_i \\tilde{P}(t,T_i) + N \\tilde{P}(t,T_n) \\\\\n& = B(t,T) - \\sum_{i=1}^n c_i l(t,T_i) v_{T_i}(t) - N l(t,T_n) v_{T_n}(t)  \\\\\n& \\quad - \\sum_{i=1}^n c_i K(t,T_i) y e^{-\\rho t} - N K(t,T_n) ye^{-\\rho t} \\\\\n& \\quad - \\gamma \\int_0^t e^{-\\rho (t-s)} \\left(\\sum_{i=1}^n c_i K(t,T_i) v_{T_i}(s) + N K(t,T_n) v_{T_n}(s) \\right) ds.\n\\end{align*}\n\\end{subequations}\nLet us now assume that $l=\\kappa K$ at all times and for all maturities, where $\\kappa>0$ is a constant. Then, the impacted coupon bond dynamics can be written as\n\\begin{align}\n\\begin{split}\n\\tilde{B}(t,T) & = B(t,T) - y e^{-\\rho t} \\left[\\sum_{i=1}^n c_i K(t,T_i) + N K(t,T_n) \\right] \\\\\n& \\quad - \\int_0^t e^{-\\rho (t-s)} \\kappa\\delta(s-t) \\left[\\sum_{i=1}^n c_i K(t,T_i) v_{T_i}(s) + N K(t,T_n) v_{T_n}(s) \\right] ds \\\\\n& \\quad - \\gamma \\int_0^t e^{-\\rho (t-s)} \\left[\\sum_{i=1}^n c_i K(t,T_i) v_{T_i}(s) + N K(t,T_n) v_{T_n}(s) \\right] ds,\n\\end{split}\n\\end{align}\nwhere $\\delta$ denotes the Dirac delta. Notice that under this assumption the impacted zero-coupon bond dynamics defined in \\eqref{impacted_bond} boils down to\n\\begin{equation}\\label{impacted_bond_simplified}\n\\tilde{P}(t,T) = P(t,T) - K(t,T) \\left[y e^{-\\rho t} + \\int_0^t e^{-\\rho (t-s)} v_T(s) \\left(\\gamma + \\kappa\\delta(s-t)\\right) ds \\right].\n\\end{equation}\nThis suggest we can define\n\\begin{equation*}\nK^B(t,T) := \\sum_{i=1}^n c_i K(t,T_i) + N K(t,T_n).\n\\end{equation*}\nand the trading speed relative to the coupon bond as\n\\begin{equation}\nv^B(t,s) := \\frac{1}{K^B(t,T)} \\left[\\sum_{i=1}^n c_i K(t,T_i) v_{T_i}(s) + N K(t,T_n) v_{T_n}(s) \\right].\n\\label{trading_speed_coupon_bond}\n\\end{equation}\nTherefore, we obtain the following price impact model for the coupon bond: \n\\begin{equation}\\label{impacted_coupon_bond_simplified}\n\\tilde{B}(t,T) = B(t,T) - K^B(t,T) \\left[y e^{-\\rho t} + \\int_0^t e^{-\\rho (t-s)} v^B(t,s) \\left(\\gamma +  \\kappa \\delta(s-t)\\right) ds \\right].\n\\end{equation}\nInterestingly, under the simplifying assumption that the functions $l$ and $K$ are equal up to some constant, we observe that the impacted zero-coupon bond $\\tilde{P}(t,T)$ in \\eqref{impacted_bond_simplified} and the impacted coupon bond $\\tilde{B}(t,T)$ in \\eqref{impacted_coupon_bond_simplified} are described by the same kind of dynamics.\n\nThis is particularly useful because, provided enough data on traded coupon bonds are available, one might attempt to use \\eqref{trading_speed_coupon_bond} and \\eqref{impacted_coupon_bond_simplified} to bootstrap the trading speeds $v_{T_i}$ relative to the zero-coupon bonds. Using the price impact model \\eqref{impacted_bond}, it would be then possible to price impacted zero-coupon bonds consistently with market data. Finally, using these impacted zero-coupon bonds as building blocks, it would be possible to price, consistently with market data, more complicated and less liquid impacted interest rate derivatives, as discussed in Section \\ref{subsec:pricing}.  \n\n\\subsection{HJM framework}\\label{subsec:HJM}\nIn this section we turn our discussion to incorporating price impact into the Heath, Jarrow and Morton framework \\cite{heath1992bond}, in order to model the forward curve. Notice that this approach, although it may look different, has some common aspects to the framework developed in Section \\ref{subsec:def}. Namely, we start by adding artificially a price impact term to the forward rate dynamics. This corresponds to adding price impact to zero-coupon bonds in Section \\ref{subsec:def}. The important difference is that, here, we are creating an impacted interest rate, which was not done in Section \\ref{subsec:def}. Then we will develop the connection between the price impact of zero-coupon bonds and the price impact term incorporated into the forward rate, in order to reveal the financial interpretation of the latter. Note that both the zero-coupon bonds and the forward rate can be used as building blocks for the whole interest rates theory. We are therefore interested in showing the connection between the two in the presence of price impact. For a thorough discussion on the HJM framework in the classic interest rate theory, we refer to Chapter 6 of the book by Filipovic \\cite{filipovic2009term}. \n\nGiven an integrable initial forward curve $T \\to \\tilde{f}(0,T)$, we assume that the impacted forward rate process $\\tilde{f}(\\cdot,T)$ is given by \n\\begin{equation}\n\\tilde{f}(t,T) = \\tilde{f}(0,T) + \\int_0^t\\left( \\alpha(s,T) + J^f(s,T)\\right) ds + \\int_0^t \\sigma(s,T) d W^{\\mathbb{P}}(s),\n\\label{impacted_forward_rate}\n\\end{equation}\nfor any $0\\leq t \\leq T$ and each maturity $T>0$. Here $W^{\\mathbb{P}}$ is a Brownian motion under the measure $\\mathbb{P}$ and $\\alpha(\\cdot,T)$, $J^f(\\cdot,T)$ and $\\sigma(\\cdot,T)$ are assumed to be progressively measurable processes and satisfy for  any $T>0$\n\\begin{align*}\n\\begin{split}\n\\int_0^T \\int_0^T (|\\alpha(s,t)| + |J^f(s,t)| )ds dt & < \\infty, \\\\\n\\sup_{s,t \\leq T} |\\sigma(s,t) | & < \\infty.\n\\end{split}\n\\end{align*}\nWhile the roles of $\\alpha(\\cdot,T)$ and $\\sigma(\\cdot,T)$ above are as in standard HJM model, the stochastic process $J^f$ represents the impact density relative to the forward rate and accounts for the fact that the forward curve is affected by the trading activity. From a modelling perspective, it plays a completely analogous role as the quantity $J_T$ defined in \\eqref{impact_density} for the impacted zero-coupon bond. In fact, in Proposition \\eqref{prop:relationship_Jf_Jp} we will determine the mathematical relationship linking these two quantities. Such relationship will allow us to understand how the forward curve is impacted by trading zero-coupon bonds. \n\nIn this framework, the impacted short rate model is given by\n\\begin{equation}\n\\tilde{r}(t) := \\tilde{f}(t,t) = \\tilde{f}(0,t) + \\int_0^t\\left( \\alpha(s,t) + J^f(s,t)\\right)  ds + \\int_0^t \\sigma(s,t) d W^{\\mathbb{P}}(s),\n\\label{impacted_short_rate}\n\\end{equation}\nand the impacted zero-coupon bond is defined as follows \n\\begin{equation}\n\\tilde{P}(t,T) = e^{-\\int_t^T \\tilde{f}(t,u) du}.\n\\label{impacted_bond_HJM}\n\\end{equation}\n\nNext we derive the explicit dynamics of $\\{\\tilde{P}(t,T)\\}_{0\\leq t\\leq T}$. The following corollary is an impacted version of Lemma 6.1 in \\cite{filipovic2009term}.\n\n\\begin{corollary}[Impacted zero-coupon bond in HJM framework]\\label{cor:impacted_zcb_HJM} \nFor every maturity $T$ the impacted zero-coupon bond defined in \\eqref{impacted_bond_HJM} follows the dynamics\n\\begin{equation}\n\\tilde{P}(t,T) = \\tilde{P}(0,T) + \\int_0^t \\tilde{P}(s,T) \\left(\\tilde{r}(s) + \\tilde{b}(s,T) \\right) ds + \\int_0^t \\tilde{P}(s,T) \\nu(s,T) d W^{\\mathbb{P}}(s),\\ \\ \\ t \\leq T,\n\\label{impaced_ZCB_HJM}\n\\end{equation}\nwhere $\\tilde{r}$ is the impacted short rate defined in \\eqref{impacted_short_rate} and \n\\begin{equation} \\label{b-v-rel}\n\\begin{aligned}\n\\nu(s,T) & := - \\int_s^T \\sigma(s,u) du, \\\\\n\\tilde{b}(s,T) & := - \\int_s^T \\alpha(s,u) du - \\int_s^T J^f(s,u) du + \\frac{1}{2} \\nu^2(s,T).\n\\end{aligned}\n\\end{equation} \n\\end{corollary}\n\nWe now show that the impact $J^f$ can be expressed in terms of the impact relative to the zero-coupon bond, and vice versa. In order to show this correspondence in terms of agent's trading speed, we need to make an additional assumption on the trading speeds on zero-coupon bonds. We assume that the price impact in the forward curve is a result of trading by one or many agents over a continuum of zero-coupon bonds with maturities $T$ and trading speeds $\\{T \\geq 0 \\, : \\,  v_T \\in \\mathcal A_T\\}$ so that \n\\begin{equation} \\label{sp-der}  \n|\\partial_T v_T(t)| <\\infty, \\quad \\textrm{for all }  0\\leq t \\leq T,  \\quad \\mathbb{P}-\\textrm{a.s.} \n\\end{equation} \nNote that this assumption in fact makes sense in bond trading, which has discrete maturities, as it claims that when there is a highly traded $T_i$-bond, you would find that also the neighbouring $T_{i-1}$, $T_{i+1}$ are liquid. Assumption \\eqref{sp-der} implies that $\\partial_T I_T(t)$ is well defined as needed in the following Proposition.  We recall that $f$ represents the unimpacted forward rate which is given by setting $J^{f} \\equiv 0$ in \\eqref{impacted_forward_rate}.\n \n\\begin{proposition}[Forward rate and zero-coupon bond price impact relation] \\label{prop:relationship_Jf_Jp}\nLet $I_T(t)$ be the overall impact defined in \\eqref{def:overall_price_impact} and $\\tilde{P}(\\cdot,T)$ the impacted zero-coupon bond price in \\eqref{impacted_bond_HJM}. Assume $\\tilde{f}(0,t) =  f(0,t)$, meaning that the initial value of the forward curve is not affected by trading. Then, the forward rate impact $J^f$ introduced in \\eqref{impacted_forward_rate} is given by\n\\begin{equation}\\label{relationship_Jf_Jp}\nJ^f(t,T) = - \\frac{\\partial}{\\partial T} \\log \\left( 1- \\frac{I_T(t)}{P(t,T)} \\right) , \\quad \\textrm{for all } 0\\leq t \\leq T \\ \\textrm{such that }\\tilde{P}(t,T) >0.  \n\\end{equation}\n\\end{proposition}\n\nThe proof of Proposition \\ref{prop:relationship_Jf_Jp} is given in Section \\ref{sec:proofs}.\n\n\\begin{remark} Note that the requirement that $\\tilde{P}(t,T)>0$ ensures that the logarithm on the right-hand side of \\eqref{relationship_Jf_Jp} is well defined, as \\eqref{bla} in the proof suggests. The proof also gives another relation between $J^{f}(\\cdot, T)$ and $I_{T}$ which always holds but is perhaps not as direct. \n\\end{remark} \n\nA well known feature of the classic HJM framework is that, under the risk neutral measure, the drift of the forward rate is completely specified by the volatility through the so called \\emph{HJM condition}. In order to understand how this condition is affected by the introduction of price impact, we will follow Theorem 6.1 of \\cite{filipovic2009term}. In particular, we have the following key result.\n\n\\begin{theorem}[HJM condition with price impact]\\label{hjm_condition_market_impact}\nLet $\\mathbb{P}$ be the real world measure under which the impacted forward rate as in \\eqref{impacted_forward_rate}. Let $\\tilde{\\mathbb{Q}} \\sim \\mathbb{P}$ be an equivalent probability measure of the form\n\\begin{equation}\\label{def:Q_tilde_rnd_HJM}\n\\frac{d \\tilde{\\mathbb{Q}}}{d \\mathbb{P}} = \\exp \\left\\{\\int_0^t \\tilde{\\gamma}(s) d W^{\\mathbb{P}}(s) - \\frac{1}{2} \\int_0^t \\tilde{\\gamma}^2(s) ds \\right\\},\n\\end{equation}\nfor some progressively measurable stochastic process $\\tilde{\\gamma}= \\{\\tilde \\gamma(t) \\}_{t\\geq 0}$ such that $\\int_0^t \\tilde{\\gamma}^2(s) ds < \\infty$, for all $t>0$, $\\mathbb P$-a.s. Then, $\\tilde{\\mathbb{Q}}$ is an equivalent (local) martingale measure if and only if \n\\begin{equation} \n\\tilde{b}(t,T) = - \\nu(t,T) \\tilde{\\gamma}(t), \\quad \\textrm{for all } 0\\leq t \\leq T. \n\\label{HJM_condition}\n\\end{equation}\nwith $\\tilde{b}(\\cdot,T)$ and $\\nu(\\cdot,T)$ defined as in \\eqref{b-v-rel}. In this case, the dynamics of the impacted forward rate under the measure $\\tilde{\\mathbb{Q}}$ is given by \n\\begin{equation}\n\\tilde{f}(t,T) = \\tilde{f}(0,T) + \\int_0^t \\left( \\sigma(s,T) \\int_s^T \\sigma(s,u) du \\right) ds + \\int_0^t \\sigma(s,T) d W^{\\tilde{\\mathbb{Q}}}(s).\n\\label{impacted_forward_rate_Q_tilde}\n\\end{equation}\nMoreover, the prices of impacted zero-coupon bonds are\n\\begin{equation}\n\\tilde{P}(t,T) = \\tilde{P}(0,T) + \\int_0^t \\tilde{P}(s,T) \\tilde{r}(s) ds + \\int_0^t \\tilde{P}(s,T) \\nu(s,T) d W^{\\tilde{\\mathbb{Q}}}(s).\n\\label{impacted_zc_bond_Q_tilde}\n\\end{equation}\n\\end{theorem}\n\nThe proof of Theorem \\ref{hjm_condition_market_impact} is given in Section \\ref{sec:proofs}.\n\nIn our context such a measure $\\tilde{\\mathbb{Q}}$ would be clearly interpreted as an impacted risk-neutral measure, completely analogous to the measure defined in \\eqref{radon_nikodym_der_Q_tilde}. In fact, the stochastic process $\\tilde{\\gamma}$ in the HJM condition \\eqref{HJM_condition} is the counterpart in the HJM framework, of the impacted market price of risk $\\tilde{\\lambda}$ defined in Section \\ref{sec:main}. Indeed, combining equations \\eqref{market_price_of_risk} and \\eqref{impaced_ZCB_HJM} we obtain for $0\\leq t\\leq T$,\n\\begin{equation*}\n\\tilde{\\lambda}(t) = \\frac{\\tilde{P}(t,T) \\left(r(t) - \\tilde{b}(t,T) \\right) - r(t) \\tilde{P}(t,T)}{\\tilde{P}(t,T) \\nu(t,T)} = - \\frac{-\\tilde{b}(t,T)}{\\nu(t,T)} = \\tilde{\\gamma}(t).\n\\end{equation*}\n\nThe HJM framework adjusted with price impact discussed in this section is therefore perfectly consistent with the price impact model for zero-coupon bonds introduced in Section \\ref{subsec:def}.\n\nWe remark once again that, in the classic theory of interest rates, the meaning of the HJM condition lies in the fact that the drift of the forward rate is constrained under the risk neutral measure. Similarly, looking at the market price of risk, we notice that a constraint is present for the drift of the zero-coupon bond. In particular, its drift, under the risk neutral measure, has to be precisely the risk free interest rate. The interesting point we would like to make here is that, once we incorporate price impact, the same kind of constraints holds, only under the newly defined impacted measure $\\tilde{\\mathbb{Q}}$.\n\nWe conclude this section by making two remarks. We first address the question of when the measure defined in Theorem \\ref{hjm_condition_market_impact} is an equivalent martingale measure, instead of just local martingale measure. The second remark concerns the Markov property of the impacted short rate. In both cases, we see that the classic results carry over to the price impact framework, thanks to the key fact that the impact component affects only the drift of the forward rate.\n\n\\begin{remark}[Impacted risk neutral measure is an EMM]\nLet $\\nu(t,T)$ be defined as in \\eqref{b-v-rel}. From Corollary 6.2 of \\cite{filipovic2009term} it follows that the measure $\\tilde{\\mathbb{Q}}$ defined in Theorem \\ref{hjm_condition_market_impact} is an equivalent martingale measure if either\n\\begin{equation*}\n\\mathbb{E}^{\\tilde{\\mathbb{Q}}} \\left[ e^{\\frac{1}{2} \\int_0^T \\nu^2(t,T) dt }\\right] < \\infty, \\quad \\text{for all} \\ T \\geq 0, \n\\end{equation*}\nor \n\\begin{equation*}\nf(t,T) \\geq 0, \\quad \\textrm{for all } 0\\leq t\\leq T. \n\\end{equation*}\n\\end{remark}\n\n\\begin{remark}[Markov property of the short rate]\nAs pointed out in Chapter 5 of \\cite{brigo2007interest}, one of the main drawbacks of HJM theory is that the implied short rate dynamics is usually not Markovian. Here we simply remark that, since the volatility of the forward rate $\\sigma(t,T)$ is not affected by price impact, if the Markov property of the short rate $r(t)$ is ensured under the measure $\\mathbb{Q}$ when there is no trading, hence no price impact, then it is also preserved in the presence of price impact under the $\\tilde{\\mathbb{Q}}$.\n\\end{remark}\n\n\n\\section{Examples}{\\label{sec:examples}}\t\n\n\\subsection{Pricing impacted Eurodollar futures with Vasicek model}{\\label{subsec:vasicek_example}}\nIn this section we illustrate the argument outlined in Section \\ref{subsec:pricing} by computing the explicit price of a Eurodollar-futures contract when the underlying short rate follows an Ornstein-Uhlenbeck process \\cite{vasicek1977equilibrium}. The dynamics under the risk neutral measure $\\mathbb{Q}$ is given by\n\\begin{equation}\nd r(t) = k (\\theta - r(t)) dt + \\sigma d W^\\mathbb{Q}(t),\n\\label{sde:vasicek}\n\\end{equation}\nwith $k,\\theta,\\sigma$ positive parameters. The dynamics of the short rate under the real world measure $\\mathbb{P}$ can be expressed as\n\\begin{equation}\nd r(t) = k (\\theta - r(t)) dt + \\sigma (d W^\\mathbb{P}(t) - \\lambda(t) dt),\n\\label{short_rate_P_lambda}\n\\end{equation}\nwhere we highlight the classic market price of risk process $\\lambda$ defined in \\eqref{market_price_of_risk}. Another representation for $r(t)$ under $\\mathbb{P}$ is \n\\begin{equation}\nd r(t) = \\tilde{k} (\\tilde{\\theta} - r(t)) dt + \\sigma (d W^\\mathbb{P}(t) - \\tilde{\\lambda}(t) dt),\n\\label{short_rate_P_lambda_tilde}\n\\end{equation}\nwhere $\\tilde{\\lambda}$ is the impacted market price of risk defined in \\eqref{def:lambda_tilde} and $\\tilde k, \\tilde \\theta$ are positive constants. Combining the two equivalent representations \\eqref{short_rate_P_lambda} and \\eqref{short_rate_P_lambda_tilde}, we see that the following holds for any $t\\geq 0$\n\\begin{equation}\nk \\theta - k r(t) - \\sigma \\lambda(t) = \\tilde{k} \\tilde{\\theta} - \\tilde{k} r(t) - \\sigma \\tilde{\\lambda}(t).