KYLE BACK EQUILIBRIUM MODELS AND LINEAR CONDITIONAL MEAN-FIELD SDEs

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1 SIAM J. CONTROL OPTIM. Vol. 56, No. 2, pp c 218 Sociey for Indusrial and Applied Mahemaics KYLE BACK EQUILIBRIUM MODELS AND LINEAR CONDITIONAL MEAN-FIELD SDEs JIN MA, RENTAO SUN, AND YONGHUI ZHOU Absrac. In his paper we sudy he Kyle Back sraegic insider rading equilibrium model in which he insider has insananeous informaion on an asse, assumed o follow an Ornsein Uhlenback-ype dynamics ha allows possible influence by he marke price. Such a model exhibis some furher inerplay beween an insider s informaion and he marke price, and i is he firs ime being pu ino a rigorous mahemaical framework of he recenly developed condiional mean-field sochasic differenial equaion (CMFSDE). Wih he help of he reference probabiliy measure concep in filering heory, we shall firs prove a general well-posedness resul for a class of linear CMFSDEs, which is new in he lieraure of boh filering heory and mean-field SDEs and will be he foundaion for he underlying sraegic equilibrium model. Assuming some furher Gaussian srucures of he model, we find a closed form of opimal inensiy of rading sraegy as well as he dynamic pricing rules, and we subsaniae he well-posedness of he resuling opimal closed-loop sysem, whence he exisence of Kyle Back equilibrium. Key words. sraegic insider rading, Kyle Back equilibrium, condiional mean-field SDEs, reference measures, opimal closed-loop sysem AMS subjec classificaions. 6H1, 91G8, 6G35, 93E11 DOI /15M12558X 1. Inroducion. In his seminal paper, Kyle [21] firs proposed a sequenial equilibrium model of asse pricing wih asymmeric informaion. The model was hen exended by Back [2] o he coninuous ime version and has since been known as he Kyle Back sraegic insider rading equilibrium model. Roughly speaking, in such a model i is assumed ha here are wo ypes of raders in a (risk neural) marke: one informed (insider) rader versus many uninformed (noise) raders. The insider sees boh (possibly fuure) value of he fundamenal asse as well as is marke value, priced by he marke maker(s), and acs sraegically in a noncompeiive manner. The noise raders, on he oher hand, ac independenly wih only marke Informaion of he asse. Finally, he marke makers se he price of he asse, in a Berrand compeiion fashion, based on he hisorical informaion of he collecive marke acions of all raders, no being able o idenify he insider. The so-called Kyle Back equilibrium is a closed-loop sysem in which he insider maximizes his/her expeced reurn in a marke efficien manner (i.e., following he given marke pricing rule). There has been much lieraure on his opic. We refer o, e.g., [2, 4, 9, 14, 15, 17, 18, 19, 2, 21] and he references herein for boh discree and coninuous ime models. I is noed, however, ha in mos of hese works only he case of saic informaion is considered, ha is, i is assumed ha informaion ha he insider could observe is ime-invarian, ofen as he fundamenal price a a given fuure momen. Received by he ediors June 12, 216; acceped for publicaion (in revised form) January 19, 218; published elecronically March 27, 218. hp:// Funding: The firs auhor is suppored in par by U.S. NSF gran DMS The hird auhor is suppored by Chinese Scholarship Council and Chinese NSF gran Deparmen of Mahemaics, Universiy of Souhern Californlia, Los Angeles, CA 989 (jinma@usc.edu, renaosu@usc.edu). Corresponding auhor. School of Big Daa and Compuer Science, Guizhou Normal Universiy, Guiyang, 551, People s Republic of China (yonghuizhou@gznu.edu.cn). 1154

2 KYLE BACK MODELS AND LCMF SDEs 1155 Mahemaically, his amouns o saying ha he insider has he knowledge of a given random variable whose value canno be deeced from he marke informaion a he curren ime. I is ofen assumed ha he sysem has a cerain Gaussian srucure (e.g., he fuure price is a Gaussian random variable), so ha he opimal sraegies can be calculaed explicily. The siuaion will become more complicaed when he fundamenal prices progress as a sochasic process {v, and he insider is able o observe prices dynamically in a nonanicipaive manner. The asymmeric informaion naure of he problem has conceivably led o he use of filering echniques in he sudy of he Kyle Back model, and we refer o, e.g., [1] and [6] for he saic informaion case and o, e.g., [9] and [17] for he dynamic informaion case. I is noed ha in [6] i is furher assumed ha he acions of noise raders may have some memory, so ha he observaion process in he filering problem is driven by a fracional Brownian moion, adding he echnical difficulies in a differen aspec. We also noe ha he Kyle Back model has been coninuously exended in various direcions. For example, in a saic informaion seing, [16] recenly considered he case when noise rading volailiy is a sochasic process, and in he dynamic informaion case [1, 11, 12] sudied he Kyle Back equilibrium for he defaulable underlying asse via dynamic Markov bridges, exhibiing furher heoreical poenial of he problem. In his paper we are ineresed in a generalized Kyle Back equilibrium model in a dynamic informaion seing, in which he asse dynamics is of he form of an Ornsein Uhlenback SDE whose drif also reflecs he marke senimen (e.g., supply and demand, earning base, ec.), quanified by he marke price. The problem is hen naurally embedded ino a (linear) filering problem in which boh sae and observaion dynamics conain he filered signal process (see secion 2 for deails). We noe ha such a srucure is no covered by he exising filering heory, and hus i is ineresing in is own righ. In fac, under he seing of his paper he signal-observaion dynamics form a coupled (linear) condiional mean-field sochasic differenial equaion (CMFSDE) whose well-posedness, o he bes of our knowledge, is new. The main objecive of his paper is hus wofold. Firs, we shall look for a rigorous framework on which he well-posedness of he underlying CMFSDE can be esablished. The main device of our approach is he reference probabiliy measure ha is ofen seen in nonlinear filering heory (see, e.g., [24]). Roughly speaking, we give he marke maker s observaion a prior probabiliy disribuion so ha i is a Brownian moion ha is independen of he maringale represening he aggregaed rading acions of he noisy raders, and we hen prove ha he original signal-observaion SDEs have a weak soluion. More imporanly, we shall prove ha he uniqueness in law holds among all weak soluions ha are absoluely coninuous wih respec o he reference probabiliy measure. We should noe ha such a uniqueness in paricular resolves a long-sanding issue on he Kyle Back equilibrium model: he idenificaion of he oal raded volume movemens and he innovaion process of he corresponding filering problem, which has been argued only heurisically, or by economic insinc, in he lieraure (see, e.g., [1]). The second goal of his paper is o solve he opimizaion problem (maximum expeced reurn for he insider) under he generalized Kyle Back equilibrium model. Uilizing he Gaussian srucure and he lineariy of he CMFSDEs, we esablish he firs order necessary condiions on opimal inensiy as well as he marke prices. In he case when he dynamics of he underlying asse does no depend on he marke price, we give he explici soluions o he insider s rading inensiy and we shall jusify he well-posedness of he closed-

