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Conditional mean-fields stochastic differential equation and their application

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Conditional mean-fields stochastic differential equation and their application

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Content CONDITIONAL MEAN-FIELDS STOCHASTIC DIFFERENTIAL EQUATION AND THEIR APPLICATION by Rentao Sun A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (APPLIED MATHEMATICS) December 2017 Copyright 2017 Rentao Sun To my family 2 Acknowledgment I would like to express my deepest appreciation to my committee chair Professor Jin Ma for the continuous support of my Ph.D study and related research, for his patience, motivation, and immense knowledge. I have learned not only valuable academic ideas from him in research, but also wisdom in every aspect in life which will denitely benet me a lot in the future. Without his guidance and persistent help this dissertation would not have been possible. I would like to sincerely thank my dissertation committee member Prof. Jianfeng Zhang, Prof. Jinchi Lv, and guidance committee member Prof. Sergey Lototsky and Prof. Remigijus Mikulevicius. Not only for their insightful comments and encourage- ment in my defense, but also for providing me great special topic courses throughout the last ve years. I also would like to thank Professor Yonghui Zhou, my colleagues Weisheng Xie, Xiaojing Xing, Cong Wu and Eunjung Noh for valuable discussions. Thanks to all my roommates who shared unforgettable memory with me. Finally, my most heartfelt thanks to my parents Guoming Sun and Yufang Sun for all the greatest support in my life. 3 Table of Contents Dedication 2 Acknowledgment 3 Chapter 1: Insider Trading Problem and Linear Conditional Mean-eld SDEs 5 1 Introduction to Insider Trading Problem . . . . . . . . . . . . . . . . 5 2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 The Linear Conditional Mean-eld SDEs . . . . . . . . . . . . . . . . 14 3.1 The General Result . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Deterministic Coecient Cases . . . . . . . . . . . . . . . . . 20 4 Proof of Theorem 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5 Necessary Conditions for Optimal Trading Strategy . . . . . . . . . . 35 6 Worked-out Cases and Examples . . . . . . . . . . . . . . . . . . . . 45 7 A Linear Quadratic Optimization Problem . . . . . . . . . . . . . . . 51 7.1 Description of the model . . . . . . . . . . . . . . . . . . . . . 51 7.2 Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 8 A Simple Non-linear Case of CMFSDEs . . . . . . . . . . . . . . . . 59 8.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 8.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Chapter 2: Non-linear Conditional Mean-eld SDEs 64 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4 A Stochastic Control Problem . . . . . . . . . . . . . . . . . . . . . . 80 5 Variational Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6 Stochastic Maximum Principle . . . . . . . . . . . . . . . . . . . . . . 99 Bibliography 112 4 Chapter 1 Insider Trading Problem and Linear Conditional Mean-eld SDEs 1 Introduction to Insider Trading Problem In his seminal paper, A.S. Kyle [28] rst proposed a sequential equilibrium model of asset pricing with asymmetric information. The model was then extended by K. Back [2] to the continuous time version, and has since known as the Kyle-Back strategic insider trading equilibrium model. Roughly speaking, in such a model it is assumed that there are two types of traders in a risk neutral market: one informed (insider) trader vs. many uninformed (noise) traders. The insider \sees" both (possibly future) value of the fundamental asset as well as its market value, priced by the market makers, and acts strategically in a non-competitive manner. The noise traders, on the other hand, act independently with only market information of the asset. Finally, the market makers set the price of the asset, in a Bertrand competition fashion, based on the historical information of the collective market actions of all traders, without knowing identify the insider. The so-called Kyle-Back equilibrium is a closed-loop system in which the insider maximizes his/her expected return in a market ecient manner (i.e., following the given market pricing rule). 5 There has been a large number of literature on this topic. We refer to, e.g., [2, 3, 11, 13, 17, 22{25, 28] and the references therein for both discrete and continuous time models. It is noted, however, that in most of these works only the case of static information is considered, that is, it is assumed that information that the insider could observe is \time-invariant", often as the fundamental price at a given future moment. Mathematically, this amounts to saying that the insider has the knowledge of a given random variable whose value cannot be detected from the market information at current time. It is often assumed that the system has a certain Gaussian structure (e.g., the future price is a Gaussian random variable), so that the optimal strategies can be calculated explicitly. The situation will become more complicated when the fundamental price progresses as a stochastic processfv t ;t 0g and the insider is able to observe the price process dynamically in a \non-anticipative" manner. The asym- metric information nature of the problem has conceivably led to the use of ltering techniques in the study of the Kyle-Back model, and we refer to, e.g., [1] and [5] for the static information case, and to, e.g., [11] and [22] for the dynamic information case. It is noted that in [5] it is further assumed that the actions of noise traders may have some \memory", so that the observation process in the ltering problem is driven by a fractional Brownian motion, adding technical diculties in a dierent aspect. We also note that the Kyle-Back model has been continuously extended in various direc- tions. For example, in a static information setting, [18] recently considered the case when noise trading volatility is a stochastic process, and in the dynamic information case [14{16] studied the Kyle-Back equilibrium for the defaultable underlying asset via dynamic Markov bridges, exhibiting further theoretical potential of the problem. In this paper we are interested in a generalized Kyle-Back equilibrium model in a dynamic information setting, in which the asset dynamics is of the form of an Ornstein-Uhlenback SDE whose drift also contains market sentiment (e.g., supply and 6 demand, earning base, etc.), quantied by the market price. The problem is then nat- urally imbedded into a (linear) ltering problem in which both state and observation dynamics contain the ltered signal process (seex2 for details). We note that such a structure is not covered by the existing ltering theory, and thus it is interesting in its own right. In fact, under the setting of this paper the signal-observation dynam- ics form a \coupled" (linear) conditional mean-eld stochastic dierential equations (CMFSDEs, for short) whose well-posedness, to the best of our knowledge, is new. The main objective of this section is thus two-fold. First, we shall look for a rigorous framework on which the well-posedness of the underlying CMFSDE can be established. The main device of our approach is the \reference probability measure" that is often seen in the nonlinear ltering theory (see, e.g., [33]). Roughly speaking, we give the observable market movement a \prior" probability distribution so that it is a Brownian motion that is independent of the martingale representing the aggregated trading actions of the noisy traders, and we then prove that the original signal- observation SDEs have a weak solution. More importantly, we shall prove that the uniqueness holds among all weak solutions whose laws are absolutely continuous with respect to the reference probability measure. We should note that such a uniqueness in particular resolves a long-standing issue on the Kyle-Back equilibrium model: the identication of the total traded volume movements and the innovation process of the corresponding ltering problem, which has been argued either only heuristically or by economic instinct in the literature (see, e.g., [1]). The second goal of this paper is to identify the Kyle-Back equilibrium, that is, the optimal closed-loop system for this new type of partially observable optimization problem. Assuming a Gaussian structure and the linearity of the CMFSDEs, we give the explicit solutions to the insider's trading intensity and justify the well-posedness of the closed-loop system 7 (whence the \existence" of the equilibrium). These solutions in particular cover many existing results as special cases. The rest of the paper is organized as follows. In section 2 we give the preliminaries of the Kyle-Back equilibrium model and formulate the strategic insider trading prob- lem. In section 3 we formulate a general form of linear CMFSDE and introduce the notion of its solutions and their uniqueness. We state the main well-posedness result, calculate its explicit solutions in the case of deterministic coecients, and discuss an important extension to the unbounded coecients case. Section 4 will be devoted to the proof of the main well-posedness theorem. In section 5 we characterize the optimal trading strategy, and give a rst order necessary condition for the optimal intensity, and nally in section 6 we focus on the synthesis analysis, and validate the Kyle-Back equilibrium. In some spacial cases, we give the closed-form solutions, and compare them to the existing results. 2 Problem Formulation In this section we describe a continuous time Kyle-Back equilibrium model that will be investigated in this paper, as well as related technical settings. We begin by assuming that all randomness of the market comes from a common complete probability space ( ;F;P) on which is dened 2-dimensional Brownian motion B = (B v ;B z ), where B v =fB v t : t 0g represents the noise of the fun- damental value dynamics, and B z =fB z t : t 0g represents the collective action of the noise traders. For notational clarity, we denote F v =fF B v t : t 0g and F z 4 =fF B z t : t 0g to be the ltrations generated by B v and B z , respectively, and denote F = F v _F z , with the usual P-augmentation such that it satises the usual hypotheses (cf. e.g., [32]). 8 Further, throughout the paper we will denote, for a generic Euclidean space X, regardless of its dimension,h;i andjj to be its inner product and norm, respectively. We denote the space of continuous functions dened on [0;T ] with the usual sup-norm byC([0;T ];X), and we shall make use of the following notations: For any sub--eldGF T and 1 p <1, L p (G;X) denotes the space of all X-valued,G-measurable random variables such that Ejj p <1. As usual, 2L 1 (G;X) means that it isG-measurable and bounded. For 1p<1, L p F ([0;T ];X) denotes the space of allX-valued,F-progressively measurable processes satisfyingE R T 0 j t j p dt<1. The meaning ofL 1 F ([0;T ];X) is dened similarly. Throughout this paper we assume that all the processes are 1-dimensional, but higher dimensional cases can be easily deduced without substantial diculties. Therefore, we will often drop X(= R) from the notation. Also, throughout the paper we shall denote all \L p -norms" bykk p , regardless its being L p (G) or L p F ([0;T ]), when the context is clear. Consider a given stock whose fundamental value (or its return) isV =fV t :t 0g traded on a nite time interval [0;T ]. There are three types of agents in the market: (i) The insider, who directly observes the realization of the value V t at any time t2 [0;T ] and submits his order X t at t2 [0;T ]. (ii) The noise traders, who have no direct information of the given asset, and (col- lectively) submit an order Z t at t2 [0;T ] in the form Z t = Z t 0 z t dB z t ; t 0; (2.1) 9 where z =f z t :t 0g is a given continuous deterministic function satisfying z t > 0. (iii) The market makers, who observe only the total traded volume Y t =X t +Z t ; t 0; (2.2) and set the market price of the underlying asset at each time t, denoted by P t , based on the observed informationF Y t 4 = fY s ;s tg. We denote S t 4 = E[(V t P t ) 2 ] to be the measure of the discrepancy between market price and fundamental value of the asset. We now give a more precise description of the two main ingredients in the model above: the dynamics of market price P and the fundamental value V . First, in the same spirit of the original Kyle-Back model, we can assume that the market price P is the result of a Bertrand-type competition among the market makers (cf. e.g., [3]), and therefore should be taken as P t =E[V t jF Y t ], at each time t 0. Mathematically speaking, this amounts to saying that the market price is set to minimize the error S t , among allF Y t -measurable random variables inL 2 ( ), hence the \best estimator" that the market maker is able to choose given the informationF Y t . It is easy to see that under such a choice one must haveE[(V t P t )Y t ] = 0, that is, the market makers should expect a zero prot at each time t2 [0;T ]. It is worth noting that, however, unlike the static case (i.e., V t v), the process P is no longer a (P;F Y )-martingale in general. Next, in this paper we shall also assume that the dynamics of the value of the stock V =fV t g takes the form of an It^ o process: dV t = F t dt + v t dB v t , t 0 (this would easily be the case if, e.g., the interest rate is non-zero). Furthermore, we shall assume 10 that the drift F t = F (t;V t ;P t ), t 0. Here, the dependence of F on the market priceP t is based on the following rationale: the value of the stock is often aected by factors such as supply and demand, the earnings base (cash ow per share), or more generally, the market sentiment, which all depend on the market price of the stock. Consequently, taking the Gaussian structure into consideration, in what follows we shall assume that the process V satises the following linear SDE: 8 > < > : dV t = (f t V t +g t P t +h t )dt + v t dB v t = (f t V t +g t E[V t jF Y t ] +h t )dt + v t dB v t ; t 0; V 0 N(v 0 ;s 0 ): (2.3) where the functions f t ;g t ;h t and v t are all deterministic continuous dierentiable functions with respect to time t2 [0;T ], and N(v 0 ;s 0 ) is a normal random variable with mean v 0 and standard deviation s 0 . Continuing, given the Gaussian structure of the dynamics, it is reasonable to assume that the insider's optimal trading strategy (in terms of \number of shares") is of the form (see, e.g., [1{3,22,28]): dX t = t (V t P t )dt; t 0; (2.4) where t > 0 is a deterministic continuous dierentiable function with respect to time t in [0, T), known as the insider trading intensity. Consequently, it follows from (2.2) that the total traded volume process observed by the market makers can be expressed as dY t = t (V t P t )dt + z t dB z t = t (V t E[V t jF Y t ])dt + z t dB z t ; t 0: (2.5) 11 We note that SDEs (2.3) and (2.5) form a (linear) conditional mean-eld SDE (CMFSDE), which is beyond the scope of the traditional ltering theory. Such SDE have been studied in [7] and [12] in general nonlinear forms, but none of them covers the one in this form. In fact, if we further allow the function h in (2.3) to be an F Y -adapted process, as many stochastic control problems do, then (2.3) and (2.5) would become a fully convoluted CMFSDE whose well-posedness, to the best of our knowledge, has not been studied in the literature, even in the linear form. We should mention that the equation (2.5) in the case when V t v was already noted in [1] and [5], but without addressing the uniqueness of the solution. In the next sections we shall establish a mathematical framework so these SDEs can be