\n\\label{relationship_vasicek_pars}\n\\end{equation}\nSimilarly to what is done in the standard theory (see Brigo and Mercurio \\cite{brigo2007interest}, section 3.2.1), we assume the short rate $r(t)$ has the same kind of dynamics under the measures $\\mathbb{P}$, $\\mathbb{Q}$ and $\\tilde{\\mathbb{Q}}$, that is \n\\begin{equation}\n\\lambda(t) = \\lambda r(t), \\ \\ \\ \\tilde{\\lambda}(t) = \\tilde{\\lambda} r (t),\n\\label{lambda_lambda_t_vasicek}\n\\end{equation}\nwith $\\lambda,\\tilde{\\lambda}$ constants. The whole impact is then encapsulated in the constant $\\tilde{\\lambda}$. By plugging \\eqref{lambda_lambda_t_vasicek} into \\eqref{relationship_vasicek_pars}, we deduce\n\\begin{align}\\label{tilde_pars}\n\\begin{split}\n\\tilde{k} & = k - \\sigma (\\tilde{\\lambda} - \\lambda), \\\\ \n\\tilde{\\theta} & = \\frac{k \\theta}{k - \\sigma (\\tilde{\\lambda} - \\lambda)}. \n\\end{split}\n\\end{align}\nClearly, in order to ensure all parameters are positive, we must require\n\\begin{equation*}\nk > \\sigma (\\tilde{\\lambda} - \\lambda).\n\\end{equation*}\n\nIn this way, the short rate $r(t)$ is normally distributed under all three measures. In particular, plugging the Girsanov transformation from the measure $\\mathbb{P}$ to the measure $\\mathbb{\\tilde{Q}}$, defined in \\eqref{radon_nikodym_der_Q_tilde}, into equation \\eqref{short_rate_P_lambda_tilde}, the short rate dynamics under $\\tilde{\\mathbb{Q}}$ can be conveniently rewritten as\n\\begin{equation} \\label{vasicek_under_Q_tilde}\nd r(t) = \\tilde{k}(\\tilde{\\theta} - r(t)) dt + \\sigma d W^{\\tilde{\\mathbb{Q}}}(t).\n\\end{equation}\n\nSince the short rate under $\\tilde{\\mathbb{Q}}$ is Gaussian, $\\{\\int_t^T r(s) ds\\}_{t\\geq 0}$ is also a Gaussian process. At the same time, we recall the well known fact that if $X$ is a normal random variable with mean $\\mu_X$ and variance $\\sigma^2_X$, then $\\mathbb{E}(\\exp(X)) = \\exp (\\mu_X + \\frac{1}{2} \\sigma_X^2)$. Following the same argument as in (Brigo and Mercurio \\cite{brigo2007interest}, Chapters 3.2.1, 3.3.2 and Chapter 4), we can use \\eqref{vasicek_under_Q_tilde} in order to  express the impacted zero-coupon bond price as follows\n\\begin{equation*}\n\\tilde{P}(t,T) = A(t,T) e^{-B(t,T) r(t)}\n\\end{equation*}\nwhere\n\\begin{align}\\label{A_B_coeff_vasicek}\n\\begin{split}\nA(t,T) & = \\exp \\left\\{ \\left(\\tilde{\\theta} - \\frac{\\sigma^2}{2 \\tilde{k}^2} \\right) [B(t,T) - T + t] - \\frac{\\sigma^2}{4 \\tilde{k}} B^2(t,T) \\right\\}, \\\\\nB(t,T) & = \\frac{1}{\\tilde{k}} \\left(1 - e^{-\\tilde{k} (T-t)} \\right). \n\\end{split}\n\\end{align}\nHence, the key expectation needed to compute the impacted Eurodollar future fair price in equation \\eqref{fair_price_impacted_Eurodollar} is equal to\n\\begin{equation} \\label{eu-d}\n\\mathbb{E}^{\\tilde{\\mathbb{Q}}} \\left[ \\frac{1}{\\tilde{P}(t,T)} \\right] = \\frac{1}{A(t,T)} \\mathbb{E}^{\\tilde{\\mathbb{Q}}} \\left[e^{B(t,T) r(t)} \\right].\n\\end{equation}\nSince $r(t)$ is normally distributed, $B(t,T) r(t)$ will be normally distributed as well with mean and variance respectively equal to (see Brigo and Mercurio \\cite{brigo2007interest}, Eq. (3.7))\n\\begin{subequations}\n\\begin{align*}\n\\mathbb{E}^{\\tilde{\\mathbb{Q}}}[B(t,T) r(t)] & = B(t,T) \\left[r(0) e^{-\\tilde{k}t} + \\theta (1-e^{-\\tilde{k}t}) \\right], \\\\\n\\text{Var}^{\\tilde{\\mathbb{Q}}}[B(t,T) r(t)] & = B^2(t,T) \\left[ \\frac{\\sigma^2}{2 \\tilde{k}} (1 - e^{-2 \\tilde{k} t}) \\right].\n\\end{align*}\n\\end{subequations}\nTherefore in order to get the impacted price of a Eurodollar-future  contract we need to compute the expectation in the right hand side of \\eqref{eu-d} which can be written explicitly as\n\\begin{multline*}\n\\mathbb{E}^{\\tilde{\\mathbb{Q}}} \\left[ \\frac{1}{\\tilde{P}(t,T)} \\right] = \\frac{1}{A(t,T)} \\times \\\\\n\\times \\exp \\left\\{ B(t,T) [r(0) e^{- \\tilde{k} t} + \\theta (1-e^{- \\tilde{k}t})] + \\frac{1}{2} B^2(t,T) \\left[ \\frac{\\sigma^2}{2\\tilde{k}} (1-e^{-2 \\tilde{k} t}) \\right] \\right\\}.\n\\end{multline*}\nThe main conclusion here is that defining the short rate under the impacted risk neutral measure preserves analytical tractability of interest rate derivatives precises. \n\n\\subsection{Pricing impacted Eurodollar futures with Hull White model}{\\label{subsec:hullwhite_example}}\nIn this section we compute the explicit price of a Eurodollar-futures contract when the underlying short rate follows a Hull White model \\cite{hull1990pricing}. We start with the classic framework where there is not price impact. In this case the short rate is given by \n\\begin{equation*}\nd r(t) = \\left[ \\theta(t) - a r(t) \\right] dt + \\sigma d W^{\\mathbb{Q}}(t),\n\\end{equation*}\nwhere $a$ and $\\sigma$ are positive constants and the function $\\theta$ is chosen in order to fit exactly the term structure of interest rates being currently observed in the market. Denoting by $P^M(0,T)$ the unimpacted market discount factor for the maturity $T$ and defining the (unimpacted) market instantaneous forward rate at time $0$ for the maturity $T$\n\\begin{equation*}\nf^M(0,T) := - \\frac{\\partial}{\\partial T} \\ln P^M(0,T),\n\\end{equation*}\nthe function $\\theta$ is given by (see e.g. Brigo and Mercurio \\cite{brigo2007interest}, Chapter 3, Eq. (3.34))\n\\begin{equation*}\n\\theta(t) = \\frac{\\partial f^M(0,t)}{\\partial T} + a f^M(0,t) + \\frac{\\sigma^2}{2 a} \\left(1-e^{-2 a t} \\right),\n\\end{equation*}\nwhere $\\frac{\\partial f^M(0,t)}{\\partial T}$ denotes the partial derivative of $f^M$ with respect to its second variable. We start by computing the price under the classic risk neutral measure $\\mathbb{Q}$. According to eq. (3.36)--(3.37) in Chapter 3 of \\cite{brigo2007interest}, the short rate is normally distributed with mean and variance respectively equal to \n\\begin{subequations}\n\\begin{align*}\n\\mathbb{E}^\\mathbb{Q}[r(t) | \\mathcal{F}_s] & = r(s) e^{-a (t-s)} + \\alpha(t) - \\alpha(s) e^{-a (t-s)} \\\\\n\\text{Var}^\\mathbb{Q}[r(t) | \\mathcal{F}_s] & = \\frac{\\sigma^2}{2 a} \\left[1 - e^{-2 a (t-s)} \\right],\n\\end{align*}\n\\end{subequations}\nwhere \n\\[\n\\alpha(t) := f^M(0,t) + \\frac{\\sigma^2}{2 a^2} (1-e^{-a t})^2.\n\\]\nAs before, we notice that the integral of the short rate will be normally distributed as well, hence the price of a zero-coupon bond under the classic risk neutral measure is given by (see eq. (3.39) in Chapter 3 of \\cite{brigo2007interest}), \n\\begin{equation*}\nP(t,T) = A(t,T) e^{-B(t,T) r(t)},\n\\end{equation*}\nwhere\n\\begin{align*}\nA(t,T) &= \\frac{P^M(0,T)}{P^M(0,t)} \\exp \\left\\{B(t,T) f^M(0,t) - \\frac{\\sigma^2}{4 a} (1-e^{-2 a t}) B^2(t,T)\\right\\},\\\\\nB(t,T) &= \\frac{1}{a} \\left[1-e^{-a (T-t)} \\right].\n\\end{align*}\nMoreover, the term $B(t,T) r(t)$ is still normally distributed and we immediately have\n\\begin{subequations}\n\\begin{align*}\n\\mathbb{E}^\\mathbb{Q}[ B(t,T) r(t) | \\mathcal{F}_s] & = B(t,T) \\left(r(s) e^{-a (t-s)} + \\alpha(t) - \\alpha(s) e^{-a (t-s)} \\right), \\\\\n\\text{Var}^\\mathbb{Q}[ B(t,T) r(t) | \\mathcal{F}_s] & = B^2(t,T) \\frac{\\sigma^2}{2 a} \\left[1 - e^{-2 a (t-s)} \\right].\n\\end{align*}\n\\end{subequations}\nThis implies that the expectation we are interested in, under the classic risk neutral measure $\\mathbb{Q}$, can be written explicitly as (see Section 13.12.1 in \\cite{brigo2007interest})\n\\begin{multline*}\n\\mathbb{E}^\\mathbb{Q} \\left[ \\frac{1}{P(t,T)} \\right] = \\frac{1}{A(t,T)} \\exp \\left\\{B(t,T) \\mathbb{E}^\\mathbb{Q}[r(t)] + \\frac{1}{2} B^2(t,T) \\text{Var}^\\mathbb{Q}[r(t)] \\right\\}.\n\\end{multline*}\nNext we derive the corresponding expression under the impacted risk neutral measure $\\tilde{\\mathbb{Q}}$ in \\eqref{radon_nikodym_der_Q_tilde}. We assume as in Section \\ref{subsec:vasicek_example} that the market price of risk and impacted market price of risk are given by\n\\[\n\\lambda(t) = \\lambda r(t), \\ \\ \\ \\tilde{\\lambda}(t) = \\tilde{\\lambda} r(t),\n\\]\nfor some constants $\\lambda,\\tilde{\\lambda}$. Using the Girsanov change of measure from $\\mathbb{Q}$ to $\\tilde{\\mathbb{Q}}$ defined in Section \\ref{subsec:rnm}, it follows that the short rate under $\\tilde{\\mathbb{Q}}$ is given by  \n\\begin{subequations}\n\\begin{align*}\nd r(t) & = \\left[ \\theta(t) - a r(t) \\right] dt + \\sigma d W^{\\mathbb{Q}}(t) \\\\\n& = \\left[\\theta(t)- a r(t) \\right] dt + \\sigma d W^{\\tilde{\\mathbb{Q}}}(t) + \\sigma(\\tilde{\\lambda} - \\lambda) r(t) dt \\\\\n& = \\left[ \\theta(t) - (a - \\sigma (\\tilde{\\lambda} - \\lambda)) r(t) \\right] dt + \\sigma d W^{\\tilde{\\mathbb{Q}}}(t).\n\\end{align*}\n\\end{subequations}\nHence, we can define the impacted parameter\n\\begin{equation*}\n\\tilde{a} := a - \\sigma (\\tilde{\\lambda} - \\lambda).\n\\end{equation*}\nThe pricing formula for $\\mathbb{E}^{\\tilde {\\mathbb{Q}}}\\left[ \\frac{1}{P(t,T)} \\right]$ is then derived by following the same steps as in the classic case. Similarly to the Vasicek model, analytical tractability is preserved.\n\n\\section{Numerical results}{\\label{sec:numerical_results}}\nIn this section we give a few numerical examples for the behaviour of the yield curve under price impact in the framework of short-rate affine models, which was described in Section \\ref{subsec:yield}. In order to compute the cross price impact, we need the drift and the volatility of the zero-coupon bond. For the sake of simplicity, we assume the short rate is described by a Vasicek model \\eqref{sde:vasicek}\n\\begin{equation*}\nd r(t) = k (\\theta - r(t)) dt + \\sigma d W^\\mathbb{Q}(t),\n\\end{equation*}\nwith $k,\\theta,\\sigma$ positive parameters. Then, the drift and the diffusion coefficients of the unimpacted zero-coupon bond are given by \\eqref{affine:drift_vol_zero_coupon_bond}:\n\\begin{align*}\n\\mu_T(t,r(t)) &= e^{-B(t,T) r(t)} \\bigg[\\frac{\\partial A}{\\partial t} - A(t,T) \\frac{\\partial B}{\\partial t} r(t) + \\frac{1}{2} A(t,T) B^2(t,T) \\sigma^2& \\\\\n&\\quad - A(t,T) B(t,T) k (\\theta -r(t)) \\bigg], \\\\\n\\sigma_T(t,r(t))& = - \\sigma B(t,T) A(t,T) e^{-B(t,T) r(t)},\n\\end{align*}\nwhere the functions $A,B$ are given as in \\eqref{A_B_coeff_vasicek} and their derivatives are given by\n\\begin{equation*}\n\\frac{\\partial B}{\\partial t} = - e^{-k (T-t)}, \\ \\ \\ \\\n\\frac{\\partial A}{\\partial t} = A(t,T) \\left[ \\left(\\theta - \\frac{\\sigma^2}{2 k^2} \\right) \\left(\\frac{\\partial B}{\\partial t} + 1 \\right) - \\frac{\\sigma^2}{2 k} B(t,T) \\frac{\\partial B}{\\partial t} \\right].\n\\end{equation*}\nWe can then plug all these quantities in equation \\eqref{sde:cross_impacted_bond_S} to determine the dynamics of the cross-impacted zero-coupon bond and therefore the corresponding impacted yield. We set the following values for the parameters in \\eqref{sde:vasicek}:\n\\begin{equation*}\nk=0.20, \\quad \\theta = 0.10, \\quad \\sigma = 0.05, \\quad r_0 = 0.01. \n\\end{equation*}\nWe consider zero-coupon bonds with maturities $ \\mathbb{T} := \\left\\{1,2,5,10,15 \\right\\}$ years and assume that an agent is trading on the bond with maturity $T=5$ years. All the other zero-coupon bonds experience cross price impact during the trading period. We fix the execution time horizon to be $\\tau = 10$ days. All bonds are simulated over the time interval $[0, 9\\ \\text{months}]$, discretized in $N=365$ subintervals with time step $\\Delta t = 1\/365$. The short rate $r$ defined in \\eqref{sde:vasicek} is simulated via Euler-Maruyama scheme. Since we are going to describe the average behaviour of the yield curve under market impact, we also set the number of Monte Carlo simulations to $M=10.000$. As we shall see below in the detailed algorithm, for each realization of the short rate, we will have a corresponding impacted yield curve. The idea is then to plot the average of such curves.\n\nFor the sake of simplicity, we discuss the benchmark trading strategy\n\\begin{equation}\\label{def:benchmark_strategy}\nv_T(s):=\n\\begin{cases} c, & \\mbox{if } s \\leq \\tau \\\\ 0, & \\mbox{otherwise }\n\\end{cases}\n\\end{equation}\nwith $c$ some positive constant if we buy, negative if we sell. In our simulations we choose $c=2$. The transient impact defined in \\eqref{def:transient_impact} reads as\n\\begin{equation}\n\\Upsilon_T^v(t) = y e^{-\\rho t} + \\gamma e^{-\\rho t} \\int_0^t e^{\\rho s} c \\mathbbm{1}_{s \\leq \\tau} ds,\n\\label{benchmark_transient_impact}\n\\end{equation}\nwhere the  parameters are set to\n\\begin{equation*}\n\\rho = 2, \\ \\ \\ \\gamma = 1, \\ \\ \\ y = 0.01.\n\\end{equation*}\nThe functions $l,K$ introduced in \\eqref{impacted_bond} are assumed to be of the form\n\\begin{equation*}\nl(t,T) = \\kappa \\left(1-\\frac{t}{T}\\right)^\\alpha, \\ \\ \\ K(t,T) =  \\left(1-\\frac{t}{T}\\right)^\\beta\n\\end{equation*}\nwith $\\kappa \\geq 0, \\alpha,\\beta \\geq 1$. In particular, we choose\n\\begin{equation*}\n\\alpha = 1, \\ \\ \\beta=1, \\ \\ \\kappa =0.01. \n\\end{equation*}\nFollowing \\eqref{impacted_bond_differential_form}, the price of the impacted bond in $T=5$y is \n\\begin{equation*}\n\\tilde{P}(t,T) = P(t,T) + \\int_0^t J_T(s) ds,\n\\end{equation*}\nwhere $J_T$, which was defined in \\eqref{impact_density}, is specified to be\n\\begin{equation*}\nJ_T(t) = - \\frac{\\kappa}{T} v_T(t) + \\left(1-\\frac{t}{T} \\right) \\left[-\\rho \\Upsilon_T^v + v_T(t) \\right] - \\Upsilon_T^v(t)\n\\end{equation*}\nThe algorithm we implemented to simulate the impacted yield curve consists of the following steps.\n\\begin{steps}\n\\item Simulate a path of the short rate $r(t)$ given in \\eqref{sde:vasicek} for $t \\in [0,9\\ \\text{months}]$. \n\\item Compute the unimpacted zero-coupon bond price $P(t,T)$ for the trading maturity  $T=5$ years using equation \\eqref{affine:bond_price} for $t \\in [0,9\\ \\text{months}]$.\n\\item Compute the unimpacted yield $Y(t,T)$ by plugging $P(t,T)$ in \\eqref{def:classic_yield} for $t \\in [0,9\\ \\text{months}]$.\n\\item Compute the (directly) impacted zero-coupon bond $\\tilde{P}(t,T)$ using \\eqref{impacted_bond_differential_form} for $t \\in [0,9\\ \\text{months}]$.\n\\item Compute the (directly) impacted yield $\\tilde{Y}(t,T)$ by plugging $\\tilde{P}(t,T)$ into \\eqref{def:impacted_yield} for $t \\in [0,9\\ \\text{months}]$.\n\\item For all other maturities $S=1,2, 10, 15$ years, compute the cross impacted zero-coupon bond price $\\tilde{P}(t,S)$ using equation \\eqref{sde:cross_impacted_bond_S} for $t \\in [0,9\\ \\text{months}]$.\n\\item Compute the cross impacted yield $\\tilde{Y}(t,S)$ by plugging $\\tilde{P}(t,S)$ into \\eqref{def:impacted_yield} for $t \\in [0,9\\ \\text{months}]$.\n\\item Repeat these steps $M=10.000$ times and compare the average of $Y(t,T)$ with the average of $\\tilde{Y}(t,T)$.\n\\end{steps}\n\nIn Figure \\ref{fig:imp_unimp_yield_average} we visualize for all maturities the average classic yield $\\mathbb{E}[Y(t,T)]$ versus the average impacted yield $\\mathbb{E}[\\tilde{Y}(t,T)]$ at times $t=5$ days (middle of trading), $t=11$ days (right after trading is ended) and $t=270$ days (after $9$ months). \n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=.50]{imp_unimp_yield_average_intermediate.png}\n\\caption{Trading zero-coupon bond with maturity $T=5$ years. Average unimpacted yield curve and average impacted yield curve in the middle of trading (top left panel), right after trading is concluded (top right panel) and after nine months (bottom panel).}\n\\label{fig:imp_unimp_yield_average}\n\\end{figure}\nIn the top panels we see that the yield has decreased over all maturities as result of trading. This is consistent with the fact that a buy strategy of bonds pushes their prices up due to price impact, hence the yield decreases. Clearly, the almost parallel shift of the yield curve is a just a consequence of the very simple (constant) trading strategy we defined in equation \\eqref{def:benchmark_strategy}. We expect to observe much more complicated behaviours when implementing more sophisticated strategies. In the bottom panel, instead, we observe that, roughly nine months after performing the trades, the two yield curves pretty much coincide. This is due to the transient component in the price impact model, which induces impacted yield curve to converge to its classic counterpart as time goes by. When analysing price impact due to zero-coupon bond trading, one aspect that certainly can't be ignored is the special nature of the assets we are trading. Unlike what happens with stocks, the time evolution of zero-coupon bonds is constrained, specifically by the fact that they must reach value $1$ at maturity. It therefore appears that two fundamental forces are in play: the intrinsic \\emph{pull to par} effect, which makes both the impacted and unimpacted bond price go to $1$, hence the corresponding yields to $0$, and the \\emph{price impact} effect, which induces the bond price to first increase (if we buy) or decrease (if we sell), and then revert back to its unimpacted value. Interestingly, when trading stocks, it will take the impacted asset forever to converge to the unimpacted counterpart, as the transient impact converges to $0$ as time $t$ goes to infinity. When trading bonds, though, this convergence occurs in finite time. In order to better understand the role played by price impact, in Figure \\ref{fig:imp_classic_bonds} we compare directly the behaviour of the impacted bond $\\tilde{P}(t,T)$ and of the classic bond $P(t,T)$ for different maturities $T$.