3 1156 JIN MA, RENTAO SUN, AND YONGHUI ZHOU loop sysem (whence he exisence of he equilibrium) using he CMFSDE heory esablished in his paper. These soluions in paricular cover many exising resuls as special cases. The res of he paper is organized as follows. In secion 2 we give he preliminaries of he Kyle Back equilibrium model and formulae he sraegic insider rading problem. In secion 3 we formulae a general form of linear CMFSDE and inroduce he noion of is soluions and heir uniqueness. We sae he main well-posedness resul and, in he case of deerminisic coefficien, calculae is soluions. Secion 4 will be devoed o he proof of he main well-posedness heorem. In secion 5 we characerize he opimal rading sraegy and give a firs order necessary condiion for he opimal inensiy, and finally in secion 6 we give some closed-form soluion o some special cases and compare hem o he exising resuls. 2. Problem formulaion. In his secion we describe a coninuous ime Kyle Back equilibrium model ha will be invesigaed in his paper, as well as relaed echnical seings. We begin by assuming ha all randomness of he marke comes from a common complee probabiliy space (Ω, F, P) on which is defined wo-dimensional Brownian moion B = (B v, B z ), where B v = {B v : represens he noise of he fundamenal value dynamics for he insider, and B z = {B z : represens he collecive acion of he noise raders. For noaional clariy, we denoe F v = {F Bv : and F z = {F Bz : o be he filraions generaed by B v and B z, respecively, and denoe F = F v F z, wih he usual P-augmenaion such ha i saisfies he usual hypoheses (cf., e.g., [25]). Furher, hroughou he paper we will denoe, for a generic Euclidean space X, regardless of is dimension,, and o be is inner produc and norm, respecively. We denoe he space of coninuous funcions defined on [, T ] wih he usual sup-norm by C([, T ]; X), and we shall make use of he following noaion: For any sub-σ-field G F T and 1 p <, L p (G; X) denoes he space of all X-valued, G-measurable random variables ξ such ha E ξ p <. As usual, ξ L (G; X) means ha i is G-measurable and bounded. For 1 p <, L p F ([, T ]; X) denoes he space of all X-valued, F-progressively measurable processes ξ saisfying E ([, T ]; ξ p d <. The meaning of L F X) is defined similarly. Throughou his paper we assume ha all he processes are one-dimensional, bu higher dimensional cases can be easily deduced wihou subsanial difficulies. Therefore, we will ofen drop X(= R) from he noaion. Also, hroughou he paper we shall denoe all L p -norms by p, wheher for L p (G) or for L p F ([, T ]), when he conex is clear. Consider a given sock whose fundamenal value (or is reurn) is V = {V : raded on a finie ime inerval [, T ]. There are hree ypes of agens in he marke: (i) he insider, who direcly observes he realizaion of he value V a any ime [, T ] and submis his/her order X a [, T ]; (ii) he noise raders, who have no direc informaion of he given asse and (collecively) submi an order Z a [, T ] in he form (2.1) Z = σ z db z,,

4 KYLE BACK MODELS AND LCMF SDEs 1157 where σ z = {σ z : is a given coninuous deerminisic funcion, and we assume ha σ z for all [, T ]; (iii) he marke makers, who can observe only he oal raded volume (2.2) Y = X + Z,, and ses he price of he asse a ime, denoed by P,, based on he observed informaion F Y = σ{y s, s. We denoe S = E[(V P ) 2 ] o be he error variance of P. We now give a more precise descripion of he wo main ingrediens in he model above: he dynamics of marke price P and he fundamenal value V. Firs, in he same spiri of he original Kyle Back model, we can sill assume ha he marke price P is he resul of a Berrand-ype compeiion among he marke makers (cf., e.g., [4]) and herefore should be aken as he condiional expecaion of V given he informaion F Y a each ime. Mahemaically speaking, his amouns o saying ha he marke price should be se, a each ime, as P = E[V F Y ] which, as he projecion of V L 2 (Ω) ono he subspace L 2 (F Y ), is he bes esimaor ha he marke maker is able o choose given he informaion F Y. I should be noed ha, however, unlike he saic case (i.e., V v), he process P is no longer a (P, F Y )-maringale in general. Nex, in his paper we shall also assume ha he dynamics of he value of he sock V = {V akes he form of an Iô process: dv = F d+σ v db v, (his would easily be he case if, e.g., he ineres rae is nonzero). Furhermore, we shall assume ha he drif F = F (, V, P ),. Here, he dependence of F on he marke price P is based on he following raionale: he value of he sock is ofen affeced by facors such as supply and demand, he earnings base (cash flow per share), or, more generally, he marke senimen, which all depend on he marke price of he sock. Consequenly, aking he Gaussian srucure ino consideraion, in wha follows we shall assume ha he process V saisfies he following linear SDE: (2.3) dv = (f V + g P + h )d + σ v db v = (f V + g E[V F Y ] + h )d + σ v db v,, V N(v, s ), where he funcions f, g, h, and σ v are all deerminisic coninuous differeniable funcions wih respec o ime [, T ], and N(v, s ) is a normal random variable wih mean v and sandard deviaion s. Coninuing, given he Gaussian srucure of he dynamics, i is reasonable o assume ha he insider s opimal rading sraegy (in erms of number of shares ) is of he form (see, e.g., [1, 2, 4, 17, 21]) (2.4) dx = β (V P )d,, where β > is a deerminisic coninuous differeniable funcion wih respec o ime in [, T), known as he insider rading inensiy. Consequenly, i follows from (2.2) ha he oal raded volume process observed by he marke maker can be expressed as (2.5) dy = β (V P )d + σ z db z = β (V E[V F Y ])d + σ z db z,. We noe ha SDEs (2.3) and (2.5) form a (linear) CMFSDE, which is beyond he scope of he radiional filering heory. Such SDEs have been sudied in [8] and [13]

5 1158 JIN MA, RENTAO SUN, AND YONGHUI ZHOU in general nonlinear forms, bu none of hem covers he SDEs in he form. In fac, if we allow he funcion h in (2.3) o be an F Y -adaped process, as many sochasic conrol problems do, hen o he bes of our knowledge, he well-posedness of he fully convolued CMFSDE such as (2.3) and (2.5) has no been sudied in he lieraure, even in he linear form. We should menion ha (2.5) was already noed in [1] and [6] in he case when V v, bu wihou addressing he uniqueness of he soluion. The well-posedness of he soluion o (2.3) and (2.5), especially he uniqueness, is acually quie involved; in he nex secions we shall esablish a mahemaical framework so hese SDEs can be sudied rigorously. Given he dynamics (2.3) and (2.5), our main purpose is o find an opimal rading inensiy β for he insider o maximize his/her expeced wealh. More specifically, denoe he wealh process of he insider by W = {W :, and assume ha he sraegy is self-financing (cf., e.g., [7]); hen he oal wealh of he insider over ime duraion [, T ], based on he marke price made by he marke makers, should be (2.6) W T = = X dp = X T P T β (V P )(P T P )d. β (V P )P d Here in he above we used a simple inegraion by pars and definiion (2.4). Thus he opimizaion problems can be described as (2.7) sup β E[W T ] = sup β J(β) = sup β β E[(V P )(P T P )]d. Remark 2.1. We should remark ha he simple form of opimizaion problem (2.7) is due largely o he lineariy of he dynamics (2.3) and (2.5), as well as he Gaussian assumpion on he iniial sae v. These lead o a Gaussian srucure, whence he rading sraegy (2.4). The general nonlinear and/or non-gaussian Kyle Back model requires furher sudy of CMFSDE and an associaed filering problem, and one should seek opimal conrol from a larger class of admissible conrols. In ha case he firs order condiion sudied in his paper will become a Ponryagin ype sochasic maximum principle (see, for example, [8]), and he soluion is expeced o be much more involved. We will address such general problems in our fuure publicaions. We end his secion by noing ha he main idea for solving he CMFSDE is o inroduce he so-called reference probabiliy space in he nonlinear filering lieraure (see, e.g., [24]), which can be described as follows. Assumpion 2.2. There exiss a probabiliy space (Ω, F, Q ) on which he process (B v, Y ), [, T ], is a wo-dimensional coninuous maringale, where B v is a sandard Brownian moion and Y is he observaion process wih quadraic variaion Y = (σz s) 2 ds. The probabiliy measure Q will be referred o as he reference measure. We remark ha Assumpion 2.2 amouns o saying ha we are giving a prior disribuion o he price process Y = {Y : ha he marke maker is observing, which is no unusual in saisical modeling, and will faciliae he discussion grealy. A naural example is he canonical space: Ω = C ([, T ]; R 2 ), he space of all wodimensional coninuous funcions null a zero; F = B(Ω ); F = B (Ω ) = σ{ω( ) : ω Ω, [, T ]; and (B v, Y ) is he canonical process. In he case σ 2 = 1, Q is he Wiener measure.