studied rigorously. Given the dynamics (2.3) and (2.5), our main purpose now is to nd an optimal trading intensity for the insider to maximize his/her expected wealth, whence the Kyle-Back equilibrium. More specically, denote the wealth process of the insider by W =fW t :t 0g, and assume that the strategy is self-nancing (cf. e.g., [6]), then the total wealth of the insider over time duration [0;T ], based on the market price made by the market makers, should be W T = Z T 0 X t dP t =X T P T Z T 0 t (V t P t )P t dt = Z T 0 t (V t P t )(P T P t )dt: (2.6) Here in the above we used a simple integration by parts and denition (2.4). Thus the optimization problems can be described as sup E[W T ] 4 = sup J() = sup Z T 0 t E[(V t P t )(P T P t )]dt: (2.7) Remark 2.1. We should remark that the simple form of optimization problem (2.7) is due largely to the linearity of the dynamics (2.3) and (2.5), as well as the Gaussian 12 assumption on the initial state v. These lead to a Gaussian structure, whence the trading strategy (2.4). The general nonlinear and/or non-Gaussian Kyle-Back model requires further study of CMFSDE and associated ltering problem, and one should seek optimal control from a larger class of \admissible controls". In that case the rst order condition studied in this paper will become a Pontryagin type stochastic maximum principle (see, for example, [7]), and the solution is expected to be much more involved. We will address such general problems in our future publications. We end this section by noting that the main idea for solving the CMFSDE is to introduce the so-called reference probability space in nonlinear ltering literature (see, e.g., [33]), which can be described as follows. Assumption 2.2. There exists a probability space ( 0 ;F 0 ;Q 0 ) on which the process (B v t ;Y t ), t2 [0;T ], is a 2-dimensional continuous martingale, whereB v is a standard Brownian motion and Y is the observation process with quadratic variationhYi t = R t 0 ( z s ) 2 ds. The probability measureQ 0 will be referred to as the reference measure. We remark that Assumption 2.2 amounts to saying that we are giving a prior distribution to the price processY =fY t :t 0g that the market maker is observing, which is not unusual in statistical modeling, and will facilitate the discussion greatly. A natural example is the canonical space: 0 4 = C 0 ([0;T ];R 2 ), the space of all 2- dimensional continuous functions null at zero;F 0 4 =B( 0 );F 0 t 4 =B t ( 0 ) =f!(^ t) :!2 0 g, t2 [0;T ]; and (B v ;Y ) is the canonical process. In the case z 1,Q 0 is the Wiener measure. 13 3 The Linear Conditional Mean-eld SDEs In this section we study the linear conditional mean-eld SDEs (CMFSDE) (2.3) and (2.5) that play an important role in this paper. In fact, let us consider a slightly more general case that is useful in applications: 8 > < > : dX t =ff t X t +g t E[X t jF Y t ] +h t gdt + 1 t dB 1 t ; X 0 =v; dY t =fH t X t +G t E[X t jF Y t ]gdt + 2 t dB 2 t ; Y 0 = 0; (3.1) where B 4 = (B 1 ;B 2 ) is a standard Brownian motion dened on a given probability space ( ;F;P), and v N(v 0 ;s 0 ) is independent of B. In light of Assumption 2.2, throughout this section we shall assume the following: Assumption 3.1. (i) The coecients f, g, 1 , 2 , G, and H are all deterministic, continuous functions; and i t > 0, i = 1; 2, for all t2 [0;T ]; (ii) there exists a probability space ( 0 ;F 0 ;Q 0 ) on which the process (B 1 t ;Y t ), t2 [0;T ], is a 2-dimensional continuous martingale, such that B 1 is a standard Q 0 - Brownian motion, andhYi t = R t 0 j 2 s j 2 ds, t2 [0;T ],Q 0 -a.s.; (iii) the coecient h is anF Y -adapted, continuous process, such that E Q 0 h sup 0tT jh t j 2 i <1: Remark 3.2. The Assumption 3.1-(iii) amounts to saying that the process h is dened on the reference probability space ( 0 ;F 0 ;Q 0 ), and adapted to the Brownian ltrationF Y , as we often see in the stochastic control with partial observations (cf. [4]). 14 3.1 The General Result To simplify notations in what follows we shall assume that 1 = 2 1. We rst introduce two denitions of the solution to CMFSDE (3.1). Let P(R) denote all probability measures on (R;B(R)), whereB(R) is the Borel -eld of R, and N(v 0 ;s 0 )2P(R) denotes the normal distribution with mean v 0 and variance s 0 . Denition 3.3. Let 2P(R) be given. An eight-tuple ( ;F;F;P;X;Y;B 1 ;B 2 ) is called a weak solution to CMFSDE (3.1) with initial distribution if (i) (B 1 ;B 2 ) is anF-Brownian motion underP; (ii) (X;Y;B 1 ;B 2 ) satises (3.1),P-a.s.; (iii) X 0 ; and is independent of (B 1 ;B 2 ) underP. Denition 3.4. A weak solution ( ;F;F;P;X;Y;B 1 ;B 2 ) is called aQ 0 -weak solution to CMFSDE (3.1) if (i) there exists a probability measureP 0 on ( 0 ;F 0 ), and processes (X 0 ;Y 0 ;B 1;0 ;B 2;0 ) dened on ( 0 ;F 0 ;P 0 ), whose law underP 0 is the same as that of (X;Y;B 1 ;B 2 ) under P; and (ii)P 0 Q 0 . In what follows for any Q 0 -weak solution, we shall consider only its copy on the reference measurable space ( 0 ;F 0 ), and we shall still denote the solution by (X;Y;B 1 ;B 2 ). The uniqueness of the solutions to CMFSDE (3.1) is a more delicate issue. In fact, even the weak uniqueness (in the usual sense) for CMFSDE (2.3) and (2.5) is not clear. However, we have a much better hope, at least in the linear case, for Q 0 -solutions. We rst introduce the following \Q 0 -pathwise uniqueness". 15 Denition 3.5. The CMFSDE (3.1) is said to have \Q 0 -pathwise uniqueness" if for any twoQ 0 -weak solutions ( 0 ;F 0 ;F 0 ;P i ;X i ;Y i ;B 1;i ;B 2;i ), i = 1; 2, such that (i) X 1 0 =X 2 0 ; and (ii)Q 0 f(B 1;1 t ;Y 1 t ) = (B 1;2 t ;Y 2 t );8t2 [0;T ]g = 1, then it holds thatQ 0 f(X 1 t ;B 2;1 t ) = (X 2 t ;B 2;2 t );8t2 [0;T ]g = 1, andP 1 =P 2 . Theorem 3.6. Assume that Assumption 3.1 is in force, and further thath is bounded. LetN(v 0 ;s 0 ) be given. Then CMFSDE (3.1) possesses a weak solution with initial distribution , denoted by ( ;F;F;P;X;Y;B 1 ;B 2 ). Moreover, if we denote P t =E P [X t jF Y t ], t2 [0;T ], then P satises the following SDE: 8 > < > : dP t = [(f t +g t )P t +h t ]dt +S t H t fdY t [H t +G t ]P t dtg; t2 [0;T ]; P 0 =v 0 : (3.2) where S t =Var(P t ) satises the Riccati equation: dS t = [1 + 2f t S t H 2 t S 2 t ]dt; S 0 =s 0 : (3.3) Furthermore, the weak solution can be chosen as Q 0 -weak solution, and the Q 0 - pathwise uniqueness holds. We remark that Theorem 3.6 does not imply that CMFSDE (3.1) has a strong solution, as not every weak solution is aQ 0 -weak solution. The proof of Theorem 3.6 is a bit lengthy, we shall defer it to next section. We nevertheless present a lemma below, which will be frequently used in our discussion, so as to facilitate the argument in the next section. 16 To begin with, we consider any ltered probability space ( ;F;F;P) on which is dened a standard Brownian Motion (B 1 t ;B 2 t ). We assume that F = F (B 1 ;B 2 ) . For any 2L 2 F ([0;T ]) we dene L to be the solution to the following SDE, dL t =L t t dB 2 t ; t 0; L 0 = 1: (3.4) In other words, L is a local martingale in the form of the Dol eans-Dade stochastic exponential: L t = exp n Z t 0 s dB 2 s 1 2 Z t 0 j s j 2 ds o : (3.5) Next let 2L 2 F ([0;T ]) and consider the following SDE: dY t = ( t +h(Y ) t )dt +dB 2 t ; Y 0 = 0; (3.6) where h : [0;T ]C([0;T ])7!R is \progressively measurable" in the sense that, it is a measurable function such that for each t2 [0;T ], h(y) t =h(y ^t ) t for y2C([0;T ]). (A simple case would be h(y) t = ~ h(y t ), where ~ h is a measurable function.) We should note that in general the well-posedness of SDE (3.6) is non-trivial without any specic conditions on h, but in what follows we shall assume a priori that (3.6) has a (weak) solution on some probability space ( ;F;P). We say thath2L 2 F Y ([0;T ]) if h t =h(Y ) t , t2 [0;T ], such thatE R T 0 jh(Y ) t j 2 dt<1. We have the following lemma. Lemma 3.7. Suppose that the SDE (3.6) has a solution Y t , t2 [0;T ], for given t 2 L 2 F B 1 ([0;T ]) and h2 L 2 F Y ([0;T ]) on some probability space ( ;F;P). Let be given by d t = t dt +dB 2 t ; t 0; 0 = 0: (3.7) 17 Assume further that L (+h) , the solution to (3.4) with =( +h), is an (F;P)- martingale. Then, for any t2 [0;T ], it holds that E P [ t jF Y t ] =E P [ t jF t ]; 8t2 [0;T ]; P-a.s. (3.8) Proof. Clearly, it suces to proveE P [ T jF Y T ] =E P [ T jF T ], as the cases fort<T are analogous. To this end, we dene a new probability measureQ on ( ;F T ) by dQ dP F T =L (+h) T ; where L (+h) is the solution to the SDE (3.4) with =( +h), and it is a true martingale on [0;T ] by assumption. By Girsanov Theorem, the process (B 1 ;Y ) is a standard Brownian motion on [0;T ] underQ. Now dene L t = 1=L (+h) t , then L satises the following SDE on ( ;F T ;Q): d L t = L t ( t +h t )dY t ; t2 [0;T ]; L 0 = 1: (3.9) Furthermore, by the Kallianpur-Striebel formula, we have E P [ T jF Y T ] = E Q [ T L T jF Y T ] E Q [ L T jF Y T ] : (3.10) On the other hand, L has the explicit form: L T = exp n Z T 0 [ t +h t ]dY t 1 2 Z T 0 [ t +h t ] 2 dt o (3.11) = exp n Z T 0 h t dY t 1 2 Z T 0 jh t j 2 dt + Z T 0 t dY t 1 2 Z T 0 [j t j 2 + 2h t t ]dt o 4 = L 0 T T ; 18 where L 0 T 4 = exp n Z T 0 h t dY t 1 2 Z T 0 jh t j 2 dt o ; T 4 = exp n Z T 0 t dY t 1 2 Z T 0 [[ t ] 2 + 2h t t ]dt o : Note that h isF Y -adapted, so is L 0 T . We derive from (3.10) that E P [ T jF Y T ] = E Q [ T T jF Y T ] E Q [ T jF Y T ] : (3.12) Now dene Y 1 t = R t 0 h s ds. Since h is F Y -adapted, so is Y 1 , and consequently t = Y t Y 1 t , t 0 isF Y -adapted. Moreover, since (B 1 ;Y ) is a standard Brownian motion under Q, and is F B 1 -adapted, we conclude that t is independent of Y t underQ. Therefore, using integration by parts we obtain that T = exp n Z T 0 t d t 1 2 Z T 0 j t j 2 dt o (3.13) = exp n T T Z T 0 t d t 1 2 Z T 0 j t j 2 dt o : Since is independent ofY underQ, and t ,t2 [0;T ] isF Y T -measurable, a Monotone Class argument shows thatE Q [ T jF Y T ] isF T measurable; and similarly,E Q [ T T jF Y T ] is alsoF T measurable. ConsequentlyE P [ T jF Y T ] isF T measurable, thanks to (3.10). Finally, notingF F Y we have E P [ T jF Y T ] =E P fE P [ T jF Y T ]jF T g =E P [ T jF T ]; (3.14) proving the lemma. 19 3.2 Deterministic Coecient Cases An important special case is when all the coecients in the linear CMFSDE (3.1) are deterministic. In this case we expect that the solution (X;Y ) is Gaussian, and it can be solved in a much more explicit way. The following linear CMFSDE will be useful in the study of insider trading equilibrium model in the latter half of the paper. 8 > < > : dX t = [f t X t +g t E[X t jF Y t ] +h t ]dt + 1 t dB 1 t ; X 0 =v; dY t =H t (X t E[X t jF Y t ])dt + 2 t dB 2 t ; Y 0 = 0; (3.15) wherevN(v 0 ;s 0 ) and is independent of (B 1 ;B 2 ) and all the coecients are assumed to be deterministic. Bounded Coecients Case In light of Theorem 3.6 let us introduce the following functions: k t = H 2 t S t j 2 t j 2 ; l t = H t S t 2 t ; t 0: (3.16) where S is the solution to the following Riccati equation dS t dt = ( 1 t ) 2 + 2f t S t l 2 t ; t 0; S 0 =s 0 : (3.17) We have the following result. Proposition 3.8. Let Assumption 3.1 be in force, and assume further that the process h in (3.1) is also a deterministic and continuous function. Let (X;Y ) be the solution 20 of (3.15) on the probability space ( ;F;P), and denote P t =E P [X t jF Y t ], t 0. Then X and P have the following explicit forms respectively: for t 0, it holdsP-a.s. that X t = P t + 1 (t; 0) h vv 0 + Z t 0 1 (0;r)( 1 r dB 1 r l r dY r ) i ; (3.18) P t = 2 (t; 0) n v 0 + Z t 0 2 (0;r)h r dr + (vv 0 ) 3 (t; 0) (3.19) + Z t 0 1 r 1 (0;r) 3 (t;r)dB 1 r + Z t 0 [ 2 (0;r)l r 1 (0;r) 3 (t;r)l r ]dB 2 r o ; where, for 0rt, 8 > > > < > > > : 1 (t;r) = exp n Z t r (f u k u )du o ; 2 (t;r) = exp n Z t r (f u +g u )du o ; 3 (t;r) = Z t r 1 (u; 0) 2 (0;u)k u du: (3.20) Proof. We rst note that the SDE (3.15) is a special case of (3.1) with G =H. Then, following the same argument of Theorem 3.6 one can show that when 1 > 0 and 2 > 0 are not equal to 1, the SDE (3.2) for the process P t =E P [X t jF Y t ] reads dP t = [(f t +g t )P t +h t ]dt + H t S t ( 2 t ) 2 dY t ; P 0 =v 0 ; (3.21) and S t satises a Riccati equation dS t dt = ( 2 t ) 2 + 2f t S t h H t S t 2 t i 2 ; S 0 =s 0 : (3.22) Now applying the Girsanov transformation we can dene a new probability mea- sure Q under which (B 1 ;Y ) is a continuous martingale, such that B 1 is a standard 21 Brownian motion, anddhYi t =j 2 t j 2 dt. Then, underQ, the dynamic ofV t 4 =X t P t can be written as dV t = h f t H 2 t S t ( 2 t ) 2 i V t dt + 1 t dB 1 t H t S t 2 t dY t = [f t k t ]V t dt + 1 t dB 1 t l t dY t ; X 0 P 0 =vv 0 : It then follows that the identity (3.18) holds Q-almost surely, and hence P-almost surely. Similarly, applying the constant variation formula for the linear SDE (3.21) and noting (3.1) we obtain that, with 2 (t;r) = exp( R t r (f u +g u )du), for 0r;tT , P t = 2 (t; 0) n v 0 + Z t 0 2 (0;r)h r dr + Z t 0 2 (0;r) H r S r ( 2 r ) 2 dY r o (3.23) = 2 (t; 0) n v 0 + Z t 0 2 (0;r)h r dr + Z t 0 2 (0;r) H r S r ( 2 r ) 2 [H r (X r P r )dr + 2 r dB 2 r ] o : Now plugging (3.18) into (3.23), and applying Fubini, we have P t = 2 (t; 0) n v 0 + Z t 0 2 (0;r)h r dr + (vv 0 ) Z t 0 1 (r; 0) 2 (0;r)k r dr + Z t 0 1 (0;r) 1 r Z t r 1 (u; 0) 2 (0;u)k u dudB 1 r + Z t 0 [ 2 (0;r)l r 1 (0;r)l r Z t r 1 (u; 0) 2 (0;u)k u du]dB 2 r o (3.24) = 2 (t; 0) n v 0 + Z t 0 2 (0;r)h r dr + (vv 0 ) 3 (t; 0) + Z t 0 1 (0;r) 1 r 3 (t;r)dB 1 r + Z t 0 [ 2 (0;r)l r 1 (0;r) 3 (t;r)l r ]dB 2 r o ; where 3 (t;r) 4 = R t r 1 (u; 0) 2 (0;u)k u du. This proves (3.19), whence the proposition. 22 Unbounded Coecients Case We note that Theorem 3.6 as well as the discussion so far rely heavily on the assump- tion that all the coecients are bounded, especially H and G (see Assumption 3.1). However, in our applications we will see that the coecients H =G = , where is the insider trading intensity which, at least in the optimal case, will satisfy lim t!T t = +1, violating Assumption 3.1. In other words, the closed-loop system will exhibit a certain Brownian \bridge" nature (see also, e.g., [3, 14, 15]), for which the well-posedness result of Theorem 3.6 actually does not apply. To overcome such a con ict, we introduce the following relaxed version of Assump- tion 3.1. Assumption 3.9. There exists a sequencefT n g n1 , with 0<T n %T , and a sequence of probability measuresfQ n g n1 on ( 0 ;F 0 ), satisfying (i) Assumption 3.1 holds for each ( 0 ;F 0 ;Q n ) over [0;T n ], n 1 (ii)Q n+1 F 0 Tn =Q n , n 1. We shall refer to the sequence of probability measuresQ 0 :=fQ n g n1 as the reference family of probability measures, and the associated sequencefT n g n1 as the announcing sequence. Clearly, if the reference measureQ 0 exists, thenQ n =Q 0 j F 0 Tn , n 1. It is known, however, that in the dynamic observation case the Kyle-Back equilibrium may only exist on [0;T ) (see, e.g. [16] and the references cited therein). In such a case the reference family would play a fundamental role. A reasonable extension of the notion ofQ 0 -weak solution over [0;T ) is as follows. 23 Denition 3.10. LetQ 0 be a reference family of probability measures, with announc- ing sequencefT n g. A sequencef( 0 ;F 0 ;P n ;X n ;Y n ;B 1;n ;B 2;n )g n1 is called aQ 0 - weak solution of (3.1) on [0;T ) if for each n 1, ( 0 ;F 0 ;P n ;X n ;Y n ;B 1;n ;B 2;n ) is a Q n -weak solution on [0;T n ]. It is worth noting that if the coecients of CMFSDE (3.1) satisfy Assump- tion 3.1 on each sub-interval [0;T n ], then one can apply Theorem 3.6 for each n to get aQ 0 -solution. Furthermore, since the solutions will be pathwisely unique under each Q n over [0;T n ], it is easy to check that (X n+1 t ;Y n+1 t ;B 1;n+1 t ;B 2;n+1 t ) = (X n t ;Y n t ;B 1;n t ;B 2;n t ), t2 [0;T n ], Q n -a.s. We can then dene a process (X;Y;B 1 ;B 2 ) on [0;T ) by simply setting (X t ;Y t ;B 1 t ;B 2 t ) = (X n t ;Y n t ;B 1;n t ;B 2;n t ), for t 2 [0;T n ], n 1, and we shall refer to such a process as theQ 0 -solution on [0;T ). TheQ 0 - pathwise uniqueness on [0;T ) can be dened in an obvious way. We have the following extension of Theorem 3.6, whose proof is left for the interested reader. Theorem 3.11. Assume that Assumption 3.9 is in force, and letQ 0 be the family of reference measures with announcing sequencefT n g. Assume further that Assumption 3.1 holds for each Q n on [0;T n ]. Then CMFSDE (3.1) possesses aQ 0 -weak solution on [0;T ), and it isQ 0 -pathwisely unique on [0;T ). 