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=.50]{imp_classic_bonds.png}\n\\caption{Trading bond with maturity $T=5$ years. Averaged impacted zero-coupon bond vs averaged classic zero-coupon bond for maturities $1$ year (top left panel), $2$ years (top right panel), $5$ years (middle left panel), $10$ years (middle right panel), $15$ (bottom panel). All curves are observed over the interval $[0, 1\\ \\text{year}]$.}\n\\label{fig:imp_classic_bonds}\n\\end{figure}\nWe observe that, over one year, the pull-to-par effect is somehow stronger than the transient impact effect in bonds with short maturity $(T=1,2)$. By this, we mean that the unimpacted and impacted bonds meet at, or very close to, maturity. \\footnote{It can be observed that the price of the cross-impacted zero-coupon bond with maturity $S=1$ year is not $1$ at expiration, as it should be, but slightly higher (top left panel). This is not a numerical error, but rather a consequence of our model not being able to ensure the cross-impacted bonds reach value precisely $1$ at their respective maturities. See Remark \\ref{rem:cross_impacted_bonds_at_maturity}.} For bonds with long maturity ($T=5,10,15$), instead, the transient effect is prominent. This causes the impacted bond curve and the unimpacted bond curve to cross each other significantly before their maturity. In fact, we can numerically compute the first instant the two curves meet and we observe that the longer the maturity, the sooner this happens. This is illustrated in Figure \\ref{fig:first_hitting_time}.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=.50]{first_hitting_time.png}\n\\caption{Trading bond with maturity $T=5$ years. First instant (in days) impacted bond curve and unimpacted bond curve cross for maturities $5,10,15$ years. All curves are observed over the interval $[0, 1\\ \\text{year}]$.}\n\\label{fig:first_hitting_time}\n\\end{figure}\n\nThe interplay between the cross price impact effect, averaged over $10.000$ realizations, and the pull to par effect is demonstrated in Figure \\ref{fig:average_imp_bond_1_rho} for the price of a zero-coupon bond with maturity $S=1$ year when trading a bond with maturity $T=5$ years. Trading takes place on the first 10 days of the year, while the time scale in the graph is of one year. We illustrate this effect for various values of the transient impact parameter $\\rho$ in equation \\eqref{benchmark_transient_impact}.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=.50]{average_imp_bond_1_rho.png}\n\\caption{Averaged cross price impact effect vs. pull to par effect over $10.000$ realizations is demonstrated for the price of a bond with maturity $S=1$ year when trading a bond with maturity $T=5$ years, for various values of $\\rho$. Trading takes place on the first 10 days of the year, while the time scale in the graph is of one year.}\n\\label{fig:average_imp_bond_1_rho}\n\\end{figure}\nIt can be observed that the higher $\\rho$, the more aggressively the price is \"pulled down\" close to the original price before the trades. At the beginning, far from maturity, the transient impact component dominates and the price decreases. After some time, though, the bond intrinsic nature takes over and the price starts to increase.\n\nAnother phenomenon which is revealed in our framework is the interplay between the mean reversion of the short rate model and the price impact. Recall that in Section \\ref{subsec:vasicek_example} we found that the mean reversion speed $k$ under the measure $\\mathbb{Q}$ and the mean reversion $\\tilde{k}$ under the price-impacted measure $\\tilde{\\mathbb{Q}}$ are linked by \\eqref{relationship_vasicek_pars} as follows\n\\[\n\\tilde{k} = k - \\sigma(\\tilde{\\lambda}-\\lambda),\n\\]\nwith $\\tilde{\\lambda}, \\lambda$ representing the impacted market price of risk and the classic market price of risk respectively. We stress that the higher $k$, the faster the short rate $r$ under $\\mathbb{Q}$ and its counterpart under $\\tilde{\\mathbb{Q}}$ converge to their respective stationary distributions. At the same time, since the variance of the stationary distribution is $\\sigma^2\/(2k)$, large values of $k$ reduce the overall variance of the model, thereby making the two types of rates that we consider closer to each other. This, in turn, implies that after a long time ($T=10,15$ years) the zero-coupon bond $P$ and impacted zero-coupon bond $\\tilde{P}$, hence their yields, will be closer to each other. Conversely, if $k$ is small, the two short rates are quite far from each other and the overall variance of the model is large. Furthermore, looking again at \\eqref{relationship_vasicek_pars}, we notice that the larger $k$, the less significant the impact component $-\\sigma (\\tilde{\\lambda} - \\lambda)$, and vice versa. In a way, the speed of mean reversion works in an opposite direction to the price impact. We demonstrate this in Figure \\ref{fig:yield_curve_mean_reversion} for $k=0.01$ (top panel) and for $k=0.20$ (bottom panel). As above, we trade the zero-coupon bond with maturity $T=5$ years, trading occurs for the first $10$ days and the yields are observed after $9$ months. The difference in behaviour is evident for long maturities $(T=5,10,15)$. While in the bottom panel unimpacted yield and impacted yield are really close to each other (as in Figure \\eqref{fig:imp_unimp_yield_average}, right panel), in the top panel the distance between the two yields is rather significant.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=.50]{yield_curve_mean_reversion.png}\n\\caption{Impacted and unimpacted yield curves for $k=0.01$ (top panel) and for $k=0.20$ (bottom panel) when trading zero-coupon bond with maturity $T=5$ years. Trading occurs during the first $10$ days. Yield curves are observed after nine months.}\n\\label{fig:yield_curve_mean_reversion}\n\\end{figure}\n\n\\section{Optimal execution of bonds in presence of price impact }\\label{sec:optexec}\nIn this section we consider a problem of an agent who tries to liquidate a large inventory of $T$-bonds within a finite time horizon $[0,\\tau]$ where $\\tau < T$. We assume that the agent's transactions create both temporary and transient price impact and that the performance of the agent is measured by a revenue-cost functional that captures the transaction costs which result by price impact, and the risk of holding inventory for long time periods.  Our optimal execution framework is closely related to the framework which was proposed for execution of equities in Section \\ref{subsec:def} of \\cite{neuman2020optimal}. The main difference between the two frameworks is that in our framework the price impact has to vanish at the bond's maturity in order to satisfy the boundary condition $\\tilde P(T,T)=1$. \n\nLet $T>0$ denote the bond's maturity. We assume that the unimpacted bond price $P(\\cdot,T)$ is given by \\eqref{class-b} and we consider the canonical decomposition $P(\\cdot,T)= A(\\cdot,T) + \\bar{M}(\\cdot,T)$, where \n\\begin{equation*}\nA(t,T) := \\int_0^t \\mu_T(s,r(s)) ds, \\quad 0\\leq t\\leq T, \n\\end{equation*}\nis a predictable finite-variation process and  \n\\begin{equation*}\n\\bar{M}(t,T) := \\int_0^t \\sigma_T(s,r(s)) d W^{\\mathbb{P}}(s) , \\quad 0\\leq t\\leq T, \n\\end{equation*}\nlocal martingale. We assume that the coefficients $\\sigma_{T},\\mu_{T}$ in \\eqref{class-b} are such that we have \n\\begin{equation} \\label{def: mathcal_h_squared} \n\\mathbb{E}[\\langle \\bar{M}(\\cdot,T) \\rangle_\\tau] + \\mathbb{E} \\left[ \\left( \\int_0^\\tau |d A (\\cdot,T)| \\right)^2 \\right] < \\infty.\n\\end{equation} \nIn this case we say that a bond price $\\{P(t,T)\\}_{t \\in [0,T]}$ is in $\\mathcal{H}^2$.\n\nThe initial position of the agent's inventory is denoted by $x>0$ and the number of shares the agent holds at time $t\\in [0,\\tau]$ is given by \n\\begin{equation} \\label{def:X}\nX^{v_T}(t)\\triangleq x-\\int_0^t v_T(s)ds\n\\end{equation}\nwhere $\\{v_T(t)\\}_{t \\in [0,\\tau]}$ denotes the trading speed. We say that the trading speed is admissible if it belongs to the following class of admissible strategies \n\\begin{equation} \\label{def:admissset} \n\\mathcal A \\triangleq \\left\\{ v_T \\, : \\, v_T \\textrm{ progressively measurable s.t. } \\mathbb E\\left[ \\int_0^\\tau v^2_T(s) ds \\right] <\\infty \\right\\}.\n\\end{equation} \nWe assume that the trader's trading activity causes price impact on the bond's price as described by $\\{\\tilde P(t,T)\\}_{t \\in [0,T]}$ in \\eqref{impacted_bond}. \n\nAs in Section 2 of \\cite{neuman2020optimal}, we now suppose that the trader's optimal trading objective is to unwind her initial position $x>0$ in the presence of temporary and transient price impact through maximizing the following performance functional \n\\begin{equation}  \\label{j-fun}\n\\begin{aligned}\n\\mathcal{J}(v) := \\mathbb{E} \\bigg [\\int_0^\\tau \\bigg(P(t,T) - K(t,T) \\Upsilon_T^v(t) \\bigg) v_T(t) dt - \\int_0^\\tau l(t,T) v_T^2(t) dt + X_T^v(\\tau) P(\\tau,T)  \\\\\n- \\phi \\int_0^\\tau (X_T^v(t))^2 dt - \\varrho (X_T^v(\\tau))^2 \\bigg].\n\\end{aligned}\n\\end{equation} \nThe first, second and third terms in $\\mathcal J$ represent the trader's terminal wealth, meaning the final cash position including the accrued trading costs induced by temporary and transient price impact, as well as the remaining final risky asset position's book value. The fourth and fifth terms, instead, account for the penalties $\\phi,\\varrho>0$ on the trader's running penalty (i.e. the risk aversion term) and the penalty of holding any terminal inventory, respectively. \n\nSince $T$ is fixed, for the sake of readability we will omit the subscripts $T$ for the rest of this section. \nThe main result of this section is the derivation of the unique optimal admissible strategy, namely \n\\begin{equation}\n\\mathcal{J}(v) \\to \\max_{v \\in \\mathcal{A}}.\n\\label{opt_stoch_control}\n\\end{equation}\nand exhibiting an explicit expression for the optimal trading strategy. We define\n\\begin{equation}\nA(t) :=\n\\left(\n\\begin{matrix}\n0 & 0 & -1 & 0 \\\\\n0 & -\\rho & \\gamma & 0 \\\\\n-2 \\phi \\Lambda(t) & \\begin{matrix} \\rho K(t,T) \\Lambda(t)\\\\ \\hfill{} - \\Lambda'(t) K(t,T) - \\Lambda(t) \\partial_t K(t,T) \\end{matrix}  & 0 & \\Lambda'(t) + \\rho \\Lambda(t) \\\\\n0 & 0 & K(t,T) \\gamma & \\rho\n\\end{matrix}\n\\right),\n\\label{def:time_dep_matrix_A}\n\\end{equation}\nwhere\n\\begin{equation}\n\\Lambda(t) := \\frac{1}{2 l(t,T)}. \n\\label{def:Lambda_optimal_exe}\n\\end{equation}\nNote that $\\Lambda(t)$ is well defined for $0\\leq t \\leq \\tau$ since $l(t,T)>0$ on this interval by \\eqref{l-pos}. Let $\\Phi$ be the fundamental solution of the matrix-valued ordinary differential equation\n\\begin{align}\\label{matrix_ODE}\n\\begin{split}\n\\frac{d}{dt} \\Phi(t) & = A(t) \\Phi(t), \\\\\n\\Phi(0) & = \\text{Id}.\n\\end{split}\n\\end{align}\nLet us define the matrix\n\\begin{equation}\\label{def:psi_matrix}\n\\Psi(t,\\tau) := \\Phi^{-1}(\\tau) \\Phi(t). \n\\end{equation}\nWe also define the vector $G$:\n\\begin{align}\\label{def:G_components}\n\\begin{split}\nG^1(t,\\tau) & := \\frac{\\varrho}{l(\\tau,T)} \\Psi^{11}(t,\\tau) - \\frac{K(\\tau,T)}{2 l(\\tau,T)} \\Psi^{21}(t,\\tau) - \\Psi^{31}(t,\\tau) ,\\\\\nG^2(t,\\tau) & := \\frac{\\varrho}{l(\\tau,T)} \\Psi^{12}(t,\\tau) - \\frac{K(\\tau,T)}{2 l(\\tau,T)} \\Psi^{22}(t,\\tau) - \\Psi^{32}(t,\\tau)  ,\\\\\nG^3(t,\\tau) & := \\frac{\\varrho}{l(\\tau,T)} \\Psi^{13}(t,\\tau) - \\frac{K(\\tau,T)}{2 l(\\tau,T)}\\Psi^{23}(t,\\tau) - \\Psi^{33}(t,\\tau) , \\\\\nG^4(t,\\tau) & := \\frac{\\varrho}{l(\\tau,T)} \\Psi^{14}(t,\\tau) - \\frac{K(\\tau,T)}{2 l(\\tau,T)}\\Psi^{24}(t,\\tau) - \\Psi^{34}(t,\\tau) . \n\\end{split}\n\\end{align}\nNext, we define the process\n\\begin{equation}\n\\Gamma^{\\hat{v}}(t) := \\frac{\\Lambda'(t)}{\\Lambda(t)} \\left(P(t,T) + \\tilde{M}(t) - 2 \\phi \\int_0^t X^{\\hat{v}}(u) du \\right),\n\\label{def:Gamma_quantity}\n\\end{equation}\nwhere $\\tilde{M}$ is the square integrable martingale \n\\begin{equation}\\label{def:square_int_mart_M_tilde}\n\\tilde{M}(s) := \\mathbb{E}_s \\bigg[2 \\phi \\int_0^\\tau X^{\\hat{v}}(u) du + 2 \\varrho X^{\\hat{v}}(\\tau) - P(\\tau,T) \\bigg], \n\\end{equation}\nand $\\mathbb{E}_t$ denotes the expectation conditioned on the filtration $\\mathcal{F}_t$ for all $t \\in [0,\\tau]$. Finally we define the following functions on $0\\leq t \\leq \\tau$,\n\\begin{align}\\label{def:v_terms}\n\\begin{split}\nv_0(t,\\tau) & :=  \\left(1-\\frac{G^4(t,\\tau) \\Psi^{43}(t,\\tau)}{G^3(t,\\tau) \\Psi^{44}(t,\\tau)} \\right)^{-1}, \\\\\nv_1(t,\\tau) & := \\left(\\frac{G^4(t,\\tau) \\Psi^{41}(t,\\tau)}{G^3(t,\\tau) \\Psi^{44}(t,\\tau)} - \\frac{G^1(t,\\tau)}{G^3(t,\\tau)} \\right), \\\\\nv_2(t,\\tau) & := \\left(\\frac{G^4(t,\\tau) \\Psi^{42}(t,\\tau)}{G^3(t,\\tau) \\Psi^{44}(t,\\tau)} - \\frac{G^2(t,\\tau)}{G^3(t,\\tau)} \\right) ,\\\\\nv_3(t,\\tau) & := \\frac{G^4(t,\\tau)}{G^3(t,\\tau)}.\n\\end{split}\n\\end{align}\nIn order for the optimal strategy to be well defined, we will need additional assumptions. Note that if $l, K$ are positive constants these assumptions translate to Assumption 3.1 and Lemma 5.5 in \\cite{neuman2020optimal}. \n\\begin{assumption} \\label{ass-opt}  \n\\begin{enumerate}[label=(A.\\arabic*), ref=A.\\arabic*] We assume that the following hold:\n\\item \\label{A.1}\n\\[\n\\inf_{0\\leq t \\leq \\tau} |G^4(t,\\tau) \\Psi^{43}(t,\\tau) - G^3(t,\\tau) \\Psi^{44}(t,\\tau)| > 0,\n\\] \n\\item \\label{A.2}\n\\[\n\\sup_{0 \\leq t \\leq \\tau} |\\Psi^{4j}(t,\\tau)| < \\infty, \\ \\ \\sup_{0 \\leq t \\leq \\tau} |G^j(t,\\tau)| < \\infty, \\ \\ \\ j \\in \\left\\{1,2,3,4\\right\\}\n\\]\n\\item \\label{A.3}\n\\[\n\\inf_{0\\leq t\\leq \\tau}  |\\Psi^{44}(t,\\tau)| >0, \\quad  \\inf_{0\\leq t\\leq \\tau} | G^3(t,\\tau)|>0.\n\\]\n\\end{enumerate}\n\\end{assumption}\n\\begin{remark}\nAt this point we stress the fact that the conditions in Assumption \\ref{ass-opt} are not very general, however the purpose of this section is to show how to incorporate optimal execution into the impacted bonds framework. Future work may improve the theoretical results on this topic. \n\\end{remark} \nNext we present the main result of this section, which derives the unique optimal trading speed.\n\n\\begin{theorem}[Optimal trading strategy]\n\\label{linear_feedback_form}\nUnder Assumption \\ref{ass-opt}, there exists a unique optimal strategy $\\hat{v} \\in \\mathcal{A}$ which maximises \\eqref{opt_stoch_control} and it is given by the following feedback form \n\\begin{align}\\label{optimal_strategy_linear_feedback}\n\\begin{split}\nv(t) & = v_0(t,\\tau) \\Bigg( v_1(t,\\tau) X^v(t) + v_2(t,\\tau) \\Upsilon^v(t) \\\\\n& \\qquad \\qquad + v_3(t,\\tau) \\mathbb{E}_t \\left[ \\int_t^\\tau \\frac{\\Lambda(s) \\Psi^{43}(s,\\tau)}{\\Psi^{44}(t,\\tau)} (\\mu(s) + \\Gamma^{v}(s)) ds \\right] \\\\\n&  \\qquad \\qquad - \\mathbb{E}_t \\left[\\int_t^\\tau \\Lambda(s) \\frac{G^3(s,\\tau)}{G^3(t,\\tau)} (\\mu(s) + \\Gamma^{v}(s)) ds \\right] \\Bigg), \n\\end{split}\n\\end{align}\nfor all $t \\in (0,\\tau)$.\n\\end{theorem}\nThe proof Theorem \\ref{linear_feedback_form} is given in Section \\ref{sec:proof_linear_feedback}.\n\n\\section{Proofs of the results from Section \\ref{sec:main}}{\\label{sec:proofs}}\n\n\\begin{proof} [Proof of Theorem \\ref{thm_impacted_market_price}]\n\nWe adapt the argument by Bjork in Section 3.2 of \\cite{bjork1997interest} to our case. We fix two maturities $T$ and $S$, and we consider a portfolio $V$ consisting of $S$-bonds and $T$-bonds. \nWe further assume that both bonds are traded with admissible trading speeds $v_T$ and $v_S$ which correspond by \\eqref{impact_density} to impact densities $J_T$ and $J_S$.\n\nFrom \\eqref{class-b} and \\eqref{impacted_bond_differential_form} we can write the dynamics of the impacted bonds as follows: \n\\begin{equation}\\label{dyn_imp_bonds}\n\\begin{aligned}\nd \\tilde{P}(t,T) = & \\mu_T(t,r(t)) dt + J_T(t) dt + \\sigma_T(t,r(t)) d W^\\mathbb{P}(t), \\\\\nd \\tilde{P}(t,S) = & \\mu_S(t,r(t)) dt + J_S(t) dt + \\sigma_S(t,r(t)) d W^\\mathbb{P}(t).\n\\end{aligned}\n\\end{equation}\nLet $\\tilde{h}_T,\\tilde{h}_S$ by locally bounded predictable processes representing the weights of the $T$ and $S$ bonds, respectively. We denote by $\\tilde{V}(t)$ the portfolio value process, i.e.\n\\begin{equation*}\n\\tilde{V}(t) \\equiv \\tilde{V}(t;\\tilde{h}) := \\tilde{h}_T(t) \\tilde{P}(t,T) + \\tilde{h}_S(t) \\tilde{P}(t,S).\n\\end{equation*}\nSince, by assumption, the impacted-portfolio is self-financing, it holds at any time $t$ (see Definition \\ref{def:self_financing})\n\\begin{equation*}\nd \\tilde{V}(t;\\tilde{h}) = \\tilde{h}_T(t) d \\tilde{P}(t,T) + \\tilde{h}_S(t) d \\tilde{P}(t,S).\n\\end{equation*}\nIt is convenient to define the relative (impacted) weights\n\\begin{equation*}\n\\alpha_{T_i}(t) := \\frac{\\tilde{h}_{T_i}(t) \\tilde{P}(t,T_i)}{\\tilde{V}(t;\\tilde{h})},\\ \\ T_i \\in \\left\\{T,S\\right\\},\n\\end{equation*}\nWe conclude that if the impacted portfolio is self financing, then\n\\begin{equation}\n\\frac{d \\tilde{V}(t)}{\\tilde{V}(t)} = \\alpha_T(t) \\frac{d \\tilde{P}(t,T)}{\\tilde{P}(t,T)} + \\alpha_S(t) \\frac{d \\tilde{P}(t,S)}{\\tilde{P}(t,S)}.