6 KYLE BACK MODELS AND LCMF SDEs The linear condiional mean-field SDEs. In his secion we sudy he linear CMFSDEs (2.3) and (2.5) ha play an imporan role in his paper. In fac, le us consider a slighly more general case ha is useful in applicaions: { dx = {f X + g E[X F Y ] + h d + σ 1 db 1, X = v; (3.1) dy = {H X + G E[X F Y ]d + σ 2 db 2, Y =, where B = (B 1, B 2 ) is a sandard Brownian moion defined on a given probabiliy space (Ω, F, P), and v N(v, s ) is independen of B. In ligh of Assumpion 2.2, hroughou his secion we shall assume he following. Assumpion 3.1. (i) The coefficiens f, g, σ 1, σ 2, G, and H are all deerminisic, coninuous funcions, and σ i >, i = 1, 2, for all [, T ]. (ii) There exiss a probabiliy space (Ω, F, Q ) on which he process (B 1, Y ), [, T ], is a wo-dimensional coninuous maringale, such ha B 1 is a sandard Q -Brownian moion, and Y = σ2 s 2 ds, [, T ], Q -a.s. (iii) The coefficien h is an F Y -adaped, coninuous process such ha [ ] E Q sup h 2 <. T Remark 3.2. Assumpion 3.1(iii) amouns o saying ha he process h is defined on he reference probabiliy space (Ω, F, Q ) and adaped o he Brownian filraion F Y, as we ofen see in he sochasic conrol wih parial observaions (cf. [5]) The general resul. To simplify noaion in wha follows we shall assume ha σ 1 = σ 2 1. We firs inroduce wo definiions of he soluion. Le P(R) denoe all probabiliy measures on (R, B(R)), where B(R) is he Borel σ-field of R, and µ N(v, s ) P(R) denoe he normal disribuion wih mean v and variance s. Definiion 3.3. Le µ P(R) be given. An eigh-uple (Ω, F, F, P; X, Y, B 1, B 2 ) is called a weak soluion o CMFSDE (3.1) wih iniial disribuion µ if (i) (B 1, B 2 ) is an F-Brownian moion under P; (ii) (X, Y, B 1, B 2 ) saisfies (3.1), P-a.s.; (iii) X µ; and is independen of (B 1, B 2 ) under P. Definiion 3.4. A weak soluion (Ω, F, F, P; X, Y, B 1, B 2 ) is called a Q -weak soluion if (i) here exiss a probabiliy measure P on (Ω, F ), and processes (X, Y, B 1,, B 2, ) defined on (Ω, F, P ), whose law under P is he same as ha of (X, Y, B 1, B 2 ) under P; and (ii) P Q. In wha follows for any Q -weak soluion, we shall consider only is copy on he reference measurable space (Ω, F ), and we shall sill denoe he soluion by (X, Y, B 1, B 2 ). The uniqueness of he soluions o he CMFSDE (3.1) is a more delicae issue. In fac, even he weak uniqueness (in he usual sense) for CMFSDEs (2.3) and (2.5) is no clear. However, we have a much beer hope, a leas in he linear case, for Q -soluions. We firs inroduce he following Q -pahwise uniqueness. Definiion 3.5. The CMFSDE (3.1) is said o have Q -pahwise uniqueness if for any wo Q -weak soluions (Ω, F, F, P i ; X i, Y i, B 1,i, B 2,i ), i = 1, 2, such ha

7 116 JIN MA, RENTAO SUN, AND YONGHUI ZHOU (i) X 1 = X 2 ; and (ii) Q {(B 1,1, Y 1 ) = (B 1,2, Y 2 ) for all [, T ] = 1, i holds ha Q {(X 1, B 2,1 ) = (X 2, B 2,2 ) for all [, T ] = 1, and P 1 = P 2. Theorem 3.6. Assume ha Assumpion 3.1 is in force and furher ha h is bounded. Le µ N(v, s ) be given. Then he CMFSDE (3.1) possesses a weak soluion wih iniial disribuion µ, denoed by (Ω, F, F, P; X, Y, B 1, B 2 ). Moreover, if we denoe P = E P [X F Y ], [, T ], hen P saisfies he following SDE: (3.2) { dp = [(f + g )P + h ]d + S H {dy [H + G ]P d, [, T ], P = v, where S =Var(P ) saisfies he Riccai equaion: (3.3) ds = [1 + 2f S H 2 S 2 ]d, S = s. Furhermore, he weak soluion can be chosen as a Q -weak soluion, and he Q -pahwise uniqueness holds. We remark ha Theorem 3.6 does no imply ha he CMFSDE has a srong soluion, since one canno fix he soluion probabiliy space (Ω, F, P) arbirarily. Since he proof of Theorem 3.6 is a bi lenghy, we defer i o he nex secion. We shall neverheless presen a lemma below, which will be frequenly used in our discussion, so as o faciliae he argumen in he nex secion. To begin wih, we consider any filered probabiliy space (Ω, F, F, P) on which is defined a sandard Brownian moion (B 1, B 2 ). We assume ha F = F (B1,B 2). For any η L 2 F ([, T ]) we define Lη o be he soluion o he following SDE: (3.4) dl = L η db 2,, L = 1. In oher words, L η is a local maringale in he form of he Doléans Dade sochasic exponenial: (3.5) { L η = exp η s dbs η s 2 ds. Nex le α L 2 F ([, T ]) and consider he following SDE: (3.6) dy = (α + h(y ) )d + db 2, Y =, where h : [, T ] C([, T ]) R is progressively measurable in he sense ha i is a measurable funcion such ha for each [, T ], h(y) = h(y ) for y C([, T ]). (A simple case would be h(y) = h(y ), where h is a measurable funcion.) We should noe ha in general he well-posedness of SDE (3.6) is nonrivial wihou any specific condiions on h, bu in wha follows we shall assume a priori ha (3.6) has a (weak) soluion on some probabiliy space (Ω, F, P). We say ha h L 2 F Y ([, T ]) if h = h(y ), [, T ], such ha E h(y ) 2 d <. We have he following lemma.

8 KYLE BACK MODELS AND LCMF SDEs 1161 Lemma 3.7. Suppose ha he SDE (3.6) has a soluion Y, [, T ], for given α L 2 F B1 ([, T ]) and h L 2 F Y ([, T ]) on some probabiliy space (Ω, F, P). Le β be given by (3.7) dβ = α d + db 2,, β =. Assume furher ha L (α+h), he soluion o (3.4) wih η = (α + h), is an (F, P)- maringale. Then, for any [, T ], i holds ha (3.8) E P [α F Y ] = E P [α F β ] [, T ], P-a.s. Proof. Clearly, i suffices o prove E P [α T FT Y ] = EP [α T F β T ], as he cases for < T are analogous. To his end, we define a new probabiliy measure Q on (Ω, F T ) by dq dp FT = L (α+h) T, where L (α+h) is he soluion o he SDE (3.4) wih η = (α + h), and i is a rue maringale on [, T ] by assumpion. By he Girsanov heorem, he process (B 1, Y ) is a sandard Brownian moion on [, T ] under Q. Now define L = 1/L (α+h) ; hen L saisfies he following SDE on (Ω, F T, Q): (3.9) d L = L (α + h )dy, [, T ], L = 1. Furhermore, by he Kallianpur Sriebel formula, we have (3.1) E P [α T F Y T ] = EQ [α T LT F Y T ] E Q [ L T F Y T ]. On he oher hand, L has he explici form { L T = exp [α + h ]dy 1 T (3.11) [α + h ] 2 d 2 { = exp h dy 1 2 = L T Λ T, h 2 d + α dy 1 2 [ α 2 + 2h α ]d where L T { = exp h dy 1 T h 2 d ; 2 { Λ T = exp α dy 1 2 [[α ] 2 + 2h α ]d. Noe ha h is F Y -adaped, and so is L T. We derive from (3.1) ha (3.12) E P [α T F Y T ] = EQ [α T Λ T F Y T ] E Q [Λ T F Y T ]. Now define Y 1 = h sds. Since h is F Y -adaped, so is Y 1, and consequenly β = Y Y 1, is F Y -adaped. Moreover, since (B 1, Y ) is a sandard Brownian