4 Proof of Theorem 3.6 In this section we prove Theorem 3.6. We begin by making the following reduction: it suces to consider the SDE (3.1) where the initial stateX 0 =vv 0 is deterministic, that is, s 0 = 0. Indeed, suppose that (X x ;Y x ) is a weak solution of (3.1) along with some probability space ( ;F;P) andP-Brownian motion (B 1 ;B 2 ), andv is any 24 random variable dened on (R;B(R)), with normal distribution 4 = N(v 0 ;s 0 ), we dene the product space ~ 4 = R; ~ F 4 =F B(R); ~ P 4 =P ; and write generic element of ~ !2 ~ as ~ ! = (!;x). Then for each t 0, the mapping ~ !7!X x t (!) denes a random variable on ( ~ ; ~ F; ~ P), and x7!X x 0 4 =v(x) is a normal random variable with distribution N(v 0 ;s 0 ) and is independent of (B 1 ;B 2 ), by de- nition. Bearing this in mind, throughout the section we shall assume that the initial state X 0 =x is deterministic. 4.1 Existence Our main idea to prove the existence of the weak solution is to \decouple" the state and observation equations in (3.1) by considering the dynamics of the ltered state process P t 4 =E P [X t jF Y t ], t 0, which is known to satisfy an SDE, thanks to linear (Kalman-Bucy) ltering theory. To be more precise, we consider the following system of SDEs on the reference probability space ( ;F;Q 0 ), on which (B 1 ;Y ) is a Brownian motion: 8 > > > > > > > > > > < > > > > > > > > > > : dX t = [f t X t +g t P t +h t ]dt +dB 1 t ; X 0 =x; dB 2 t =dY t [H t X t +G t P t ]dt; B 2 0 = 0; dP t = [(f t +g t )P t +h t ]dt +S t H t fdY t [H t +G t ]P t dtg; P 0 =x; dS t = [2f t S t H 2 t S 2 t + 1]dt; S 0 = 0: (4.1) 25 We note that by Assumption 3.1, all coecients f;g;H;G are deterministic and h2 L 2 F Y (C([0;T ])), it is easy to see that the linear system (4.1) has a (pathwisely) unique solution (X t ;B 2 t ;P t ) on ( ;F;Q 0 ). Now let L =fL t g t0 be the solution to the SDE: dL t =L t (H t X t +G t P t )dY t ; L 0 = 1: (4.2) ThenL is a positiveQ 0 -local martingale, hence aQ 0 -supermartingale withE Q 0 [L t ] L 0 = 1. Furthermore, L can be written as the Dol eans-Dade exponential: L t = expf Z t 0 (H s X s +G s P s )dY s 1 2 Z t 0 jH s X s +G s P s j 2 ds o ; t2 [0;T ]: (4.3) We have the following lemma. Lemma 4.1. Assume that Assumption 3.1 holds, and further thath is bounded. Then the process L =fL t ;t 0g is a true (F;Q 0 )-martingale on [0;T ]. Proof. We follow the idea of that in [4]. Since L is a supermartingale with E Q 0 [L t ] 1, we need only show that E Q 0 [L t ] = 1, for all t 0. To this end, we dene, for any "> 0, L " t 4 = L t 1 +"L t ; t2 [0;T ]: Then clearly 0L " t L t ^ 1 " , and an easy application of It^ o's formula shows that dL " t = "L 2 t [H t X t +G t P t ] 2 (1 +"L t ) 3 dt + L t [H t X t +G t P t ] (1 +"L t ) 2 dY t ; t 0; L " 0 = 1 1 +" : (4.4) 26 Since for each xed "> 0, L t [H t X t +G t P t ] (1 +"L t ) 2 2 = "L t [H t X t +G t P t ] "(1 +"L t ) 2 2 [H t X t +G t P t ] 2 " ; we see that the stochastic integral on the right hand side of (4.4) is a true martingale. It then follows that E Q 0 [L " t ] = 1 1 +" E Q 0 h Z t 0 "L 2 t [H t X t +G t P t ] 2 (1 +"L t ) 3 dt i : (4.5) Next, we observe that L t > 0, and 0 "L 2 t [H t X t +G t P t ] 2 (1 +"L t ) 3 = ("L t )L t [H t X t +G t P t ] 2 (1 +"L t ) 3 L t [H t X t +G t P t ] 2 : Note that L " is bounded. By sending "! 0 on both sides of (4.5) and applying Dominated Convergence Theorem we can then conclude thatE Q 0 [L t ] = 1, provided E Q 0 h Z T 0 L t [H t X t +G t P t ] 2 dt i <1: (4.6) It remains to check (4.6). To this end, let us dene X t =X 1 t + t , where 8 > < > : d t =f t t dt +dB 1 t ; 0 =x; dX 1 t = [f t X 1 t +g t P t +h t ]dt; X 1 0 = 0: (4.7) By Gronwall's inequality, it is readily seen that jX 1 t jC Z t 0 jg s P s +h s jdsC h 1 + Z t 0 jP s jds i ; t2 [0;T ]: (4.8) 27 Here and in the sequel C > 0 denotes a generic constant depending only on the bounds of the coecients f;g;H;G, h, and the duration T > 0, which is allowed to vary from line to line. Now, noting that L t is a super-martingale with L 0 = 1, we deduce from (4.8) that E Q 0 h Z T 0 L t jX 1 t j 2 dt i C n 1 +E Q 0 h Z T 0 Z t 0 L t jP s j 2 dsdt io (4.9) C n 1 +E Q 0 h Z T 0 L s jP s j 2 ds io : Consequently we have E Q 0 h Z T 0 L t [H t X t +G t P t ] 2 dt i CE Q 0 h Z T 0 L t jX 1 t j 2 +j t j 2 +jP t j 2 dt i (4.10) C n 1 +E Q 0 h Z T 0 L t (j t j 2 +jP t j 2 )dt io : Continuing, let us recall that the processesP and satisfy (4.1) and (4.7), respec- tively. By It^ o's formula we see that 8 > < > : dj t j 2 = [2f t j t j 2 + 1]dt + 2 t dB 1 t ; djP t j 2 = 2M t jP t j 2 + 2P t h t +S 2 t H 2 t dt + 2S t H t P t dY t ; (4.11) whereM t 4 = (f t +g t )S t H t (H t +G t ),t 0. Next, we dene, for> 0 andt2 [0;T ], X t = X t [1 +jX t j 2 ] 1=2 ; P t = P t [1 +jP t j 2 ] 1=2 : 28 ThenjX t jjX t j^ 1 2 andjP t jjP t j^ 1 2 ,8t; and it is not hard to show that lim !0 X =X, lim !0 P =P , uniformly on [0;T ], in probability. Now, dene dL t =L t [H t X t +G t P t ]dY t ; L 0 = 1: (4.12) Since X and P are now bounded, L is a martingale and E Q 0 [L t ] = 1, t2 [0;T ]. Furthermore, by the stability of SDEs one shows that, possibly along a subsequence, L t converges to L t ,Q 0 -a.s., t2 [0;T ]. Noting (4.11) and applying It^ o's formula we have, for t2 [0;T ] L t j t j 2 =x 2 + Z t 0 L s [2f s j s j 2 + 1]ds + Z t 0 2L s s dB 1 s + Z t 0 L s j s j 2 [H s X s +G s P s ]dY s : Since has nite moments for all orders (see (4.7)), the boundedness of X and P then renders the two stochastic integrals on the right hand side above both true martingales. Thus, taking expectations on both sides above, and applying Gronwall's inequality, we get E Q 0 L t j t j 2 C; 8t2 [0;T ]; (4.13) where C is a constant independent of . Applying Fatou's Lemma we then get that E Q 0 L t j t j 2 lim !0 E Q 0 [L t j t j 2 ]C: (4.14) Finally, noting (4.11) and applying It^ o's formula again we have dL t jP t j 2 = L t 2M t jP t j 2 + 2P t h t +S 2 t H 2 t dt + 2S t H t L t P t dY t (4.15) +L t jP t j 2 [H t X t +G t P t ]dY t + 2S t H t L t P t [H t X t +G t P t ]dt: 29 By similar arguments as before, and noting thatjX t jjX t j andjP jjP t j, one shows E Q 0 [L t jP t j 2 ] C n 1 +E Q 0 h Z t 0 L s [jP s j 2 +jX s j 2 ]ds io C n 1 +E Q 0 h Z t 0 L s [jP s j 2 +jX 1 s j 2 +j s j 2 ]ds io ; t2 [0;T ]: This, together with (4.9), implies that E Q 0 [L t jP t j 2 ]C n 1 +E Q 0 h Z t 0 L s [jP s j 2 +j s j 2 ]ds io ; t2 [0;T ]: Applying the Gronwall inequality and recalling (4.13) we then obtain E Q 0 [L t jP t j 2 ]C n 1 +E Q 0 h Z t 0 L s j s j 2 ds io C; t2 [0;T ]: (4.16) By Fatou's lemma one again shows that E Q 0 [L t jP t j 2 ] C, for all t2 [0:T ]. This, together with (4.14) and (4.10), leads to (4.6). The proof is now complete. We can now complete the proof of existence. Since L t is a (F;Q 0 ) martingale, we dene a probability measure P by dP dQ 0 F T = L T , and apply the Girsanov Theorem so that (B 1 ;B 2 ) is a P-Brownian motion on [0;T ]. Now, by looking at the rst two equations of (4.1), we see that ( ;F;P;X;Y;B 1 ;B 2 ) would be a weak solution to (3.1) if we can show that P t =E P [X t jF Y t ]; t2 [0;T ]; P-a.s. (4.17) 30 To prove (4.17) we proceed as follows. We consider the following linear ltering problem on the space ( ;F;P): 8 > < > : d t =f t t dt +dB 1 t ; 0 =x; d t =H t t dt +dB 2 t ; 0 = 0: (4.18) Denote b t = E P [ t jF t ], t 0. Then by linear ltering theory, we know that b satises the following SDE: db t =f t b t dt +S t H t fd t H t b t dtg; b 0 =x; (4.19) where S t satises (3:3) (or (4.1)). On the other hand, from (4.18) we see that is F B 1 -adapted, and from (4.1) we see that P is F Y -adapted, therefore we can apply Lemma 3.7 to conclude thatE P [ t jF Y t ] =E P [ t jF t ] =b t , t2 [0;T ]. Now let us dene e P t = E P [X t jF Y t ], t 0. Recall that X = X 1 +, where X 1 satises a randomized ODE (4.7) and is obviously F Y -adapted, we see that e P = X 1 +b , and it satises the SDE: d e P t = [f t e P t +g t P t +h t ]dt +S t H t fd t H t b t dtg; t 0; e P 0 =x: (4.20) Note that X =X 1 + and ~ P =X 1 +b we see that d t H(t)b t dt =H t ( t b t )dt +dB 2 t =dY t [H t e P t +G t P t ]dt: Then (4.20) implies that d e P t = [f t e P t +g t P t +h t ]dt +S t H t fdY t [H t e P t +G t P t ]dtg: (4.21) 31 Dene P t =P t e P t , then it follows from (4.1) and (4.21) that dP t = [f t S t H 2 t ]P t dt; P 0 = 0: Thus P t 0,P-a.s., for any t 0. That is, (4.17) holds, proving the existence. It is worth noting that the weak solution that we have constructed is actually a Q 0 -weak solution. 4.2 Uniqueness Again we need only consider the solutions with deterministic initial state. We rst note that if ( 0 ;F 0 ;P;F 0 ;X;Y;B 1 ;B 2 ) is a Q 0 -weak solution to (3.1), then we can assume without loss of generality that F 0 = F B 1 ;B 2 , hence Brownian. Next, we can dene e P t =E P [X t jF Y t ]. We are to show that e P t satises an SDE of the form as that in (4.1) underQ 0 , from which we shall derive theQ 0 -pathwise uniqueness. To this end, we recall that, as a Q 0 -weak solution, one has P Q 0 . Dene a P-martingale Z t 4 = E P dQ 0 dP F t , t 0. Since (B 1 ;Y ) is a P-semi-martingale with decomposition: B 1 t =B 1 t ; Y t = Z t 0 (H s X s +G s e P s )ds +B 2 t ; t 0: (4.22) By Girsanov-Meyer Theorem (see, e.g., [32, Theorem III-20]), it is aQ 0 -semi-martingale with the decomposition (B 1 t ;Y t ) = (N 1 t ;N 2 t ) + (A 1 t ;A 2 t ), whereN = (N 1 ;N 2 ) is aQ 0 - local martingale of the form N 1 t =B 1 t Z t 0 1 Z s d[Z;B 1 ] s ; N 2 t =B 2 t Z t 0 1 Z s d[Z;B 2 ] s ; t 0; 32 and A = (A 1 ;A 2 ) is a nite variation process. Since by assumption (B 1 ;Y ) is Q 0 - Brownian motion, we have A 0. In other words, it must hold that B 1 t =B 1 t Z t 0 1 Z s d[Z;B 1 ] s ; Y t =B 2 t Z t 0 1 Z s d[Z;B 2 ] s ; t 0: (4.23) Consider now a (F;P)-martingale dM t = Z 1 t dZ t . Since F is Brownian, applying Martingale Representation Theorem we see that there exists a process = ( 1 ; 2 )2 L 2 F ([0;T ]) such thatdM t = 1 t dB 1 t + 2 t dB 2 t ,t2 [0;T ]. Thus (4.23) amounts to saying that [M;B 1 ] t = Z t 0 1 s ds 0; Y t =B 2 t [M;B 2 ] t =B 2 t Z t 0 2 s ds; t 0: Comparing this to (4.22) we have 1 0 and 2 (HX +G e P ). That is, Z = L (HX+G e P ) , the solution to the SDE: dZ t =Z t dM t =Z t (H t X t +G t e P t )dB 2 t ; t2 [0;T ]; Z 0 = 1; and hence it can be written as the Dol eans-Dade stochastic exponential: Z t = exp n Z t 0 (H s X s +G s e P s )dB 2 s 1 2 Z t 0 jH s X s +G s e P s j 2 ds o : (4.24) Let us now consider again the following ltering problem on probability space ( ;F;P). 8 > < > : d t =f t t dt +dB 1 t ; 0 =x: d t = [H t t ]dt +dB 2 t ; 0 = 0: (4.25) 33 As before, we know that b t =E P [ t jF t ] satises the SDE: db t =f t b t dt +S t H t fd t H t b t dtg; b 0 =x; (4.26) whereS t satises (3.3). SinceL (HX+G e P ) =Z is aP-martingale, we can apply Lemma 3.7 again to conclude thatE P [ t jF Y t ] =E P [ t jF t ] =b t . Now let X t = t +X 1 t , and Y t = t +Y 1 t as before, where X 1 satises the ODE (4.7), and Y 1 satises the ODE dY 1 t = [H t X 1 t +G t e P t ]dt; Y 1 0 = 0: (4.27) Furthermore, since X 1 t is F Y -adapted, we have e P t = X 1 t +b t . Combining (4.7) and (4.26) we see that e P t satises the SDE: d e P t = [(f t +g t ) e P t +h t ]dt +S t H t fd t H t b t dtg; e P 0 =x: (4.28) Since =YY 1 , we derive from (4.27) that d t H t b t dt =dY t dY 1 t H t b t dt =dY t [H t e P t +G t e P t ]dt; and (8.17) becomes d e P t = [(f t +g t ) e P t +h t ]dt+S t H t fdY t [(H t +G t ) e P t ]dtg; t2 [0;T ]; e P 0 =x: (4.29) That is, e P t satises the same SDE asP t does in (4.1) on the reference space ( ;F;Q 0 ). To nish the argument, let ( ;F;P i ;F;X i ;Y i ;B 1;i ;B 2;i ), i = 1; 2 be any twoQ 0 - weak solutions, and dene e P i t 4 = E P i [X i t jF Y i t ], t 0, i = 1; 2. Then the arguments 34 above show that (X i ;B 2;i ; e P i ), i = 1; 2, are two solutions to the linear system of SDEs (4.1), underQ 0 . Thus if (B 1;1 ;Y 1 ) (B 1;2 ;Y 2 ) underQ 0 , then we must have (X 1 ;B 2;1 ; ~ P 1 ) (X 2 ;B 2;2 ; ~ P 2 ), underQ 0 , which in turn shows, in light of (4.24), that P 1 =P 2 . This proves theQ 0 -pathwise uniqueness of solutions to (3.1). 5 Necessary Conditions for Optimal Trading Strat- egy In this section we study the optimization problem (2.7). We still denote the price dynamics observable by the insider to beV =fV t :t 0g, and assume that it satises the SDE: 8 > < > : dV t = [f t V t +g t E P [V t jF Y t ] +h t ]dt + v t dB v t ; t2 [0;T ]; V 0 =vN(v 0 ;s 0 ); (5.1) and we assume that the demand dynamics observable by the market makers, denoted by Y =fY t :t 0g, satises the SDE dY t = [ t (V t E P [V t jF Y t ])]dt + z t dB z t ; t2 [0;T ]; Y 0 = 0: (5.2) We should note that in (5.1) and (5.2) the probabilityP should be understood as one dened on the canonical space ( 0 ;F 0 ;F 0 ), on which the solution to (5.1) and (5.2) isQ 0 -pathwisely unique. For notational simplicity, from now on we shall denote E =E P , when there is no danger of confusion. Moreover, note that E[P t (V t P t )] =E[E[(V t P t )P t jF Y t ]] = 0; 35 and that all the coecients are now assumed to be deterministic, we can apply Propo- sition 3.8 to write the problem (2.7) as J() = Z T 0 t E[(V t P t )P T ]dt = 2 (T; 0) Z T 0 t 1 (t; 0) n s 0 3 (T; 0) (5.3) + Z t 0 [(l 2 r + ( v r ) 2 ) 2 1 (0;r) 3 (T;r) 1 (0;r) 2 (0;r)l 2 r ]dr o dt; where i , i = 1; 2; 3, and l, k are dened by (3.20) and (3.16), respectively, with H =. Now by integration by parts we can easily check that Z t 0 [(l 2 r + ( v r ) 2 ) 2 1 (0;r) 3 (T;r)]dr = Z t 0 3 (T;r)d[S r 2 1 (0;r)] = 3 (T;t)S t 2 1 (0;t) 3 (T; 0)s 0 + Z t 0 1 (0;r) 2 (0;r)l 2 r dr; we thus have J() = 2 (T; 0) R T 0 t S t 1 (0;t) 3 (T;t)dt, and the original optimal con- trol problem (2.7) is equivalent to the following sup J() = sup Z T 0 t S t 1 (0;t) 3 (T;t)dt: (5.4) Before we proceed any further let us specify the \admissible strategy" and the standing assumptions on the coecients that will be used throughout this section. We note that the assumptions will be slightly stronger than Assumption 3.1. Assumption 5.1. (i) All coecients f, g, h, v , and z are deterministic, continu- ous functions on [0;T ], such that z t c, v t c for all t2 [0;T ] for some constant c> 0; 36 (ii) the trading intensity is continuous on [0;T ), t > 0 for all t2 [0;T ), and lim t!T t > 0 exists (it may be +1). Consequently, 4 = inf t2[0;T ] t > 0. Remark 5.2. (i) In practice it is not unusual to assume that lim t!T t =1, which amounts to saying that the insider is desperately trying to maximize the advantage of the asymmetric information (cf. e.g., [1]). We shall actually prove that this is the case for the optimal strategy, provided the Assumption 5.1-(ii) holds. In what follows we say that a trading intensity is admissible if it satises Assumption 5.1-(ii). By a slight abuse of notations we still denote all admissible trading intensities byU ad . (ii) For 2 U ad , the well-posedness of CMFSDEs (5.1) and (5.2) should be understood in the sense of Theorem 3.11, and we shall consider its (unique)Q 0 - solution. We note that the solution of the Riccati equation (3.17)S, as well as the functions 1 and 3 dened by (3.20), depends on the choice of trading intensity . We shall at times denote them byS , 1 , and 3 , respectively, to emphasize their dependence on . The following lemma is simple but useful for our analysis. Lemma 5.3. Let Assumption 5.1 be in force. Then for any 2U ad , (i) the Riccati equation (3.17) has a solution S =S dened on [0;T ), such that S t > 0, for all t2 [0;T ). Furthermore, there exists a constant C > 0, depending on the bounds of the coecients and in Assumption 5.1, such that S t s 0 +C t, t2 [0;T ]; (ii)kS k 1 e KT (s 0 +KT ), where K 4 =k v k 2 1 + 2kfk 1 ; (iii)S T 4 = lim t!T S t <1, that is, the solutionS can be extended continuously to [0;T ]; (iv) S T = 0 if and only if lim t!T 1 (t; 0) = 0. 