\n\\label{self_fin_imp_portfolio_relative}\n\\end{equation}\nIn order to ease the notation, we suppress the dependence on $r(t)$ in the drift and volatility. Substituting the dynamics \\eqref{dyn_imp_bonds} into \\eqref{self_fin_imp_portfolio_relative}, we have\n\\begin{multline}\n\\frac{d \\tilde{V}(t)}{\\tilde{V}(t)} = \\frac{\\alpha_T(t)}{\\tilde{P}(t,T)} (\\mu_T(t) - J_T(t)) dt + \\frac{\\alpha_S(t)}{\\tilde{P}(t,S)} (\\mu_S(t) - J_S(t)) dt + \\\\\n+ \\left( \\alpha_S(t) \\frac{\\sigma_S(t)}{\\tilde{P}(t,S)} + \\alpha_T(t) \\frac{\\sigma_T(t)}{\\tilde{P}(t,T)} \\right) d W^\\mathbb{P}(t).\n\\label{relative_dynamics}\n\\end{multline}\nAt this point, we choose the relative weights so that the diffusive part of the equation above vanishes, that is, \n\\begin{equation}\\label{sys1}\n\\begin{aligned}\n\\alpha_T(t) + \\alpha_S(t) & = 1, \\\\\n\\alpha_T(t) \\frac{\\sigma_T(t)}{\\tilde{P}(t,T)} + \\alpha_S(t) \\frac{\\sigma_S(t)}{\\tilde{P}(t,S)} & = 0.\n\\end{aligned}\n\\end{equation}\nSolving this system gives \n\\begin{equation}\\label{expression_relative_weights}\n\\begin{aligned}\n\\alpha_S(t) & = \\frac{\\sigma_T(t)\/\\tilde{P}(t,T)}{\\sigma_T(t)\/\\tilde{P}(t,T) - \\sigma_S(t)\/\\tilde{P}(t,S)}, \\\\\n\\alpha_T(t) & = - \\frac{\\sigma_S(t)\/\\tilde{P}(t,S)}{\\sigma_T(t)\/\\tilde{P}(t,T) - \\sigma_S(t)\/\\tilde{P}(t,S)}.\n\\end{aligned}\n\\end{equation}\nNotice that the above expressions are well defined. Indeed, if the denominator was approaching zero, then the sum of the two weights would be zero and this would contradict \\eqref{sys1}. Next, we substitute \\eqref{expression_relative_weights} into \\eqref{relative_dynamics}. Following again Bjork's argument, we use the fact that our impacted portfolio is locally risk-free (as in Definition \\ref{def:locally_risk_free}) by assumption and deduce the following relationship must hold:\n\\begin{multline*}\n\\frac{\\mu_T(t) - J_T(t)}{\\tilde{P}(t,T)} \\left(- \\frac{\\sigma_S(t)\/\\tilde{P}(t,S)}{\\sigma_T(t)\/\\tilde{P}(t,T) - \\sigma_S(t)\/\\tilde{P}(t,S)} \\right) + \\\\\n+ \\frac{\\mu_S(t) - J_S(t)}{\\tilde{P}(t,S)} \\left( \\frac{\\sigma_T(t)\/\\tilde{P}(t,T)}{\\sigma_T(t)\/\\tilde{P}(t,T) - \\sigma_S(t)\/\\tilde{P}(t,S)} \\right) = r(t).\n\\end{multline*}\nMultiplying both sides by the term\n\\begin{equation*}\n\\frac{\\sigma_T(t)}{\\tilde{P}(t,T)} - \\frac{\\sigma_S(t)}{\\tilde{P}(t,S)},\n\\end{equation*}\nwe obtain\n\\begin{equation*}\n\\left(\\frac{\\mu_S(t) - J_S(t)}{\\tilde{P}(t,S)} - r(t) \\right) \\left(\\frac{\\sigma_T(t)}{\\tilde{P}(t,T)} \\right) = \\left(\\frac{\\mu_T(t) - J_T(t)}{\\tilde{P}(t,T)} - r(t) \\right) \\left(\\frac{\\sigma_S(t)}{\\tilde{P}(t,S)} \\right).\n\\end{equation*}\nIt follows that,\n\\begin{equation*}\n\\left( \\frac{\\mu_S(t) - J_S(t)}{\\tilde{P}(t,S)} - r(t) \\right) \\left( \\frac{\\tilde{P}(t,S)}{\\sigma_S(t)}\\right) = \\left(\\frac{\\mu_T(t) - J_T(t)}{\\tilde{P}(t,T)} - r(t) \\right) \\left( \\frac{\\tilde{P}(t,T)}{\\sigma_T(t)}\\right),\n\\end{equation*}\nand rearranging we deduce\n\\begin{equation} \\label{eq:lambda_tilde_independent_of_mat}\n\\frac{\\mu_S(t) - J_S(t) - r(t) \\tilde{P}(t,S)}{\\sigma_S(t)} = \\frac{\\mu_T(t) - J_T(t) - r(t) \\tilde{P}(t,T)}{\\sigma_T(t)}.\n\\end{equation}\nNotice that the left hand side of \\eqref{eq:lambda_tilde_independent_of_mat} depends on $S$ but not on $T$, while the right hand side of \\eqref{eq:lambda_tilde_independent_of_mat} depends on $T$, but not on $S$. Since $S$ and $T$ are arbitrary, we conclude that both sides of \\eqref{eq:lambda_tilde_independent_of_mat} depend only on $t$ and $r(t)$.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm_absence_arbitrage}]\nThe proof is similar to the proof of Proposition 1.1 in Chapter 1.2 of \\cite{bjork1997interest} (see also Harrison and Kreps \\cite{harrison1979martingales} Theorem 2 and relative Corollary in Section 3 and Harrison and Pliska \\cite{harrison1981martingales}, Theorem 2.7, Section 2). For the sake of completeness, we give the proof here, translated in our price impact environment. Let $T<+\\infty$ be some finite maturity. Let $\\tilde{h}$ be an arbitrage portfolio and $\\tilde{V}$ the corresponding value process. Then, given the positivity of the discount factor (bank account) defined in \\eqref{def:bank_account} and the equivalence between the real world measure $\\mathbb{P}$ and the impacted risk neutral measure $\\tilde{\\mathbb{Q}}$, we immediately deduce\n\\begin{equation}\n\\tilde{\\mathbb{Q}} \\left( \\frac{\\tilde{V}(T)}{B(T)} \\geq 0 \\right) = 1, \\ \\ \\ \\tilde{\\mathbb{Q}} \\left( \\frac{\\tilde{V}(T)}{B(T)} >0 \\right) > 0.\n\\label{Q_tilde_probs}\n\\end{equation}\nMoreover we have\n\\begin{equation*}\n0 = \\tilde{V}(0) = \\frac{\\tilde{V}(0)}{B(0)} = \\mathbb{E}^{\\tilde{\\mathbb{Q}}} \\left[\\frac{\\tilde{V}(T)}{B(T)} \\right] > 0,\n\\end{equation*}\nwhere the first equality comes from the definition of arbitrage, the second from the fact that $B(0)=1$ and the third from the fact that $\\tilde{V}(t)\/B(t)$ is a martingale under $\\tilde{\\mathbb{Q}}$. Finally, the positivity of the expectation is a consequence of \\eqref{Q_tilde_probs}. We get a contradiction so we conclude that absence of arbitrage must hold.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{prop:relationship_Jf_Jp}]\nWe start by writing the impacted forward rate defined in \\eqref{impacted_forward_rate} as\n\\begin{equation*}\n\\tilde{f}(t,T) = f(t,T) + \\int_0^t J^f(s,T) ds, \n\\end{equation*}\nwhere $f$ represents the unimpacted forward rate (see e.g. Chapter 6, of \\cite{filipovic2009term}) and we used the assumption $\\tilde{f}(0,t) = f(0,t)$. Then, using \\eqref{impacted_bond_HJM}, we deduce\n\\begin{align}\\label{eq:dummy}\n\\begin{split}\n\\tilde{P}(t,T) & = \\exp \\left\\{-\\int_t^T \\tilde{f}(t,u) du\\right\\} \\\\\n& = \\exp \\left\\{-\\int_t^T f(t,u) du - \\int_t^T J^f(s,u) du \\right\\} \\\\\n& = P(t,T) \\exp \\left\\{-\\int_t^T J^f(s,u) du \\right\\},\n\\end{split}\n\\end{align}\nwhere $P$ denotes the unimpacted zero-coupon bond and we used the well known relation between $P(t,T)$ and $f(t,T)$. From \\eqref{impacted_bond} and \\eqref{def:overall_price_impact} we have \n\\begin{equation} \\label{bla} \n\\tilde{P}(t,T) = P(t,T) - I_T(t).\n\\end{equation}\nSubstituting this last expression into \\eqref{eq:dummy} and rearranging, we obtain\n\\begin{equation*}\n\\exp \\left\\{-\\int_t^T J^f(s,u) du \\right\\} = \\frac{\\tilde{P}(t,T)}{\\tilde{P}(t,T) + I_T(t)}.\n\\end{equation*}\nBy taking logarithms on both sides yields and using \\eqref{bla} we get \n\\begin{equation*}\n\\begin{aligned} \n\\int_t^T J^f(s,u) du &= - \\log \\left( \\frac{\\tilde{P}(t,T)}{\\tilde{P}(t,T) + I_T(t)} \\right) \\\\\n&= - \\log \\left( 1- \\frac{I_T(t)}{P(t,T)} \\right).\n\\end{aligned}  \n\\end{equation*}\nDifferentiating with respect to maturity, we get \\eqref{relationship_Jf_Jp}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{hjm_condition_market_impact} ]\nLet $B(t)$ be the bank account defined in \\eqref{def:bank_account} and let the impacted zero-coupon bond $\\tilde{P}$ follow the dynamics \\eqref{impaced_ZCB_HJM}. By applying Ito's formula to the discounted impacted zero-coupon bond price, we immediately find\n\\begin{equation*}\nd \\frac{\\tilde{P}(t,T)}{B(t)} = \\frac{\\tilde{P}(t,T)}{B(t)} \\tilde{b}(t,T) dt + \\frac{\\tilde{P}(t,T)}{B(t)} \\nu(t,T) d W^{\\mathbb{P}}(t),\n\\end{equation*}\nwith $\\tilde{b}$ and $\\nu$ defined as in \\eqref{b-v-rel}. Changing measure form the real world $\\mathbb{P}$ to the impacted risk neutral $\\tilde{\\mathbb{Q}}$ as in \\eqref{def:Q_tilde_rnd_HJM} implies \n\\begin{equation*}\nd \\frac{\\tilde{P}(t,T)}{B(t)} = \\frac{\\tilde{P}(t,T)}{B(t)} \\left(\\tilde{b}(s,T) + \\nu(t,T) \\tilde{\\gamma}(t) \\right) dt + \\frac{\\tilde{P}(t,T)}{B(t)} \\nu(t,T) d W^{\\tilde{\\mathbb{Q}}}(t).\n\\end{equation*}\nTherefore, we clearly see that\n\\begin{equation*}\n\\frac{\\tilde{P}(t,T)}{B(t)} \\ \\ \\text{local martingale under}\\ \\ \\tilde{\\mathbb{Q}} \\iff \\tilde{b}(s,T) = - \\nu(t,T) \\tilde{\\gamma}(t).\n\\end{equation*}\nThis is our new HJM condition. Notice also that since both functions $\\nu$ and $\\tilde{b}$ are continuous with respect to $T$, this condition is equivalent to saying that the impacted measure $\\tilde{\\mathbb{Q}}$ is an equivalent local martingale measure. Following Theorem  6.1 in \\cite{filipovic2009term}, Chapter 6, we can plug in the explicit expression for $\\tilde b$ in \\eqref{b-v-rel} and write the HJM condition \\eqref{HJM_condition} as\n\\begin{equation}\\label{gh0} \n-\\int_s^T \\alpha(s,u) du - \\int_s^T J^f(s,u) du + \\frac{1}{2} \\nu^2(s,T)  = - \\nu(t,T) \\tilde{\\gamma}(t).\n\\end{equation} \nDifferentiating both sides with respect to the maturity $T$ yields the equation \n\\begin{equation*}\n- \\alpha(t,T) + \\sigma(t,T) \\int_t^T \\sigma(t,u) du - J^f(t,T) = \\sigma(t,T) \\tilde{\\gamma}(t),\n\\end{equation*}\nthat is\n\\begin{equation} \\label{gh1} \n\\alpha(t,T) + J^f(t,T) = \\sigma(t,T) \\int_t^T \\sigma(t,u)du - \\sigma(t,T) \\tilde{\\gamma}(t).\n\\end{equation}\nSubstituting \\eqref{gh1} in the dynamics of the forward rate \\eqref{impacted_forward_rate} and using Girsanov yields \\eqref{impacted_forward_rate_Q_tilde}. Using \\eqref{HJM_condition} along with \\eqref{impaced_ZCB_HJM} and Girsanov gives \\eqref{impacted_zc_bond_Q_tilde}.\n\\end{proof}\n\n\\section{Proof of Theorem \\ref{linear_feedback_form}}{\\label{sec:proof_linear_feedback}}\nThe uniqueness of the optimal strategy follows by a standard convexity argument for the performance functional \\eqref{j-fun}. Hence we only need to derive the optimal strategy. \n  \nWe start by deriving a system of coupled forward-backward stochastic differential equations (FBSDEs) which is satisfied by the solution to the stochastic control problem. \n\n\\begin{lemma}[FBSDE system]\n\\label{FBSDE_system}\nA control $\\hat{v} \\in \\mathcal{A}$ solves the optimization problem \\eqref{opt_stoch_control} if and only if the processes $(X^{\\hat{v}},\\Upsilon^{\\hat{v}},\\hat{v},Z^{\\hat{v}})$ satisfy the coupled forward-backward stochastic differential equations  \n\\begin{equation}\n\\begin{cases}\nd X^{\\hat{v}}(t) &= - \\hat{v}(t) dt, \\ \\ \\ X^{\\hat{v}}(0) = x, \\\\\nd \\Upsilon^{\\hat{v}}(t)& = - \\rho \\Upsilon^{\\hat{v}}(t) dt + \\gamma \\hat{v}(t) dt, \\ \\ \\ \\Upsilon^{\\hat{v}}(0) = y, \\\\\nd \\hat{v}(t) &= \\Lambda(t) d P(t,T) - 2 \\Lambda(t) \\phi X^{\\hat{v}}(t) dt \\\\\n& \\quad + \\Upsilon^{\\hat{v}}(t) \\left[-\\Lambda'(t) K(t,T) - \\Lambda(t) \\partial_t K(t,T) + \\rho K(t,T) \\Lambda(t) \\right] dt \\\\\n& \\quad + Z^{\\hat{v}}(t) \\left[\\Lambda'(t) + \\rho \\Lambda(t) \\right] dt + \\Lambda(t) \\Gamma^{\\hat{v}}(t) dt + d M(t), \\\\\n &\\qquad \\qquad \\qquad \\qquad \\qquad\\qquad\\qquad \\hat{v}(\\tau) = \\frac{\\varrho}{l(\\tau,T)} X^{\\hat{v}}(\\tau) - \\frac{K(\\tau,T)}{2 l(\\tau,T)} \\Upsilon^{\\hat{v}}(\\tau), \\\\\nd Z^{\\hat{v}}(t)& = \\left(\\rho Z^{\\hat{v}}(t) + K(t,T) \\gamma \\hat{v}(t) \\right) dt + d N(t), \\ \\ Z^{\\hat{v}}(\\tau) = 0,\n\\end{cases}\n\\end{equation}\nfor two suitable square integrable martingales $M=(M(\\cdot,T))_{0 \\leq t \\leq \\tau}$ and $N=(N(\\cdot,T))_{0 \\leq t \\leq \\tau}$, where the $\\Lambda, \\Gamma^{\\hat{v}}$ and $\\tilde{M}$ are defined in \\eqref{def:Lambda_optimal_exe}, \\eqref{def:Gamma_quantity} and \\eqref{def:square_int_mart_M_tilde} respectively.\n\\begin{proof}\nThe proof follows the same lines as Lemmas 5.1 and 5.2 in \\cite{neuman2020optimal}. Since for all $v \\in \\mathcal{A}$ the map $v \\to \\mathcal{J}(v)$ is strictly concave, we can study the unique critical point at which the Gateaux derivative of $\\mathcal{J}$, which is defined as\n\\begin{equation*}\n\\langle \\mathcal{J}'(v), \\alpha \\rangle := \\lim_{\\epsilon \\to 0} \\frac{\\mathcal{J}(v+\\epsilon \\alpha) - \\mathcal{J}(v)}{\\epsilon},\n\\end{equation*}\nis equal to $0$ for any $\\alpha \\in \\mathcal{A}$. This derivative can be computed analytically as follows. Let $\\epsilon>0$ and $v,\\alpha \\in \\mathcal{A}$. Since for all $t \\in [0,\\tau]$,\n\\begin{align} \\label{bb1}\n\\begin{split}\nX^{v + \\epsilon \\alpha}(t) & = x - \\int_0^t (v(s) + \\epsilon \\alpha(s) )ds = X^v(t) - \\epsilon \\int_0^t \\alpha(s) ds \\\\\n\\Upsilon^{v+\\epsilon \\alpha}(t) & = \\Upsilon^v(t) + \\epsilon \\gamma \\int_0^t e^{-\\rho (t-s)} \\alpha(s) ds,\n\\end{split}\n\\end{align}\nFrom \\eqref{j-fun} and \\eqref{bb1} we have\n\\begin{align*}\n&\\mathcal{J}(v+\\epsilon \\alpha) = \\\\\n&= \\mathbb{E} \\Bigg[ \\int_0^\\tau \\left( P(t,T) - K(t,T) \\Upsilon^v(t) - K(t,T) \\epsilon \\gamma \\int_0^t e^{-\\rho (t-s)} \\alpha(s) ds \\right) \\left(v(t) + \\epsilon \\alpha(t) \\right) dt  \\\\\n&\\qquad - \\int_0^\\tau l(t,T) v^2(t) + \\epsilon^2 l(t,T) \\alpha_t^2 + 2 l(t,T) v(t) \\epsilon \\alpha(t) dt + X^v(\\tau) P(\\tau,T) - \\epsilon P(\\tau,T) \\int_0^\\tau \\alpha(s) ds  \\\\\n&\\qquad - \\phi \\int_0^\\tau (X^v(t))^2 + \\epsilon \\left(\\int_0^t \\alpha(s) ds \\right)^2 - 2 X^v(t) \\epsilon \\int_0^t \\alpha(s) ds dt  \\\\\n&\\qquad - \\varrho \\left( (X^v(\\tau))^2 + \\epsilon^2 \\left(\\int_0^\\tau \\alpha(s) ds \\right)^2 - 2 X^v(\\tau) \\epsilon \\int_0^\\tau \\alpha(s) ds \\right) \\Bigg]. \n\\end{align*}\nIt follows that \n\\begin{align*}\n\\mathcal{J}(v+\\epsilon \\alpha) - \\mathcal{J}(v) &= \\epsilon \\mathbb{E} \\Bigg[ \\int_0^\\tau \\left( P(\\tau,T) - K(t,T) \\Upsilon^v(t) \\right) \\alpha(t) dt \\\\\n&\\qquad - \\int_0^\\tau K(t,T) v(t) \\int_0^t \\gamma e^{-\\rho (t-s)} \\alpha(s) ds dt - 2 \\int_0^\\tau l(t,T) v(t) \\alpha(t) dt \\\\\n&\\qquad + 2 \\phi \\int_0^\\tau X^v(t) \\int_0^t \\alpha(s) ds dt + 2 \\varrho X^v(\\tau) \\int_0^\\tau \\alpha(s) ds\n- P(\\tau,T) \\int_0^\\tau \\alpha(s) ds \\Bigg]  \\\\\n&\\qquad+ \\epsilon^2 \\mathbb{E} \\Bigg[ \\gamma \\int_0^\\tau K(t,T) \\alpha(t) \\int_0^t e^{-\\rho (t-s)} \\alpha(s) ds dt - \\int_0^\\tau l^2(t,T) \\alpha^2(t) dt  \\\\\n&\\qquad - \\phi \\int_0^\\tau \\left( \\int_0^t \\alpha(s) ds \\right)^2 dt - \\varrho \\left(\\int_0^\\tau \\alpha(s) ds \\right)^2 \\Bigg].\n\\end{align*}\nNote that all the terms above are finite since $\\ell$ and $K$ are bounded functions and since $\\alpha, v \\in \\mathcal A$. Applying Fubini's theorem twice, we obtain\n\\begin{multline*}\n\\langle \\mathcal{J}'(v), \\alpha \\rangle = \\mathbb{E} \\Bigg[ \\int_0^\\tau \\alpha(s) \\Bigg( P(s,T) - K(s,T) \\Upsilon^v(s) - \\int_s^\\tau K(t,T) e^{-\\rho(t-s)} \\gamma v(t) dt + \\\\\n- 2 l(s,T) v(s) + 2 \\phi \\int_s^\\tau X^v(t) dt + 2 \\varrho X^v(\\tau) - P(\\tau,T) \\Bigg) ds \\Bigg],\n\\end{multline*}\nfor any $\\alpha \\in \\mathcal{A}$. We get the following condition on the optimal strategy\n\\begin{equation}\n\\begin{aligned}\n\\mathbb{E} \\Bigg[ \\int_0^\\tau \\alpha(s) \\Bigg( P(s,T) - K(s,T) \\Upsilon^v(s) - \\int_s^\\tau K(t,T) e^{-\\rho(t-s)} \\gamma v(t) dt  \\\\\n- 2 l(s,T) v(s) + 2 \\phi \\int_s^\\tau X^v(t) dt + 2 \\varrho X^v(\\tau) - P(\\tau,T) \\Bigg) ds \\Bigg] = 0.\n\\label{first_order_condition}\n\\end{aligned}\n\\end{equation}\nNext we show that given the optimal strategy $\\hat{v} \\in \\mathcal{A}$, the vector $(X^{\\hat{v}},\\Upsilon^{\\hat{v}})$ satisfies the first order condition \\eqref{first_order_condition} if and only if the vector $(X^{\\hat{v}},\\Upsilon^{\\hat{v}},\\hat{v},Z^{\\hat{v}})$ solves a FBSDE system, for some auxiliary process $Z$.\n\nFor any $s>0$ we denote by $\\mathbb{E}_s$ the conditional expectation with respect to the filtration $\\mathcal F_s$. Assume $\\hat{v} \\in \\mathcal{A}$ maximizes the functional $\\mathcal{J}$. Applying the optional projection theorem we obtain  \n\\begin{align*}\n\\mathbb{E} \\Bigg[ \\int_0^\\tau \\alpha(s) \\bigg( P(s,T) - K(s,T) \\Upsilon^v(s) - \\mathbb{E}_s \\bigg[\\int_s^\\tau K(t,T) e^{-\\rho (t-s)} \\gamma \\hat{v}(t) dt \\bigg] - 2 l(s,T) \\hat{v}(s)  \\\\\n+ \\mathbb{E}_s \\bigg[2 \\phi \\int_s^\\tau X^{\\hat{v}}(t) dt + 2 \\varrho X^{\\hat{v}}(\\tau) - P(\\tau,T) \\bigg] \\bigg) ds \\Bigg] = 0,\n\\end{align*}\nfor all $\\alpha \\in \\mathcal{A}$. This implies\n\\begin{equation}  \\label{explicit_sol_BSDE_v_hat} \n\\begin{aligned}\n&P(s,T) - K(s,T) \\Upsilon^{\\hat{v}}(s) - e^{\\rho s} \\mathbb{E}_s \\bigg[\\int_s^\\tau K(t,T) e^{-\\rho t} \\gamma \\hat{v}(t) dt \\bigg] - 2 l(s,T) \\hat{v}(s) \\\\\n&\\qquad+ \\mathbb{E}_s \\bigg[2 \\phi \\int_s^\\tau X^{\\hat{v}}(t) dt + 2 \\varrho X^{\\hat{v}}(\\tau) - P(\\tau,T) \\bigg] \\\\\n&= P(s,T) - K(s,T) \\Upsilon^{\\hat{v}}(s) \\\\\n&\\qquad - e^{\\rho s} \\left( \\mathbb{E}_s \\bigg[ \\int_0^\\tau K(t,T) e^{-\\rho t} \\gamma \\hat{v}(t) dt \\bigg] - \\int_0^s K(t,T) e^{-\\rho t} \\gamma \\hat{v}(t) dt \\right) \\\\\n&\\qquad - 2 l(s,T) \\hat{v}(s) + \\mathbb{E}_s \\bigg[2 \\phi \\int_0^\\tau X^{\\hat{v}}(t) dt + 2 \\varrho X^{\\hat{v}}(\\tau) - P(\\tau,T) \\bigg] - 2 \\phi \\int_0^s X^{\\hat{v}}(t) dt \\\\\n&= 0, \\ \\ \\ d \\mathbb{P} \\otimes ds\\ \\ \\text{a.e. on} \\ \\ \\Omega \\times [0,\\tau]. \\\\\n\\end{aligned}\n\\end{equation} \nNext, we define the square-integrable martingale\n\\begin{equation}\\label{def:square_int_mart_N_tilde}\n\\tilde{N}(s) := \\mathbb{E}_s \\left[\\int_0^{\\tau} K(t,T) e^{-\\rho t} \\gamma \\hat{v}(t) dt \\right]\n\\end{equation}\nand the auxiliary square-integrable process\n\\begin{equation*}\nZ^{\\hat{v}}(s) := e^{\\rho s} \\bigg( \\int_0^s K(t,T) e^{-\\rho t} \\gamma \\hat{v}(t) dt - \\tilde{N}(s) \\bigg),\n\\end{equation*}\nfor all $s \\in [0,\\tau]$. Note that since both $l$ and $K$ are assumed to be uniformly bounded and $v \\in \\mathcal A$, we have that $P(\\tau,T) \\in L^2(\\Omega,\\mathcal{F}_\\tau,\\mathbb{P})$. Therefore, we obtain\n\\begin{equation} \\label{fgr}\nP(s,T) - K(s,T) \\Upsilon^{\\hat{v}}(s) + Z^{\\hat{v}}(s) - 2 l(s,T) \\hat{v}(s) + \\tilde{M}(s) - 2 \\phi \\int_0^s X^{\\hat{v}}(t) dt = 0\n\\end{equation}\nalmost everywhere on $\\Omega \\times [0,\\tau]$, where $\\tilde{M}$ is the square-integrable martingale defined in \\eqref{def:square_int_mart_M_tilde}, and we immediately see that the process $Z^{\\hat{v}}$ satisfies the BSDE\n\\begin{equation*}\nd Z^{\\hat{v}}(t) = \\left(\\rho Z^{\\hat{v}}(t) + K(t,T) \\gamma \\hat{v}(t) \\right) dt - e^{\\rho t} d \\tilde{N}(t), \\quad Z^{\\hat{v}}(\\tau) = 0.