9 1162 JIN MA, RENTAO SUN, AND YONGHUI ZHOU moion under Q, and α is F B1 -adaped, we conclude ha α is independen of Y under Q. Therefore, using inegraion by pars we obain ha { Λ T = exp α dβ 1 T (3.13) α 2 d 2 { = exp α T β T β dα 1 2 α 2 d Since α is independen of Y under Q, and β, [, T ], is FT Y -measurable, a monoone class argumen shows ha E Q [Λ T FT Y ] is F β T measurable, and similarly, EQ [α T Λ T FT Y ] is also F β T measurable. Consequenly EP [α T FT Y ] is F β T measurable, hanks o (3.1). Finally, noing F β F Y we have (3.14) E P [α T F Y T ] = E P {E P [α T F Y T ] F β T = EP [α T F β T ], proving he lemma Deerminisic coefficien cases. An imporan special case is when all he coefficiens in he linear CMFSDE (3.1) are deerminisic. In his case we expec ha he soluion (X, Y ) is Gaussian, and i can be solved in a much more explici way. The following linear CMFSDE will be useful in he sudy of insider rading equilibrium model in he laer half of he paper: { dx = [f X + g E[X F Y ] + h ]d + σ 1 db 1, X = v; (3.15) dy = H (X E[X F Y ])d + σ 2 db 2, Y =, where v N(v, s ) and is independen of (B 1, B 2 ) and all he coefficiens are assumed o be deerminisic Bounded coefficiens case. In ligh of Theorem 3.6 le us inroduce he following funcions: (3.16) k = H2 S σ 2 2, l = H S,, where S is he soluion o he following Riccai equaion: (3.17) ds d = (σ1 ) 2 + 2f S l 2,, S = s. We have he following resul. Proposiion 3.8. Le Assumpion 3.1 be in force, and assume furher ha he process h in (3.1) is also a deerminisic and coninuous funcion. Le (X, Y ) be he soluion of (3.15) on he probabiliy space (Ω, F, P), and denoe P = E P [X F Y ],. Then X and P have he following explici form, respecively: for, i holds P-a.s. ha (3.18) (3.19) [ X = P + φ 1 (, ) v v + { P = φ 2 (, ) v + + σ 1 rφ 1 (, r)φ 3 (, r)db 1 r + σ 2 ] φ 1 (, r)(σrdb 1 r 1 l r dy r ) ; φ 2 (, r)h r dr + (v v )φ 3 (, ) [φ 2 (, r)l r φ 1 (, r)φ 3 (, r)l r ]db 2 r.,

10 KYLE BACK MODELS AND LCMF SDEs 1163 where, for r, { { φ 1 (, r) = exp (f u k u )du ; φ 2 (, r) = exp (f u + g u )du ; (3.2) r r φ 3 (, r) = φ 1 (u, )φ 2 (, u)k u du. r Proof. We firs noe ha he SDE (3.15) is a special case of (3.1) wih G = H. Then, following he same argumen of Theorem 3.6 one can show ha when σ 1 > and σ 2 > are no equal o 1, he SDE (3.2) for he process P = E P [X F Y ] reads (3.21) dp = [(f + g )P + h ]d + H S (σ 2 ) 2 dy, P = v, and S saisfies a Riccai equaion (3.22) ds d = (σ2 ) 2 + 2f S [ ] 2 H S, S = s. Now applying he Girsanov ransformaion we can define a new probabiliy measure Q under which (B 1, Y ) is a coninuous maringale, such ha B 1 is a sandard Brownian moion, and d Y = σ 2 2 d. Then, under Q, he dynamic of V = X P can be wrien as [ ] dv = f H2 S (σ 2 ) 2 V d + σ 1 db 1 H S σ 2 dy = [f k ]V d + σ 1 db 1 l dy, X P = v v. I hen follows ha he ideniy (3.18) holds Q-almos surely and hence P-almos surely. Similarly, applying he consan variaion formula for he linear SDE (3.21) and noing (3.1) we obain ha, wih φ 2 (, r) = exp( r (f u + g u )du), for r, T, (3.23) P = φ 2 (, ) { v + { = φ 2 (, ) v + φ 2 (, r)h r dr + φ 2 (, r)h r dr + σ 2 φ 2 (, r) H rs r (σr) 2 2 dy r φ 2 (, r) H rs r (σ 2 r) 2 [H r(x r P r )dr + σ 2 rdb 2 r ] Now plugging (3.18) ino (3.23), and applying Fubini, we have { P = φ 2 (, ) v + φ 2 (, r)h r dr + (v v ) φ 1 (r, )φ 2 (, r)k r dr (3.24) + + = φ 2 (, ) + φ 1 (, r)σr 1 φ 1 (u, )φ 2 (, u)k u dudbr 1 r [φ 2 (, r)l r φ 1 (, r)l r φ 1 (u, )φ 2 (, u)k u du]dbr 2 r { v + φ 2 (, r)h r dr + (v v )φ 3 (, ) φ 1 (, r)σ 1 rφ 3 (, r)db 1 r + [φ 2 (, r)l r φ 1 (, r)φ 3 (, r)l r ]db 2 r,.