37 Proof. Let 2U ad be given, and denote S = S and 1 = 1 , etc., throughout the proof for simplicitly. (i) First note that if S t = 0 for some t<T , we dene = infft2 [0;T );S t = 0g. Then, from (3.17) we see that at it holds that dSt dt j t= =j v j 2 > 0. But on the other hand by denition of we must have SS h h = S h h < 0 for h> 0 small enough, a contradiction. That is, S t > 0,8t2 [0;T ). Next, let us denote, for (t;s;)2 [0;T ] (0;1) [0;1), the right side of (3.17) by G(t;s;) 4 = ( v t ) 2 + 2f t s h s z t i 2 : Then for any 2U ad , it holds that G(t;s; t ) = 2 t j z t j 2 h s f t j z t j 2 2 t i 2 + f 2 t j z t j 2 2 t +j v t j 2 max t2[0;T ] n f 2 t j z t j 2 2 +j v t j 2 o 4 =C ; (5.5) thanks to Assumption 5.1. Thus S t s 0 +C t, for all t2 [0;T ], proving (i). (ii) To nd the bound that is independent of the choice of , we note that dS t dt =G(t;S t ; t ) ( v t ) 2 + 2jf t jS t K(1 +S t ); 8t2 [0;T ]; whereK 4 =k v k 2 1 +2kfk 1 . Thus the result then follows from the Gronwall's inequal- ity. (iii) Since G is quadratic in s, and lim s!1 G(t;s;) =1, it is easy to see from (5.5) that, for any given 2U ad , max t;s G + (t;s; t ) = max t;s G(t;s; t )C : (5.6) 38 On the other hand, we write S t s 0 = Z t 0 G(r;S r ; r )dr = Z t 0 G + (r;S r ; r )dr Z t 0 G (r;S r ; r )dr 4 =I + (t)I (t); whereI are dened in an obvious way. SinceI + () andI () are monotone increas- ing, both limitsI + (T ) andI (T ) exist, which may be +1. But (5.6) implies that I + (T )<1, and by (i), I (t) =I + (t)S t +s 0 <I + (t) +s 0 , for all t2 [0;T ), we conclude that I (T )<1 as well. That is, S T 0 exists. (iv) We rewrite the equation (3.17) as follows (recall the denition of 1 (3.20)), S t = exp(logS t ) =s 0 exp n Z t 0 dS t S t o =s 0 exp n Z t 0 (2f t + ( v t ) 2 S t 2 t S t ( z t ) 2 )dt o = s 0 1 (t; 0) exp n Z t 0 (f t + ( v t ) 2 S t )dt o : (5.7) Thus, the result follows easily from (iii). This completes the proof. In the rest of the section we shall try to solve the optimization problem (5.4). We rst note that by denition the quantity S and hence 1 and 3 all depend on the choice of trading intensity function . Therefore (5.4) is essentially a problem of calculus of variation. We shall proceed by rst looking for the rst order necessary conditions, and then nd the conditions that are sucient for us to determine the optimal strategy. To begin with, let us denote, for any dierentiable functional F : C([0;T ])7! C([0;T ]) and any;2C([0;T ]), the directional derivative ofF at in the direction by r F () t = d dy F ( t +y t )j y=0 : (5.8) 39 We rst give some useful directional derivatives that will be used frequently in the sequel. Recall the solutionS to the Riccati equation (3.17), and the functions 1 and 3 , dened by (3.20). Note that they are all functionals of the trading intensity 2C([0;T ]). Lemma 5.4. Let =f t g be an arbitrary continuous function on [0;T ]. Then the following identities hold, provided all the directional derivatives exist: (i)r t = t ; (ii)r S t = 2 1 (t; 0) Z t 0 r r r S r 2 1 (0;r)dr; where t = 2tSt ( z t ) 2 ; (iii)r 1 (t; 0) = 1 (t; 0) Z t 0 [r r r +r S r r ]dr; where t = 2 t ( z t ) 2 ; (iv)r 1 (0;t) = 1 (0;t) Z t 0 [r r r +r S r r ]dr; (v)r 3 (T;t) = Z T t fr 1 (r; 0) 2 (0;r)k r + 1 (r; 0) 2 (0;r)[r r r +r S r r ]gdr. Proof. (i) is obvious. (iii){(v) follows directly from chain rule. We only prove (ii). To see this, recall (3.17). We have S t ( +y) =s 0 + Z t 0 ( v r ) 2 dr + Z t 0 n 2f r S r ( +y) h S r ( +y)( r +y r ) z t i 2 o dr; and thus r S t = Z t 0 d dy n 2f r S r ( +y) h S r ( +y)( r +y r ) z r i 2 o y=0 dr = Z t 0 n 2f r r S r 2S r r z r h r r S r z r + S r r z r io dr: Denote S r 4 =r S, we see that it satises an ODE d dt S r t = 2 h f t 2 t S t ( z t ) 2 i S r t 2 t S 2 t t ( z t ) 2 = 2[f t k t ]S r t 2 t S 2 t t ( z t ) 2 ; S r 0 = 0: 40 Solving it and noting that =r and t = 2tSt ( z t ) 2 we obtain S r t = Z t 0 2 r S 2 r r ( z r ) 2 exp n Z t r 2(f u k u )du o dr = 2 1 (t; 0) Z t 0 r r r S r 2 1 (0;r)dr; proving (ii), whence the lemma. We are now ready to prove the following necessary conditions of optimal strategies for the original control problem (2.7). Theorem 5.5. Assume that Assumption 5.1 is in force. Suppose that 2U ad is an optimal strategy of the problem (2.7), then (1) it holds that 1 (t; 0)( z t ) 2 2 t S t 3 (T;t) + 1 S t t = Z T t [ r 1 (r; 0) 3 (T;r) + 2 r ( z t ) 2 r ]dr where t = 2 1 (t; 0) Z t 0 r s r 1 (0;r)dr Z T t g r 1 (r; 0) 2 (0;r)dr: (2) Furthermore, lim t!T t =1, and consequently, lim t!T S t = 0. In particu- lar, P T =E P [V T jF Y T ] =V T ; P-a.s. Proof. Suppose that 2U ad is an optimal strategy of the problem (2.7). Then it is also an optimal of the problem (5.4). Thus for any function 2 C([0;T ]), it holds that r J() = d J( +y) dy y=0 = 0; 41 or equivalently, 0 = Z T 0 [r t S t 1 (0;t) 3 (T;t) + t r S t 1 (0;t) 3 (T;t) + t S t r 1 (0;t) 3 (T;t) + t S t 1 (0;t)r 3 (T;t)]dt: (5.9) Then substitutingr ;r S, andr i , i = 1; 2; 3 in Lemma 5.4 into (5.9) and changing the order of integration if necessary we obtain that, formally, 1 (t; 0)( z t ) 2 2 t S t 3 (T;t) + 1 S t t = Z T t [ r 1 (r; 0) 3 (T;r) + 2 r ( z t ) 2 r ]dr (5.10) where t = 2 1 (t; 0)[ 1 (t; 0) 2 (0;t) 3 (T;t)] Z t 0 r S r 1 (0;r)dr: (5.11) To justify the identity (5.10) we now show that both sides of (5.10) are nite for any 2U ad (note that it is possible that t !1, ast!T ). To this end, we rst note that 1 (t; 0) and 2 (r;s) are bounded, and 3 (T;t) = 1 (t; 0) 2 (0;t) 1 (T; 0) 2 (0;T ) Z T t g r 1 (r; 0) 2 (0;r)dr; (5.12) thus it is also bounded, and clearly lim t!T 3 (T;t) = 0. Furthermore, we rewrite (4.16) as t = 2 1 (t; 0)G(t;T ) Z t 0 r S r 1 (0;r)dr; (5.13) 42 whereG(t;T ) 4 = 1 (t; 0) 2 (0;t) 3 (T;t) = 1 (T; 0) 2 (0;T )+ R T t g r 1 (r; 0) 2 (0;r)dr, thanks to (5.12). We claim that the following integral Z T 0 [ r 1 (r; 0) 3 (T;r) + 2 r ( z t ) 2 r ]dr 4 =I 1 +I 2 ; (5.14) where I 1 and I 2 are dened in an obvious way, is well-dened. Indeed, Assumption 5.1 and the boundedness of S imply that, modulo a universal constant, t 1 (t; 0) 3 (T;t) t exp n Z t 0 f u 2 u S u j z u j 2 du o t exp( R t 0 2 u du) : (5.15) On the other hand, by (5.13) it is easy to see that 2 t t 2 t 2 1 (t; 0) R t 0 r S r 1 (0;r)dr, and Z t 0 r S r 1 (0;r)dr = Z t 0 r S r exp n Z r 0 h f u + 2 u S u j z u j 2 i du o dr C Z t 0 r S r exp n Z r 0 2 u S u j z u j 2 du o dr (5.16) = C Z t 0 j z r j 2 r d h exp n Z r 0 2 u S u j z u j 2 du oi C exp n Z t 0 2 u S u j z u j 2 du o : Here in the above C > 0 is a generic constant, which depends only on the bounds of the coecients and in Assumption 5.1, and is allowed to vary from line to line. Thus, similar to (5.15), we derive from (5.13) and (5.16) that 2 t t C 2 t 1 (t; 0)G(t;T ) exp n Z t 0 2 u S u j z u j 2 du o 2 t 1 (t; 0) 2 t exp( R t 0 2 r dr) : (5.17) But notice that t 1 + 2 t , and that for any > 0, we have Z T 0 2 t expf R t 0 2 u dug dt 1e R T 0 2 u du 1; 43 we can easily derive from (5.15) and (5.17) that both I 1 and I 2 in (5.14) are nite, proving the claim. We can now use the identity (5.10) to prove both conclusions of the theorem. We begin by observing that lim t!T S t = 0 must hold. In fact, multiplying S t on both sides of (5.10) and then taking limits t!T , and noting that lim t!T 3 (T;t) = 0, we conclude that lim t!T t = 0. But then the equation (4.16), together with the fact that lim t!T 3 (T;t) = 0 but lim t!T 2 (0;t)6= 0, implies that lim t!T 1 (t; 0) = 0, and hence lim t!T S t = 0, thanks to Lemma 5.3. We now claim that lim t!T t =1. Indeed, suppose not, then we have lim t!T dS t dt = lim t!T G(t;S t ; t ) = lim t!T ( v t ) 2 c> 0; which is a contradiction, since S t > 0,8t2 [0;T ), and lim t!T S t = 0, proving the claim. Now note that S t is the variance of the process V t P t , t 0, the facts that lim t!T S t = 0 and that both processesV andP are continuous lead to thatV T =P T , P-a.s.. This completes the proof of part (2). It remains to prove part (1). But note that 1 (T; 0) = 0, we see from (5.13) that t = t , as G(t;T ) = R T t g r 1 (r; 0) 2 (0;r)dr. Thus (1) follows directly from (5.10). Remark 5.6. (i) Theorem 5.5 amounts to saying that, similar to the static infor- mation case (cf. e.g., [1, 22]), the optimal strategy for the optimization problem (2.7) should also maximize the advantage on the asymmetry of information near the terminal time T , i.e., lim t!T t =1. Also, in equilibrium, such asymmetry of the information will disappear at the terminal time since S T = lim t!T S t = 0, that is, 44 P T = E P [V T jF Y T ] = V T , P-a.s., despite the fact that the insider only observes V t at time t<T (see also, e.g., [3,14{16]). (ii) Although Theorem 5.5 gives only the necessary condition of the optimal strat- egy, the well-posedness of the optimal closed-loop system is guaranteed by Theorem 3.11. In other words, combining Theorem 5.5 and Theorem 3.11 we have proved the existence of the Kyle-Back equilibrium for the problem (2.3)|(2.7) under our assumptions. 6 Worked-out Cases and Examples In general, it is not easy to nd out an closed-form optimal strategy for the original problem (2.7), although we somehow predicted its behavior in Theorem 5.5. In this subsection we consider a special case for which the optimal strategy can be found explicitly. We show that it does possess the properties that we presented in the last sections, which in a sense justies our results. More precisely, we shall consider a case where the market price does not impact the underlying asset price, namely, g t 0 in equation (3.15). Recall from Lemma 5.4-(ii) that t = 2tSt ( z t ) 2 , t2 [0;T ]. Theorem 6.1. Assume that all the assumptions of Theorem 5.5 are in force and further that the coecient g t 0 in the dynamics (3.15). Dene 0 0 so that 2 0 4 4 = s 0 + R T 0 2 2 (0;r)( v r ) 2 dr R T 0 ( z t ) 2 dt : Then, the solution to the optimization problem (2.7) is given as follows: (i) the optimal strategy is given by t = 0 2 (t; 0)( z t ) 2 2S t ; 45 (ii) the error variance of price P t S t = 2 2 (t; 0) Z T t 2 0 4 ( z r ) 2 2 2 (0;r)( v r ) 2 dr; t2 [0;T ]; (6.1) (iii) the corresponding expected payo is given by J() = 0 2 (T; 0) 2 Z T 0 ( z t ) 2 dt; (iv) the market price is given by P t =E P [V t jF Y t ] = 2 (t; 0) h v 0 + Z t 0 2 (0;r)h r dr + 0 2 Y t i ; t2 [0;T ]; (v) nally, it holds that lim t!T t =1, lim t!T S t = 0, and in particular, P T =E P [V T jF Y T ] =V T ; P-a.s. (6.2) Proof. Let2U ad be an optimal strategy. Then by Theorem 5.5, we should have lim t!T S t = 0, and hence 1 (T; 0) = lim t!T 1 (t; 0) = 0, thanks to Lemma 5.3. Since g t 0, we have 3 (T;t) = 1 (t; 0) 2 (0;t) 1 (T; 0) 2 (0;T ) = 1 (t; 0) 2 (0;t); (6.3) and (5.10) now reads 2 1 (t; 0) 2 (0;t)( z t ) 2 2 t S t = Z T t r 2 1 (r; 0) 2 (0;r)dr: (6.4) 46 Now recall from Lemma 5.4-(ii) that t = 2tSt ( z t ) 2 , we can rewrite (6.4) as 12 (t) = t Z T t r 12 (r)dr; t2 [0;T ]; (6.5) where 12 (t) 4 = 2 1 (t; 0) 2 (0;t). Now dierentiating with respect to t on both sides of (6.5) we obtain that d 12 (t) dt = t t 12 (t) + d t dt Z T t r 12 (r)dr = h t t + 1 t d t dt i 12 (t): (6.6) Now since g 0, we see from (3.20) that 1 (t; 0) = exp( R t 0 (f r k r )dr), 2 (0;t) = exp( R t 0 f r dr), where, by (3.16), k t = 2 t St ( z t ) 2 . We can easily compute that d 12 (t) dt = 2[f t k t ] 12 (t)f t 12 (t) = [f t 2k t ] 12 (t): Plugging this into (6.6) and noting that, by denition, 2k t = t t , we obtain that d t dt =f t t ; 0 = 2 0 s 0 ( z 0 ) 2 : (6.7) Solving the above ODE we get 2 t S t ( z t ) 2 = t = 0 exp n Z t 0 f r dr o = 0 2 (t; 0): (6.8) Consequently we obtain that t = 0 2 (t; 0)( z t ) 2 2S t , proving (i). 47 (ii) Note that S satises the ODE (3.17) and 2 (0;t) = 1 2 (t; 0) = 0 = t , where satises (6.7), it is easy to check that d dt ( 2 2 (0;t)S t ) = 2 2 (0;t) h ( v t ) 2 + 2f t S t t S t z t 2 i +S t 2 0 d dt h 1 2 t i = 2 2 (0;t) h ( v t ) 2 + 2f t S t t S t z t 2 i 2f t S t 2 2 (0;t) = 2 2 (0;t)( v t ) 2 2 0 ( z t ) 2 4 : Integrating both sides from t to T , and noting that S T = 0 and 2 (0;t) = 1 2 (t; 0), we derive (6.1). Furthermore, setting t = 0 in (6.1), one can then solve 2 0 4 4 = S 0 + R T 0 2 (0;r)( v r ) 2 dr R T 0 ( z t ) 2 dt ; proving (ii). (iii) Combining the expression of , S, as well as equations (5.3) and (6.3), the corresponding expected payo is J() = 2 (T; 0) J() = 2 (T; 0) Z T 0 t S t 1 (0;t) 3 (T;t)dt = 2 (T; 0) Z T 0 t S t 1 (0;t) 1 (t; 0) 2 (0;t) dt = 2 (T; 0) Z T 0 0 2 (t; 0)( z t ) 2 2S t S t 2 (0;t)dt = 0 2 (T; 0) 2 Z T 0 ( z t ) 2 dt: (iv) It follows from (3.23) and (6.13) that the market price follows the dynamics P t =E P [V t jF Y t ] = 2 (t; 0) h v 0 + Z t 0 2 (0;r)h r dr + 0 2 Y t i ; t2 [0;T ]: 48 (v) Finally, again by Lemma 5.3, one has lim t!T t =1 and lim t!T S t = 0. In particular, P T =E[V t jF Y T ] =V T ,P-a.s. We note that Theorem 6.1 contains several previously known results as special cases. We list them as follows. 1. Static case. In this case V t v, where vN(v 0 ;S 0 ). Setting f 0;g 0;h 0; v 0 in Theorem 6.1 we have 2 (t; 0) 1; and 0 = 4S 0 R T 0 ( z t ) 2 dt 1=2 = 2; where 4 = p S 0 f R T 0 ( z t ) 2 dtg 1=2 is the so-called price sensitivity or Kyle's (cf. [1]). The optimal strategy is given by t = 0 ( z t ) 2 2S t = R T 0 ( z t ) 2 dt 1=2 ( z t ) 2 S 1=2 0 R T t ( z r ) 2 dr ; and the corresponding expected payo is given by J() =S 1=2 0 Z T 0 ( z t ) 2 dt 1=2 : Furthermore, one can easily check that, for t2 [0;T ], the corresponding market price is given by P t =E P [V t jF Y t ] =v 0 +Y t ; and the corresponding mean square error is S t = S 0 R T t ( z r ) 2 dr R T 0 ( z t ) 2 dt : 49 In particular, S T = 0, which implies thatV 0 =P T ,P-a.s. These results coincide with those of [1]. If we further assume that T = 1 and z t , where > 0 is a constant, then the optimal trading intensity becomes t = p S 0 (1t) ; 0t< 1; the corresponding expected payo is J() = p S 0 , and the corresponding market price is P t =v 0 + p S 0 Y t , 0t<T . We recover the results of Back [2]. 2. Long-lived information case. We now compare our results with that of Back-Pedersen [3] (see also Danilova [22]), in which the insider continuously observes the dynamics of V t that is assumed to be a martingale. By setting T = 1, f 0, g 0, h 0 and z 1, Theorem 6.1 implies 2 0 4 = 1, assuming S 0 4 = 1 R 1 0 ( v s ) 2 ds, and the optimal trading intensity is t = 1 1t R 1 t ( v s ) 2 ds ; 0t< 1: The corresponding expected payo is J() = 1, and the corresponding market price is P t =v 0 +Y t , 0t<T . 50 7 A Linear Quadratic Optimization Problem 7.1 Description of the model In this section, we study a linear quadratic optimization problem based on the lin- ear CMFSDEs. Consider the following linear model on a given probability space ( ;F;F;P) with standard Brownian Motion (B 1 ;B 2 ). 