\n\\end{equation*}\nFrom \\eqref{def:transient_impact} we get that $\\Upsilon^{\\hat{v}}$ satisfies   \n\\begin{equation*}\nd \\Upsilon^{\\hat{v}}(t) = - \\rho \\Upsilon^{\\hat{v}}(t) dt + \\gamma \\hat{v}(t) dt, \\quad \\Upsilon^{\\hat{v}}(0) = y.\n\\end{equation*}\nRecall that $\\Lambda$ was defined in \\eqref{def:Lambda_optimal_exe}. From \\eqref{fgr} it follows that $\\hat{v}$ satisfies the backward stochastic differential equation \n\\begin{align*}\n\\begin{split}\nd \\hat{v}(s) & = \\Lambda'(s) \\left(P(s,T) - K(s,T) \\Upsilon^{\\hat{v}}(s) + Z^{\\hat{v}}(s) + \\tilde{M}(s) - 2 \\phi \\int_0^s X^{\\hat{v}}(u) du \\right)ds \\\\\n& \\quad + \\Lambda(s) \\bigg(d P(s,T) - \\partial_s K(s,T) \\Upsilon^{\\hat{v}}(s) ds - K(s,T) d \\Upsilon^{\\hat{v}}(s) + d Z^{\\hat{v}}(s) \\\\\n& \\quad \\quad + d \\tilde{M}(s) - 2 \\phi X^{\\hat{v}}(s) ds \\bigg) \\\\\n& = \\Lambda'(s) \\left(P(s,T) - K(s,T) \\Upsilon^{\\hat{v}}(s) + Z^{\\hat{v}}(s) + \\tilde{M}(s) - 2 \\phi \\int_0^s X^{\\hat{v}}(u) du \\right)ds \\\\\n& \\quad + \\Lambda(s) d P(s,T) - \\Lambda(s) \\partial_s K(s,T) \\Upsilon^{\\hat{v}}(s) ds + \\rho K(s,T) \\Upsilon^{\\hat{v}}(s) \\Lambda(s) ds  \\\\\n& \\quad   + \\Lambda(s) \\rho Z^{\\hat{v}}(s) ds  - 2 \\Lambda(s) \\phi X^{\\hat{v}}(s) ds + \\Lambda(s) d \\tilde{M}(s) - \\Lambda(s) e^{\\rho s} d \\tilde{N}(s)  \\\\\n\\hat{v}(\\tau) & = \\frac{\\varrho}{l(\\tau,T)} X^{\\hat{v}}(\\tau) - \\frac{K(\\tau,T)}{2 l(\\tau,T)} \\Upsilon^{\\hat{v}}(\\tau),\n\\end{split}\n\\end{align*}\nPutting these equations together with \\eqref{inv}, we obtain the FBSDE system \\eqref{FBSDE_system} with $M,N$ square-integrable martingales defined as\n\\begin{align*}\n\\begin{split}\nM(t) & := \\int_0^t \\Lambda(s) d \\tilde{M}(s) - \\int_0^t \\Lambda(s) e^{\\rho s} d \\tilde{N}(s) \\\\\nN(t) & := - \\int_0^t e^{\\rho s} d \\tilde{N}(s).\n\\end{split}\n\\end{align*}\nIn order to check the integrability of $M$, recall that $\\Lambda$ was defined in \\eqref{def:Lambda_optimal_exe}. Since $l$ is bounded away from $0$ on $[0,\\tau] $ (see \\eqref{l-pos}) we have \n\\[\n\\sup_{0 \\leq t \\leq \\tau} | \\Lambda(t) | < \\infty.\n\\]\nThen, it holds\n\\begin{align*}\n\\begin{split}\n\\mathbb{E}[M^2(t)] & \\leq \\mathbb{E} \\left[ \\int_0^t \\Lambda^2(s) d [\\tilde M]_s \\right] + \\mathbb{E} \\left[ \\int_0^t\\Lambda^2(s) e^{2\\rho s} d[\\tilde N]_s \\right] \\\\\n& \\leq C_1 \\mathbb{E} [\\tilde M]_T + C_2 \\mathbb{E} [ \\tilde N]_T \\\\\n&< \\infty\n\\end{split}\n\\end{align*}\nfor some constants $C_1,C_2$, where in the last inequality we used the fact that both $\\tilde{M}$ and $\\tilde{N}$ are square integrable martingales.\n\nNext, assume that  $(\\hat{v},X^{\\hat{v}},\\Upsilon^{\\hat{v}},Z^{\\hat v})$ is a solution to the FBSDE system \\eqref{FBSDE_system} and $\\hat{v} \\in \\mathcal{A}$. We will show that $\\hat{v}$ satisfies the first order condition \\eqref{first_order_condition}, hence it maximizes the cost functional \\eqref{j-fun}. First, note that the BSDE for $\\hat{v}$ can be solved explicitly and the solution is indeed given in \\eqref{explicit_sol_BSDE_v_hat}\n\\begin{align*}\n\\begin{split}\n\\hat{v}(s) & = \\frac{1}{2 l(s,T)} \\Bigg(P(s,T)- K(s,T) \\Upsilon^{\\hat{v}}(s) + Z^{\\hat{v}}(t) + \\tilde{M}(s) - 2 \\phi \\int_0^s X^{\\hat{v}}(t) dt \\Bigg) \\\\\n& = \\frac{1}{2l(s,T)} \\Bigg(P(s,T) - K(s,T) \\Upsilon^{\\hat{v}}(s) - e^{\\rho s} \\left(\\tilde{N}(s) - \\int_0^s K(t,T) e^{-\\rho t} \\gamma \\hat{v}(t) dt \\right) \\\\\n& \\quad + \\tilde{M}(s) - 2 \\phi \\int_0^s X^{\\hat{v}}(u) du \\Bigg),\n\\end{split}\n\\end{align*}\nwith $\\tilde{N},\\tilde{M}$ defined in \\eqref{def:square_int_mart_N_tilde} and \\eqref{def:square_int_mart_M_tilde}, respectively. Plugging this into the first order condition \\eqref{first_order_condition} yields\n\\begin{align*}\n&\\mathbb{E} \\Bigg[\\int_0^\\tau \\bigg(e^{\\rho s} \\left(\\tilde{N}(s) - \\int_0^\\tau K(t,T) e^{-\\rho t} \\gamma \\hat{v}(t) dt \\right) - \\tilde{M}(s) \\\\\n&\\qquad + 2 \\phi \\int_0^\\tau X(t) dt + 2 \\varrho X^{\\hat{v}}(\\tau) - P(\\tau,T) \\bigg)ds \\Bigg] \\\\\n&= \\mathbb{E} \\Bigg[ \\int_0^\\tau \\alpha(s) \\bigg( e^{\\rho s} (\\tilde{N}(s) - \\tilde{N}(\\tau)) - \\tilde{M}(s) + \\tilde{M}(\\tau) \\bigg) ds \\Bigg] \\\\\n&= \\mathbb{E} \\Bigg[ \\int_0^\\tau \\alpha(s) \\bigg(e^{\\rho s} (\\tilde{N}(s) - \\mathbb{E}_s[\\tilde{N}(\\tau)]) - \\tilde{M}(s) + \\mathbb{E}_s[\\tilde{M}(\\tau)] \\bigg)ds \\Bigg] \\\\\n& = 0,\n\\end{align*}\nfor all $\\alpha \\in \\mathcal{A}$. Since $\\tilde{N},\\tilde{M}$ are martingales, hence the first order condition \\eqref{first_order_condition} is satisfied and $\\hat{v} \\in \\mathcal{A}$ is the optimal strategy.\n\\end{proof}\n\\end{lemma}\n\nBefore giving the proof of our main theorem, we will need the following Lemma, which will help us to show the optimal strategy in \\eqref{linear_feedback_form} is indeed admissible.\n\n\\begin{lemma}\\label{lem:Gamma_quantity}\nLet $\\Gamma^{\\hat{v}}$ be defined as in \\eqref{def:Gamma_quantity}. Then, there exist constants $C_1, C_2 > 0$ such that \n\\begin{equation*}\n\\mathbb{E} \\left[  \\int_0^\\tau \\big(\\Gamma^{\\hat{v}}(s) \\big)^{2}  ds \\right] \\leq C_1 + C_2\\mathbb{E} \\left[\\int_0^\\tau v^2(s) ds \\right].\n\\end{equation*}\n\\begin{proof}\nFirstly, by the assumptions on $l$ (see \\eqref{l-pos} and \\eqref{k-assump}) it follows that\n\\begin{equation}\\label{bounded_ratio_lambda_lambda_prime}\n\\sup_{0 \\leq t \\leq \\tau} \\bigg | \\frac{\\Lambda'(t)}{\\Lambda(t)} \\bigg| = \\sup_{0 \\leq t \\leq \\tau} \\bigg| \\frac{\\partial_t l(t,T)}{l(t,T)} \\bigg| < \\infty,\n\\end{equation}\nwhere $\\Lambda$ is given in \\eqref{def:Lambda_optimal_exe}. Therefore, from \\eqref{bounded_ratio_lambda_lambda_prime},  \\eqref{def:Gamma_quantity} and Jensen's inequality we get that there exist constants $C_1,C_2>0$ such that\n\\begin{align*} \n \\mathbb{E} \\left[  \\int_0^\\tau \\Gamma^{\\hat{v}}(s)^2  ds \\right] \n&\\leq \\mathbb{E} \\left[  \\int_0^\\tau \\Big(\\frac{\\Lambda'(s)}{\\Lambda(s)}\\Big)^2 \\left(P^2(s,T) + \\tilde{M}^2(s) + 4 \\phi^2 \\Big(\\int_0^s X^{\\hat{v}}(u) du \\Big)^2 \\right) ds \\right] \\\\ \n&\\leq C_1 \\mathbb{E} \\left[ \\int_0^\\tau \\left(  P^2(s,T) + \\tilde{M}^2(s) +  4 \\phi^2  \\Big(\\int_0^s X^{\\hat{v}}(u) du \\Big)^2 \\right)ds \\right] \\\\ \n&\\leq C_2 +  4 \\phi^2  \\mathbb{E} \\left[ \\int_0^\\tau \\Big(\\int_0^s X^{\\hat{v}}(u) du \\Big)^2 ds \\right],\n\\end{align*}\nwhere we used \\eqref{def: mathcal_h_squared} and the fact that the martingale $\\tilde{M}$ defined in \\eqref{def:square_int_mart_M_tilde} is square-integrable. \n\nNext, using the definition of $X^{\\hat{v}}$ in \\eqref{def:X} and Jensen's inequality twice, we deduce\n\\begin{align*} \n\\mathbb{E} \\left[ \\int_0^\\tau \\Big(\\int_0^s X^{\\hat{v}}(u) du \\Big)^2 ds\\right] & = \n\\mathbb{E} \\left[ \\int_0^\\tau \\Big(\\int_0^s \\big(x-\\int_0^u  \\hat v(y) dy  \\big)du\\Big)^2 ds \\right] \\\\\n&\\leq C_1 + C_2\\mathbb{E} \\left[\\int_0^\\tau \\int_0^s \\int_0^u \\hat v^2(y) dy du ds \\right] \\\\\n&\\leq C_1 + C_2 \\mathbb{E} \\left[ \\int_0^\\tau \\int_0^\\tau \\int_0^\\tau \\hat v^2(y) dy du ds  \\right] \\\\\n&\\leq C_1 + C_2 \\mathbb{E} \\left[\\int_0^\\tau  \\hat v^2(y) ds \\right], \n\\end{align*} \nfor some constants $C_1,C_2$, and we are done. \n\\end{proof}\n\\end{lemma}\nWe are now ready to prove Theorem \\ref{linear_feedback_form}. \n\n\\begin{proof}[Proof of Theorem \\ref{linear_feedback_form}] \nDefine\n\\begin{equation*}\n\\mathbf{X}^{{v}}(t) := \n\\left(\n\\begin{matrix}\nX^{\\hat v}(t) \\\\\n\\Upsilon^{\\hat v}(t) \\\\\n\\hat v(t) \\\\\nZ^{\\hat v}(t)\n\\end{matrix}\n\\right), \\ \\ \\ \n\\mathbf{M}(t) :=\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\nP(t,T) + \\int_0^t \\Gamma^{\\hat v}(s) ds + \\int_0^t \\Lambda^{-1}(s) d M(s) \\\\\n\\int_0^t \\Lambda^{-1}(s) d N(s)\n\\end{matrix}\n\\right),\n\\end{equation*}\nwhere $\\Lambda$ and $\\Gamma^{\\hat v}$ are defined in \\eqref{def:Lambda_optimal_exe} and \\eqref{def:Gamma_quantity} respectively. The FBSDE system \\eqref{FBSDE_system} can be written as\n\\begin{equation*}\nd \\mathbf{X}_t^{\\hat v}= A(t) \\mathbf{X}_t^{\\hat v} dt + \\Lambda(t) d \\mathbf{M}(t), \\ \\ \\ 0 \\leq t \\leq \\tau,\n\\end{equation*}\nwhere the matrix $A(t)$ is defined in \\eqref{def:time_dep_matrix_A}, with initial conditions\n\\begin{equation*}\n\\mathbf{X}^{{\\hat v},1}(0) = x, \\ \\ \\ \\mathbf{X}^{{\\hat v},2}(0) = y,\n\\end{equation*}\nand terminal conditions\n\\begin{equation} \\label{term}\n\\left(\\frac{\\varrho}{l(\\tau,T)}, - \\frac{K(\\tau,T)}{2 l(\\tau,T)}, -1, 0 \\right) \\mathbf{X}^{\\hat v}(\\tau) = 0, \\ \\ \\ (0,0,0,1) \\mathbf{X}^{\\hat v}(\\tau) = 0.\n\\end{equation}\nExploiting linearity, the unique solution can be expressed as\n\\begin{equation*}\n\\mathbf{X}^{\\hat v}(\\tau) = \\Phi(\\tau) \\Phi^{-1}(t) \\mathbf{X}^{\\hat v}(t) + \\int_t^\\tau \\Phi(\\tau) \\Phi^{-1}(s) \\Lambda(s) d \\mathbf{M}(s),\n\\end{equation*}\nwhere $\\Phi$ solves the ODE \\eqref{matrix_ODE}. Moreover, it can be immediately seen that the first terminal condition in \\eqref{term} yields\n\\begin{align*}\n\\begin{split}\n0 & = G^1(t,\\tau) X^{\\hat v}(t) + G^2(t,\\tau) \\Upsilon^{\\hat v}(t) + G^3(t,\\tau) \\hat v(t) + G^4(t,\\tau) Z^{\\hat v}(t) \\\\\n& \\quad + \\int_t^\\tau \\Lambda(s) \\left(G^3(s,\\tau) \\left(d P(s,T) + \\Gamma^{\\hat v}(s) ds + \\Lambda^{-1}(s) d M(s) \\right) + G^4(s,\\tau) \\Lambda^{-1}(s) d N(s) \\right)\n\\end{split}\n\\end{align*}\nwith $G = (G^1,G^2,G^3,G^4)$ defined in \\eqref{def:G_components}. Solving for the trading speed $v$, taking expectations and using that $P \\in \\mathcal{H}^2$, together with the fact that both $M$ and $N$ are square integrable martingales, implies\n\\begin{align}\\label{trading_speed_u_with_z}\n\\begin{split}\n\\hat v(t) & = - \\frac{G^1(t,\\tau)}{G^3(t,\\tau)} X^{\\hat v}(t) - \\frac{G^2(t,\\tau)}{G^3(t,\\tau)} \\Upsilon^{\\hat v}(t) - \\frac{G^4(t,\\tau)}{G^3(t,\\tau)} Z^{\\hat v}(t) \\\\\n& \\quad - \\mathbb{E}_t \\left[\\int_t^\\tau \\Lambda(s) \\frac{G^3(s,\\tau)}{G^3(t,\\tau)}(\\mu(s) + \\Gamma^{\\hat v}(s)) ds \\right].\n\\end{split}\n\\end{align}\nRecall that $\\Psi$ was defined in \\eqref{def:psi_matrix}. Then the second terminal condition in \\eqref{term} implies\n\\begin{align*}\n\\begin{split}\n0 & = (0,0,0,1) \\Psi(t,\\tau) \\mathbf{X}^{\\hat v}(t) + (0,0,0,1) \\int_t^\\tau \\Psi(s,\\tau) \\Lambda(s) d \\mathbf{M}(s) \\\\\n& = \\Psi^{41}(t,\\tau) X^{\\hat v}(t) + \\Psi^{42}(t,\\tau) \\Upsilon^{\\hat v}(t) + \\Psi^{43}(t,\\tau) \\hat v(t) + \\Psi^{44}(t,\\tau) Z^{\\hat v}(t) \\\\\n& \\quad + \\int_t^\\tau \\Lambda(s) \\left(\\Psi^{43}(s,\\tau) \\left(d P(s,T) + \\Gamma^{\\hat v}(s) ds + \\Lambda^{-1}(s) d M(s) \\right) + \\Psi^{44}(s,\\tau) \\Lambda^{-1}(s) d N(s) \\right).\n\\end{split}\n\\end{align*}\nHence, taking expectation and solving for $Z^u$ yields\n\\begin{align}\\label{aux_process_Z}\n\\begin{split}\nZ^{\\hat v}(t) & = - \\frac{\\Psi^{41}(t,\\tau)}{\\Psi^{44}(t,\\tau)} X^{\\hat v}(t) - \\frac{\\Psi^{42}(t,\\tau)}{\\Psi^{44}(t,\\tau)} \\Upsilon^{\\hat v}(t) - \\frac{\\Psi^{43}(t,\\tau)}{\\Psi^{44}(t,\\tau)} {\\hat v}(t) \\\\\n& \\quad - \\mathbb{E}_t \\left[\\int_t^\\tau \\frac{\\Lambda(s) \\Psi^{43}(s,\\tau)}{\\Psi^{44}(t,\\tau)} (\\mu(s) + \\Gamma^{\\hat v}(s)) ds \\right].\n\\end{split}\n\\end{align}\nTherefore, plugging \\eqref{aux_process_Z} into \\eqref{trading_speed_u_with_z} gives\n\\begin{align*}\n\\begin{split}\n\\hat v(t) & = - \\frac{G^1(t,\\tau)}{G^3(t,\\tau)} X^{\\hat v}(t) - \\frac{G^2(t,\\tau)}{G^3(t,\\tau)} \\Upsilon^{\\hat v}(t) + \\frac{G^4(t,\\tau) \\Psi^{41}(t,\\tau)}{G^3(t,\\tau) \\Psi^{44}(t,\\tau)} X^{\\hat v}(t) \\\\\n& \\quad + \\frac{G^4(t,\\tau) \\Psi^{42}(t,\\tau)}{G^3(t,\\tau) \\Psi^{44}(t,\\tau)} \\Upsilon^{\\hat v}(t) + \\frac{G^4(t,\\tau) \\Psi^{43}(t,\\tau)}{G^3(t,\\tau) \\Psi^{44}(t,\\tau)} {\\hat v}(t) \\\\\n& \\quad + \\frac{G^4(t,\\tau)}{G^3(t,\\tau)} \\mathbb{E}_t \\left[\\int_t^\\tau \\frac{\\Lambda(s) \\Psi^{43}(s,\\tau)}{\\Psi^{44}(t,\\tau)} (\\mu(s) + \\Gamma^{\\hat v}(s)) ds \\right] \\\\\n& \\quad - \\mathbb{E}_t \\left[\\int_t^\\tau \\Lambda(s) \\frac{G^3(s,\\tau)}{G^3(t,\\tau)} (\\mu(s) + \\Gamma^{\\hat v}(s)) ds \\right]. \n\\end{split}\n\\end{align*}\nRearranging and using the Definitions \\ref{def:v_terms}, we obtain the linear feedback form \\eqref{optimal_strategy_linear_feedback}. Finally, we prove that the optimal trading strategy is admissible, that is, $\\hat{v} \\in \\mathcal{A}$, as defined in \\eqref{def:admissset}. Thanks to assumptions  \\eqref{A.1} and \\eqref{A.2}, we immediately see that\n\\begin{equation*}\n\\sup_{0 \\leq t \\leq \\tau} |v_0(t,\\tau)| < \\infty.\n\\end{equation*}\nSimilarly, from assumptions \\eqref{A.1}-\\eqref{A.3} we deduce that $v_1$ and $v_2$ are both bounded on $[0,\\tau]$. Exploiting again assumptions \\eqref{A.1}-\\eqref{A.3}, together with \\eqref{def: mathcal_h_squared} we get that  \n\\begin{align*}\n& \\sup_{0 \\leq t \\leq \\tau} \\bigg| \\mathbb{E}_t \\bigg[\\int_t^\\tau \\frac{\\Lambda(s) \\Psi^{43}(s,\\tau)}{\\Psi^{44}(t,\\tau)} (\\mu(s) + \\Gamma^{\\hat v}(s)) ds \n - \\mathbb{E}_t \\int_t^\\tau \\Lambda(s) \\frac{G^3(s,\\tau)}{G^3(t,\\tau)} (\\mu(s) + \\Gamma^{\\hat v}(s)) ds \\bigg] \\bigg|\\\\\n& \\leq C \\mathbb{E} \\left[\\int_0^\\tau( |\\mu(s) |+ |\\Gamma^{\\hat v}(s)| )ds \\right] \\\\\n& \\leq \\tilde{C}_1 + \\tilde{C}_2\\left( \\mathbb{E} \\left[\\int_0^\\tau \\Gamma^{\\hat v}(s)^{2}ds \\right] \\right)^{1\/2} \\\\\n& \\leq \\tilde{C}_1 +\\tilde C_2\\left(\\mathbb{E} \\left[\\int_0^\\tau \\hat{v}^2(s)ds \\right] \\right)^{1\/2}, \n\\end{align*}\nwhere we have used Jensen's inequality and Lemma \\ref{lem:Gamma_quantity} in the last two inequalities. \nUsing the above bound, together with  equations \\eqref{def:X} and \\eqref{def:transient_impact} we get from  \\eqref{optimal_strategy_linear_feedback} that\n\\begin{equation*}\n\\mathbb{E}[\\hat{v}^2(t)] \\leq C_1 + C_2 \\int_0^\\tau \\mathbb{E}[\\hat{v}^2(s)] ds, \\ \\ \\ 0 \\leq t \\leq \\tau,\n\\end{equation*}\nfor some positive constants $C_1,C_2$, where we used again Jensen's inequality. Thanks to Gronwall's lemma, we get that\n\\begin{equation*}\n\\sup_{0 \\leq t \\leq \\tau} \\mathbb{E}[\\hat{v}^2(t)] < \\infty,\n\\end{equation*}\nwhich implies\n\\begin{equation*}\n\\int_0^\\tau \\mathbb{E}[\\hat v^2(s)] ds < \\infty.\n\\end{equation*}\nHence Fubini's theorem, we conclude that $\\hat v\\in \\mathcal A$. \n\\end{proof}\n\t\n\\section*{Acknowledgements}\nWe are very grateful to an anonymous referees for  careful reading of the manuscript,\nand for a number of useful comments and suggestions that significantly improved this paper.\n\n\n\\newpage\n \n\t\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWhen Tunka-Rex started its operation in 2012 with 18 antennas it had two main goals. \nFirst, a cross-calibration with the Tunka-133 photomultiplier array using the established non-imaging air-Cherenkov technique for air showers \\cite{Tunka133_NIM2014}. \nBy this the precision of the radio measurements for the energy and the depth of the shower maximum, $X_\\mathrm{max}$, could be determined experimentally \\cite{TunkaRex_Xmax2016}. \nThis goal is complementary to the comparison of radio to air-fluorescence measurements at the Pierre Auger Observatory \\cite{Holt_AERA_ICRC2017}. \nIt also gives additional confidence to the measurements of radio arrays whose analysis is based mainly on Monte Carlo simulations m\\cite{BuitinkLOFAR_Xmax2014}. \nSecond, the practical demonstration that the radio technique is cost-effective -- at least when the antennas are attached as extension to other detectors in order to improve the total accuracy. \n\nMeanwhile both goals have been achieved and the new main goal of Tunka-Rex is a mass-sensitive measurement of the energy spectrum between $10^{17}$ and $10^{18}\\,$eV, since in this energy range the transition from galactic to extra-galactic cosmic-rays is assumed, but not yet understood. \nFor this purpose the Tunka-Rex array has been extended to 63 antennas that measure in coincidence with particles and air-Cherenkov detectors (the latter only during clear nights).  \nIn combination with other arrays at different locations of the Earth, Tunka-Rex can search for anisotropies of different mass components and consequent differenced of the energy spectrum between the northern and southern hemispheres. \nA precise measurement of the energy spectrum can be used to study whether the second knee really exists at a few $100\\,$PeV and whether it is a distinct feature from the heavy knee discovered by KASCADE-Grande \\cite{KGheavyKnee2011}. \nFinally, Tunka-Rex remains a testbed for future developments of the radio technique. \nWhile the physics of the radio emission by air showers is understood sufficiently well \\cite{HuegeReview2016, SchroederReview2016}, significant technical developments have to be done for sparse and large arrays of the next-generation, such as GRAND \\cite{GRAND_ICRC2017}.\n\n\\begin{figure*}[t]\n  \\centering\n  \\includegraphics[width=0.7\\linewidth]{TunkaRexMap2017.pdf}\n  \\caption{Map of the cosmic-ray detectors of TAIGA: the Tunka-133 air-Cherenkov array consisting of 25 clusters of seven photomultipliers each and a local data acquisition (DAQ) at each cluster center, the Tunka-Grande particle-detector array with one detector station at each of the 19 inner cluster centers, and the Tunka-Rex radio array, with 3 antenna stations at each of the 19 inner clusters triggered by both, Tunka-133 and Tunka-Grande, and one antenna stations at each outer cluster triggered by Tunka-133.}\n  \\label{fig_tunkaRexMap}\n\\end{figure*}\n\n\\begin{figure*}[t]\n  \\centering\n  \\includegraphics[width=0.99\\linewidth]{TunkaRexCluster5.jpg}\n  \\caption{Photo of one cluster center with three antenna stations, a particle-detector station with surface scintillators in the gray shelter and underground scintillators in a concrete tunnel below the longish hump attached to the shelter, and a photomultiplier detector (silver cylinder) next to the local data-acquisition of this cluster (white box).}\n  \\label{fig_tunkaPhoto}\n\\end{figure*}\n\n\\section{Status of Tunka-Rex}\nSince October 2016 Tunka-Rex consists of 63 antenna stations distributed over a total area of about $3\\,$km$^2$ (figure \\ref{fig_tunkaRexMap}) \\cite{Schroeder_TunkaRex_ARENA2016}. \nEach of the 19 inner clusters of the cosmic-ray detectors of TAIGA (Tunka Advanced Instrument for cosmic ray physics and Gamma Astronomy) features 3 antenna stations (Tunka-Rex), 7 photomultiplier detectors for the air-Cherenkov light (Tunka-133), and 1 scintillator station with surface and underground particle detectors for electron and muon measurements (Tunka-Grande). \nThe 6 outer clusters feature only 1 antenna station and 7 photomultiplier detectors, each.\nDuring clear nights, when the air-Cherenkov array Tunka-133 operates it triggers all radio antennas, and during the remaining time the particle-detector array Tunka-Grande triggers all inner antennas. \nThis means that for every air shower all 57 inner antennas are read out, and additionally the 6 outer antennas exclusively when the air-Cherenkov detector operates. \n\nEach radio station consists of a set of two perpendicularly aligned SALLAs with low-noise amplifiers integrated directly in the antenna \\cite{AERAantennaPaper2012}. \nAfter passing a another filter-amplifier, the radio signal is digitized by the same digital data-acquisition used for the other cosmic-ray detectors of TAIGA (figure~\\ref{fig_tunkaPhoto}) \\cite{TAIGA_2014}. \nUsing the core-position of the denser Tunka-133 as input, the radio data are analyzed by a special version of the Offline software frame work developed by the Pierre Auger Collaboration \\cite{RadioOffline2011}, and afterwards compared to the Tunka-133 measurements. \nA combined analysis with the particle signals of Tunka-Grande is planned, too. \n\nTunka-Rex is calibrated on an absolute scale using the same external reference source that was used first by LOPES and later by LOFAR \\cite{TunkaRex_NIM_2015, 2015ApelLOPES_improvedCalibration, NellesLOFAR_calibration2015}. \nHowever, the antennas and the analog electronics of the subsequent signal chain are from three different production series, i.e., each of the inner clusters has one antenna of each series (figure~\\ref{fig_exampleEvent}). \nWhile all three series use the same electronics scheme, the gain and phase response varies slightly from series to series, and the inter-calibration is not yet completed. \nThus, at the moment most of the analyses are still based on almost 200 events detected by Tunka-Rex during the first two years of data taking with only one antenna per cluster triggered by the Tunka-133 air-Cherenkov detector. \n\n\n\n\\section{Event Reconstruction}\nThe reconstruction of the air-shower parameters consists of several steps described in more detail in the given references. \nFirst, the detector response is deconvoluted from the raw data using the phase and gain obtained by calibration measurements\\footnote{ \nLately we found that the correction for the phase responses was insufficient, i.e., delays were calculated too small and the distortion of the radio pulses due to dispersion was not accounted for properly. \nNevertheless, the pulse distortion by the signal chain is only a second-order effect of about $2\\,\\%$ on the maximum pulse amplitude we use in our standard analyses. \nThe necessary corrections on our previous results have been partially applied already: the figures presented here are still preliminary and we expect further changes that are small compared to other uncertainties, though. \nTogether with other minor improvements, e.g., in fitting procedures, this is the main reason why the results in this proceedings slightly differ from the ones published earlier. \n}.\nThen, we search for pulses with a signal-to-noise ratio $SNR > 10$ (in power) in a signal windows depending on the trigger time by Tunka-133 \/ Tunka-Grande \\cite{TunkaRex_NIM_2015}, and correct the pulse amplitude for the average bias due to noise \\cite{TunkaRex_Xmax2016}. \nThe threshold is tuned such that pure noise has a pass-chance of slightly below $5\\,\\%$. \nPhase and gain of the antenna response depend on the arrival direction of the radio signal and are deconvoluted in an iterative process while reconstructing the direction of the air shower from the pulse arrival times in the individual antenna stations. \nAt the moment we still use a plane-wave approximation for this purpose, since we need the arrival direction mainly as quality cut, where the expected difference to the more accurate hyperbolic wavefront of the order of $1^\\circ$ is unimportant \\cite{2014ApelLOPES_wavefront}.\nNonetheless, this simple method yields an accuracy of the arrival direction of about $1^\\circ$ for showers with zenith angles $\\theta < 50^\\circ$, and all events for which the Tunka-Rex direction disagrees by more than $5^\\circ$ from Tunka-133 are rejected as false-positive. \nThis removes almost all events passing the SNR cut by chance, and the potentially remaining ones have not deteriorated our main analysis results, since we do not observe any obvious outliers. \n\nEnergy and $X_\\mathrm{max}$ are reconstructed in a subsequent step from the lateral-distribution of the radio amplitude \\cite{KostuninTheory2015}. \nIn order to fit a simple, one dimensional lateral distribution function (LDF), we correct the amplitude in each station for the azimuthal asymmetry of the radio footprint. \nFor the correction we simply assume that the strength of the Askaryan effect accounts for a constant $8.5\\,\\%$ of the maximum geomagnetic amplitude (i.e, more for showers that are not perpendicular to the geomagnetic field), since more complicated correction formulas did not significantly improved the subsequent accuracy for the reconstructed energy and $X_\\mathrm{max}$.\nFinally we fit a Gaussian LDF and use the amplitude at $120\\,$m as estimator for the shower energy and the slope at $180\\,$m for the reconstruction of $X_\\mathrm{max}$ (see figure \\ref{fig_exampleEvent} for an example event). \nAll parameters, i.e., the proportionality coefficient for the energy and the function used for $X_\\mathrm{max}$, have been tuned against CoREAS simulations and not to the Tunka-133 measurements we compare to. \n\n\n\\begin{figure*}[t]\n  \\centering\n  \\includegraphics[width=0.99\\linewidth]{exampleEventCombined.pdf}\n  \\caption{Example event after correction of the amplitudes in individual antennas for the geomagnetic angle and for the azimuthal asymmetry caused by the interference of the Askaryan and the geomagnetic effects. \n  Gray crosses are antenna below threshold, the small outer crosses are clusters without particle detectors that were not operating when the event was recorded.}\n  \\label{fig_exampleEvent}\n\\end{figure*}\n\n\\section{Results}\n\\label{section_results}\nUsing the standard method described above we can determine the energy of every event. \nFor $X_\\mathrm{max}$ the uncertainties are usually too large for events with only three or four antennas above threshold which is easy to understand: \nWhile the axis distance of $120\\,$m used for energy estimation is close to the maximum of the LDF, the amplitude is lower and closer to the noise level around the axis distance of $180\\,$m used for $X_\\mathrm{max}$ determination. \nConsequently, the threshold for $X_\\mathrm{max}$ is about $0.2$ higher in $\\lg E$ and less than one third of our events have sufficient quality to measure $X_\\mathrm{max}$. \nFigure \\ref{fig_energyXmax} shows the correlation of the radio and air-Cherenkov measurements of the same showers. \n\nAs cross-check that the correlation is real and not due to any unwanted implicit tuning, we kept the Tunka-133 reconstruction of half of the events blind to the persons working on the Tunka-Rex analysis. \nThe collaboration members responsible for the Tunka-133 reconstruction at first revealed only the shower axis and kept the energy and $X_\\mathrm{max}$ secret until they had received the corresponding Tunka-Rex values. \nThus, we consider the observed correlations an experimental proof that radio measurements can be used to measure not only the energy, but also $X_\\mathrm{max}$, as indicated earlier by LOPES \\cite{2012ApelLOPES_MTD}.\nThe Tunka-Rex precisions we have derived from the deviations to the Tunka-133 values are better than $15\\,\\%$ for the energy and about $40\\,$g\/cm$^2$ for $X_\\mathrm{max}$. \n\nAs next step, we aim at a lower threshold for $X_\\mathrm{max}$ by further developing the computing-intensive analysis method used by LOFAR \\cite{BuitinkLOFAR_Xmax2014, Kostunin_TopDownXmax_ICRC2017}.\nWe have already shown that the threshold can be lowered for the energy.\nThe energy can be reconstructed with slightly worse precision even for events with only a single antenna station above threshold when using the shower axis measured by Tunka-133 or Tunka-Grande.\nThis is possible because shower-to-shower fluctuations have only a small impact on the amplitude at the reference distance of $120\\,$m so that we can reconstruct the energy by using the average shape of the lateral distribution \\cite{Hiller_ARENA2016}. \nMoreover, we will check whether we can further improve the accuracy of our energy reconstruction by implementing ideas used by AERA \\cite{AERAenergyPRL_PRD_combined2016}, and we work on a calculation of the time-, energy- and direction-dependent efficiency as basis for an energy spectrum \\cite{Fedorov_TunkaRexEfficiency_ICRC2017}.\n\n\\begin{figure*}[t]\n  \\centering\n  \\includegraphics[width=0.48\\linewidth]{TunkaRexEnergyCorrelation.pdf}\n  \\hfill\n  \\includegraphics[width=0.50\\linewidth]{TunkaRexXmaxCorrelation.pdf}\n  \\caption{Correlation of the energy (left) and the distance to the shower maximum (right) between Tunka-Rex radio measurements and the Tunka-133 air-Cherenkov measurements of the same air-showers \\cite{TunkaRex_Xmax2016}.}\n  \\label{fig_energyXmax}\n\\end{figure*}\n\n\nMotivated by the high accuracy of the energy reconstruction we have applied our radio measurements on an important open issue in cosmic-ray physics: the absolute energy scale of different experiments \\cite{TunkaRexScale2016}. \nWhile the absolute scale accuracy of Tunka-Rex of $20\\,\\%$ is not better than the one of the host experiment Tunka-133, we were able to make a relative comparison to KASCADE-Grande with a higher accuracy of about $10\\,\\%$, since Tunka-Rex was calibrated by exactly the same reference source as LOPES, the radio extension of KASCADE-Grande. \nThe comparison of the energy scales has been done in two different ways, both using the primary energy reconstructed by the host experiments and the radio amplitudes measured by Tunka-Rex and LOPES. \n\nThe first method uses the ratio between the energy reconstructed by the host experiments and the radio amplitude at a common reference distance after correction for the local strengths of the geomagnetic field. \nNeglecting smaller effects, e.g., due to different observation levels and various peculiarities of the two radio arrays, different ratios between the radio amplitude and the energy directly translate into corresponding differences of the absolute energy scales of the host experiments. \nThe second method is based on CoREAS simulations using the energy reconstructed by the host experiments as true input. \nThus, the ratio between the simulated and observed radio amplitudes corresponds to the ratio of the energy scales of the host experiments. \nSince the real composition is unknown, we simulated both, a pure proton and a pure iron composition, as extreme cases with almost identical results. \nThe general advantage of the second method is that it takes the specific differences between the LOPES and Tunka-Rex arrays into account because the detector simulation is included, while the advantage of the first method is that it does not depend on the theoretical models implicit in the Monte Carlo simulations. \n\nBoth methods yield the same final result within their (correlated) systematic uncertainties. \nThe energy scales of Tunka-133 and KASCADE-Grande are equal within a systematic uncertainty of about $10\\,\\%$ (figure \\ref{fig_scaleComparison}).\nThe small and not significant difference of the energy scales measured by the radio detectors is the same as the one obtained directly from the energy spectra of the host experiments \\cite{2012ApelKGenergySpectrum, Tunka133_NIM2014}.\nThis is remarkable since Tunka-133 and KASCADE-Grande use completely different detection techniques, namely the measurement of air-Cherenkov light and the detection of secondary particles at ground, respectively. \n\n\\begin{figure*}[t]\n  \\centering\n  \\includegraphics[width=0.39\\linewidth]{TunkaRex_vs_CoREAS.pdf}\n  \\hfill\n  \\includegraphics[width=0.59\\linewidth]{energyScaleResult.pdf}\n  \\caption{Left: Comparison of the radio amplitudes in individual antenna stations measured by Tunka-Rex and simulated by CoREAS using the Tunka-133 energy as input. \n  They agree within uncertainties for both extreme cases of a pure proton or a pure iron composition. \n  Right: Ratio of the Tunka-133 and KASCADE-Grande energy scales determined with different methods via their radio extension Tunka-Rex and LOPES; within a total uncertainty of about $10\\,\\%$ both experiments have the same absolute scale \\cite{TunkaRexScale2016}.}\n  \\label{fig_scaleComparison}\n\\end{figure*}\n\n\\section{Conclusion}\nTunka-Rex has shown that radio arrays can provide a cost-effective enhancement for existing air-shower arrays in order to increase their total accuracy. \nThe radio measurements can compete with the established optical air-fluorescence and air-Cherenkov techniques regarding the absolute accuracy of the energy, which enables several science cases, e.g., the comparison of the absolute energy scales of different experiments. \nMoreover, radio measurements provide an measurement of $X_\\mathrm{max}$ during day time. \nThe $X_\\mathrm{max}$ resolution of the first Tunka-Rex measurements, however, cannot yet compete with the optical techniques. \nSince these first results are based on only one antenna station per cluster, we will soon study how much the resolution improves with the current configuration of three antenna stations per cluster. \nFinally, the combination of the radio measurements with the muon measurements by Tunka-Grande will provide an additional sensitivity to the mass composition.\n\n\\clearpage\n\n\\section*{Acknowledgements}\nThe construction of Tunka-Rex was funded by the German Helmholtz association and the Russian Foundation for Basic Research (grant HRJRG-303).\nThis work has been supported by the Helmholtz Alliance for Astroparticle Physics (HAP),\nby Deutsche Forschungsgemeinschaft (DFG grant SCHR 1480\/1-1),\nby the Russian Federation Ministry of Education and Science (projects 14.B25.31.0010, 2017-14-595-0001-003, No3.9678.2017\/8.9, No3.904.2017\/4.6, 3.6787.2017\/7.8, 1.6790.2017\/7.8), \nby the Russian Foundation for Basic Research (grants 16-02-00738, 16-32-00329, 17-02-00905),\nand by grant 15-12-20022 of the Russian Science Foundation (section~\\ref{section_results}).