11 1164 JIN MA, RENTAO SUN, AND YONGHUI ZHOU This proves (3.19), whence he proposi- where φ 3 (, r) = r φ 1(u, )φ 2 (, u)k u du. ion Unbounded coefficiens case. We noe ha Theorem 3.6 as well as he discussion so far rely heavily on he assumpion ha all he coefficiens are bounded, especially H and G (see Assumpion 3.1). However, in our applicaions we will see ha he coefficiens H = G = β, where β is he insider rading inensiy which, in he opimal case, will only be defined on [, T ), and lim T β = +, violaing Assumpion 3.1. In oher words, he closed-loop sysem will exhibi a cerain Brownian bridge naure (see also, e.g., [4, 1, 11]), for which he well-posedness resul of Theorem 3.6 acually does no apply. To overcome such a conflic, we inroduce he following relaxed version of Assumpion 3.1. Assumpion 3.9. There exiss a sequence {T n n 1, wih < T n T, and a sequence of probabiliy measures {Q n n 1 on (Ω, F ), saisfying he following: (i) Assumpion 3.1 holds for each (Ω, F, Q n ) over [, T n ], n 1, (ii) Q n+1 F = Q n, n 1. Tn We shall refer o he sequence of probabiliy measures Q := {Q n n 1 as he reference family of probabiliy measures and o he associaed sequence {T n n 1 as he announcing sequence. Clearly, if he reference measure Q exiss, hen Q n = Q F Tn, n 1. I is known, however, ha an equivalen maringale measure may no exis on [, T ] for a general Kyle Back model (see, e.g., [12]). In such a case he reference family would play a similar role, bu over [, T ). A reasonable exension of he noion of Q -weak soluion over [, T ) is as follows. Definiion 3.1. Le Q be a reference family of probabiliy measures wih announcing sequence {T n. A sequence {(Ω, F, P n, X n, Y n, B 1,n, B 2,n ) n 1 is called a Q -weak soluion of (3.1) on [, T ) if for each n 1, (Ω, F, P n, X n, Y n, B 1,n, B 2,n ) is a Q n -weak soluion on [, T n ]. I is worh noing ha if he coefficiens of CMFSDE (3.1) saisfy Assumpion 3.1 on each subinerval [, T n ], hen one can apply Theorem 3.6 for each n o ge a Q -soluion. Furhermore, since he soluions will be pahwisely unique under each Q n over [, T n ], i is easy o check ha (X n+1, Y n+1, B 1,n+1, B 2,n+1 ) = (X n, Y n, B 1,n, B 2,n ), [, T n ], Q n -a.s. We can hen define a process (X, Y, B 1, B 2 ) on [, T ) by simply seing (X, Y, B 1, B 2 ) = (X n, Y n, B 1,n, B 2,n ), for [, T n ], n 1, and we shall refer o such a process as he Q -soluion on [, T ). The Q - pahwise uniqueness on [, T ) can be defined in an obvious way. We have he following exension of Theorem 3.6, whose proof is lef for he ineresed reader. Theorem Assume ha Assumpion 3.9 is in force, and le Q be he family of reference measures wih announcing sequence {T n. Assume furher ha Assumpion 3.1 holds for each Q n on [, T n ]. Then he CMFSDE (3.1) possesses a Q -weak soluion on [, T ), and i is Q -pahwisely unique on [, T ). 4. Proof of Theorem 3.6. In his secion we prove Theorem 3.6. We begin by making he following reducion: i suffices o consider he SDE (3.1) where he iniial sae X = v v is deerminisic, ha is, s =. Indeed, suppose ha (X x, Y x ) is a weak soluion of (3.1) along wih some probabiliy space (Ω, F, P) and P-Brownian moion (B 1, B 2 ), and v is any random variable defined on (R, B(R)), wih normal

12 KYLE BACK MODELS AND LCMF SDEs 1165 disribuion µ = N(v, s ); we define he produc space Ω = Ω R, F = F B(R), P = P µ, and wrie he generic elemen of ω Ω as ω = (ω, x). Then for each, he mapping ω X x (ω) defines a random variable on ( Ω, F, P), and x X x = v(x) is a normal random variable wih disribuion N(v, s ) and is independen of (B 1, B 2 ), by definiion. Bearing his in mind, hroughou he secion we shall assume ha he iniial sae X = x is deerminisic Exisence. Our main idea o prove he exisence of he weak soluion is o decouple he sae and observaion equaions in (3.1) by considering he dynamics of he filered sae process P = E P [X F Y ],, which is known o saisfy an SDE, hanks o linear (Kalman Bucy) filering heory. To be more precise, we consider he following sysem of SDEs on he reference probabiliy space (Ω, F, Q ), on which (B 1, Y ) is a Brownian moion: (4.1) dx = [f X + g P + h ]d + db 1, X = x; db 2 = dy [H X + G P ]d, B 2 = ; dp = [(f + g )P + h ]d + S H {dy [H + G ]P d, P = x; ds = [2f S H 2 S 2 + 1]d, S =. We noe ha by Assumpion 3.1, all coefficiens f, g, H, G are deerminisic, and h L 2 F Y (C([, T ])), i is easy o see ha he linear sysem (4.1) has a (pahwisely) unique soluion (X, B 2, P ) on (Ω, F, Q ). Now le L = {L be he soluion o he SDE (4.2) dl = L (H X + G P )dy, L = 1. Then L is a posiive Q -local maringale, hence a Q -supermaringale wih E Q [L ] L = 1. Furhermore, L can be wrien as he Doléans Dade exponenial: { L = exp (H s X s + G s P s )dy s 1 (4.3) H s X s + G s P s 2 ds, [, T ]. 2 We have he following lemma. Lemma 4.1. Assume Assumpion 3.1. Assume furher ha h is bounded. Then he process L = {L ; is a rue (F, Q )-maringale on [, T ]. Proof. We follow he idea of ha in [5]. Since L is a supermaringale wih E Q [L ] 1, we need only show ha E Q [L ] = 1 for all. To his end, we define, for any ε >, L ε = L 1 + εl, [, T ]. Then clearly L ε L 1 ε, and an easy applicaion of Iô s formula shows ha (4.4) dl ε = εl2 [H X + G P ] 2 (1 + εl ) 3 d + L [H X + G P ] (1 + εl ) 2 dy, ; L ε = ε. Since for each fixed ε >, L [H X + G P ] (1 + εl ) 2 2 = εl [H X + G P ] ε(1 + εl ) 2 2 [H X + G P ] 2, ε

13 1166 JIN MA, RENTAO SUN, AND YONGHUI ZHOU we see ha he sochasic inegral on he righ-hand side of (4.4) is a rue maringale. I hen follows ha (4.5) E Q [L ε ] = 1 [ 1 + ε εl 2 EQ [H X + G P ] 2 ] (1 + εl ) 3 d. Nex, we observe ha L >, and εl2 [H X + G P ] 2 (1 + εl ) 3 = (εl )L [H X + G P ] 2 (1 + εl ) 3 L [H X + G P ] 2. Noe ha L ε is bounded. By sending ε on boh sides of (4.5) and applying he dominaed convergence heorem we can hen conclude ha E Q [L ] = 1, provided [ ] T (4.6) E Q L [H X + G P ] 2 d <. I remains o check (4.6). To his end, le us define X = X 1 + α, where { dα = f α d + db 1, α = x, (4.7) dx 1 = [f X 1 + g P + h ]d, X 1 =. By Gronwall s inequaliy, i is readily seen ha (4.8) X 1 C [ g s P s + h s ds C 1 + ] P s ds, [, T ]. Here and in wha follows C > denoes a generic consan depending only on he bounds of he coefficiens f, g, H, G, h and he duraion T >, which is allowed o vary from line o line. Now, noing ha L is a super-maringale wih L = 1, we deduce from (4.8) ha [ ] { [ T ] (4.9) E Q L X 1 2 d C 1 + E Q L P s 2 dsd { [ ] T C 1 + E Q L s P s 2 ds. Consequenly we have [ ] [ T ] T E Q L [H X +G P ] 2 [ d CE Q L X 1 (4.1) 2 + α 2 + P 2] d { [ ] T C 1 + E Q L ( α 2 + P 2 )d. Coninuing, le us recall ha he processes P and α saisfy (4.1) and (4.7), respecively. By Iô s formula we see ha (4.11) { d α 2 = [2f α 2 + 1]d + 2α db 1 ; d P 2 = [ ] 2M P 2 + 2P h + S 2 H 2 d + 2S H P dy,