8 > < > : dX t = [F (t)X t +g(t) b X t +c(t)v t +f(t)]dt +dB 1 t ; X 0 =x 0 ; dY t = [H(t)X t +h(t) b X t ]dt +dB 2 t ; Y 0 = 0; (7.1) where b X t =E P [X t jF Y t ]; fv t g is a process in the admissible control set, which is dened as all square- integrableF Y -adapted processes; F;g;c;f;H;h are deterministic and continuous functions on [0;T ]. From the linear conditional mean-eld theory developed in previous sections, we know that (7.1) has an uniqueQ 0 solution, and the process b X t has the following dynamics: d b X t = [F (t) b X t +f(t) +g(t) b X t +c(t)v t ]dt +P t H(t)fdY t [H(t) b X t +h(t) b X t ]dtg; (7.2) where P t satises the following Riccatti equation. dP t = [2F (t)P t H(t) 2 P 2 t + 1]dt; P 0 = 0: (7.3) 51 Consider the following general quadratic payo: J(v()) = E n Z T 0 [M(t)jX t j 2 +N(t)jv t j 2 + 2m(t)X t + 2n(t)v t ]dt +M(T )jX T j 2 + 2m(T )X T o (7.4) Then we can state the optimization problem: minimize the payo (7.4) among the set of admissible controls. 7.2 Optimality We shall now proceed to solve the linear quadratic optimal stochastic control problems with linear CMFSDEs. Dene " t =X t b X t , then we have d" t =F (t)" t +dB 1 t P t H(t)fdY t [H(t) +h(t)] b X t dtg (7.5) Noting dY t = [H(t)X t +h(t) b X t ]dt +dB 2 t , we have the SDE for " t . d" t = [F (t)P t H(t) 2 ]" t dt +dB 1 t P t H(t)dB 2 t ;" 0 = 0 One can easily check that " t is a gaussian process with mean 0 and covariance P t , under ProbabilityP. NotingE[ b X t " t ] = 0 andE[" 2 t ] =P t , we can transform the payo function into the following. J(v(:)) = Ef Z T 0 [M(t)( b X t +" t ) 2 +N(t)jv t j 2 + 2m(t)( b X t +" t ) + 2n(t)v t ]dt +M(T )( b X T +" T ) 2 + 2m(T )X T g = Ef Z T 0 [M(t)j b X t j 2 +N(t)jv t j 2 + 2m(t) b X t + 2n(t)v t ]dt (7.6) +M(T )j b X T j 2 + 2m(T ) b X T g + Z T 0 [M(t)P t ]dt +M(T )P T : 52 We now dene an auxiliary cost functional that contains only the control v t , which will facilitate our future discussion. J 1 (v()) = Ef Z T 0 [M(t)j b X t j 2 +N(t)jv t j 2 + 2m(t) b X t + 2n(t)v t ]dt (7.7) +M(T )j b X T j 2 + 2m(T ) b X T g: Next, let us consider the innovation process I v (t) =Y t Z t 0 [H(s) +h(s)] b X s ds = t Z t 0 H(s)b s ds; t 0: It is well-known (cf. e.g., [4]) that, under probability P, I v (t) is anfF t g-Wiener process. One can also check that it is anfF Y t g-Wiener process. So, we have d b X t = [F (t) b X t +f(t) +g(t) b X t +c(t)v t ]dt +P t H(t)dI v (t): (7.8) Consequently, the original optimal control problem is equivalent to the following: 8 > > > > < > > > > : d b X t = [F (t) b X t +f(t) +g(t) b X t +c(t)v t ]dt +P t H(t)dI v (t); b X 0 =x 0 J 1 (v()) = Ef R T 0 [M(t)j b X t j 2 +N(t)jv t j 2 + 2m(t) b X t + 2n(t)v t ]dt +M(T )j b X T j 2 + 2m(T ) b X T g: Letv t =N(t) 1 [n(t)+c(t)((t) b X t +r(t))]+w t ,t 0,w t is a new control and (t), r(t) are functions of t. Let t be the solution of the following SDE. d t = f[F (t) +g(t)N 1 (t)c(t) 2 (t)] t +f(t)c(t)N 1 (t)[n(t) +c(t)r(t)]dtg +P t H(t)dI v ; 0 = x 0 : 53 Let us dene e X t = b X t t . Then, e X satisfying the following SDE. d e X t =f[F (t) +g(t)N 1 (t)c(t) 2 (t)] e X t +c(t)w t gdt Dene t =N 1 (t)[n(t) +c(t)((t) t +r(t))] Noting the dierence between t and v t , we can write v t in the following way. v t = t N 1 (t)c(t)(t) e X t +w t : (7.9) Hence payo function J 1 () can be written in the following. J 1 (v()) = Ef Z T 0 [M(t)j e X t + t j 2 +N(t)j t N 1 (t)c(t)(t) e X t +w t j 2 +2m(t)( e X t + t ) + 2n(t)( t N 1 (t)c(t)(t) e X t +w t )]dt +M(T )j e X T + T j 2 + 2m(T )( e X T + T )g = Ef Z T 0 [M(t)j t j 2 +N(t)j t j 2 + 2m(t) t + 2n(t) t ]dt +M(T )j T j 2 + 2m(T ) T g +Ef Z T 0 [M(t)j e X t j 2 +N(t)[N 1 (t)c(t)(t) e X t +w t ] 2 ]dt +M(T )j e X T j 2 g +2I 0 ; where I 0 = E n Z T 0 h M(t) e X t t +N(t) t [N 1 (t)c(t)(t) e X t +w t ] +m(t) e X t +n(t)[N 1 (t)c(t)(t) e X t +w t ] i dt +M T e X T T +m(T ) e X T o : (7.10) 54 We want to makeI 0 = 0, by choosing proper (t) andr(t). IntroduceS t ,k(t) andq t s.t. S t =k(t) t +q(t) and k(T ) =M(T ) , q(T ) =m(T ). Then one can write I 0 in the following way. I 0 = E n Z T 0 [M(t) e X t t +N(t) t [N 1 (t)c(t)(t) e X t +w t ] +m(t) e X t +n(t)[N 1 (t)c(t)(t) e X t +w t ]]dt +S T e X T o Let us replace S T e X T by its integral form. Applying Ito's formula, we have d e X t S t = n e X t [F (t) +g(t)N 1 (t)c(t) 2 (t)]S t +c(t)w t S t + e X t [ dk(t) dt t + dq(t) dt ] + e X t k(t)[F (t) +g(t)N 1 (t)c(t) 2 (t)] t +f(t)c(t)N 1 (t)[n(t) +c(t)r(t)] o dt +k(t)P t H(t)dI v (t) Plugging (7.11) into I 0 , then I 0 will read as follows: I 0 = E n Z T 0 h M(t) e X t t +N(t) t [N 1 (t)c(t)(t) e X t +w t ] +m(t) e X t +n(t)[N 1 (t)c(t)(t) e X t +w t ] + e X t [F (t) +g(t)N 1 (t)c(t) 2 (t)]S t +c(t)w t S t + e X t [ dk(t) dt t + dq(t) dt ] + e X t k(t)[(F (t) +g(t) N 1 (t)c(t) 2 (t)) t +f(t)c(t)N 1 (t)(n(t) +c(t)r(t))] i dt o 55 By setting the coecient of e X t and w t equal 0, we have 0 = M(t) t c(t) t t +m(t)N 1 (t)n(t)c(t)(t) +[F (t) +g(t)N 1 (t)c(t) 2 (t)]S t + [ dk(t) dt t + dq t dt ] +k(t)[F (t) +g(t)N 1 (t)c(t) 2 (t)] t +k(t)f(t) (7.11) k(t)c(t)N 1 (t)[n(t) +c(t)r(t)] and 0 =N(t) t +n(t) +c(t)S t : (7.12) Noting that t =N 1 (t)(c t +cr +n) and S t = k t +q, and by setting the coecient of t and constant term equal 0, we obtain the following identities: 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : M(t) +N 1 (t)c(t) 2 (t) 2 + 2[F (t) +g(t)N 1 (t)c(t) 2 (t)]k(t) + dk(t) dt = 0; N 1 (t)c(t) 2 (t)r(t) +m(t) + [F (t) +g(t)N 1 (t)c(t) 2 (t)]q(t) + dq(t) dt +k(t)[f(t)c(t)N 1 (t)(n(t) +c(t)r(t))] = 0; c(t)(t) +c(t)k(t) = 0; c(t)r(t) +c(t)q(t) = 0: Clearly, above equations indicate that k and qr, and the following ODEs for and r. 8 > > > > > > < > > > > > > : d(t) dt +M(t) + 2[F (t) +g(t)](t)N 1 (t)c(t) 2 (t) 2 = 0; (T ) =M(T ); dr(t) dt +m(t) + [F (t) +g(t)]r(t) + (t)[f(t)c(t)N 1 (t)n(t)c(t) 2 N 1 (t)r(t)] = 0; r(T ) =m(T ): 56 Therefore, by choosing and r as the solutions of above ODEs, we can make sure that I 0 = 0. Hence, one can deduce that J 1 (v()) E n Z T 0 h M(t)j t j 2 +N(t)j t j 2 + 2m(t) t + 2n(t) t i dt +M(T )j T j 2 + 2m(T ) T o : (7.13) Applying Ito's formula, one can see that d[(t) 2 t + 2r(t) t ] = n [M(t) 2 t + 2m(t) t +N(t) 2 t + 2n(t) t ] +2f(t)r(t)N 1 (t)[n(t) +c(t)r(t)] 2 + (t)P 2 t H(t) 2 o dt +2[(t) t +r(t)]P t H(t) dI v (t): Then we can rewrite the right hand side of inequality (7.13), and we obtain J 1 (v()) Z T 0 f2f(t)r(t)N 1 (t)[n(t) +c(t)r(t)] 2 + (t)P 2 t H(t) 2 gdt: (7.14) Note that the RHS of above inequality is deterministic. More precisely, it does not depend on control v or probability P. Introduce a process b Z t , which is the solution to the following SDE. d b Z t = n [F (t) +g(t)P t H(t)(H(t) +h(t))] b Z t +f(t) c(t)N 1 (t)[n(t) +c(t)[(t) b Z t +r(t)]] o dt +P t H(t) dY t ; b Z 0 =x 0 : 57 Dene a control u by u t =N 1 (t)fn(t) +c(t)[(t) b Z t +r(t)]g: (7.15) Note that b Z t is adapted toF Y t , so u t is an admissible control. Introduce process Z t , which has the following dynamic. dZ t = [F (t)Z t +g(t) b Z t +c(t)u t +f(t)]dt +dB 1 t ; Z 0 =x 0 : Dene t by d t = t [H(t)Z t +h(t) b Z t ]dY t ; 0 = 1 One can check easily that t is a Q 0 martingale. So, we can dene a probability P u by the following. dP u dQ = T (7.16) By Girsanov theory, one can check that b B 2 t is a Brownian Motion under P u . d b B 2 t =dY t [H(t)Z t +h(t) b Z t ]dt; b B 2 0 = 0: We also have the innovation process I u (t) =Y t Z t 0 [H(s) +h(s)] b Z s ds: (7.17) Plug I u (t) into the dynamic of b Z t , then we can rewrite the the SDE of b Z t in the following way. d b Z t = [F (t) b Z t +f(t) +g(t) b Z t +c(t)u t ]dt +P t H(t)dI u (t); b Z 0 =x 0 : 58 Plugging b Z t into the payo function J 1 (), we can write J 1 () in the following way. J 1 (u(:)) = E P u n Z T 0 [M(t)j b Z t j 2 +N(t)u(t) 2 + 2m(t) b Z t + 2n(t)u(t)] dt +M(T ) b Z 2 T + 2m(T ) b Z T o = Z T 0 f2f(t)r(t)N 1 (t)[n(t) +c(t)r(t)] 2 + (t)P 2 t H(t) 2 gdt Therefore, u 0 t =N 1 (t)fn(t) +c(t)[(t) b X t +r(t)]g is the optimal control. 8 A Simple Non-linear Case of CMFSDEs In this section, let us consider a simple non-linear case of CMFSDE, where the dynamic is still linear to X t but non-linear to E P [X t jF Y t ] and Y t . More precisely, the model is described in the following. 8 > < > : dX t = [F (t)X t +g(E P [X t jF Y t ];Y t ;t)]dt +dB 1 t ; X 0 = dY t = [H(t)X t +h(E P [X t jF Y t ];Y t ;t)]dt +dB 2 t ; Y 0 = 0: (8.1) Assumption 8.1. The functions (u;y;t)7!g(u;y;t) andh(u;y;t) are bounded and continuous, for (u;y;t)2RR [0;T ]; g(u;y;t) and h(u;y;t) are Lipschitz continuous in u and y. F (t);H(t) are both continuous on [0;T ]. Note, the proof of well posedness is similar to the proof of linear case in Section 4. That's because that the SDEs in (8.1) are linear to X t , so linear ltering theory can be applied. If the linear condition doesn't hold any more, the well posedness problem will become complicated. Non-linear ltering theory may be helpful, but it has not been fully explored yet. 59 8.1 Existence To prove the existence of solution to (8.1), let us consider the following SDE system. 8 > > > > > > > > > > < > > > > > > > > > > : dX t = [F (t)X t +g(U t ;Y t ;t)]dt +dB 1 t ; X 0 =; dY t = [H(t)X t +h(U t ;Y t ;t)]dt +dB 2 t ; Y 0 = 0; dU t = [F (t)U t +g(U t ;Y t ;t)]dt +P t H(t)fdY t [H(t)U t +h(U t ;Y t ;t)]dtg; U 0 =E; dP t = [2F (t)P t H(t) 2 P 2 t + 1]dt; P 0 = 0: (8.2) Thanks to assumption 8.1, one can easily show that the above SDEs system has unique solution (X t ;Y t ;U t ). If we can show that U t is actually equal to E[X t jF Y t ], then the existence of solution to (8.1) is obtained. Lemma 8.2. U t =E P [X t jF Y t ]; Pa:s: (8.3) Proof of Lemma 8.2. Split X t into 2 parts, X t = t +X 1 t , where d t = F (t) t dt +dB 1 t ; 0 =; (8.4) dX 1 t = [F (t)X 1 t +g(U t ;Y t ;t)]dt; X 1 0 = 0: (8.5) Let us also split Y t , Y t = t +Y 1 t , where d = [H(t) t ]dt +dB 2 t ; 0 = 0; (8.6) dY 1 t = [H(t)X 1 t +h(U t ;Y t ;t)]dt; Y 1 0 = 0: (8.7) 60 Denoteb t =E P [ t jF t ]. Let us consider the following ltering problem. 8 > < > : d t =F (t) t dt +dB 1 t ; 0 =: d t = [H(t) t ]dt +dB 2 t ; 0 = 0: By Kalman lter theory, we have the following SDE about b t . db t =F (t)b t dt +P t H(t)fd t H(t)b t dtg; b 0 =E[] where P t satises (3:3). By Lemma 3.7, we have E P [ t jF Y t ] = b t . Denote e U t = E P [X t jF Y t ]. Noting that X 1 t is adapted toF Y t , we have e U t =X 1 t +b t (8.8) Hence, e U t satises the following SDE. d e U t = [F (t) e U t +g(U t ;Y t ;t)]dt +P t H(t)fd t H(t)b t dtg; and e U 0 =E[]. Since d t H(t)b t dt = dY t dY 1 t H(t)b t dt = dY t [H(t) e U t +h(U t ;Y t ;t)]dt; then we can write the dynamic of e U t in the following way. d e U t = [F (t) e U t +g(U t ;Y t ;t)]dt +P t H(t)fdY t [H(t) e U t +h(U t ;Y t ;t)]dtg: (8.9) 61 Set U t =U t e U t , then U 0 = 0 and dU t = [F (t)P t H(t) 2 ]U t dt (8.10) Clearly, ODE (8.10) indicates that U t 0, which means that U t = e U t a:s:, for any t. 8.2 Uniqueness Suppose that (X t ;Y t ) is a solution of SDE system (8.1). ThenE P [X t jF Y t ] is uniquely dened, denoted by e U t . Let us try to nd the SDE of e U t . Split X t = t +X 1 t , where d t = F (t) t dt +dB 1 t ; 0 =; (8.11) dX 1 t = [F (t)X 1 t +g( e U t ;Y t ;t)]dt; X 1 0 = 0: (8.12) Split Y t = t +Y 1 t , where d = [H(t) t ]dt +dB 2 t ; 0 = 0; (8.13) dY 1 t = [H(t)X 1 t +h( e U t ;Y t ;t)]dt; Y 1 0 = 0: (8.14) Denoteb t =E P [ t jF t ]. Consider the following ltering problem. 8 > < > : d t =F (t) t dt +dB 1 t ; 0 =: d t = [H(t) t ]dt +dB 2 t ; 0 = 0: By Kalman lter theory, we have the following SDE about b t . db t =F (t)b t dt +P t H(t)fd t H(t)b t dtg; b 0 =E[] (8.15) 62 where P t satises (3:3). By Lemma 3.7, we haveE P [ t jF Y t ] =b t . Noting that X 1 t is adapted toF Y t , we have e U t =X 1 t +b t (8.16) Hence, e U t satises the following SDE. d e U t = [F (t) e U t +g( e U t ;Y t ;t)]dt +P t H(t)fd t H(t)b t dtg; and e U 0 =E[]. Since d t H(t)b t dt = dY t dY 1 t H(t)b t dt = dY t [H(t) e U t +h( e U t ;Y t ;t)]dt; then d e U t = [F (t) e U t +g( e U t ;Y t ;t)]dt +P t H(t)fdY t [H(t) e U t +h( e U t ;Y t ;t)]dtg; (8.17) Note e U t has the same SDE withU t in (8.2). So, a solution (X t ;Y t ) of SDE system (8.1) must also be a solution of SDE system (8.2), which only has one solution. Therefore, the SDE system (8.1) only has one solution. 63 Chapter 2 Non-linear Conditional Mean-eld SDEs 1 Introduction In this section, we focus on the following mean-eld-type stochastic control problem, on a given ltered probability space ( ;F;P;F =fF t g t0 ): 8 > < > : dX t =Efb(t;' ^t ;E[X t jG t ];u)gj'=X; u=u t dt +Ef(t;' ^t ;E[X t jG t ];u)gj'=X; u=u t dB t ; X 0 =x; (1.1) where B is an F-Brownian motion, b and are measurable functions satisfying rea- sonable conditions,' ^t andX ^t denote the continuous function and process, respec- tively, \stopped" at t; G 4 =fG t g t0 is a given ltration that could involve the infor- mation of X itself, and u =fu t : t 0g is the \control process", assumed to be adapted to a ltration H =fH t g t0 , whereH t F X t _G t , t 0. We note that ifG t = f; g, for all t 0 (i.e., the conditional expectation in (1.1) becomes expectation),H t =F X t , and coecients are \Markovian" (i.e., ' ^t = ' t ), then the problem becomes a stochastic control problem with McKean-Vlasov dynamics and/or a Mean-eld game (see, for example, [19{21] in its \forward" form, and [8{10] in its 64 \backward" form). On the other hand, when G is a given ltration, this is the so- called conditional mean-eld SDE (CMFSDE for short) studied in [12]. We note that in that case the conditioning is essentially \open-looped". The problem that this paper is particularly focusing on is whenG t =F Y t , t 0, where Y is an \observation process" of the dynamics of X, i.e., the case when the pair (X;Y ) forms a \close-looped" or \coupled" CMFSDE. More precisely, we shall consider the following partially observed controlled dynamics (assuming b = 0 for notational simplicity): 8 > > > > > > < > > > > > > : dX t =Ef(t;' ^t ;E[X t jF Y t ];u)gj'=X; u=u t dB 1 t +Efb (t;' ^t ;E[X t jF Y t ];u)gj'=X; u=u t dB 2 t ; X 0 =x; dY t =Efh(t;' t ;E[X t jF Y t ])gj '=X dt + ~ dB 2 t ; Y 0 = 0: (1.2) Here X is the \signal" process that can only be observed through Y , (B 1 ;B 2 ) is a standard Brownian motion, and ~ is a constant. We should note that in SDEs (1.2) the conditioning ltration F Y now depends on X itself, therefore it is much more convoluted than the CMFSDE we have seen in the literature. Furthermore, the path- dependent nature of the coecients makes the SDE essentially non-Markovian. Such form of CMFSDEs, to the best of our knowledge, has not been explored fully in the literature. Our study of the CMFSDE (1.2) is strongly motivated by the following variation of the mean eld game in a nance context, which would result in a McKean-Vlasov stochastic control problem with partial observation as we are proposing (see, e.g., [20] or [26] for a more detailed background). Consider a rm whose fundamental value, 65 under the risk neutral measure P 0 with zero interest, evolves as the following SDE with \stochastic volatility" =(t;!), (t;!)2 [0;1) : X t =x + Z t 0 (s;)dB 1 s ; t 0; (1.3) whereB 1 is the intrinsic noise from inside the rm. We assume that such fundamental value process cannot be observed directly, but can be observed through a stochastic dynamics (e.g., its stock value) via an SDE: Y t = Z t 0 h(s;)ds +B 2 t ; t 0; (1.4) whereB 2 is the noise from the market, which we assume is independent ofB 1 (this is by no means necessary, we can certainly consider the ltering problem with correlated noises). Now let us assume that the volatility in (1.3) is aected by the actions of a large number of investors, and all can only make decisions based on the information from the process Y . Therefore, following the argument of [20] (or [26]) we begin by considering N individual investors, and that i-th investor's private state dynamics is of the form: dU i t = i (t;U i ^t ; N t ; i t )dB 1;i t ; t 0; 1iN; (1.5) whereB 1;i 's are independent Brownian motions, and N t denotes the empirical condi- tional distribution of U = (U 1 ; ;U N ), given the (common) observation Y =fY t : 66 t 0g, that is, N t 4 = 1 N P N j=1 E[U j t jF Y t ] , where x denotes the Dirac