\n\n\\bibliographystyle{JHEP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $\\Sigma_{g,p}$ denote a connected orientable surface of genus-$g$ with $p$-punctures, when $p=0$ we write  $\\Sigma_{g}$. \nThe \\textit{mapping class group} of $\\Sigma_{g,p}$ is the group of isotopy classes of orientation-preserving homeomorphisms of $\\Sigma_{g,p}$ preserving the set of punctures. \n\nHere is a brief history of generating sets for ${\\rm Mod}(\\Sigma_{g,p})$:\nDehn~\\cite{de} showed that  ${\\rm Mod}(\\Sigma_{g})$ can be generated by $2g(g-1)$ Dehn twists. About a quarter century later, Lickorish~\\cite{l3} gave a generating set  consisting of $3g-1$ Dehn twists. \nLater, Humphries~\\cite{H} reduced the number of Dehn twist generators to $2g+1$. He also proved that  the number $2g+1$ is minimal for $g\\geq 2$. \nJohnson~\\cite{j} proved that  the same set of Dehn twists also generates ${\\rm Mod}(\\Sigma_{g,1})$. \nIn the presence of multiple punctures, Gervais~\\cite{G} proved that ${\\rm Mod}(\\Sigma_{g,p})$ can be generated by $2g+p$ Dehn twists for $p\\geq1$.\n\nIf it is not required that the generators are Dehn twists, then it is possible to obtain smaller generating sets for ${\\rm Mod}(\\Sigma_{g,p})$:   \nFor $g\\geq 3$ and $p=0$, Lu~\\cite[Theorem~$1.3$]{lu} proved that  \n${\\rm Mod}(\\Sigma_{g})$ can be generated by three elements\nFor $g\\geq 1$ and $p=0$ or $1$, \na minimal (since the group is not abelian) generating set of two elements, a product of two Dehn twists  and a product of $2g$ Dehn twists,   \nwas first given by Wajnryb~\\cite{w}.  Korkmaz~\\cite[Theorem~$5$]{mk2} improved this result by showing that one of these two generators can be taken as a Dehn twist. \nHe also showed that this group is generated by two elements of finite order ~\\cite[Theorem~$14$]{mk2}.  \nFor $g\\geq 3$, Kassabov obtained a generating set of  involution elements where the number of generators depends on $g$ and the parity of $p$  (see ~\\cite[Theorem~1]{ka}).\nLater, Monden~\\cite{m1} removed the parity condition on $p$ for $g=7$ and $g=5$. For $g\\geq1$ and $p\\geq 2$, Monden~\\cite{m2} also gave a generating set for ${\\rm Mod}(\\Sigma_{g,p})$ \nconsisting of three elements.  Recently, he~\\cite{m3} gave a minimal generating set for ${\\rm Mod}(\\Sigma_{g,p})$ containing two elements for $g\\geq3$.\n\n\nNote that any infinite group generated by two involutions must be isomorphic to the infinite dihedral group whose subgroups are either cyclic or dihedral.  Since \n${\\rm Mod}(\\Sigma_{g,p})$ contains nonabelian free groups, it cannot be generated by two involutions.  In this paper, we obtain the following result (cf. ~\\cite[Remark~5]{ka}):\n\n\\begin{thma}\\label{thma}\nFor every even integer $p\\geq 8$ and $g\\geq 14$, ${\\rm Mod}(\\Sigma_{g,p})$ can be generated by three involutions.\nMoreover, for every even integer $p\\geq 4$ and for $g=3, 4, 5$ or $6$,  ${\\rm Mod}(\\Sigma_{g,p})$ can be generated by four involutions.\n\\end{thma}\n\nAt the end of the paper, we also show that Theorem~A also \nholds for the cases $p=2$ or $p=3$.  For surfaces with odd number of punctures, we have the following result:\n\n\\begin{thmb}\\label{thmb}\nFor every odd integer $p\\geq 9$ and $g\\geq 13$, ${\\rm Mod}(\\Sigma_{g, p})$ is generated by four involutions.\nMoreover, for every odd integer $p\\geq 5$ and for $g=3, 4, 5$ or $6$,  ${\\rm Mod}(\\Sigma_{g,p})$ can be generated by five involutions.\n\\end{thmb}\n\nThe paper is organized as follows: In Section~\\ref{S2}, we quickly provide the necessary background on mapping class groups. \nThe proofs of Theorem~A  and Theorem~B  are given in Section~\\ref{S3}.  \n\n\n\\medskip\n\n\\noindent\n{\\bf Acknowledgements.}\nThis work was supported by the Scientific and Technological Research Council of Turkey (T\\\"{U}B\\.{I}TAK)[grant number 120F118].\n\n\n\\par  \n\\section{Background and Results on Mapping Class Groups} \\label{S2}\n\n Let $\\Sigma_{g,p}$ denote a connected orientable surface of genus $g$ with $p$ punctures specified by the set $P=\\lbrace z_1,z_2,\\ldots,z_p\\rbrace$ of $p$ distinguished points. If $p$ is zero then we omit it from the notation and write $\\Sigma_{g}$. {\\textit{The mapping class group}} \n ${\\rm Mod}(\\Sigma_{g,p})$ of the surface $\\Sigma_{g,p}$ is defined to be the group of the isotopy classes of orientation preserving\n self-diffeomorphisms of $\\Sigma_{g,p}$ which fix the set $P$. {\\textit{The mapping class group}} ${\\rm Mod}(\\Sigma_{g,p})$ of the surface $\\Sigma_{g,p}$ is defined to be the group of isotopy classes of all orientation preserving self-diffeomorphisms of $\\Sigma_{g,p}$ which fix the set $P$. Let ${\\rm Mod}_{0}(\\Sigma_{g,p})$ denote the subgroup of ${\\rm Mod}(\\Sigma_{g,p})$ consisting of elements which fix the set $P$ pointwise. It is obviuos that we have the following exact sequence:\n \\[\n1\\longrightarrow {\\rm Mod}_{0}(\\Sigma_{g,p}) \\longrightarrow {\\rm Mod}(\\Sigma_{g,p}) \\longrightarrow Sym_{p}\\longrightarrow 1,\n\\]\nwhere $Sym_p$ denotes the symmetric group on the set $\\lbrace1,2,\\ldots,p\\rbrace$ and the last projection is given by the restriction of the isotopy class of a diffeomorphism to its action on the punctures. \\par\nLet $\\beta_{i,j}$ be an embedded arc that joins two punctures $z_i$ and $z_j$ and does not intersect $\\delta$ on $\\Sigma_{g,p}$. \nLet $D_{i,j}$ denote a closed regular neighbourhood of $\\beta_{i,j}$, which is a disk with two punctures. \nThere exists a diffeomorphism $H_{i,j}: D_{i,j} \\to D_{i,j}$, which interchanges the punctures such that $H_{i,j}^{2}$ is equal to the right handed Dehn twist about $\\partial D_{i,j}$ and is the identity on the complement of the interior of $D_{i,j}$. Such a diffeomorphism is said to be \\textit{the (right handed) half twist} about $\\beta_{i,j}$. It can be extended to a diffeomorphism of ${\\rm Mod}(\\Sigma_{g,p})$. Throughout the paper we do not distinguish a \n diffeomorphism from its isotopy class. For the composition of two diffeomorphisms, we\nuse the functional notation; if $f$ and $g$ are two diffeomorphisms, then\nthe composition $fg$ means that $g$ is applied first and then $f$.\\\\\n\\indent\n For a simple closed \ncurve $a$ on $\\Sigma_{g,p}$, following ~\\cite{apy,mk1}, we denote the right-handed \nDehn twist $t_a$ about $a$ by the corresponding capital letter $A$.\nLet us also remind the following basic facts of Dehn twists that we use frequently throughout  the paper: Let $a$ and $b$ be \nsimple closed curves on $\\Sigma_{g,p}$ and $f\\in {\\rm Mod}(\\Sigma_{g,p})$.\n\\begin{itemize}\n\\item  If $a$ and $b$ are disjoint, then $AB=BA$ (\\textit{Commutativity}).\n\\item If $f(a)=b$, then $fAf^{-1}=B$ (\\textit{Conjugation}).\n\\end{itemize}\nLet us finish this section by noting that we  denote the conjugation relation $fgf^{-1}$ by $f^{g}$ for any $f,g \\in {\\rm Mod}(\\Sigma_{g,p})$.\n\n\n\n\n\\section{Involution generators for ${\\rm Mod}(\\Sigma_{g,p})$}\\label{S3}\nLet us start this section by recalling the following set of generators given by Korkmaz~\\cite[Theorem~$5$]{mk1}.\n\n\\begin{theorem}\\label{thm1}\nIf $g\\geq3$, then the mapping class group ${\\rm Mod}(\\Sigma_g)$ can be generated by the four elements $R$, $A_1A_{2}^{-1}$, $B_1B_{2}^{-1}$, $C_1C_{2}^{-1}$.\n\\end{theorem}\n\nLet us also recall the following well known result from algebra.\n\\begin{lemma}\\label{lemma1}\nLet $G$ and $K$ be groups. Suppose that the following short exact sequence holds,\n\\[\n1 \\longrightarrow N \\overset{i}{\\longrightarrow}G \\overset{\\pi}{\\longrightarrow} K\\longrightarrow 1.\n\\]\nThen the subgroup $\\Gamma$ contains $i(N)$ and has a surjection to $K$ if and only if $\\Gamma=G$.\n\\end{lemma}\n\\par\n\nIn our case where $G={\\rm Mod}(\\Sigma_{g,p})$ and $N={\\rm Mod}_{0}(\\Sigma_{g,p})$,\nwe have the following short exact sequence:\n\\[\n1\\longrightarrow {\\rm Mod}_{0}(\\Sigma_{g,p})\\longrightarrow {\\rm Mod}(\\Sigma_{g,p}) \\longrightarrow Sym_{p}\\longrightarrow 1.\n\\]\nTherefore, we obtain the following useful result which follows immediately from Lemma~\\ref{lemma1}. Let $\\Gamma$ be a subgroup of ${\\rm Mod}(\\Sigma_{g,p})$. If the subgroup $\\Gamma$ contains ${\\rm Mod}_{0}(\\Sigma_{g,p})$ and has a surjection to $Sym_p$ then $\\Gamma={\\rm Mod}(\\Sigma_{g,p})$.\n\n\n\\begin{figure}[hbt!]\n\\begin{center}\n\\scalebox{0.3}{\\includegraphics{gevenpeven.png}}\n\\caption{The involutions $\\rho_1$ and $\\rho_2$ if $g=2k$ and $p=2b$.}\n\\label{EE}\n\\end{center}\n\\end{figure}\n\n\nThroughout the paper, we consider the embeddings of $\\Sigma_{g,p}$ into $\\mathbb{R}^{3}$ in such a way that it is invariant under the rotations \n$\\rho_1$ and $\\rho_2$.  Here, $\\rho_1$ and $\\rho_2$ are the rotations by $\\pi$ about the $z$-axis  (see Figures~\\ref{EE} and~\\ref{OE}). \nNote that ${\\rm Mod}(\\Sigma_{g,p})$ contains the element $R=\\rho_1\\rho_2$ which satisfies the following:\n\\begin{itemize}\n\\item [(i)]   $R(a_i)=a_{i+1}$, $R(b_i)=b_{i+1}$ for $i=1,\\ldots,g-1$ and $R(b_g)=b_{1}$,\n\\item [(ii)]  $R(c_i)=c_{i+1}$ for $i=1,\\ldots,g-2$,\n\\item [(iii)] $R(z_1)=z_p$ and $R(z_i)=z_{i-1}$ for $i=2,\\ldots,p$.\n\\end{itemize}\n\n\n\n\nWe want to note here that, in the following lemmata, where we present generating sets for surfaces with even number of punctures, we mainly follow the proof \nof \\cite[Theorem~$5$]{mk1}.   We use them in the proof of Theorem A and then for surfaces with odd number of punctures we explain how our arguments are modified.\n\n\\begin{lemma}\\label{lemeven}\nFor every even integer $g=2k\\geq14$ and every even integer $p=2b\\geq 10$, the subgroup of ${\\rm Mod}(\\Sigma_{g,p})$ generated by the elements \n\\[\n\\rho_1, \\rho_2 \\textrm{ and }\\rho_1H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}B_{k-1}A_k\nA_{k+2}^{-1}B_{k+3}^{-1}C_{k+4}^{-1}\\]\ncontains the Dehn twists $A_i$, $B_i$ and $C_i$  for $i=1,\\ldots,g$.\n\\end{lemma}\n\n\\begin{proof}\n\n\nConsider the models of $\\Sigma_{g,p}$ depicted in Figure~\\ref{EE}. Let $F_1:=H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}B_{k-1}A_k\nA_{k+2}^{-1}B_{k+3}^{-1}C_{k+4}^{-1}$ and let $\\Gamma$ be the subgroup of ${\\rm Mod}(\\Sigma_{g,p})$ generated by the elements \n$\\rho_1$, $\\rho_2$ and $\\rho_1F_1$. One can see that the elements $R=\\rho_1\\rho_2$ and $F_1=\\rho_1 \\rho_1 F_1$ are contained in\nthe subgroup $\\Gamma$.\nLet $F_2$ be the element obtained by the conjugation of $F_1$ by $R^{-3}$. Since \n\\[\nR^{-3}(c_{k-3}, b_{k-1}, a_k, a_{k+2}, b_{k+3}, c_{k+4}) = (c_{k-6}, b_{k-4}, a_{k-3}, a_{k-1}, b_{k}, c_{k+1})\n\\]\nand\n\\[\nR^{-3}(z_{b-1},z_{b},z_{b+1})=(z_{b+2},z_{b+3},z_{b+4}),\n\\]\n\\[F_2=F_{1}^{R^{-3}}=H_{b+2,b+3}H_{b+4,b+3}^{-1}C_{k-6}B_{k-4}A_{k-3}A_{k-1}^{-1}B_{k}^{-1}C_{k+1}^{-1}\\in \\Gamma. \n\\]\n\\noindent \nLet $F_3$ be the element $F_1^{F_1F_2^{-1}}$, that is\n$\nF_3=H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}A_{k-1}B_k\nA_{k+2}^{-1}B_{k+3}^{-1}C_{k+4}^{-1} \\in \\Gamma.\n$\n\nSince we use repeatedly similar calculations in the remaining parts of the paper, let us provide  some details here. It can be shown that the diffeomorphism $F_1F_2^{-1}$ maps the curves $\\lbrace c_{k-3},b_{k-1},a_k,a_{k+2},b_{k+3},c_{k+4} \\rbrace$ to the curves $\\lbrace c_{k-3},b_{k-1},a_k,b_{k+2},a_{k+3},c_{k+4}  \\rbrace$, respectively. Also it follows from the factorizations of half twists $H_{b-1,b}H_{b+1,b}^{-1}$ and $H_{b-4,b-3}H_{b-2,b-3}^{-1}$ commute and  we get\n\\begin{eqnarray*}\nF_3&=&F_1^{F_1F_2^{-1}}\\\\\n&=&(F_1F_2^{-1})(H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}B_{k-1}A_k\nA_{k+2}^{-1}B_{k+3}^{-1}C_{k+4}^{-1})(F_1F_2^{-1})^{-1}\\\\\n&=&H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}A_{k-1}B_k\nA_{k+2}^{-1}B_{k+3}^{-1}C_{k+4}^{-1}.\n\\end{eqnarray*}\n The subgroup $\\Gamma$ contains the following elements:\n\\begin{eqnarray*}\nF_4&=&F_{3}^{R^{-3}}=H_{b+2,b+3}H_{b+4, b+3}^{-1}C_{k-6}A_{k-4}B_{k-3}A_{k-1}^{-1}B_{k}^{-1}C_{k+1}^{-1},\\\\\nF_5&=&F_{3}^{F_3F_4}=H_{b-1,b}H_{b+1,b}^{-1}B_{k-3}A_{k-1}B_k\nA_{k+2}^{-1}B_{k+3}^{-1}C_{k+4}^{-1}.\n\\end{eqnarray*} \n From these, we obtain the element \n $F_5F_3^{-1}=B_{k-3}C_{k-3}^{-1}$, which is contained in $\\Gamma$. By conjugating this element with powers of $R$, we conclude that  \n \\[\n B_{i}C_{i}^{-1}\\in \\Gamma \\  \\textrm{for}  \\ i=1, \\ldots, g-1.\n \\]\nThe subgroup $\\Gamma$ also contains the element $F_1F_3^{-1}=B_{k-1}A_{k}B_{k}^{-1}A_{k-1}^{-1}$.\nAfter conjugating with $R^3$ and considering the inverse, we have $A_{k+2}B_{k+3}A_{k+3}^{-1}B_{k+1}^{-1} \\in \\Gamma$.\nThis in turn  implies that for $i=1,\\ldots,g-1$, the elements\n\\[\nA_iB_{i+1}A_{i+1}^{-1}B_i^{-1}\\in \\Gamma.\n\\]\nWe also have the following elements in $\\Gamma$:\n\\begin{eqnarray*}\nF_6&=&F_1 (A_{k+2}B_{k+3}A_{k+3}^{-1}B_{k+2}^{-1})=H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}B_{k-1}A_{k}\nB_{k+2}^{-1}A_{k+3}^{-1}C_{k+4}^{-1},\\\\\nF_7&=&F_{6}^{R^{-3}}=H_{b+2,b+3}H_{b+4,b+3}^{-1}C_{k-6}B_{k-4}A_{k-3}\nB_{k-1}^{-1}A_{k}^{-1}C_{k+1}^{-1} \\textrm{ and }\\\\\nF_8&=&F_{6}^{F_6F_7}=H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}B_{k-1}A_k\nC_{k+1}^{-1}A_{k+3}^{-1}C_{k+4}^{-1},\n\\end{eqnarray*}\nHence, we can conclude that $F_8^{-1}F_6=C_{k+1}B_{k+2}^{-1}\\in \\Gamma$. Again conjugating with $R$  implies that\n\\[\nC_iB_{i+1}^{-1}\\in \\Gamma, \\ \\textrm{for all} \\  i=1,\\ldots,g-1. \n\\]\nFurthermore, we can see that  $\\Gamma$  contains the following elements: \n\\begin{eqnarray*}\nF_9&=&(B_{k-3}C_{k-3}^{-1})F_3(C_{k+4}B_{k+5}^{-1})=H_{b-1,b}H_{b+1,b}^{-1}B_{k-3}A_{k-1}B_kA_{k+2}^{-1}B_{k+3}^{-1}B_{k+5}^{-1},\\\\\nF_{10}&=&F_{9}^{R^{-3}}=H_{b+2,b+3}H_{b+4,b+3}^{-1}B_{k-6}A_{k-4}B_{k-3}A_{k-1}^{-1}B_{k}^{-1}B_{k+2}^{-1},\\\\\nF_{11}&=&F_{9}^{F_9F_{10}}=H_{b-1,b}H_{b+1,b}^{-1}B_{k-3}A_{k-1}B_kB_{k+2}^{-1}B_{k+3}^{-1}B_{k+5}^{-1}.\n\\end{eqnarray*}\nFrom these, we obtain $F_{11}F_{9}^{-1}=B_{k+2}^{-1} A_{k+2} \\in \\Gamma$. By the action of $R$, we can conclude that\n\\[\nA_iB_i^{-1}\\in \\Gamma \\ \\textrm{for all}  \\ i=1, \\ldots, g. \n\\]\nThis completes the proof by  Theorem~\\ref{thm1} since the subgroup $\\Gamma$ contains the elements\n\\begin{eqnarray*}\nA_1A_{2}^{-1}&=&(A_1B_1^{-1})(B_1C_1^{-1})(C_1B_{2}^{-1})(B_{2}A_{2}^{-1}),\\\\\nB_1B_{2}^{-1}&=&(B_1C_1^{-1})(C_1B_{2}^{-1}) \\textrm{ and }\\\\\nC_1C_{2}^{-1}&=&(C_1B_{2}^{-1})(B_{2}C_{2}^{-1}).\n\\end{eqnarray*}\n\\end{proof}\n\n\\begin{figure}[hbt!]\n\\begin{center}\n\\scalebox{0.2}{\\includegraphics{goddpeven.png}}\n\\caption{The involutions $\\rho_1$ and $\\rho_2$ for $g=2k+1$ and $p=2b$.}\n\\label{OE}\n\\end{center}\n\\end{figure}\n\nIf $g$ is  odd and $p$ is even, we have the following result:\n\\begin{lemma}\\label{lemodd}\nFor every odd integer $g=2k+1 \\geq15$ and even integer $p=2b \\geq 10$, the subgroup of ${\\rm Mod}(\\Sigma_{g,p})$  generated by the elements\n\\[\n\\rho_1,\\rho_2 \\textrm{ and }\\rho_1H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}B_kA_{k+1}A_{k+2}^{-1}B_{k+3}^{-1}C_{k+5}^{-1}\n\\]\n contains the Dehn twists $A_i$, $B_i$ and $C_i$  for $i=1,\\ldots,g$.\n\\end{lemma}\n\n\\begin{proof}\nConsider the models for $\\Sigma_{g,p}$ as shown in Figure~\\ref{OE}. Let $\\Gamma$ denote the subgroup of ${\\rm Mod}(\\Sigma_{g,p})$ \ngenerated by the elements $\\rho_1$, $\\rho_2$ and $\\rho_1G_1$, where $G_1=H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}B_kA_{k+1}A_{k+2}^{-1}B_{k+3}^{-1}C_{k+5}^{-1}$. \nThe elements $R=\\rho_1\\rho_2$ and $G_1=\\rho_1 \\rho_1 G_1$ belong to the subgroup $\\Gamma$. \nLet $G_2$ denote the conjugation of $G_1$ by $R^{-3}$,\n\\[\nG_2=G_{1}^{R^{-3}}=H_{b+2,b+3}H_{b+4,b+3}^{-1}C_{k-6}B_{k-3}A_{k-2}A_{k-1}^{-1}B_{k}^{-1}C_{k+2}^{-1}.\n\\]\nIt is easy to verify that the element\n\\begin{eqnarray*}\nG_3&=&G_{1}^{G_1G_2}\\\\\n&=&H_{b-1,b}H_{b+1,b}^{-1}B_{k-3}B_kA_{k+1}A_{k+2}^{-1}C_{k+2}^{-1}C_{k+5}^{-1}\n\\end{eqnarray*}\nis contained in $\\Gamma$.\nLet\n\\[\nG_4=G_{3}^{R^{-3}}=H_{b+2,b+3}H_{b+4,b+3}^{-1}B_{k-6}B_{k-3}A_{k-2}A_{k-1}^{-1}C_{k-1}^{-1}C_{k+2}^{-1}.\n\\]\nThus we get the element\n\\begin{eqnarray*}\nG_5=G_3^{G_3G_4^{-1}}\n=H_{b-1,b}H_{b+1,b}^{-1}B_{k-3}C_{k-1}A_{k+1}A_{k+2}^{-1}C_{k+2}^{-1}C_{k+5}^{-1},\n\\end{eqnarray*}\nwhich is contained in $G$. This implies that $G_3G_5^{-1}=B_kC_{k-1}^{-1}\\in \\Gamma$. By conjugating $B_kC_{k-1}^{-1}$ with powers of $R$, we see that\n\\[\nB_{i+1}C_{i}^{-1}\\in \\Gamma,\n\\]\nfor all $i=1,\\ldots,g-1$. In particular, the element $C_{k+5}B_{k+6}^{-1}\\in \\Gamma$. Hence, the subgroup $\\Gamma$ contains the following element:\n\\[\nG_6=G_1(C_{k+5}B_{k+6}^{-1})=H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}B_kA_{k+1}A_{k+2}^{-1}B_{k+3}^{-1}B_{k+6}^{-1}.\n\\]\nThen, we see that the elements\n\\begin{eqnarray*}\nG_7&=&G_6^{R^{-3}}=H_{b+2,b+3}H_{b+4,b+3}^{-1}C_{k-6}B_{k-3}A_{k-2}A_{k-1}^{-1}B_{k}^{-1}B_{k+3}^{-1} \\textrm{ and}\\\\\nG_8&=&G_6^{G_6G_7}=H_{b-1,b}H_{b+1,b}^{-1}B_{k-3}B_kA_{k+1}A_{k+2}^{-1}B_{k+3}^{-1}B_{k+6}^{-1}\n\\end{eqnarray*}\nare contained in $\\Gamma$, which implies that the subgroup $\\Gamma$ contains the element $G_6G_8^{-1}=C_{k-3}B_{k-3}^{-1}$. By the action of $R$ we see that\n\\[\nC_iB_i^{-1} \\in \\Gamma\n\\]\nfor all $i=1,\\ldots, g-1$.  Moreover, we get \n\\begin{eqnarray*}\nG_9&=&(B_{k-2}C_{k-3}^{-1})G_6=H_{b-1,b}H_{b+1,b}^{-1}B_{k-2}B_kA_{k+1}A_{k+2}^{-1}B_{k+3}^{-1}B_{k+6}^{-1} \\in \\Gamma,\\\\\nG_{10}&=&G_9^{R^{-3}}=H_{b+2,b+3}H_{b+4,b+3}^{-1}B_{k-5}B_{k-3}A_{k-2}A_{k-1}^{-1}B_{k}^{-1}B_{k+3}^{-1}\\in \\Gamma \\textrm{ and}\\\\\nG_{11}&=&G_9^{G_9G_{10}}=H_{b-1,b}H_{b+1,b}^{-1}A_{k-2}B_kA_{k+1}A_{k+2}^{-1}B_{k+3}^{-1}B_{k+6}^{-1}\\in \\Gamma.