14 KYLE BACK MODELS AND LCMF SDEs 1167 where M = (f +g ) S H (H +G ),. Nex, we define, for δ > and [, T ], X δ = X [1 + δ X 2 ] ; P δ P 1/2 = [1 + δ P 2 ]. 1/2 Then X δ X δ 1 2 and P δ P δ 1 2 for all, and i is no hard o show ha lim δ X δ = X, lim δ P δ = P, uniformly on [, T ], in probabiliy. Now, define (4.12) dl δ = L δ [H X δ + G P δ ]dy, L δ = 1. Since X δ and P δ are now bounded, L δ is a maringale and E Q [L δ ] = 1, [, T ]. Furhermore, by he sabiliy of SDEs one shows ha, possibly along a subsequence, L δ converges o L, Q -a.s., [, T ]. Noing (4.11) and applying Iô s formula we have, for [, T ], L δ α 2 = x 2 + L δ s[2f s α s 2 + 1]ds + 2L δ sα s db 1 + L δ s α s 2 [H s X δ s + G s P δ s ]dy s. Since α has finie momens for all orders (see (4.7)), he boundedness of X δ and P δ hen renders he wo sochasic inegrals on he righ-hand side above boh rue maringales. Thus, aking expecaions on boh sides above, and applying Gronwall s inequaliy, we ge (4.13) E Q[ L δ α 2] C [, T ], where C is a consan independen of δ. Applying Faou s lemma we hen ge ha (4.14) E Q[ L α 2] lim E Q [L δ α 2 ] C. δ Finally, noing (4.11) and applying Iô s formula again we have dl δ P 2 = L δ [ 2M P 2 + 2P h + S 2 H 2 ] (4.15) d + 2S H L δ P dy + L δ P 2 [H X δ + G P δ ]dy + 2S H L δ P [H X δ + G P δ ]d. By similar argumens as before, and noing ha X δ X and P δ P, one shows [ ] E Q [L δ P 2 ] C {1 + E Q L δ s[ P s 2 + X s 2 ]ds [ C {1 + E Q This, ogeher wih (4.9), implies ha [ E Q [L δ P 2 ] C {1 + E Q ] L δ [ P s 2 + Xs α s 2 ]ds, [, T ]. ] L δ s[ P s 2 + α s 2 ]ds, [, T ]. Applying he Gronwall inequaliy and recalling (4.13) we hen obain [ ] (4.16) E Q [L δ P 2 ] C {1 + E Q L δ s α s 2 ds C, [, T ]. By Faou s lemma one again shows ha E Q [L P 2 ] C for all [.T ]. This, ogeher wih (4.14) and (4.1), leads o (4.6). The proof is now complee.

15 1168 JIN MA, RENTAO SUN, AND YONGHUI ZHOU We can now complee he proof of exisence. Since L is a (F, Q ) maringale, we define a probabiliy measure P by dp FT dq = L T and apply he Girsanov heorem so ha (B 1, B 2 ) is a P-Brownian moion on [, T ]. Now, by looking a he firs wo equaions of (4.1), we see ha (Ω, F, P, X, Y, B 1, B 2 ) would be a weak soluion o (3.1) if we can show ha (4.17) P = E P [X F Y ], [, T ], P-a.s. To prove (4.17) we proceed as follows. We consider he following linear filering problem on he space (Ω, F, P): (4.18) { dα = f α d + db 1, α = x; dβ = H α d + db 2, β =. Denoe α = E P [α F β ],. Then by linear filering heory, we know ha α saisfies he following SDE: (4.19) d α = f α d + S H {dβ H α d, α = x, where S saisfies (3.3) (or (4.1)). On he oher hand, from (4.18) we see ha α is F B1 -adaped, and from (4.1) we see ha P is F Y -adaped; herefore we can apply Lemma 3.7 o ge ha E P [α F Y ] = E P [α F β ] = α, [, T ]. Now le us define P = E P [X F Y ],. Recall ha X = X 1 + α, where X 1 saisfies a randomized ODE (4.7) and is obviously F Y -adaped; we see ha P = X 1 + α, and i saisfies he SDE (4.2) d P = [f P + g P + h ]d + S H {dβ H α d, ; P = x. Noing ha X = X 1 + α and P = X 1 + α we see ha dβ H() α d = H (α α )d + db 2 = dy [H P + G P ]d. Then (4.2) implies ha (4.21) d P = [f P + g P + h ]d + S H {dy [H P + G P ]d. Define P = P P ; hen i follows from (4.1) and (4.21) ha d P = [f S H 2 ] P d, P =. Thus P, P-a.s., for any. Tha is, (4.17) holds, proving he exisence. I is worh noing ha he weak soluion ha we have consruced is acually a Q -weak soluion Uniqueness. Again we need only consider he soluions wih deerminisic iniial sae. We firs noe ha if (Ω, F, P, F, X, Y, B 1, B 2 ) is a Q -weak soluion o (3.1), hen we can assume wihou loss of generaliy ha F = F B1,B 2, hence Brownian. Nex, we can define P = E P [X F Y ]. We are o show ha P saisfies an SDE of he form of ha in (4.1) under Q, from which we shall derive he Q -pahwise uniqueness.

16 KYLE BACK MODELS AND LCMF SDEs 1169 To his end, we recall ha, as a Q -weak soluion, one has P Q. Define a P-maringale Z [ ] = E P dq dp F,. Since (B 1, Y ) is a P-semimaringale wih decomposiion: (4.22) B 1 = B 1 ; Y = (H s X s + G s Ps )ds + B 2,. By he Girsanov Meyer heorem (see, e.g., [25, Theorem III-2]), i is a Q -semimaringale wih he decomposiion (B 1, Y ) = (N 1, N 2 ) + (A 1, A 2 ), where N = (N 1, N 2 ) is a Q -local maringale of he form N 1 = B 1 1 Z s d[z, B 1 ] s, N 2 = B 2 1 Z s d[z, B 2 ] s,, and A = (A 1, A 2 ) is a finie variaion process. Since by assumpion (B 1, Y ) is a Q -Brownian moion, we have A. In oher words, i mus hold ha (4.23) B 1 = B 1 1 Z s d[z, B 1 ] s, Y = B 2 1 Z s d[z, B 2 ] s,. Consider now a (F, P)-maringale dm = Z 1 dz. Since F is Brownian, applying he Maringale represenaion heorem we see ha here exiss a process θ = (θ 1, θ 2 ) L 2 F ([, T ]) such ha dm = θ 1 db 1 + θ 2 db 2, [, T ]. Thus (4.23) amouns o saying ha [M, B 1 ] = θ 1 sds ; Y = B 2 [M, B 2 ] = B 2 θ 2 sds,. Comparing his o (4.22) we have θ 1 and θ 2 (HX + G P ). Tha is, Z = L (HX+G P ), he soluion o he SDE dz = Z dm = Z (H X + G P )db 2, [, T ], Z = 1, and hence i can be wrien as he Doléans Dade sochasic exponenial: { Z = exp (H s X s + G s Ps )dbs 2 1 (4.24) H s X s + G s Ps 2 ds. 2 Le us now consider again he following filering problem on probabiliy space (Ω, F, P): { dα = f α d + db 1, α = x, (4.25) dβ = [H α ]d + db 2, β =. As before, we know ha α = E P [α F β ] saisfies he SDE (4.26) d α = f α d + S H {dβ H α d, α = x, where S saisfies (3.3). Since L (HX+G P ) = Z is a P-maringale, we can apply Lemma 3.7 again o conclude ha E P [α F Y ] = E P [α F β ] = α. Now le X = α + X 1, and Y = β + Y 1 as before, where X 1 saisfies he ODE (4.7), and Y 1 saisfies he ODE (4.27) dy 1 = [H X 1 + G P ]d, Y 1 =.