measure at x. More precisely, the notation in (1.5) means (see, e.g., [20]), i (t;U i ^t ; N t ; i t ) 4 = Z R ~ i (t;U i ^t ;y; i t ) N t (dy) = 1 N N X j=1 Z R ~ i (t;U i ^t ;y; i t ) E[U j t jF Y t ] (dy) (1.6) = 1 N N X j=1 ~ i (t;U i ^t ;E[U j t jF Y t ]; i t ): Here, ~ i 's are the functions dened on appropriate (Euclidean) spaces. We now assume that each investor chooses an individual strategy to minimize the cost; the cost functional of the i-th agent is of the form: J i ( i ) 4 =E n i (U i T ) + Z T 0 L i (t;U i ^t ; N t ; i t )dt o ; 1iN; (1.7) Following the argument of Lasry and Lions [30] (see also [20,21,26]), if we assume that the game is symmetric, i.e., ~ i = ~ ; L i and i = are independent ofi, and that the number of investorsN converges to +1, then under suitable technical conditions, one can nd (approximate) Nash equilibriums through a limiting dynamics, and assign a representative investor the unied strategy , determined by a McKean-Vlasov type dynamics dX t =(t;X ^t ; t ; t )dB 1 t ; t 0; (1.8) where is the conditional distribution of X t givenF Y t , and (t;X ^t ; t ;u t ) 4 = Z (t;X ^t ;y;u t ) t (dy) =Ef(t;' ^t ;E[X t jF Y t ];u)gj'=X u=u t : 67 Furthermore, the value function becomes, with similar notations, V (x) = inf J() 4 =E n (X T ) + Z T 0 L(t;X ^t ; t ; t )dt o : (1.9) We note that (1.8) and (1.9), together with (1.4), form a stochastic control problem with McKean-Vlasov dynamics and partial observations, as we are proposing. The main objective of this paper is two-fold: We shall rst study the exact mean- ing as well as the well-posedness of the dynamics, and then investigate the Stochastic Maximum Principle for the corresponding stochastic control problem. For the well- posedness of (1.2) we shall use a scheme that combines the idea of [19] and the tech- niques of nonlinear ltering, and prove the existence and uniqueness of the solution to SDE (1.8) via Schauder's xed point theorem onP 2 ( ), the space of probability mea- sures with nite second moment, endowed with the 2-Wasserstein metric. We note that the important elements in this argument include the so-called reference probabil- ity space that is often seen in the nonlinear ltering theory and the Kallianpur-Striebel formula (cf. e.g., [4,33]), which enable us to dene the solution mapping. Our next task is to prove Pontryagin's Maximum Principle for our stochastic control problem. The main idea is similar to earlier works of the rst two authors ( [10,31]), with some signicant modications. In particular, since in the present case the control problem can only be carried out in a weak form, due to the lack of strong solution of CMFSDE, the existence of the common reference probability space is essen- tial. Consequently, extra eorts are needed to overcome the complexity caused by the change of probability measures, which, together with the path-dependent nature of the underlying dynamic system, makes even the rst order adjoint equation more complicated than the traditional ones. To the best of our knowledge, the resulting mean-eld backward SDE is new. 68 2 Existence We will discuss the existence of a solution of the system (2.1). 8 > > > > < > > > > : dX t =b(t;X :^t ; XjY t ;u t (!))dt +(t;X :^t ; XjY t ;u t (!))dB 1 t +b (t;X :^t ; XjY t ;u t (!))dB 2 t ; X 0 =x: dY t =h(t;X :^t ; XjY t )dt +dB 2 t ; Y 0 = 0: (2.1) where XjY t is the law ofE[X t jF Y t ] under probability measure P . b;;b ;h are functions satisfying the following. (t;X :^t ; XjY t ;u t ) = Z (t;X :^t ;y;u t ) XjY t (dy) (2.2) h(t;X :^t ; XjY t ) = Z h (t;X :^t ;y) XjY t (dy) for =b; orb . Denition 2.1. (Weak Solution). An eight-tuple ( ;F;P;F;X;Y;B 1 ;B 2 ) is called a solution to the ltering equation (2.1) if 1. ( ;F;P;F) is a ltered probability space; 2. (B 1 ;B 2 ) is a 2-dimensional Brownian motion underP; 3. (X;Y ) is anF-adapted continuous process such that (2.1) holds for allt2 [0;T ], P-almost surely. First, x (P u ;u). DenoteP 0 =P u . Sinceu is xed, we can rewriteb =b(t;!;X :^t ; t ), =(t;!;X :^t ; t ) andb =b (t;!;X :^t ; t ). 69 Assumption 2.2. The coecients b, ,b : C T P(C T )!R and h enjoys the following properties: (i) for xed (;)2 C T P(C T ), the mapping (t;!)! (b;)(t;!;';) is F- pro- gressively measurable process; (ii) j(t;!;' 1 :^t ; 1 )(t;!;' 2 :^t ; 2 )jKj sup t2[0;T ] j' 1 t ' 2 t j +W 2 ( 1 ; 2 )j (2.3) for =b;,b respectively. Assumption 2.3. (i) The mappings (t;';x;y;z)7! (t;' ^t ;y;z), b (t;' ^t ;y;z), h(t;x), f(t;' ^t ;y;z), and (x;y) are bounded and continuous, for (t;';x;y;z)2 [0;T ]C T RRU; (ii) The partial derivatives @ y , @ z , @ y b , @ z b , @ y f, @ z f, @ x h, @ x , @ y are bounded and continuous, for (';x;y;z)2C T RRU, uniformly in t2 [0;T ]; (iii) The mappings '7! (t;' ^t ;y;z);b (t;' ^t ;y;z);f(t;' ^t ;y;z), as function- als from C T to R, are Frech et dierentiable. Furthermore, there exists a family of measuresf`(t;)gj t2[0;T ] , satisfying 0 R T 0 `(t;ds) C, for all t2 [0;T ], such that both derivatives, denoted by D ' = D ' (t;' ^t ;y;z) and D ' f = D ' f(t;' ^t ;y;z), respectively, satisfy jD ' (t;' ^t ;y;z)( )j +jD ' f(t;' ^t ;y;z)( )j Z T 0 j (s)j`(t;ds); 2C T ; (2.4) uniformly in (t;';y;z); (iv) The mappingsy7!y@ y (t;' ^t ;y;z);y@ y b (t;' ^t ;y;z) are uniformly bounded, uniformly in (t;';z); (v) The mapping x7!x@ x h(t;x) is bounded, uniformly in (t;x)2 [0;T ]R; 70 (vi) The mappings x7! xh(t;x);x 2 @ x h(t;x) are bounded, uniformly in (t;x)2 [0;T ]R. We note that some of the assumptions above are merely technical and can be improved, but we prefer not to dwell on such technicalities and focus on the main ideas instead. Lemma 1. If X has continuous paths, P 0 -a.s., then so is U XjY . Proof refer to [7]. To prove the existence, we shall use Schauder's xed point theorem. Consider a subset ofP(C T ): E = n 2P(C T )j sup t2[0;T ] Z R d jyj 4 t (dy)<1 o (2.5) Clearly,E is a convex subset ofP(C T ). Note (2.1) constructs a mappingT :E! P(C T ). Denote T () = =P [U ] 1 2P(C T ): (2.6) Theorem 2.4. The solution mappingT :E!P(C T ) enjoys the following proper- ties: 1. T (E )E . 2. T (E ) is compact under Wassertein-2 metric. 3. T :E!P(C T ) is continuous under the Wasserstein-2 metric. Proof of Theorem 2.4. (1) We need to show sup t2[0;T ] Z R d jyj 4 t (dy)<1 (2.7) 71 To prove this, we note by Jensen's inequality, Z R d jyj 4 t (dy) = Z R d jyj 4 P [U ] 1 =E P [jE P [X t jF Y t ]j 4 ]E P [jX t j 4 ]: UnderQ 0 , B 1 is a Brownian motion, so it's standard to argue that sup 0tT E Q 0 [jX t j 2n ]<C(1 +jxj 2n ); 8n2N: (2.8) It's worth to note that the constant C above does not depend on . Furthermore, noting the L is an L 2 -martingale underQ 0 , we have sup t2[0;T ] Z R d jyj 4 t (dy) sup t2[0;T ] E P [jX t j 4 ] = sup 0tT E Q 0 [L T jX t j 4 ] E Q 0 [jL T j 2 ] 1=2 sup t2[0;T ] E Q 0 [jX t j 8 ] 1=2 <1 In other words, =T ()2E , proving (1). (2) We shall prove that any sequencef n gE has a subsequence, may denoted byf n g itself, such that lim n!1 T ( n ) =2T (E ) in Wasserstein-2 metric. First, let us argue that the familyfT ( n )g n1 is tight. To this end, recall that U n t =E P n [X n t jF Y t ] = S n t S n;0 t ; (2.9) where S n t =E Q 0 [L n t X n t jF Y t ], S n;0 t =E Q 0 [L n t jF Y t ], t 0. Then by FKK equation with correlated noises, one has dU n t =E P n t [b t ]dt + n E P n t [X t h t ]U n t E P n t [h t ] +E P n t [b t ] o n dY t E P n t [h t ]dt o ; (2.10) 72 whereE P n t [ : ] =E P n [ :jF Y t ]. Note that h is bounded, so isE P n [hjF Y t ]. We thus have the following estimate: E P n [jU n t U n s j 4 ] CE P n h Z t s E P n [jX n u j 2 jF Y u ]du 2 i (2.11) CE P n h Z t s jE P n [jX n u j 2 jF Y u ]j 2 du i jtsj C sup 0tT E P n [jX n t j 4 ]jtsj 2 Cjtsj 2 Thus, the sequencefU n g, as continuous processes, is relatively compact. Therefore their lawsfT ( n ) =P n [U n ] 1 ;n 1gP(C T ) is tight. Consequently, we can nd a subsequence, my assume itself, converges weakly to a limit 2 P(C(T )). Furthermore, we have the following estimate: Z Ct k'k 2 n (d') = E P n [kU n k 2 C T ] =E P n [ sup 0tT jE P n [X n t jF n t ]j 2 ] (2.12) E P n h sup 0tT E P n [ sup 0rT jX n r jjF n t ] 2 i 3 h E P n [ sup 0rT jX n r j 3 ] i 2=3 = 3 h E Q 0 [L n T sup 0rT jX n r j 3 ] i 2=3 3 h E Q 0 [(L n T ) 4 ] i 1=6 h E Q 0 [ sup 0rT jX n r j 4 ] i 1=2 < +1 Noting that h is bounded and (2.8), one can deduce that sup n1 Z Ct k'k 2 n (d')< +1 (2.13) This, together with the fact that n =T ( n ) w !, implies that W 2 ( n ;)! 0, and 2E , as n!1, where W 2 (:;:) is the Wasserstein metric onP(C T ). This proves (2). 73 (3) We now check that the mappingT :E !E is continuous. To this end, for each 2E wo consider the following SDE on the probability space ( ;F;Q 0 ): 8 > > > > < > > > > : dX t = (b t h t b t )dt + t dB 1 t +b t dY t ; X 0 =x; dB 2 t =dY t h(t;X :^t ; t )dt; B 2 0 = 0; dL t =h(t;X :^t ; t )L t dY t ; L 0 = 1: (2.14) Now let n E be any sequence such that n ! , as n!1, in Wasserstein- 2 metric, and denote (X n ;B n;2 ;L n ) be the corresponding solutions to (3.1). By assumption (2.2), we have is a functional Lipschitz deterministic function, with Lipschitz constant independent of n . Also, noting h is bounded, by standard SDE argument, we have the following convergence, as n!1: E Q 0 n sup 0tT jX n t X t j p + sup 0tT jL n t L t j p o ! 0;8p 1: (2.15) We deduce that U n t = S n t =S n;0 t converges in probability under Q 0 to E Q 0 [LtXtjF Y t ] E Q 0 [LtjF Y t ] = E P [X t jF Y t ], where dP =L T dQ 0 . Now for any '2C b (R), sending n!1, we have h';T ( n ) t i = E P n ['(E P n [X n t jF Y t ])] =E Q 0 [L n T '(E P n [X n t jF Y t ])] (2.16) ! E Q 0 [L T '(E P [X t jF Y t ])] =E P ['(E P [X t jF Y t ])] = h';P [E P [X t jF Y t ]] 1 i This implies that t = P [E P [X t jF Y t ]] 1 = T () t , for all t 2 [0;T ]. Further- more, we can prove that T ( n ) w ! . This, together with (2.13), shows that W 2 (T ( n );T ())! 0, as n! 0; proving the continuity ofT . The proof is com- plete. 74 Corollary 2.5. SDE (2.1) has at least one solution. 3 Uniqueness In this section we will investigate the uniqueness of the solution (2.1). LetQ 0 be the reference probability measure. Then, under Q 0 , (B 1 ;Y ) is a Brownian motion. We shall argue that the solution to the SDE (2.1) is pathwisely unique. 8 > > > > < > > > > : dX t = (b t h t b t )dt + t dB 1 t +b t dY t ; X 0 =x; dB 2 t =dY t h(t;X :^t ; t )dt; B 2 0 = 0; dL t =h(t;X :^t ; t )L t dY t ; L 0 = 1: (3.1) Proposition 3.1. Assume proper assumption. Let u;v2U ad be given. Then for any p> 1, there exists a constant C p > 0, such that the following estimates hold: E 0 h sup 0sT (jX u s X v s j 2 +jL u s L v s j 2 +jX u s L u s X v s L v s j 2 ) i C p kuvk 2 p=2;2;Q 0 (3.2) E 0 h sup 0sT (jX u s X v s j p i C p kuvk 2 p=2;2;Q 0 (3.3) Proof of Proposition 3.1. Split the proof into 3 step. Step 1 (Estimate for X). u (t;X u :^t ; u t ) v (t;X v :^t ; v t ) (3.4) = j Z (t;X u :^t ;y;u t ) u t (dy) Z (t;X v :^t ;y;v t ) v t (dy)j = Cfju t v t j + sup 0st jX u s X v s j + Z (t;X v :^t ;y;v t )[ u t (dy) v t (dy)]g 75 By the fact that dP u =L u T dQ 0 , we see that j Z R (t;;X v :^t ;y;v t )[ u t (dy) v t (dy)]j (3.5) = E u [(t;';E u [X u t jF Y t ];u)]E v [(t;';E u [X v t jF Y t ];u)] '=X v ;u=vt = E 0 n L u t (t;'; E 0 [L u t X u t jF Y t ] E 0 [L u t jF Y t ] ;u)L v t (t;'; E 0 [L v t X u t jF Y t ] E 0 [L v t jF Y t ] ;u) o '=X v t ;u=vt I 1 +I 2 Denote S u t =E 0 [L u t X u t jF Y t ]; S u;0 t =E 0 [L u t jF Y t ] (3.6) Then we can rewrite I 1 and I 2 in the following way. I 1 = E 0 n L u t (t;'; S u t S u;0 t ;u)L v t (t;'; S u t S v;0 t ;u) o '=X v t ;u=vt (3.7) = E 0 n S u;0 t (t;'; S u t S u;0 t ;u)S v;0 t (t;'; S u t S v;0 t ;u) o '=X v t ;u=vt and I 2 = E 0 n L v t h (t;'; S u t S v;0 t ;u)(t;'; S v t S v;0 t ;u) io '=X v t ;u=vt (3.8) = E 0 n S v;0 t h (t;'; S u t S v;0 t ;u)(t;'; S v t S v;0 t ;u) io '=X v t ;u=vt : Clearly, we have I 2 CE 0 n S v;0 t jS u t S v t j S v;0 t o CE 0 [jL u t X u t L v t X v t j] (3.9) 76 To estimate I 1 , we writee (t;';y;z) = y(t;'; S u t (!) y ;z). Assume @ y e () is uniformly bounded. Then, I 1 Cjj@ y e jj 1 E 0 jS t u; 0S v;0 t jCE 0 jL u t L v t j (3.10) Therefore, j(u)(v)jC n ju t v t j +jX u t X v t j +E 0 jL u t L v t j +E 0 [jL u t X u t L v t X v t j] o (3.11) Consider hb in the drift term. Note h; andb are uniformly bounded. h(t;;X u :^t ; u t )b (t;;X u :^t ; u t ;u t )h(t;;X v :^t ; v t )b (t;;X v :^t ; v t ;v t ) jh(u)jjb (u)b (v)j +jb (v)jjh(u)h(v)j (3.12) C n ju t v t j + sup 0st jX u s X v s j +E 0 jL u t L v t j +E 0 [jL u t X u t L v t X v t j] o Therefore, applying Gronwall's Inequality, we obtain E 0 h sup t jX u t X v t j p i CE 0 n Z t 0 ju t v t j 2 +E 0 jL u t L v t j 2 +E 0 [jL u t X u t L v t X v t j 2 ]dt o p=2 Step 2 (Estimate for L). jL u t h(t;;X u :^t ; u )L v t h(t;;X v :^t ; v )j (3.13) Z n L u t h(t;X u :^t ;y)L v t h(t;X v :^t ;y) o u t (dy) + L v t Z h(t;X v :^t ;y)[ u t (dy) v t (dy)] A 1 +A 2 : 77 A 1 = Z n L u t h(t;X u t ;y)L v t h(t;X v t ;y) o u t (dy) (3.14) Z n L u t h(t; L u t ;X u :^t L u t ;y)L v t h(t; L v t ;X v :^t L v t ;y) o u t (dy) Let ^ h(t;!;x;y) =xh(t; L v t X v t x ;y). Assume @ x ^ h is uniformly bounded. L u t h(t; L u t X u t L u t ;y)L v t h(t; L v t X v t L v t ;y) (3.15) L u t h(t; L u t X u t L u t ;y)L u t h(t; L v t X v t L u t ;y) + L u t h(t; L v t X v t L u t ;y)L v t h(t; L v t X v t L v t ;y) CjL u t X u t L v t X v t j +jj@ x ^ hjj 1 jL u t L v t j So, A 1 C[jL u t X u t L v t X v t j +jL u t L v t j] (3.16) By similar proof in Step 1, one can show A 2 CL v t [E 0 jL u t L v t j +E 0 [jL u t X u t L v t X v t j] (3.17) So, E 0 [ sup 0st jL u s L v s j 2 ]CE 0 [ Z t 0 jL u s X u s L v s X v s j 2 ds] (3.18) Step 3 (Estimate for L t X t ). dL t X t = L t dX t +X t dL t +L t h t b t dt (3.19) = L t b t dt +L t t dB 1 t +L t (b t +X t h t )dY t 78 For =b;; orb , one can show jL u t (t;X u t ; u t ;u t )L v t (t;X v t ; v t ;v t )j (3.20) CL v t (E 0 [jL u t L v t j] +E 0 [jL u t X u t L v t X v t j]) +C(jL u t L v t j +jL u t X u t L v t X v t j) +CL v t ju t v t j Now, dene ~ h(t;x;) =xh(t;x;). Assume ~ h is smooth like h. Then, similar to the proof of inequality (3.13), we have jL u t X u t h u t L v t X v t h v t j =jL u t ~ h u t L v t ~ h v t j (3.21) C n jL u t X u t L v t X v t j +jL u t L v t j +L v t E 0 jL u t L v t j +E 0 [jL u t X u t L v t X v t j] o Therefore, E 0 h sup 0st jL u s X u s L v s X v s j 2 i Cjjuvjj 2 p;2;Q +CE 0 Z t 0 jL u s L v s j 2 ds (3.22) Together with the inequalities in Step 1 and 2, applying Gronwall's inequality, the desired results are obtained. 