\n\\end{eqnarray*}\nFrom these, we have $G_9G_{11}^{-1}=B_{k-2}A_{k-2}^{-1}\\in \\Gamma$ so that \n\\[\nB_iA_i^{-1}\\in \\Gamma,\n\\]\nfor $i=1,\\ldots,g$, by the action of $R$. The remaining part of the proof can be completed as in the proof of Lemma~\\ref{lemeven}.\n\\end{proof}\n\nIn the following four lemmata, we give generating sets for smaller genera\n\\begin{lemma}\\label{lem6}\nFor $g=6$ and every even integer $p\\geq4$, the group generated by the elements\n\\[\n\\rho_1,\\rho_2, \\rho_2B_2A_3A_4^{-1}B_5^{-1} \\textrm{ and }\\rho_1H_{b-1,b}H_{b+1,b}^{-1}C_{3}C_{4}^{-1}\n\\]\n contains the Dehn twists $A_i$, $B_i$ and $C_i$  for $i=1,\\ldots,g$.\n\\end{lemma}\n\\begin{proof}\nConsider the models for $\\Sigma_{6,p}$ as shown in Figure~\\ref{EE}. Let $\\Gamma$ be the subgroup of ${\\rm Mod}(\\Sigma_{6,p})$ generated by the elements $\\rho_1$, $\\rho_2$, $\\rho_2F_1$ and $\\rho_1E_1$ where $F_1=B_2A_3A_4^{-1}B_5^{-1}$ and $E_1=H_{b-1,b}H_{b+1,b}^{-1}C_{3}C_{4}^{-1}$. Hence the elements $R=\\rho_1\\rho_2$, $F_1=\\rho_2 \\rho_2 F_1$ and $E_1=\\rho_1\\rho_1E_1$ are contained in the subgroup $\\Gamma$. \n\nThe subgroup $\\Gamma$ contains the following elements:\n\\begin{eqnarray*}\nF_2&=&F_1^{R}=B_3A_4A_5^{-1}B_6^{-1} \\in \\Gamma,\\\\\nF_{3}&=&F_{1}^{F_1F_2}=B_2B_3A_4^{-1}A_5^{-1}\\in \\Gamma\\\\\nF_4&=&F_3^{R}=B_3B_4A_5^{-1}A_6^{-1}\\in \\Gamma\n\\textrm{ and}\\\\\nF_{5}&=&F_3^{F_3F_{4}^{-1}}=B_2B_3B_4^{-1}A_5^{-1}\\in \\Gamma.\n\\end{eqnarray*}\nHence we get the element $F_5^{-1}F_3=B_4A_4^{-1}\\in \\Gamma$. By the action of $R$, for all $i=1,\\ldots,6$,\n\\[\nA_iB_i^{-1}\\in \\Gamma.\n\\]\nMoreover, we have\n\\begin{eqnarray*}\nF_6&=&E_1^{E_1F_3}=H_{b-1,b}H_{b+1,b}^{-1}B_{3}C_{4}^{-1} \\in \\Gamma\n\\textrm{ and}\\\\\nF_{7}&=&E_1^{E_1F_{1}}=H_{b-1,b}H_{b+1,b}^{-1}C_{3}B_{5}^{-1}\\in \\Gamma.\n\\end{eqnarray*}\nThis implies that $F_6E_1^{-1}=B_3C_3^{-1}\\in \\Gamma$ and $F_7^{-1}E_1=B_5C_4^{-1}\\in \\Gamma$ and so\n\\[\nB_iC_i^{-1}\\in \\Gamma \\textrm{ and } B_{i+1}C_{i}^{-1}\\in \\Gamma,\n\\]\nfor all $i=1,\\ldots,5$, by conjugating these elements with powers of $R$. The proof can be completed as in the proof of Lemma~\\ref{lemeven}.\n\\end{proof}\n\n\\begin{lemma}\\label{lem5}\nFor $g=5$ and every even integer $p\\geq4$, the group generated by the elements\n\\[\n\\rho_1,\\rho_2, \\rho_1H_{b-1,b}H_{b+1,b}^{-1}A_{3}A_{4}^{-1} \\textrm{ and }\\rho_2A_2B_2C_2C_3^{-1}B_4^{-1}A_4^{-1} \n\\]\n contains the Dehn twists $A_i$, $B_i$ and $C_i$  for $i=1,\\ldots,g$.\n\\end{lemma}\n\\begin{proof}\nConsider the models for $\\Sigma_{5,p}$ as shown in Figure~\\ref{OE}. Let $\\Gamma$ denote the subgroup of ${\\rm Mod}(\\Sigma_{5,p})$ generated by the elements $\\rho_1$, $\\rho_2$, $\\rho_1F_1$ and $\\rho_2E_1$ where $F_1=H_{b-1,b}H_{b+1,b}^{-1}A_{3}A_{4}^{-1}$ and $E_1=A_2B_2C_2C_3^{-1}B_4^{-1}A_4^{-1}$. Thus the elements $R=\\rho_1\\rho_2$, $F_1=\\rho_1 \\rho_1 F_1$ and $E_1=\\rho_2\\rho_2E_1$ are in the subgroup $\\Gamma$. \n\nOne can obtain the following elements:\n\\begin{eqnarray*}\nF_2&=&F_1^{R^{-1}}=H_{b,b+1}H_{b+2,b+1}^{-1}A_{2}A_{3}^{-1} \n\\\\\nF_{3}&=&F_2^{E_1}=H_{b,b+1}H_{b+2,b+1}^{-1}B_{2}A_{3}^{-1} \\textrm{ and}\\\\\nF_4&=&F_3^{E_1}=H_{b,b+1}H_{b+2,b+1}^{-1}C_{2}A_{3}^{-1},\n\\end{eqnarray*}\nwhich are contained in $\\Gamma$. Thus we get that $F_2F_3^{-1}=A_2B_2^{-1}\\in \\Gamma$ and $F_3F_4^{-1}=B_2C_2^{-1}\\in \\Gamma$. By conjugating these elements with powers of $R$, we see that \n\\[\nA_iB_i^{-1}\\in \\Gamma \\textrm{ and }B_jC_j^{-1}\\in \\Gamma,\n\\]\nwhich also implies that $A_iC_i^{-1}\\in \\Gamma$ for all $i=1,\\ldots, 5$ and $j=1,\\ldots, 4$. \nFinally, it can be verified that \n\\[\nE_1(a_3,c_3)=(a_3,b_4)\n\\]\nso that the group $\\Gamma$ contains the element\n\\[\n(A_3C_3^{-1})^{E_1}=A_3B_4^{-1}.\n\\]\nHence $A_iB_{i+1}^{-1}\\in \\Gamma$ for all $i=1,\\ldots,5$ by the action of $R$. The rest of the proof is similar to that of Lemma~\\ref{lemeven}.\n\\end{proof}\n\n\n\n\\begin{lemma}\\label{lem4}\nFor $g=4$ and every even integer $p\\geq4$, the group generated by the elements\n\\[\n\\rho_1,\\rho_2, \\rho_2B_1A_2A_3^{-1}B_4^{-1} \\textrm{ and }\\rho_1H_{b-1,b} H_{b+1,b}^{-1}C_{2}C_{3}^{-1} \n\\]\n contains the Dehn twists $A_i$, $B_i$ and $C_i$  for $i=1,\\ldots,g$.\n\\end{lemma}\n\\begin{proof}\nLet us consider the models for $\\Sigma_{4,p}$ as shown in Figure~\\ref{EE} and let $\\Gamma$ be the subgroup of ${\\rm Mod}(\\Sigma_{4,p})$ generated by the elements $\\rho_1$, $\\rho_2$, $\\rho_2F_1$ and $\\rho_1E_1$ where $F_1=B_1A_2A_3^{-1}B_4^{-1}$ and $E_1=H_{b-1,b} H_{b+1,b}^{-1}C_{2}C_{3}^{-1}$. Thus it is clear that the elements $R=\\rho_1\\rho_2$, $F_1=\\rho_2 \\rho_2 F_1$ and $E_1=\\rho_1\\rho_1E_1$ belong to the subgroup $\\Gamma$.   We have the element\n\\[\nF_2=E_1^{E_1F_1}=H_{b-1,b} H_{b+1,b}^{-1}C_{2}B_{4}^{-1}\\in \\Gamma.\n\\]\nThus the subgroup $\\Gamma$ contains the elements $F{_2}^{-1} E_1=B_4C_3^{-1}$ and $\\rho_1(B_4C_3^{-1})\\rho_1=B_2C_2^{-1}$. By conjugating these elements with powers of $R$, we get\n\\[\nB_{i+1}C_i^{-1}\\in \\Gamma \\textrm{ and } B_iC_i^{-1}\\in \\Gamma\n\\]\nfor all $i=1,2,3$.  One can also obtain that the subgroup $\\Gamma$ contains the following elements:\n\\begin{eqnarray*}\nF_3&=&(C_1B_1^{-1})F_1=C_1A_2A_3^{-1}B_4^{-1},\\\\\nF_4&=&F_3^{R}(B_1C_1^{-1})=C_2A_3A_4^{-1}B_1^{-1}(B_1C_1^{-1})=C_2A_3A_4^{-1}C_1^{-1}\\textrm{ and}\\\\\nF_5&=&F_3^{F_3F_4}=C_1A_2A_3^{-1}A_4^{-1}.\n\\end{eqnarray*}\nThus we obtain that $F_5F_3^{-1}=A_4B_4^{-1}\\in \\Gamma$. By the action of $R$, $A_iB_i^{-1}\\in \\Gamma$ for all $i=1,2,3,4$.\nThe remaining part of the proof is very similar to that of Lemma~\\ref{lemeven}.\n\\end{proof}\n\n\\begin{lemma}\\label{lem3}\nFor $g=3$ and every even $p\\geq4$, the group generated by the elements\n\\[\n\\rho_1,\\rho_2, \\rho_1H_{b-1,b} H_{b+1,b}^{-1}A_{2}A_{3}^{-1} \n\\textrm{ and }\\rho_2A_1B_1C_1C_2^{-1}B_3^{-1}A_3^{-1}\n\\]\n contains the Dehn twists $A_i$, $B_i$ and $C_i$  for $i=1,2,3$.\n\\end{lemma}\n\\begin{proof}\nConsider the models for $\\Sigma_{3,p}$ as shown in Figure~\\ref{OE}. Let $\\Gamma$ be the subgroup of ${\\rm Mod}(\\Sigma_{3,p})$ generated by the elements $\\rho_1$, $\\rho_2$, $\\rho_1F_1$ and $\\rho_2E_1$ where $F_1=H_{b-1,b} H_{b+1,b}^{-1}A_{2}A_{3}^{-1}$ and $E_1=A_1B_1C_1C_2^{-1}B_3^{-1}A_3^{-1}$. Thus the elements $R=\\rho_1\\rho_2$, $F_1=\\rho_1 \\rho_1 F_1$ and $E_1=\\rho_2\\rho_2E_1$ are contained in the subgroup $\\Gamma$. We get the elements\n\\begin{eqnarray*}\nF_2&=&F_1^{R^{-1}}=H_{b,b+1} H_{b+2,b+1}^{-1}A_{1}A_{2}^{-1}\\in \\Gamma,\\\\\nF_3&=&F_2^{E_1}=H_{b,b+1} H_{b+2,b+1}^{-1}B_{1}A_{2}^{-1}\\in \\Gamma \\textrm{ and}\\\\\nF_4&=&F_3^{E_1}=H_{b,b+1} H_{b+2,b+1}^{-1}C_{1}A_{2}^{-1}\\in \\Gamma.\n\\end{eqnarray*}\nFrom these, the subgroup $\\Gamma$ contains the elements $F_2F_3^{-1}=A_1B_1^{-1}$ and $F_3F_4^{-1}=B_1C_1^{-1}$, which implies that $A_1C_1^{-1}\\in \\Gamma$. Hence\n\\[\nA_iB_i^{-1}\\in \\Gamma, B_jC_j^{-1}\\in \\Gamma \\textrm{ and } A_jC_j^{-1}\\in \\Gamma\n\\]\nfor all $i=1,2,3$ and $j=1,2$, by the action of $R$.  We also have the following element\n\\[\n(A_2C_2^{-1})^{E_1}=A_2B_3^{-1},\n\\]\nwhich is contained in $\\Gamma$. This implies that \n\\[\nA_iB_{i+1}^{-1}\\in \\Gamma\n\\]\nfor $i=1,2$ by the action of $R$. One can complete the proof as in the proof of Lemma~\\ref{lemeven}.\n\\end{proof}\n\n\\begin{remark}\\label{podd}\nOur results in lemmata~\\ref{lemeven}--\\ref{lem3}  are also valid for surfaces with odd number of punctures.  To see that our proofs also work \nfor such surfaces we refer the reader to Figures~$3$ and $5$ in \\cite{apy1}.\n\\end{remark}\n\n\n\\begin{figure}[hbt!]\n\\begin{center}\n\\scalebox{0.3}{\\includegraphics{curves.png}}\n\\caption{The curves $e_{i,j}$ and $\\gamma_i$ on the surface $\\Sigma_{g,p}$.}\n\\label{C}\n\\end{center}\n\\end{figure}\n\n\n\nNow, in the remainder of the paper let $\\Gamma$ be the subgroup of ${\\rm Mod}(\\Sigma_{g,p})$\ngenerated by the elements given explicitly in lemmata~\\ref{lemeven}--\\ref{lem3} with the conditions mentioned in these lemmata.\nThe proof of the following lemma is similar to that of ~ \\cite[Lemma~$4.6$]{apy1}, nevertheless we give a proof for the sake of completeness of the paper.\n\n\\begin{lemma}\\label{lemma4}\nThe group ${\\rm Mod}_{0}(\\Sigma_{g,p})$ is contained in the group $\\Gamma$.\n\\end{lemma}\n\n\\begin{proof}\nIt follows from the subgroup $\\Gamma$ contains the elements $A_i$, $B_i$ and $C_j$ for all $i=1,\\ldots,g$ and $j=1,\\ldots,g-1$ by lemmata~\\ref{lemeven}--\\ref{lem3} that\nit is sufficient to prove that $\\Gamma$ contains the Dehn twists $E_{i.j}$ for some fixed $i$ ($j=1,2,\\ldots,p-1$). Let us first note that $\\Gamma$ contains $A_{g}$ and $R=\\rho_1\\rho_2$. Consider the models for $\\Sigma_{g,p}$ as shown in Figures~\\ref{EE} and~\\ref{OE}. By the fact that the diffeomorphism $R$ maps $a_{g}$ to $e_{1,p-1}$, we get\n\\[\nRA_{g}R^{-1}=E_{1,p-1} \\in \\Gamma.\n\\]\nThe diffeomorphism $\\phi_{i}=B_{i+1}\\Gamma_i^{-1}C_iB_i$ which maps each $e_{i,j}$ to $e_{i+1,j}$ for $j=1,2,\\ldots,p-1$ (see Figure~\\ref{C}). \nBy the proof of~ \\cite[Lemma~$4.5$]{apy1}, the group $\\Gamma$ contains the element $\\phi_{g}$. Thus we have\n\\[\n\\phi_{g-1}\\cdots \\phi_2\\phi_1E_{1,p-1}(\\phi_{g-1}\\cdots \\phi_2\\phi_1)^{-1}=E_{g,p-1}\\in H.\n\\]\nLikewise, the diffeomorphism $R$ sends $e_{g,p-1}$ to $e_{1,p-2}$. Then we obtain\n\\[\nRE_{g,p-1}R^{-1}=E_{1,p-2}\\in \\Gamma.\n\\]\nIt follows from \n\\[\n\\phi_{g-1}\\cdots \\phi_2\\phi_1E_{1,p-2}(\\phi_{g-1}\\cdots \\phi_2\\phi_1)^{-1}=E_{g,p-2}\\in \\Gamma\n\\]\n that\n \\[\n R(E_{g,p-2})R^{-1}=E_{1,p-3}\\in \\Gamma\n \\]\n Continuing in this way, we conclude that the elements $E_{1,1},E_{1,2},$ $\\ldots,E_{1,p-1}$ belong to $\\Gamma$, which completes the proof.\n\\end{proof}\n\n\\begin{proofa}\nConsider the surface $\\Sigma_{g,p}$ as in Figures~\\ref{EE} and~\\ref{OE}.\n\n\\underline{ If $g=2k\\geq 14$ and $p=2b\\geq 10$}: In this case, consider the surface $\\Sigma_{g,p}$ as in Figure~\\ref{EE}. Since\n\\[\n\\rho_1(c_{k-3})=c_{k+4}, \\rho_1(b_{k-1})=b_{k+3} \\textrm{ and }\\rho_1(a_{k})=a_{k\n+2},\n\\]\n we get\n\\[\n\\rho_1C_{k-3}\\rho_1=C_{k+4},\n\\rho_1B_{k-1}\\rho_1=B_{k+3} \\textrm{ and }\n\\rho_1A_{k}\\rho_1=A_{k+2}.\n\\]\nAlso, since  $\\rho_1H_{b-1,b}\\rho_1=H_{b+1,b}$,\nit is easy to see that $\\rho_1H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}B_{k-1}A_k\nA_{k+2}^{-1}B_{k+3}^{-1}C_{k+4}^{-1}$ is an involution. Therefore, the generators of the subgroup $\\Gamma$ given in Lemma~\\ref{lemeven} are involutions.\n\n\n\\underline{ If $g=2k+1\\geq 13$ and $p=2b\\geq10$}: In this case, consider the surface $\\Sigma_{g,p}$ as in Figure~\\ref{OE}. It follows from\n\\[\n\\rho_1(c_{k-3})=c_{k+5}, \\rho_1(b_{k})=b_{k+3} \\textrm{ and }\\rho_1(a_{k+1})=a_{k\n+2},\n\\]\n that we have\n\\[\n\\rho_1C_{k-3}\\rho_1=C_{k+5},\n\\rho_1B_{k}\\rho_1=B_{k+3} \\textrm{ and }\n\\rho_1A_{k+1}\\rho_1=A_{k+2}.\n\\]\nAlso, by the fact that $\\rho_1H_{b-1,b}\\rho_1=H_{b+1,b}$,\nit is easy to see that the element $\\rho_1H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}B_kA_{k+1}A_{k+2}^{-1}B_{k+3}^{-1}C_{k+5}^{-1}$ is an involution. \nWe conclude that the generators of the subgroup $\\Gamma$ given in Lemma~\\ref{lemodd} are involutions.\n\n\n\n\\underline{ If $g=3,4,5$ or $g=6$ and $p=2b\\geq4$}:\nIt follows from\n\\begin{itemize}\n\\item $\\rho_2(b_2)=b_5$,  $\\rho_2(a_3)=a_4$ and $\\rho_1(c_3)=c_4$ if $g=6$,\n\\item $\\rho_1(a_3)=a_4$,  $\\rho_2(a_2)=a_4,\\rho_2(b_2)=b_4$ and $\\rho_2(c_2)=c_3$ if $g=5$,\n\\item $\\rho_2(b_1)=b_4$,  $\\rho_2(a_2)=a_3$ and $\\rho_1(c_2)=c_3$ if $g=4$ \n\\item $\\rho_1(a_2)=a_3$,  $\\rho_2(a_1)=a_3,\\rho_2(b_1)=b_3$ and $\\rho_2(c_1)=c_2$ if $g=3$ and\n\\item $\\rho_1H_{b-1,b}\\rho_1=H_{b+1,b}$ if $g=3,4,5$ or $g=6$\n\\end{itemize}\nthat the following elements:\n\\begin{itemize}\n\\item  $\\rho_2B_2A_3A_4^{-1}B_5^{-1}$ and $\\rho_1H_{b-1,b}H_{b+1,b}^{-1}C_{3}C_{4}^{-1}$ if $g=6$,\n\\item $\\rho_1H_{b-1,b}H_{b+1,b}^{-1}A_{3}A_{4}^{-1}$ and $\\rho_2A_2B_2C_2C_3^{-1}B_4^{-1}A_4^{-1}$  if $g=5$,\n\\item $\\rho_2B_1A_2A_3^{-1}B_4^{-1}$ and $\\rho_1H_{b-1,b} H_{b+1,b}^{-1}C_{2}C_{3}^{-1}$ if $g=4$ and\n\\item $\\rho_1H_{b-1,b} H_{b+1,b}^{-1}A_{2}A_{3}^{-1}$\n and $\\rho_2A_1B_1C_1C_2^{-1}B_3^{-1}A_3^{-1}$ if $g=3$ \n\\end{itemize}\ngiven in lemmata~\\ref{lem6}--\\ref{lem3} are involutions.\n\nNext, we show that the subgroup $\\Gamma$ is equal to the mapping class group ${\\rm Mod}(\\Sigma_{g,p})$.\nBy Lemma~\\ref{lemma4}, the group ${\\rm Mod}_{0}(\\Sigma_{g,p})$ is contained in the group $\\Gamma$. Hence, by Lemma~\\ref{lemma1}, we need to prove that \n$\\Gamma$ is mapped surjectively onto $Sym_p$. The element $\\rho_1\\rho_2 \\in \\Gamma$ has the image $(1,2,\\ldots,p)\\in Sym_p$. \n\nAs proven above,  the Dehn twists $A_i$, $B_i$ and $C_i$ belong to the subgroup $\\Gamma$. Thus, it can be easily observed that the factorization of half twists $H_{b-1,b}H_{b+1,b}^{-1}$ are contained in subgroup $\\Gamma$. Therefore, the group $\\Gamma$ also contains the following element:\n\\[\nR^{b-2}(H_{b-1,b}H_{b+1,b}^{-1})R^{2-b}=H_{1,2}H_{3,2}^{-1},\n\\]\nwhich has the image $(1,2,3)\\in Sym_p$. This completes the proof since the elements $(1,2,\\ldots,p)$ and $(1,2,3)$ of $Sym_p$ generate the whole group $Sym_p$ if $p$ is even~\\cite[Theorem B]{iz}.\n\n\\end{proofa}\n\nWhen the number of punctures is odd, we introduce an additional involution $\\rho_3$ (depicted in Figure~\\ref{rho3})  to our generating set.  The main reason behind adding an extra involution is for generating \nthe symmetric group $Sym_p$.  We want to point out that aside from generating $Sym_p$, all of our proofs in the case of even number of punctures work for odd number of punctures.\nFor $\\rho_1$ and $\\rho_2$, we distribute punctures as in Figures $3$ and $5$ in \\cite{apy1} (see also Remark~\\ref{podd}).\n \n\\begin{figure}[hbt!]\n\\begin{center}\n\\scalebox{0.35}{\\includegraphics{rho3.png}}\n\\caption{The involution $\\rho_3$  on the surface $\\Sigma_{g,p}$ for $p=2b+1$.}\n\\label{rho3}\n\\end{center}\n\\end{figure}\n\n\n\\begin{proofb}\nFor the first part of the proof we show that  \n\\begin{itemize}\n\\item [(i)] For every even integer $g=2k\\geq14$ and every odd integer $p=2b+1\\geq 9$, the subgroup ${\\rm Mod}_{0}(\\Sigma_{g,p})$ of ${\\rm Mod}(\\Sigma_{g,p})$ generated by the elements \n\\[\n\\rho_1, \\rho_2 \\textrm{ and }\\rho_1H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}B_{k-1}A_k\nA_{k+2}^{-1}B_{k+3}^{-1}C_{k+4}^{-1}, \\rho_3. \n\\]\n\n\\item[(ii)] For every odd integer $g=2k+1 \\geq15$ and odd integer $p=2b+1 \\geq 9$, the subgroup ${\\rm Mod}_{0}(\\Sigma_{g,p})$ of ${\\rm Mod}(\\Sigma_{g,p})$  generated by the elements\n\\[\n\\rho_1,\\rho_2 \\textrm{ and }\\rho_1H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}B_kA_{k+1}A_{k+2}^{-1}B_{k+3}^{-1}C_{k+5}^{-1},  \\rho_3.\n\\]\n\\end{itemize}\nNote that, it is enough to prove that the subgroup generated by the elements above mapped surjectively onto $Sym_p$.  \nFor this, consider the images of the elements $\\rho_1$, $\\rho_2$ and $\\rho_3$ \n\\begin{align*}\n&(1, p-1) (2, p-2) \\ldots (b, b+1),\\\\\n&(1,p) (2, p-1) \\ldots (b, b+2), \\\\\n&(2, p-1) (3, p-2) \\ldots (b, b+2).\n\\end{align*}\nThis finishes the proof for the first part,since these elements generate $Sym_p$, see \\cite[Lemma 6]{m1}.  For the second part of the theorem, note that adding $\\rho_3$ to the corresponding \ngenerating set given in Theorem A, finishes the proof.\n\\end{proofb}\n\n\nAs a last observation, one can prove that Theorem~A also holds  for the cases $p=2$ or $p=3$.\nIn theses cases, the generating set of $\\Gamma$ can be chosen as \n\\[\n\\begin{array}{lll}\n\\lbrace \\rho_1,\\rho_2,\\rho_1C_{k-3}B_{k-1}A_k\nA_{k+2}^{-1}B_{k+3}^{-1}C_{k+4}^{-1} \\rbrace & \\textrm{if} & g=2k\\geq14,\\\\\n\\lbrace \\rho_1,\\rho_2,\\rho_1C_{k-3}B_kA_{k+1}A_{k+2}^{-1}B_{k+3}^{-1}C_{k+5}^{-1} \\rbrace & \\textrm{if} & g=2k+1\\geq13.\\\\\n\\lbrace \\rho_1,\\rho_2,\\rho_2B_2A_3A_4^{-1}B_5^{-1},\\rho_1C_{3}C_{4}^{-1} \\rbrace & \\textrm{if} & g=6.\\\\\n\\lbrace \\rho_1,\\rho_2,\\rho_1A_{3}A_{4}^{-1},\\rho_2A_2B_2C_2C_3^{-1}B_4^{-1}A_4^{-1} \\rbrace & \\textrm{if} & g=5.\\\\\n\\lbrace \\rho_1,\\rho_2,\\rho_2B_1A_2A_3^{-1}B_4^{-1},\\rho_1C_{2}C_{3}^{-1} \\rbrace & \\textrm{if} & g=4.\\\\\n\\lbrace \\rho_1,\\rho_2,\\rho_1A_{2}A_{3}^{-1},\\rho_2A_1B_1C_1C_2^{-1}B_3^{-1}A_3^{-1} \\rbrace & \\textrm{if} & g=3.\\\\\n\\end{array}.\n\\]\nIt can be easily proven that the group $\\Gamma$ contains ${\\rm Mod}_{0}(\\Sigma_{g,p})$ by the similar arguments in the proofs of  lemmata~\\ref{lemeven}--\\ref{lem3}. \nThe element $\\rho_1\\rho_2 \\in \\Gamma$ has the image $(1,2,\\ldots,p)\\in Sym_p$. Hence, this element generates $Sym_p$ for $p=2$. If $p=3$, we distribute the punctures \nas in ~ \\cite[Figure~$1$]{ka}. Then the element $\\rho_1$ has the image $(1,3)$, which generate $Sym_p$ together with the element $(1,2,3)$. Therefore, the group $\\Gamma$ \nis mapped surjectively onto $Sym_p$ for $p=2,3$. One can conclude that the group $\\Gamma$ is equal to ${\\rm Mod}(\\Sigma_{g,p})$.\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}