17 117 JIN MA, RENTAO SUN, AND YONGHUI ZHOU Furhermore, since X 1 is F Y -adaped, we have P = X 1 + α. Combining (4.7) and (4.26) we see ha P saisfies he SDE (4.28) d P = [(f + g ) P + h ]d + S H {dβ H α d, P = x. Since β = Y Y 1, we derive from (4.27) ha and (4.28) becomes dβ H α d = dy dy 1 H α d = dy [H P + G P ]d, (4.29) d P = [(f +g ) P +h ]d+s H {dy [(H +G ) P ]d, [, T ], P = x. Tha is, P saisfies he same SDE as P does in (4.1) on he reference space (Ω, F, Q ). To finish he argumen, le (Ω, F, P i, F, X i, Y i, B 1,i, B 2,i ), i = 1, 2, be any wo Q -weak soluions, and define P i = E Pi [X F i Y i ],, i = 1, 2. Then he argumens above show ha (X i, B 2,i, P i ), i = 1, 2, are wo soluions o he linear sysem of SDEs (4.1), under Q. Thus if (B 1,1, Y 1 ) (B 1,2, Y 2 ) under Q, hen we mus have (X 1, B 2,1, P 1 ) (X 2, B 2,2, P 2 ), under Q, which in urn shows, in ligh of (4.24), ha P 1 = P 2. This proves he Q -pahwise uniqueness of soluions o (3.1). 5. Necessary condiions for opimal rading sraegy. In his secion we sudy he opimizaion problem (2.7). We sill denoe he price dynamics observable by he insider o be V = {V : and assume ha i saisfies he SDE (5.1) { dv = [f V + g E P [V F Y ] + h ]d + σ v db v, [, T ], V = v N(v, s ), and we assume ha he demand dynamics observable by he marke makers, denoed by Y = {Y :, saisfies he SDE (5.2) dy = [β (V E P [V F Y ])]d + σ z db z, [, T ]; Y =. We should noe ha in (5.1) and (5.2) he probabiliy P should be undersood as one defined on he canonical space (Ω, F, F ), on which he soluion o (5.1) and (5.2) is Q -pahwisely unique. For noaional simpliciy, from now on we shall denoe E = E P, when here is no danger of confusion. Moreover, noing ha E[P (V P )] = E[E[(V P )P F Y ]] = and ha all he coefficiens are now assumed o be deerminisic, we can apply Proposiion 3.8 o ge ha he problem (2.7) can be wrien as (5.3) J(β) = = φ 2 (T, ) + β E[(V P )P T ]d { β φ 1 (, ) s φ 3 (T, ) [(lr 2 + (σr v ) 2 )φ 2 1(, r)φ 3 (T, r) φ 1 (, r)φ 2 (, r)lr]dr 2 d,

18 KYLE BACK MODELS AND LCMF SDEs 1171 where φ i, i = 1, 2, 3, and l, k are defined by (3.2) and (3.16), respecively, wih H = β. Now by inegraion by pars we can easily check ha [(l 2 r + (σ v r ) 2 )φ 2 1(, r)φ 3 (T, r)]dr = = φ 3 (T, )S φ 2 1(, ) φ 3 (T, )s + φ 3 (T, r)d[s r φ 2 1(, r)] φ 1 (, r)φ 2 (, r)l 2 rdr; we hus have J(β) = φ 2 (T, ) β S φ 1 (, )φ 3 (T, )d, and he original opimal conrol problem (2.7) is equivalen o he following: (5.4) sup β J(β) = sup β β S φ 1 (, )φ 3 (T, )d. Before we proceed any furher le us specify he admissible sraegy and he sanding assumpions on he coefficiens ha will be used hroughou his secion. We noe ha he assumpions will be slighly sronger han Assumpion 3.1. Assumpion 5.1. (i) All coefficiens f, g, h, σ v, and σ z are deerminisic, coninuous funcions on [, T ] such ha σ z c, σ v c for all [, T ] for some consan c >. (ii) The rading inensiy β is coninuous on [, T ), β > for all [, T ), and lim T β > exiss (i may be + ). Consequenly, β = inf [,T ] β >. Remark 5.2. (i) In pracice i is no unusual o assume ha lim T β =, which amouns o saying ha he insider is desperaely rying o maximize he advanage of he asymmeric informaion (cf., e.g., [1]). We shall acually prove ha his is he case for he opimal sraegy, provided Assumpion 5.1(ii) holds. In wha follows we say ha a rading inensiy β is admissible if i saisfies Assumpion 5.1(ii). By a sligh abuse of noaion we sill denoe all admissible rading inensiies by U ad. (ii) For β U ad, he well-posedness of CMFSDEs (5.1) and (5.2) should be undersood in he sense of Theorem 3.11, and we shall consider is (unique) Q -soluion. We noe ha he soluion of he Riccai equaion (3.17) S, as well as he funcions φ 1 and φ 3 defined by (3.2), all depends on he choice of rading inensiy β. We shall a imes denoe hem by S β, φ β 1, and φβ 3, respecively, o emphasize heir dependence on β. The following lemma is simple bu useful for our analysis. Lemma 5.3. Le Assumpion 5.1 be in force. Then we have he following: (i) For any β U ad, he Riccai equaion (3.17) has a soluion S = S β defined on [, T ), such ha S β > for all [, T ). Furhermore, here exiss a consan C β >, depending on he bounds of he coefficiens and β in Assumpion 5.1, such ha S β s + C β for all [, T ]. (ii) sup β Uad S β e KT (s + KT ), where K = σ v f. (iii) For any β U ad, S T = lim T S β < exiss. Tha is, he soluion S β can be exended coninuously o [, T ]. (iv) lim T S β = if and only if lim T φ β 1 (, ) =. Proof. Le β U ad be given, and denoe S = S β and φ 1 = φ β 1, ec., hroughou he proof for simplicily.

19 1172 JIN MA, RENTAO SUN, AND YONGHUI ZHOU (i) Firs noe ha if S = for some < T, we define τ = inf{ [, T ); S =. Then, from (3.17) we see ha a τ i holds ha ds d =τ = στ v 2 >. Bu on he oher hand by definiion of τ we mus have Sτ S τ h h = S τ h h < for h > small enough, a conradicion. Tha is, S > [, T ). Nex, le us denoe, for (, s, β) [, T ] (, ) [, ), he righ side of (3.17) by [ ] 2 G(, s, β) = βs (σ v ) 2 + 2f s. Then for any β U ad, i holds ha σ z (5.5) G(, s, β ) = β2 [s σ z 2 f σ z 2 ] 2 β 2 + f 2 σ z 2 β 2 + σ v 2 { f 2 σ z 2 max [,T ] β 2 + σ v 2 = C β, hanks o Assumpion 5.1. Thus S s + C β for all [, T ], proving (i). (ii) To find he bound ha is independen of he choice of β, we noe ha ds d = G(, S, β ) (σ v ) f S K(1 + S ) [, T ], where K = σ v f. Thus he resul hen follows from Gronwall s inequaliy. (iii) Since G is quadraic in s, and lim s G(, s, β) =, i is easy o see from (5.5) ha, for any given β U ad, (5.6) On he oher hand, we wrie max,s G+ (, s, β ) = max G(, s, β ) C β.,s S s = G(r, S r, β r )dr = G + (r, S r, β r )dr G (r, S r, β r )dr = I + () I (), where I ± are defined in an obvious way. Since I + ( ) and I ( ) are monoone increasing, boh limis I + (T ) and I (T ) exis, which may be +. Bu (5.6) implies ha I + (T ) <, and by (i), I () = I + () S + s < I + () + s, for all [, T ); we conclude ha I (T ) < as well. Tha is, S T exiss. (iv) We rewrie (3.17) as follows (recall he definiion of φ 1 (3.2)): (5.7) { S = exp(log S ) = s exp ds S { = s exp (2f + (σv ) 2 β2 S S (σ z ) 2 { = s φ 1 (, ) exp (f + (σv ) 2 S ) d ) d. Thus, he resul follows easily from (iii). This complees he proof.