79 4 A Stochastic Control Problem We are now ready to study the stochastic control problem with partial observation. We consider only theQ 0 -dynamics (X u ;Y;L u ), which satises the following SDE: 8 > > > > > < > > > > > : dX u t = (b t h t b t )dt + u (t;X u ^t ; u t )dB 1 t +b t dY t ; t 0; X u 0 =x; dB 2;u t =dY t h(t;X u t ; u t )dt; t 0; B 2;u 0 = 0; dL u t =h(t;X u t ; u t )L u t dY t ; t 0; L u 0 = 1; (4.1) where (B 1 ;Y ) is aQ 0 -Brownian motion,dP u =L u T dQ 0 , and X u jY t =P u [E P u [X t jF Y t ]] 1 . For simplicity, we denoteE u [] 4 =E P u [] andE 0 [] 4 =E Q 0 []. We recall that foru2U ad and 2P 2 (C T ), the coecient u ;b and h in (4.1) are dened by (2.2). Thus we can write the cost functional as J(u) 4 =E 0 n (X u T ; u T ) + Z T 0 f u (s;X u s ; u s )ds o : (4.2) An admissible control u 2U ad is said to be optimal if J(u ) = inf u2U ad J(u): (4.3) To this end, we letu 2U ad be an optimal control, and consider the convex variations of u : u ;v t :=u t +(v t u t ); t2 [0;T ]; 0<< 1; v2U ad : (4.4) Here, we assume that u ;v2 L 1 F Y (Q 0 ; [0;T ]). Since U is convex, u ;v t 2 U, for all t2 [0;T ], v2U ad , and 2 (0; 1). We denote (X ;v ;Y;L ;v ) to be the corresponding 80 Q 0 -dynamics that satises (4.1), with control u ;v . Applying Proposition 3.1 and noting that Y is a Brownian motion underQ 0 , we get, for p> 2, lim !0 E 0 h sup 0tT jX ;v t X u t j 2 i C p lim !0 ku ;v u k 2 p;2;Q 0 = 0; (4.5) lim !0 E 0 h sup 0tT jL ;v t L u t j 2 i = 0: (4.6) In the rest of the section we shall derive, heuristically, the \variational equations" which play a fundamental role in the study of Maximum Principle. The complete proof will be given in the next section. For notational simplicity we shall denote u = u , the optimal control, from now on, bearing in mind that all discussions will be carried out for theQ 0 -dynamics, therefore on the same probability space. Now foru 1 ;u 2 2U ad , let (X 1 ;L 1 ) and (X 2 ;L 2 ) denote the corresponding solutions of (4.1). We dene X =X 1;2 =X u 1 ;u 2 4 =X u 1 X u 2 and L =L 1;2 =L u 1 ;u 2 4 = L u 1 L u 2 , and will often drop the superscript \ 1;2 " if the context is clear.Denote b b t =b t h t b t . Then X and L satisfy the equations: 8 > > > > > > > > < > > > > > > > > : X t = Z t 0 [ b b 1 t b b 2 t ]dt + Z t 0 [ u 1 (s;X 1 ^s ; 1 s ) u 2 (s;X 2 ^s ; 2 s )]dB 1 s + Z t 0 [b 1 t b 2 t ]dY t ; L t = Z t 0 [L 1 s h(s;X 1 s ; 1 t )L 2 s h(s;X 2 s ; 2 t )]dY s : (4.7) 81 W.O.L.G, we assume that b t = b t = 0. As before, let U i t 4 = E u i [X i t jF Y t ] and i t = P u i [U i t ] 1 , t 0, i = 1; 2. We can easily check that u 1 (t;X 1 ^t ; 1 t ) u 2 (t;X 2 ^t ; 2 t ) = E 0 n L 1 t (t;' 1 ^t ;U 1 t ;z 1 )L 2 t (t;' 2 ^t ;U 2 t ;z 2 ) o ' 1 =X 1 ;' 2 =X 2 ;z 1 =u 1 t ;z 2 =u 2 t = E 0 n L 1;2 t (t;' 1 ^t ;U 1 t ;z 1 ) (4.8) +L 2 t h Z 1 0 D ' (t;' 2 ^t +(' 1 ^t ' 2 ^t );U 1 t ;z 1 )(' 1 ^t ' 2 ^t )d + Z 1 0 @ y (t;' 2 ^t ;U 2 t +(U 1 t U 2 t );z 1 )d (U 1 t U 2 t ) + Z 1 0 @ z (t;' 2 ^t ;U 2 t ;z 2 +(z 1 z 2 ))d (z 1 z 2 ) io ' 1 =X 1 ;' 2 =X 2 ;z 1 =u 1 t ;z 2 =u 2 t : Now let u 1 =u ;v and u 2 =u =u, and denote X 4 = X u;v = X ;v X u ; L 4 = L u;v = L ;v L u ; U 4 = U u;v = U ;v U u : Combining (4.7) and (4.8) we have X t = Z t 0 n E 0 f L s (s;' 1 ^s ;U ;v s ;z 1 )g ' 1 =X ;v ;z 1 =u ;v s + [D] ;u;v s ( X ^s ) +E 0 fB ;u;v (s;' 2 ^s ;z 1 ) U s g ' 2 =X u ; z 1 =u ;v s +C ;u;v (s)(v s u s ) o dB 1 s ; (4.9) where [D] ;u;v t ( ) =E 0 n L u t Z 1 0 D ' (t;' 2 ^t +(' 1 ^t ' 2 ^t );U ;v t ;z 1 )( )d o ' 1 =X ;v ;' 2 =X u ; z 1 =u ;v t ; B ;u;v (t;' 2 ^t ;z 1 ) =L u t Z 1 0 @ y (t;' 2 ^t ;U u t +(U ;v t U u t );z 1 )d; (4.10) C ;u;v (t) =E 0 n L u t Z 1 0 @ z (t;' 2 ^t ;U u t ;z 2 +(z 1 z 2 ))d o ' 2 =X u ;z 1 =u ;v t ;z 2 =ut : 82 Here the integral involving the Frech et derivative D ' is in the sense of Bochner. Noting that U ;v t = E 0 [L ;v t X ;v t jF Y t ] E 0 [L ;v t jF Y t ] and U u t = E 0 [L u t X u t jF Y t ] E 0 [L u t jF Y t ] , we can easily check that U t = E 0 [L u t jF Y t ]E 0 [L ;v t X ;v t jF Y t ]E 0 [L ;v t jF Y t ]E 0 [L u t X u t jF Y t ] E 0 [L ;v t jF Y t ]E 0 [L u t jF Y t ] (4.11) = E 0 [L u t jF Y t ]E 0 [ L t X ;v t +L u t X t jF Y t ]E 0 [ L t jF Y t ]E 0 [L u t X u t jF Y t ] E 0 [L ;v t jF Y t ]E 0 [L u t jF Y t ] = E 0 [ L t X ;v t +L u t X t jF Y t ] E 0 [L ;v t jF Y t ] E 0 [ L t jF Y t ] E 0 [L ;v t jF Y t ] U u t : Now, sending ! 0, and assuming that K t =K u;v t 4 = lim !0 X u;v t ; R t =R u;v t 4 = lim !0 L u;v t (4.12) both exist in L 2 (Q 0 ), then it follows from (4.7)-(4.11) we have, at least formally, K t = Z t 0 n E 0 [R s (s;' ^s ;U u s ;z)] '=X u ;z=us + [D] u;v s (K ^s ) (4.13) +E 0 h B u;v (s;' ^s ;z) E 0 [R s X u s +L u s K s jF Y s ] E 0 [L u s jF Y s ] E 0 [R s jF Y s ] E 0 [L u s jF Y s ] U u s i '=X u ; z=us +C u;v (s)(v s u s ) o dB 1 s ; where [D] u;v t ( ) 4 = E 0 fL u t D ' (t;' ^t ;U u t ;z)( )g '=X u ; z=u t ; B u;v (t;' ^t ;z) 4 = L u t @ y (t;' ^t ;U u t ;z); (4.14) C u;v (t) 4 = E 0 n L u t @ z (t;' ^t ;U u t ;z) o '=X u ; z=u t : 83 Observing also that U u t isF Y t -measurable, we have E 0 h B u;v (s;' ^s ;z) E 0 [R s X u s +L u s K s jF Y s ] E 0 [L u s jF Y s ] E 0 [R s jF Y s ] E 0 [L u s jF Y s ] U u s i '=X u ; z=us = E u h @ y (s;' ^s ;U u s ;z)E u f(L u s ) 1 R s [X u s U u s ] +K s jF Y s g i '=X u ; z=us (4.15) = E u h (L u s ) 1 @ y (s;' ^s ;U u s ;z)fR s [X u s U u s ] +L u s K s g i '=X u ; z=us = E 0 h @ y (s;' ^s ;U u s ;z)(R s X u s +L u s K s )U u s @ y (s;' ^s ;U u s ;z)R s i '=X u ; z=us : Consequently, if we dene (t;' ^t ;x;y;z) 4 =(t;' ^t ;y;z) +@ y (t;' ^t ;y;z)(xy); (4.16) then we can rewrite (4.18) as K t = Z t 0 n E 0 h (s;' ^s ;X u s ;U u s ;z)R s +@ y (s;' ^s ;U u s ;z)L u s K s i '=X u ; z=us (4.17) +[D] u;v s (K ^s ) +C u;v (s)(v s u s ) o dB 1 s : Similarly, we can formally write down the SDE for R: R t = Z t 0 ( R s h(s;X u s ; u s ) +L u s n [Dh] u;v s (K s ) (4.18) +E 0 h e (s;' ^s ;X u s ;U u s )R s +@ y h(s;' s ;U u s )L u s K s i '=X u o ) dY s ; 84 where [Dh] u;v t ( ) 4 = E 0 fL u t D ' h(t;' t ;U u t )( )g '=X u B u;v (t;' ^t ;z) 4 = L u t @ y (t;' ^t ;U u t ;z); (4.19) e (t;' ^t ;x;y) 4 = h(t;' ^t ;y) +@ y h(t;' ^t ;y)(xy): The following theorem is regarding the well-posedness of the SDEs (4.17) and (4.18). Theorem 4.1. Assume that Assumption 2.2 is in force, and letu;v2L 1 F Y (Q 0 ; [0;T ]) be given. Then, there is a unique solution (K;R)2L 1 F (Q 0 ;C 2 T ) to SDEs (4.17) and (4.18). Proof. Let u;v2 L 1 F Y (Q 0 ; [0;T ]) be given. We dene F 1 t (K;R) and F 2 t (K;R), t2 [0;T ], to be the right hand side of (4.17) and (5.5), respectively. We rst observe that F 1 t (0; 0) = R t 0 C u;v (s)(v s u s )dB 1 s , and F 2 t (0; 0) 0, t2 [0;T ]. Then, for any p> 2, it holds that E u h sup 0st jF 1 s (0; 0)j p i C p E u h Z t 0 jv s u s j 2 ds p=2 i ; t2 [0;T ]: (4.20) Now let (K i ;R i ) 2 L 1 F (Q 0 ;C T ), i = 1; 2. We dene e K i 4 = F 1 (K i ;R i ), e R i 4 = F 1 (K i ;R i ),i = 1; 2, and K 4 =K 1 K 2 , R 4 =R 1 R 2 , ^ K 4 = e K 1 e K 2 , and ^ R 4 = e R 1 e R 2 . Then, noting that ,@ y ,y@ y , and@ z are all bounded, thanks to Assumption 2.3, we see that j (t;' ^t ;x;y;z)jC(1 +jxj); (t;x;y;z)2 [0;T ]R 3 ; '2C T ; 85 where, and in what follows, C > 0 is some generic constant which is allowed to vary from line to line. It then follows that E 0 [ (t;' ^t ;X u t ;U u t ;z) R s +@ y (t;' ^t ;U u t ;z)L u t K t ] CE 0 [(1 +jX u t j)j R t j +jL u t K t j]C h E 0 [j K t j 2 +j R t j 2 ] i 1=2 : (4.21) Furthermore, since D ' is also bounded, we havej[D] u;v t ( )j C sup 0st j (s)j, for 2 C T . Then from the denition of ^ K and (4.21) we have, for any p 2 and t2 [0;T ], E 0 h sup 0st j ^ K s j 2p i C p Z t 0 E 0 [j R s j 2 +j K s j 2 ] p ds +C p Z t 0 E 0 h sup 0rs j K r j 2p i ds: (4.22) On the other hand, the boundedness ofh and@ x h implies that, recalling the denition of ^ R, for p 2 and t2 [0;T ], E 0 h sup st j ^ R s j p 2 C p Z t 0 E 0 [j R s j p ] 2 ds +C p Z t 0 E 0 [jL u s K s j p ] 2 ds (4.23) C p Z t 0 (E 0 [j R s j p ]) 2 ds +C p Z t 0 E 0 [j K s j 2p ]ds: Combining (4.22) and (4.23) we have, for t2 [0;T ], E 0 h sup 0st j ^ K s j 2p i + E 0 h sup 0st j ^ R s j p ] 2 C p Z t 0 E 0 h sup 0rs j K r j 2p i + E 0 h sup 0rs j R r j p i 2 ds: This, together with (4.20), enables us to apply standard SDE arguments to deduce that there is a unique solution (K;R)2L 1 F (P;C T ) of (4.17) and (5.5), such that for all p 2, it holds that E 0 kKk 2p C T +E 0 kRk 2p C T C p kv s u s k 2 p;2;Q 0: (4.24) 86 We leave it to the interested reader, and this completes the proof. 5 Variational Equations In this section we validate the heuristic arguments in the previous section and derive the variational equation of the optimal trajectory rigorously. Recall the processes X = X u;v , L = L u;v , and (K;R) dened in the previous section. Denote t 4 = X t K t ; ~ t 4 = L t R t ; t2 [0;T ]: (5.1) Our main purpose of this section is to prove the following result. Proposition 5.1. Let (P u ;u) = (P u ;u )2U ad be an optimal control, (X u ;L u ) be the corresponding solution of (4.1), and letU u t =E u [X u t jF Y t ],t 0. For anyv2U ad , let (K;R) = (K u;v ;R u;v ) be the solution of the linear equations (4.17) and (5.5). Then, for all p> 1, it holds that lim !0 E 0 [k k p C T ] = lim !0 E 0 h sup s2[0;T ] X ;v s X u s K s p i = 0; (5.2) lim !0 E 0 [k~ k p C T ] = lim !0 E 0 h sup s2[0;T ] L ;v s L u s R s p i = 0: (5.3) The proof of Proposition 5.1 is quite lengthy, we shall split it into two parts. [Proof of (5.3)]. Recall, L t = Z t 0 n (e s +R s )h(x;X ;v s ; ;v s ) +L u s h E 0 f(~ s +R s )h(s;' s ;U ;v s )g '=X ;v + [Dh] ;u;v s ( s +K s ) +E 0 fB ;u;v (s;' s ;z) U s g '=X u io dY s : (5.4) 87 where [Dh] ;u;v t ( ) = E 0 n L u t Z 1 0 D ' h(t;' 2 ^t +(' 1 ^t ' 2 ^t );U ;v t )( )d o ' 1 =X ;v ' 2 =X u : ; B ;u;v h (t;' 2 ^t ) = L u t Z 1 0 @ y h(t;' 2 ^t ;U u t +(U ;v t U u t ))d: (5.5) Furthermore, in light of (4.11), we can also write: U t = E 0 [(~ t +R t )X ;v t +L u t ( t +K t )jF Y t ] E 0 [L ;v t jF Y t ] E 0 [(~ t +R t )jF Y t ] E 0 [L ;v t jF Y t ] U u t : Then ~ t = Z t 0 n e s h(x;X ;v s ; ;v s ) +L u s h E 0 f~ s h(s;' s ;U ;v s )g '=X ;v + [Dh] ;u;v s s (5.6) +E 0 fB ;u;v (s;' s ) h E 0 [~ s X ;v s +L u s s jF Y s ] E 0 [L u s jF Y s ] E 0 [~ s jF Y s ] E 0 [L u s jF Y s ] U u s i g '=X u io dY s : +I 1; t +I 2; t +I 3; t +I 4; t +I ;5 t where I ;1 t = Z t 0 R r (h(r;X ;v r ; ;v r )h(r;X u r ; u r ))dY r ; I ;2 t = Z t 0 L u r E 0 [R r (h(r;' 1 ;U ;v r )h(r;' 2 ;U u r ))] ' 1 =X ;v r ; ' 2 =X u r dY r ; I ;3 t = Z t 0 L u r K r f[Dh] ;u;v r [Dh] u;v r gdY r ; I ;4 t = Z t 0 L u r ( E 0 n B ;u;v (s;' s ) E 0 [R s X ;v s +L u s K s jF Y s ] E 0 [L ;v s jF Y s ] E 0 [R s jF Y t ] E 0 [L ;v s jF Y s ] U u s o '=X u E 0 n B u;v (s;' s ) E 0 [R s X u s +L u s K s jF Y s ] E 0 [L u s jF Y s ] E 0 [R s jF Y s ] E 0 [L u s jF Y s ] U u s o '=X u ) dY r ; 88 I ;5 t = Z t 0 L u r ( E 0 n B ;u;v (s;' s ) E 0 [~ s X ;v s +L u s s jF Y s ] E 0 [L ;v s jF Y s ] E 0 [~ s jF Y s ] E 0 [L ;v s jF Y s ] U u s o '=X u E 0 n B ;u;v (s;' s ) E 0 [~ s X ;v s +L u s s jF Y s ] E 0 [L u s jF Y s ] E 0 [~ s jF Y s ] E 0 [L u s jF Y s ] U u s o '=X u ) dY r : Lemma 5.2. Suppose that Assumption 2.3 holds. Then, for all p> 1, lim !0 E 0 n sup 0tT jI ;i t j p o = 0; i = 1; ; 5: (5.7) Proof. We rst recall that U ;v s 4 =E ;v [X ;v s jF Y s ] and U u s 4 =E u [X u s jF Y s ]. Using the Kallianpur-Strieble formula we have E 0 Z T 0 jU ;v s U u s j p ds C p n E 0 Z T 0 E 0 [L ;v s X ;v s jF Y s ] E 0 [L ;v s jF Y s ] E 0 [L u s X u s jF Y s ] E 0 [L ;v s jF Y s ] p ds +E 0 Z T 0 E 0 [L u s X u s jF Y s ] E 0 [L ;v s jF Y s ] E 0 [L u s X u s jF Y s ] E 0 [L u s jF Y s ] p ds o (5.8) 4 = C p fJ 1 +J 2 g: We now estimate J 1 and J 2 respectively. First note that, for any p> 1, we can nd a constant C p > 0 such that for any 2 (0; 1) and u2U ad , E 0 [(L ;v s ) p ] +E 0 [(L ;v s ) p ] +E 0 [(L u s ) p ]C p : 89 Thus, applying the H older and Jensen inequalities as well as Proposition 3.1, we have, for any p> 1, and 2 (0; 1), E 0 Z T 0 E 0 [L ;v s X ;v s jF Y s ]E 0 [L u s X u s jF Y s ] E 0 [L ;v s jF Y s ] p ds Z T 0 E 0 n jL ;v s X ;v s L u s X u s j p E 0 [L ;v s jF Y s ] p o ds Z T 0 n fE 0 jL ;v s X ;v s L u s X u s j 2 g 1=2 n E 0 h jL ;v s X ;v s L u s X u s j 2p2 E 0 [L ;v s jF Y s ] 2p io 1=2 o ds (5.9) Z t 0 fE 0 jL ;v s X ;v s L u s X u s j 2 g 1=2 n E 0 [jL ;v s X ;v s L u s X u s j 2p2 ]E 0 [L ;v s ] 2p jF Y s o 1=2 ds C p kuvk 2;2;Q 0: Similarly, one can also argue that, for any p> 1, the following estimates hold: E 0 Z T 0 1 E 0 [L ;v s jF Y s ] 1 E 0 [L u s jF Y s ] p dsC p kuvk 2;2;Q 0; 2 (0; 1): (5.10) Clearly, (5.22) and (5.23) imply that J 1 +J 2 C p kuvk 2;2;Q 0, for some constant C p > 0, depending only on p, the Lipschitz constant of the coecients, and T . Therefore we have E 0 Z T 0 jU ;v s U u s j p dsC p kuvk 2;2;Q 0! 0; as ! 0. (5.11) We can now prove (5.7) fori = 1; ; 4. First, by Burkholder-Gundy-Davis inequality we have I ;1 t = Z t 0 R s n E 0 h L ;v s h(s;' 1 s ;U ;v s )L u s h(s;' 2 s ;U u s ) i ' 2 =X u ; ' 1 =X ;v o dY s (5.12) 90 E 0 [ sup 0tT jI ;1 t j 2 ] C Z T 0 E 0 n jR s j 2 h E 0 h L ;v s h(s;' 1 s ;U ;v s )L u s h(s;' 2 s ;U u s ) i ' 1 =X ;v ' 2 =X u i 2 o ds C Z T 0 E 0 n jR s j 2 h E 0 h (L ;v s L u s )h(s;' 1 s ;U ;v s ) 2 + L u s (' 1 ' 2 ) 2 + L u s (U ;v s U u s ) 2 i ' 1 =X ;v ' 2 =X u io ds (5.13) C Z T 0 E 0 [jR s j 2 ]E 0 [j(L ;v s L u s )j 2 ] +E 0 [jL u s j 2 ]E 0 [jR s (X ;v s X u s )j 2 ] +E 0 [jR s j 2 ]E 0 [ L u s (U ;v s U u s ) 2 ] iio ds Since h is bounded and Lipschitz continuous in (';y;z), it follows from Proposition 3.1 and (5.24) that lim !0 E 0 [sup 0tT jI ;1 t j 2 ] = 0. By the similar arguments using the continuity of D ' h, it is not hard to show that, for all p> 1, lim !0 E 0 [ sup 0tT jI ;i t j p ] = 0; i = 2; 3; 5: It remains to prove the convergence of I ;4 . To this end, we note that, for any p> 1, E 0 h sup s2[0;T ] jR s j p +jK s j p i C p ; (5.14) and by (5.24) we have, for p> 1, lim !0 E 0 Z T 0 E 0 B ;u;v (s;' ^s ;z)B u;v (s;' ^s ;z 1 ) 2 '=X u ;z=u ;v s z 1 =us p ds = 0: (5.15) This, together with (5.23), (5.24), an estimate similar to (5.22), and Proposition 3.1, yields that lim !0 E 0 [sup 0tT jI 3;;3 t j 2 ] = 0, proving the lemma. 91 Consequently, we have e t = Z t 0 L u s n E 0 f 1; s (' 1 ^s ;' 2 ^s )~ s g ' 1 =X ;v ; ' 2 =X u +E 0 f 2; s (' 2 ^s ;z) s g ' 2 =X u o dY s + Z t 0 n e s h(s;X ;v s ; ;v s ) + s L u s [Dh] ;u;v s o dY s +I t ; where I t = P 5 i=0 I ;i t , and 1; s (' 1 ^s ;' 2 ^s ;z) 4 = Z 1 0 D ' (s;' 2 ^s +(' 1 ^s ' 2 ^s );U ;v s ;z)(' 1 ^s ' 2 ^s )d; 2; s (' 2 ^s ;z) 4 =(s;' 2 ^s ;U ;v s ;z) + Z 1 0 @ y (s;' 2 ^s ;U u s +(U ;v s U u s );z)d(U ;v s U u s ); s (' 2 ^s ;z) 4 =L u s Z 1 0 @ y (s;' 2 ^s ;U u s +(U ;v s U u s );z)d: Notice that j 1; s (' 1 ^s ;' 2 ^s ;z)j+j 2; s (' 2 ^s ;z)jC(1+j' 1 ^s j+j' 2 ^s j+jU ;v s j+jU u s j);j s (' ^s ;z)jCL u s : Now by the Burkholder and Cauchy-Schwartz inequalities we have, for all p 2, t2 [0;T ], E 0 h sup s2[0;t] j~ s j 2p i C p n E 0 [kI k 2p C T ] +E 0 nh Z t 0 E 0 [j s j 2 +j~ s j 2 ] + sup r2[0;s] j s j 2 ds i p oo ; and from Gronwall's inequality one has E 0 h sup s2[0;t] j~ s j 2p i C p n E 0 h kI 3; k 2p C T + Z t 0 E 0 [j s j p ] 2 ds o ; t2 [0;T ]: (5.16) Recalling (5.6), we see that (5.3) follows from (5.16), provided (5.2) holds, which we now substantiate. 