20 KYLE BACK MODELS AND LCMF SDEs 1173 In he res of he secion we shall ry o solve he opimizaion problem (5.4). We firs noe ha by definiion he quaniy S and hence φ 1 and φ 3 all depend on he choice of rading inensiy funcion β. Therefore (5.4) is essenially a problem of calculus of variaion. We shall proceed by firs looking for he firs order necessary condiions, and hen find he condiions ha are sufficien for us o deermine he opimal sraegy. To begin wih, le us denoe, for any differeniable funcional F : C([, T ]) C([, T ]) and any β, ξ C([, T ]), he direcional derivaive of F a β in he direcion ξ by (5.8) ξ F (β) = d dy F (β + yξ ) y=. We firs give some useful direcional derivaives ha will be used frequenly in wha follows. Recall he soluion S o he Riccai equaion (3.17) and he funcions φ 1 and φ 3, defined by (3.2). Noe ha hey are all funcionals of he rading inensiy β C([, T ]). Lemma 5.4. Le ξ be an arbirary coninuous funcion on [, T ], Then he following ideniies hold, provided all he direcional derivaives exis: (i) ξ β = ξ ; (ii) ξ S = φ 2 1(, ) ξβ r α r S r φ 2 1(, r)dr, where α = 2βS (σ ; z)2 (iii) ξ φ 1 (, ) = φ 1 (, ) [ ξβ r α r + ξ S r ρ r ]dr, where ρ = β2 (σ ; z)2 (iv) ξ φ 1 (, ) = φ 1 (, ) [ ξβ r α r + ξ S r ρ r ]dr; (v) ξ φ 3 (T, ) = { ξφ 1 (r, )φ 2 (, r)k r +φ 1 (r, )φ 2 (, r)[ ξ β r α r + ξ S r ρ r ]dr. Proof. Iem (i) is obvious. Iems (iii) (v) follow direcly from he chain rule. We only prove (ii). To see his, recall (3.17). We have { [ ] 2 S (β + yξ) = s + (σr v ) 2 Sr (β + yξ)(β r + yξ r ) dr + 2f r S r (β + yξ) dr, and hus ξ S = = { [ ] 2 d Sr (β + yξ)(β r + yξ r ) 2f r S r (β + yξ) dy σr z dr y= { 2f r ξ S r 2S [ rβ r βr ξ S r σr z σr z + S ] rξ r σr z dr. Denoing S = ξ S, we see ha i saisfies an ODE, [ ] d d S = 2 f β2 S (σ z ) 2 S 2β S 2 ξ (σ z ) 2 = 2[f k ]S 2β S 2 ξ (σ z ) 2, S =. Solving i and noing ha ξ = ξ β and α = 2βS (σ we obain z)2 S 2β r S 2 { r ξ r = (σr z ) 2 exp 2(f u k u )du dr = φ 2 1(, ) r proving (ii), whence he lemma. σ z ξ β r α r S r φ 2 1(, r)dr,

21 1174 JIN MA, RENTAO SUN, AND YONGHUI ZHOU We are now ready o prove he following necessary condiions of opimal sraegies for he original conrol problem (2.7). Theorem 5.5. Assume ha Assumpion 5.1 is in force. Suppose ha β U ad is an opimal sraegy of he problem (2.7); hen (1) i holds ha where φ 1 (, )(σ z ) 2 2β S φ 3 (T, ) + 1 S Φ = Φ = φ 2 1(, ) [ ] β r φ 1 (r, )φ 3 (T, r) + β2 r (σ z ) 2 Φ r dr, β r s r φ 1 (, r)dr g r φ 1 (r, )φ 2 (, r)dr; (2) furhermore, lim T β =, and consequenly, lim T S =. In paricular, P T = E P [V T FT Y ] = V T, P-a.s. Proof. Suppose ha β U ad is an opimal sraegy of he problem (2.7). Then i is also an opimal of he problem (5.4). Thus for any funcion ξ C([, T ]), i holds ha d ξ J(β) J(β + yξ) = =, dy y= or equivalenly, = (5.9) [ ξ β S φ 1 (, )φ 3 (T, ) + β ξ S φ 1 (, )φ 3 (T, ) + β S ξ φ 1 (, )φ 3 (T, ) + β S φ 1 (, ) ξ φ 3 (T, )]d. Then subsiuing ξ β, ξ S, and ξ φ i, i = 1, 2, 3, in Lemma 5.4 ino (5.9) and changing he order of inegraion if necessary we obain ha, formally, (5.1) where φ 1 (, )(σ z ) 2 2β S φ 3 (T, ) + 1 S Ψ = [ ] β r φ 1 (r, )φ 3 (T, r) + β2 r (σ z ) 2 Ψ r dr, (5.11) Ψ = φ 2 1(, )[φ 1 (, )φ 2 (, ) φ 3 (T, )] β r S r φ 1 (, r)dr. To jusify he ideniy (5.1) we now show ha boh sides of (5.1) are finie for any β U ad (noe ha i is possible ha β, as T ). To his end, we firs noe ha φ 1 (, ) and φ 2 (r, s) are bounded, and (5.12) φ 3 (T, ) = φ 1 (, )φ 2 (, ) φ 1 (T, )φ 2 (, T ) g r φ 1 (r, )φ 2 (, r)dr, hus i is also bounded, and clearly lim T φ 3 (T, ) =. Furhermore, we rewrie (5.11) as (5.13) Ψ = φ 2 1(, )G(, T ) β r S r φ 1 (, r)dr,

22 KYLE BACK MODELS AND LCMF SDEs 1175 where G(, T ) = φ 1 (, )φ 2 (, ) φ 3 (T, ) = φ 1 (T, )φ 2 (, T )+ g r φ 1 (r, )φ 2 (, r)dr, hanks o (5.12). We claim ha he inegral (5.14) [β r φ 1 (r, )φ 3 (T, r) + β2 r (σ z ) 2 Ψ r]dr = I 1 + I 2, where I 1 and I 2 are defined in an obvious way, is well-defined. Indeed, Assumpion 5.1 and he boundedness of S imply ha, modulo a universal consan, { ( ) (5.15) β φ 1 (, )φ 3 (T, ) β exp f u β2 us u β σu z 2 du exp( β2 udu). On he oher hand, by (5.13) i is easy o see ha β 2 Ψ β 2 φ 2 1(, ) β rs r φ 1 (, r)dr, and { r (5.16) β r S r φ 1 (, r)dr = C = C β r S r exp [ f u + β2 us u σu z 2 { r β 2 β r S r exp us u σu z 2 du dr σr z 2 [ { r β 2 ] d exp us u β r σu z 2 du ] du dr { β 2 C exp us u σu z 2 du. Here in he above C > is a generic consan, which depends only on he bounds of he coefficiens and β in Assumpion 5.1 and is allowed o vary from line o line. Thus, similar o (5.15), we derive from (5.13) and (5.16) ha (5.17) { β 2 Ψ Cβ 2 φ 1 (, )G(, T ) exp β 2 us u σ z u 2 du β 2 φ 1 (, ) Bu noice ha β 1 + β 2, and ha for any δ >, we have δ β 2 exp{ δ β d 1 2 e u du 1; β2 udu β 2 exp( β2 r dr). we can easily derive from (5.15) and (5.17) ha boh I 1 and I 2 in (5.14) are finie, proving he claim. We can now use he ideniy (5.1) o prove boh conclusions of he heorem. We begin by observing ha lim T S = mus hold. In fac, muliplying S on boh sides of (5.1) and hen aking limis T, and noing ha lim T φ 3 (T, ) =, we conclude ha lim T Ψ =. Bu hen (5.11), ogeher wih he fac ha lim T φ 3 (T, ) = bu lim T φ 2 (, ), implies ha lim T φ 1 (, ) =, and hence lim T S =, hanks o Lemma 5.3. We now claim ha lim T β =. Indeed, suppose no; hen we have ds lim T d = lim G(, S, β ) = lim T T (σv ) 2 c >, which is a conradicion, since S > for all [, T ), and lim T S =, proving he claim. Now noe ha S is he variance of he process V P, ; he facs ha lim T S = and ha boh processes V and P are coninuous lead o ha V T = P T, P-a.s.. This complees he proof of par (2). I remains o prove par (1). Bu noing ha φ 1 (T, ) =, we see from (5.13) ha Ψ = Φ, as G(, T )= g r φ 1 (r, )φ 2 (, r)dr. Thus (1) follows direcly from (5.1).