92 [Proof of (5.2)]. This part is more involved. We rst rewrite (4.9) as follows X t = Z t 0 n E 0 f(~ s +R s )(s;' ^s ;U ;v s ;z)g '=X ;v ; z=u ;v s + [D] ;u;v s ( ^s +K ^s ) +E 0 fB ;u;v (s;' ^s ;z) U s g '=X u ; z=u ;v s +C ;u;v (s)(v s u s ) o dB 1 s : (5.17) Here [D] ;u;v ,B ;u;v , andC ;u;v are dened by (4.10). Furthermore, in light of (4.11), we can also write: U t = E 0 [(~ t +R t )X ;v t +L u t ( t +K t )jF Y t ] E 0 [L ;v t jF Y t ] E 0 [(~ t +R t )jF Y t ] E 0 [L ;v t jF Y t ] U u t : Plugging this into (5.17) we have X t = Z t 0 n E 0 f~ s (s;' ^s ;U ;v s ;z)g '=X ;v ; z=u ;v s + [D] ;u;v s ( ^s ) +E 0 n B ;u;v (s;' ^s ;z) h E 0 [~ s X ;v s +L u s s jF Y s ] E 0 [L ;v s jF Y s ] E 0 [~ s jF Y s ] E 0 [L ;v s jF Y s ] U u s io '=X u ; z=u ;v s o dB 1 s + Z t 0 n E 0 fR s (s;' ^s ;U s ;z)g '=X ;v ; z=u ;v s + [D] ;u;v s (K ^s ) +E 0 n B ;u;v (s;' ^s ;z) h E 0 [R s X ;v s +L u s K s jF Y s ] E 0 [L ;v s jF Y s ] E 0 [R s jF Y t ] E 0 [L ;v s jF Y s ] U u s io '=X ;v ; z=u ;v s +C ;u;v (s)(v s u s ) o dB 1 s : Now, recalling (4.17) (or more conveniently, (4.18)) we have t = X t K t = Z t 0 n E 0 f~ s (s;' ^s ;U ;v s ;z)g '=X ;v ; z=u ;v s + [D] ;u;v s ( ^s ) +E 0 n B ;u;v (s;' ^s ;z) h E 0 [~ s X ;v s +L u s s jF Y s ] E 0 [L ;v s jF Y s ] E 0 [~ s jF Y s ] E 0 [L ;v s jF Y s ] U u s io '=X u ; z=u ;v s o dB 1 s +I 3;;1 t +I 3;;2 t +I 3;;3 t +I 3;;4 t ; (5.18) 93 where, for t2 [0;T ], I 3;;1 t 4 = Z t 0 E 0 R s (s;' 1 ^s ;U ;v s ;z 1 )(s;' 2 ^s ;U u s ;z 2 ) ' 1 =X ;v ;z 1 =u ;v s ' 2 =X u ;z 2 =us dB 1 s ; I 3;;2 t 4 = Z t 0 E 0 [D] ;u;v s (K ^s ) [D] u;v s (K ^s ) dB 1 s ; (5.19) I 3;;3 t 4 = Z t 0 n E 0 n B ;u;v (s;' ^s ;z) E 0 [R s X ;v s +L u s K s jF Y s ] E 0 [L ;v s jF Y s ] E 0 [R s jF Y t ] E 0 [L ;v s jF Y s ] U u s o '=X u ; z=u ;v s E 0 n B u;v (s;' ^s ;z) E 0 [R s X u s +L u s K s jF Y s ] E 0 [L u s jF Y s ] E 0 [R s jF Y s ] E 0 [L u s jF Y s ] U u s o '=X u ; z=us o dB 1 s I 3;;4 t 4 = Z t 0 E 0 [C ;u;v (s)(v s u s )C u;v (s)(v s u s )]dB 1 s : We have the following lemma. Lemma 5.3. Suppose that Assumption 2.3 holds. Then, for all p> 1, lim !0 E 0 n sup 0tT jI 3;;i t j p o = 0; i = 1; ; 4: (5.20) Proof. We rst recall that U ;v s 4 =E ;v [X ;v s jF Y s ] and U u s 4 =E u [X u s jF Y s ]. Using the Kallianpur-Strieble formula we have E 0 Z T 0 jU ;v s U u s j p ds C p n E 0 Z T 0 E 0 [L ;v s X ;v s jF Y s ] E 0 [L ;v s jF Y s ] E 0 [L u s X u s jF Y s ] E 0 [L ;v s jF Y s ] p ds +E 0 Z T 0 E 0 [L u s X u s jF Y s ] E 0 [L ;v s jF Y s ] E 0 [L u s X u s jF Y s ] E 0 [L u s jF Y s ] p ds o (5.21) 4 = C p fJ 1 +J 2 g: We now estimate J 1 and J 2 respectively. First note that, for any p> 1, we can nd a constant C p > 0 such that for any 2 (0; 1) and u2U ad , E 0 [(L ;v s ) p ] +E 0 [(L ;v s ) p ] +E 0 [(L u s ) p ]C p : 94 Thus, applying the H older and Jensen inequalities as well as Proposition 3.1, we have, for any p> 1, and 2 (0; 1), E 0 Z T 0 E 0 [L ;v s X ;v s jF Y s ]E 0 [L u s X u s jF Y s ] E 0 [L ;v s jF Y s ] p ds Z T 0 E 0 n jL ;v s X ;v s L u s X u s j p E 0 [L ;v s jF Y s ] p o ds Z T 0 n fE 0 jL ;v s X ;v s L u s X u s j 2 g 1=2 n E 0 h jL ;v s X ;v s L u s X u s j 2p2 E 0 [L ;v s jF Y s ] 2p io 1=2 o ds Z t 0 fE 0 jL ;v s X ;v s L u s X u s j 2 g 1=2 n E 0 [jL ;v s X ;v s L u s X u s j 2p2 ]E 0 [L ;v s ] 2p jF Y s o 1=2 ds C p kuvk 2;2;Q 0: (5.22) Similarly, one can also argue that, for any p> 1, the following estimates hold: E 0 Z T 0 1 E 0 [L ;v s jF Y s ] 1 E 0 [L u s jF Y s ] p dsC p kuvk 2;2;Q 0; 2 (0; 1): (5.23) Clearly, (5.22) and (5.23) imply that J 1 +J 2 C p kuvk 2;2;Q 0, for some constant C p > 0, depending only on p, the Lipschitz constant of the coecients, and T . Therefore we have E 0 Z T 0 jU ;v s U u s j p dsC p kuvk 2;2;Q 0! 0; as ! 0. (5.24) We can now prove (5.20) for i = 1; ; 4. First, by Burkholder-Gundy-Davis inequality we have E 0 [ sup 0tT jI 3;;1 t j 2 ]C Z T 0 E 0 E 0 R s (s;' 1 ^s ;U ;v s ;z 1 )(s;' 2 ^s ;U u s ;z 2 ) ' 1 =X ;v ;z 1 =u ;v s ' 2 =X u ;z 2 =us 2 ds: Since is bounded and Lipschitz continuous in (';y;z), it follows from Proposition 3.1 and (5.24) that lim !0 E 0 [sup 0tT jI 3;;1 t j 2 ] = 0. By the similar arguments using 95 the continuity of D ' and that of @ z , respectively, it is not hard to show that, for all p> 1, lim !0 E 0 [ sup 0tT jI 3;;2 t j p ] = 0; lim !0 E 0 [ sup 0tT jI 3;;4 t j p ] = 0: It remains to prove the convergence of I 3;;3 . To this end, we note that, for any p> 1, E 0 h sup s2[0;T ] jR s j p +jK s j p i C p ; (5.25) and by (5.24) we have, for p> 1, lim !0 E 0 Z T 0 E 0 B ;u;v (s;' ^s ;z)B u;v (s;' ^s ;z 1 ) 2 '=X u ;z=u ;v s z 1 =us p ds = 0: (5.26) This, together with (5.23), (5.24), an estimate similar to (5.22), and Proposition 3.1, yields that lim !0 E 0 [sup 0tT jI 3;;3 t j 2 ] = 0, proving the lemma. We now continue the proof of (5.2). First we rewrite (5.18) as t = Z t 0 n E 0 f~ s (s;' ^s ;U ;v s ;z)g '=X ;v ; z=u ;v s + [D] ;u;v s ( ^s ) +E 0 n B ;u;v (s;' ^s ;z) h E 0 [~ s X ;v s +L u s s jF Y s ] E 0 [L u s jF Y s ] E 0 [~ s jF Y s ] E 0 [L u s jF Y s ] U u s io '=X u ; z=u ;v s o dB 1 s +I 3;;0 t + 4 X i=1 I 3;;i t ; (5.27) where I 3;;0 t 4 = Z t 0 E 0 n B ;u;v (s;' ^s ;z) h E 0 [~ s X ;v s +L u s s jF Y s ] E 0 [L ;v s jF Y s ] E 0 [~ s jF Y s ] E 0 [L ;v s jF Y s ] U u s io '=X u ; z=u ;v s B ;u;v (s;' ^s ;z) h E 0 [~ s X ;v s +L u s s jF Y s ] E 0 [L u s jF Y s ] E 0 [~ s jF Y s ] E 0 [L u s jF Y s ] U u s io '=X u ; z=u ;v s o dB 1 s 96 We note that with the same argument as before one shows that lim !0 E 0 [sup 0tT jI 3;;0 t j 2 ] = 0. On the other hand, similar to (4.15) one can argue that E 0 h B ;u;v (s;' ^s ;z) E 0 [~ s X ;v s +L u s s jF Y s ] E 0 [L u s jF Y s ] E 0 [~ s jF Y s ] E 0 [L u s jF Y s ] U u s i '=X u ; z=u ;v s = E 0 h Z 1 0 @ y (s;' ^s ;U u s +(U ;v s U u s );z)d (~ s X ;v s +L u s s U u s ~ s ) i '=X u ; z=u ;v s : Consequently, we have t = Z t 0 n E 0 f 1; s (' 1 ^s ;' 2 ^s ;z)~ s g ' 1 =X ;v ;' 2 =X u ; z=u ;v s +E 0 f 2; s (' 2 ^s ;z)~ s g ' 2 =X u ; z=u ;v s o dB 1 s + Z t 0 n E 0 f s (' 2 ^s ;z) s g ' 2 =X u z=u ;v s + [D] ;u;v s ( ^s ) o dB 1 s +I 3; t ; where I 3; t = P 4 i=0 I 3;;i t , and 1; s (' 1 ^s ;' 2 ^s ;z) 4 = Z 1 0 D ' (s;' 2 ^s +(' 1 ^s ' 2 ^s );U ;v s ;z)(' 1 ^s ' 2 ^s )d; 2; s (' 2 ^s ;z) 4 =(s;' 2 ^s ;U ;v s ;z) + Z 1 0 @ y (s;' 2 ^s ;U u s +(U ;v s U u s );z)d(U ;v s U u s ); s (' 2 ^s ;z) 4 =L u s Z 1 0 @ y (s;' 2 ^s ;U u s +(U ;v s U u s );z)d: Notice that j 1; s (' 1 ^s ;' 2 ^s ;z)j+j 2; s (' 2 ^s ;z)jC(1+j' 1 ^s j+j' 2 ^s j+jU ;v s j+jU u s j);j s (' ^s ;z)jCL u s : Now by the Burkholder and Cauchy-Schwartz inequalities we have, for all p 2, t2 [0;T ], E 0 h sup s2[0;t] j s j 2p i C p n E 0 [kI 3; k 2p C T ] +E 0 nh Z t 0 E 0 [j s j 2 +j~ s j 2 ] + sup r2[0;s] j s j 2 ds i p oo ; 97 and from Gronwall's inequality one has E 0 h sup s2[0;t] j s j 2p i C p n E 0 h kI 3; k 2p C T + Z t 0 E 0 [j~ s j p ] 2 ds o ; t2 [0;T ]: (5.28) On the other hand, setting I t 4 = I 1; t +I 2; t , t2 [0;T ], we have from (5.6) that, for p 2, E 0 h sup s2[0;t] j~ s j p i C p n E 0 [kI k p C T ] + Z t 0 E 0 [j~ s j p ]ds + Z t 0 E 0 [j s j 2p ] 1=2 ds o ; t2 [0;T ]: Then Gronwall's inequality leads to that E 0 h sup s2[0;t] j~ s j p i 2 C p n E 0 kI k p C T 2 + Z t 0 E 0 [j s j 2p ]ds o ; t2 [0;T ]: (5.29) Combining (5.28), (5.29) and Lemma 5.3 as well as the Gronwall inequality, we can easily deduce (5.2) by sending ! 0. Consequently, (5.3) holds as well. From Proposition 5.1, (4.11) and the above development we also obtain the fol- lowing corollary. Corollary 5.4. We assume that Assumption 2.3 holds. Then, for all p> 1, lim !0 E 0 [k UVk p C T ] = lim !0 E 0 [ sup 0sT j U ;v s U u s V s j p ] = 0; where V t 4 = E 0 [R t X u t +L u t K t jF Y t ] E 0 [L u t jF Y t ] E 0 [R t jF Y t ] E 0 [L u t jF Y t ] U u t ; t2 [0;T ]: 98 6 Stochastic Maximum Principle We are now ready to study the Stochastic Maximum Principle. The main task will be to determine the appropriate adjoint equation, which we expect to be a backward stochastic dierential equation of Mean-eld type. We begin with a simple analysis. Suppose that u = u is an optimal control, and for any v2U ad , we dene u ;v by (4.4). Then we have 0 J(u ;v )J(u) = 1 E 0 n E 0 [L ;v T (x;U ;v T )]j x=X T E 0 [L u T (x;U u T )]j x=X u T (6.1) + Z T 0 E 0 [L ;v s f(s;' ^s ;U ;v s ;z)]j '=X ;v ; z=u ;v s E 0 [L u s f(s;' ^s ;U u s ;z)]j'=X u ; z=us ds o : Now, repeating the same analysis as that in Proposition 3.1, then sending ! 0, it follows from Propositions 3.1, 5.1 and the continuity of the functions and f that 0 E 0 [K T ] +E 0 [R T ] +E 0 n Z T 0 n E 0 [R s f(s;' ^s ;U u s ;z)]j '=X u ;z=us +E 0 [@ y f(s;' ^s ;U u s ;z)(X u s U u s )R s +L u s K s ]j '=X u ;z=us (6.2) +E 0 [L u s D ' f(s;' ^s ;U u s ;z)( ^s )]j '=X u ;z=us; =K +E 0 [L u s @ z f(s;' ^s ;U u s ;z)]j '=X u ;z=us (v s u s ) o ds o ; where 4 = E 0 [L u T @ x (x;U u T )]j x=X u T +L u T E 0 [@ y (X u T ;y)]j y=U u T ; 4 = E 0 [(X u T ;y)] y=U u T + (X u T U u T )E 0 [@ y (X u T ;y)]j y=U u T : (6.3) 99 We now consider the adjoint equations that take the following form of backward SDEs on the reference space ( ;F;Q 0 ): 8 > < > : dp t = t dt +d t +q t dB 1 t +e q t dY t ; p T =; dQ t = t dt +d t +M t dB 1 t + f M t dY t ; Q T = : (6.4) Here the coecients ; as well as the two bounded variation processes and are to be determined. Applying It^ o's formula and recalling the variational equations (4.17) and (5.5), we can easily derive (denote U u t =E u [X u t jF Y t ], t2 [0;T ]) E 0 [K T ] +E 0 [R T ] = Z T 0 n E 0 [K s s ]E 0 [R s s ] +E 0 h q s E 0 [R s (s;' ^s ;U u s ;z)] '=X u ;z=us i +E 0 h q s E 0 h @ y (s;' z^s ;U u s ;z)[(X u s U u s )R s +L u s K s ] i '=X u ;z=us i +E 0 q s [D] u;v s (K ^s ) +q s C u;v (s)(v s u s ) + f M s R s h(s;X u s )] +E 0 f M s L u s f[Dh] u;v s (K s ) +E 0 [R s h(s;';;U u s ) (6.5) +@ y h(s;';U u s )((X u s U u s )R s +L u s K s )] '=X u s g o ds +E 0 n Z T 0 [K s d s +R s d s ] o ; 100 where [D] u;v and C u;v are dened by (4.14). By Fubini's Theorem we see that 8 > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > : E 0 q s E 0 [R s (s;'^s;U u s ;z)] '=X u ;z=us =E 0 R s E 0 [q s (s;X ^s ;y;u s )] y=U u s ; E 0 h q s E 0 h @ y (s;' z^s ;U u s ;z)[(X u s U u s )R s +L u s K s ] i '=X u ;z=us i =E 0 h E 0 q s @ y (s;X z^s ;y;u s )] y=U u s [(X u s U u s )R s +L u s K s ] i : E 0 f M s L u s E 0 [R s h(s;'s;U u s )] '=X u =E 0 R s E 0 [ f M s L u s h(s;X s ;y)] y=U u s ; E 0 h f M s L u s E 0 h @ y h(s;' s ;U u s )[(X u s U u s )R s +L u s K s ] i '=X u i =E 0 h E 0 f M s L u s @ y h(s;X s ;y)] y=U u s [(X u s U u s )R s +L u s K s ] i : (6.6) Furthermore, in light of denition of [D] u;v ((4.14)), if we denote, for xed (t;';z), 0 (t;' ^t ;z)() 4 =E 0 [L u t D ' (t;' ^t ;U u t ;z)]()2M [0;T ]; (6.7) whereM [0;T ] denotes all the Borel measures on [0;T ], then we can write [D] u;v t (K ^t ) =E 0 L u t D ' (t;' ^t ;U u t ;z)( )] '=X u ;z=u t ; =K ^t = Z t 0 K r 0 (r;X u ^r ;u r )(dr): (6.8) Let us now argue that a similar Fubini Theorem argument holds for the random measure 0 (t;X u ^t ;u t )(). First, for a given process q2 L 2 F (Q 0 ; [0;T ]), consider the following nite variation (FV) process (in fact, under Assumption 2.3, integrable variation (IV) process): A t 4 = Z T 0 Z t^s 0 q s 0 (s;X u ^s ;u s )(dr)ds; t2 [0;T ]: (6.9) 101 It is easy to check, as a (randomized) signed measure on [0;T ], it holds Q 0 -almost surely thatdA t = R T t q s 0 (s;X u ^s ;u s )(dt)ds. We note that being a \raw FV" process, the process A is notF-adapted. We now consider its dual predictable projection: p Z T t q s 0 (s;X u ^s ;u s )(dt)ds 4 =d[ p A t ]; t2 [0;T ]: (6.10) We remark thatd[ p A t ] is a predicable random measure that can be formally understood as d[ p A t ] =E 0 [dA t jF t ] =E 0 h Z T t q s 0 (s;X u ^s ;u s )(dt)ds F t i ; t2 [0;T ]: Using the denition of dual predicable projection and (6.8), we see that, for the continuous process K2L 2 F (Q 0 ;C T ), Z T 0 E 0 [q s [D] u;v s (K ^s )]ds = Z T 0 E 0 h q s Z s 0 K r 0 (r;X u ^r ;u r )(dr) i ds = E 0 h Z T 0 K r dA r i =E 0 h Z T 0 K r d[ p A r ] i (6.11) = E 0 h Z T 0 K r p Z T r q s 0 (s;X u ^s ;u s )(dr)ds i : Similarly, we denoteA f t 4 = R T 0 R t^s 0 0 f (s;X u ^s ;u s )(dr)ds,t2 [0;T ]; and denote its dual predicable projection by p R T t 0 f (s;X u ^s ;u s )(dt)ds =d[ p A f t ], t2 [0;T ]. 102 We now plug (6.6) and (6.11) into (6.5) to get: E 0 [K T ] +E 0 [R T ] = E 0 n Z T 0 n K s h s +L u s E 0 q s @ y (s;X u ^s ;y;u s ) y=U u s +L u s E 0 f M s L u s @ y h(s;X u s ;y) y=U u s i +R s h s +E 0 [q s (s;X ^s ;y;u s )] y=U u s + f M s h(s;X u s ; u s ) i +q s C u;v s (v s u s ) +R s E 0 q s @ y (s;X u ^s ;y;u s ) y=U u s (X u s U u s ) +R s h E 0 f M s L u s @ y h(s;X u s ;y) y=U u s (X u s U u s ) +E 0 f M s L u s h(s;X u s ;y y=U u s io ds +E 0 n Z T 0 [K s d s +R s d s ] o + Z T 0 K s d[ p A s ] + Z T 0 K s d[ p A h s ] o ; = E 0 n Z T 0 [K s ^ s R s ^ s +q s C u;v (s)(v s u s )]ds (6.12) +K s d[ p A s ] +K s d[ p A h s ] + [K s d s +R s d s ] o ; where 8 > > > > > > > > < > > > > > > > > : ^ t 4 = t L u t E 0 q t @ y (t;X u ^t ;y;u t ) y=U u t L u s E 0 f M s L u s @ y h(s;X u s ;y) y=U u s ; ^ t 4 = t E 0 [q t (t;X ^t ;y;u t )] y=U u t f M t h(t;X u t ; u s ) E 0 q t @ y (t;X u ^t ;y;u t ) y=U u t (X u t U u t ) E 0 f M s L u s @ y h(s;X u s ;y) y=U u s (X u s U u s )E 0 f M s L u s h(s;X u s ;y y=U u s : (6.13) 103 Combining (6.2) and (6.12) and using the processes dA , dA f and their dual predi- cable projections, we have 0 E 0 n Z T 0 [K s ^ s R s ^ s +q s C u;v (s)(v s u s )]ds + Z T 0 K s d[ p A s ] + Z T 0 K s d[ p A h s ] o +E 0 n Z T 0 h R s E 0 [f(s;X ^s ;y;u s )] y=U u s +E 0 @ y f(s;X u ^s ;y;u s ) y=U u s (X u s U u s ) +L u s K s E 0 @ y f(s;X u ^s ;y;u s ) y=U u s +C u;v f (s)(v s u s ) i ds + Z T 0 K s d[ p A f s ] o +E 0 n Z T 0 [K s d s +R s d s ] o ; (6.14) where C u;v f (s) 4 =E 0 [L u s @ z f(s;' ^s ;U u s ;z)]j '=X u ;z=us . Now, if we set t = 0, and ^ t = L u t E 0 @ y f(t;X u ^t ;y;u t ) y=U u t (6.15) ^ t = E 0 [f(t;X ^t ;y;u t )] y=U u t +E 0 @ y f(t;X u ^t ;y;u t ) y=U u t (X u t U u t ) d t = d[ p A t ]d[ p A h t ]d[ p A f t ]; then (6.14) becomes 0 E 0 n Z T 0 [q t C u;v (s) +C u;v f (s)](v s u s )ds o ; v2U ad : (6.16) From this we should be able to derive the maximum principle, provided that the adjoint equation (6.4) with coecients , , and determined by (6.13) and (6.15) is well-dened. Remark 6.1. 1) We remark that the process in (6.15) should be considered as a mapping from the spaceL 2 F ([0;T ] )L 2 F (;C T )L 2 F ([0;T ] ;U) toM F ([0;T ]), the space of all the random measures on [0;T ], such that (i) (t;!)7!(t;!;A) isF-progressively measurable, for all A2B([0;T ]); 104 (ii) (t;!;)2M ([0;T ]) is a nite Borel measure on [0;T ]. 2) Assumption 2.3-(iii) implies that the random measureD [q;X u ;u](t;dt) satises the following estimate: for any q2L 2 F ([0;T ] ) and u2U ad , E 0 h Z T 0 jd p A t j i = E 0 n Z T 0 p Z T t q s 0 (s;X u ^s ;u s )(dt)ds o E 0 n Z T 0 Z r 0 jq s jj 0 (s;X u ^s ;u s )(dt)jds o (6.17) E 0 n Z T 0 jq s j Z s 0 `(s;dt)ds o CE 0 n Z T 0 jq s jds o Ckqk 2;2;Q 0: The same estimate holds forD f [X u ;u](t;dt) as well. 3) Clearly, the processes A and A f are originated from the Frech et derivatives of and f, respectively, with respect to the path ' ^t . If and f are of Markovian type, then they will be absolutely continuous with respect to the Lebesgue measure. We shall now validate all the arguments presented above. To begin with, we note that the choice of , , and via by (6.13) and (6.15), together with the terminal 105 condition (; ) by (6.3), amounts to saying that the processes (p;q; ~ q) and (Q;M; ~ M) solve the BSDE: 8 > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > : dp t =L u t n E 0 @ y f(t;X u ^t ;y;u t ) y=U u t +E 0 q t @ y (t;X u ^t ;y;u t ) y=U u t +E 0 f M s L u s @ y h(s;X u s ;y) y=U u s o dtd p A t d p A h t d p A f t +q t dB 1 t +e q t dY t dQ t = n E 0 [q t (t;X u ^t ;y;u t )] y=U u t f M t h(t;X u t ; u s ) +E 0 q t @ y (t;X u ^t ;y;u t )] y=U u t (X u t U u t ) +E 0 f M s L u s @ y h(s;X u s ;y) y=U u s (X u s U u s )E 0 f M s L u s h(s;X u s ;y y=U u s +E 0 [f(t;X ^t ;y;u t )] y=U u t +E 0 @ y f(t;X u ^t ;y;u t ) y=U u t (X u t U u t ) o dt +M t dB 1 t + f M t dY t ; p T =; Q T = : (6.18) Now if we denote = (p;Q) T , W = (B 1 ;Y ) T , = h q ~ q M ~ M i , then we can rewrite (6.18) in a more abstract (vector) form: 8 > < > : d t =fA t +E 0 [G t t g(t;y)] y=U u t +H t t h t gdt ()(t;dt) 0 (t;dt) + t dW t ; T = ; (6.19) where 2 L 2 F W T ( ;Q 0 ); A;G;H and h are bounded, vector or matrix-valued F W - adapted processes with appropriate dimensions, g is anR 2 -valued progressively mea- surable random eld, andU is anF Y -adapted process. Moreover, theR 2 -valued nite variation processes ()(t;dt) and 0 (t;dt) take the form: ()(t;dt) = p Z T t r 1 r (dt)dr ; 0 (t;dt) = p Z T t 2 r (dt)dr ; (6.20) 106 wherer7! i r (),i = 1; 2, areM [0;T ]-valued measurable random processes satisfying, as measures with respect to the total variation norm, j 1 r (dt)j +j 2 r (dt)j`(r;dt); r2 [0;T ]; Q 0 a.s. (6.21) We note that ()(dt) and 0 (dt) are representing d[ p A t ] and [ p A f t ] in (6.18), respec- tively, and can be substantiated by (6.9) and (6.10). Furthermore, by Assumption 2.3, they both satisfy (6.21). To the best of our knowledge, BSDE (6.19) is beyond all the existing frameworks of BSDEs, and we shall give a brief proof for its well-posedness. Theorem 6.2. Assume that the Assumption 2.3 is in force. Then, the BSDE (6.19) has a unique solution (; ). Proof. The proof is more or less standard, we shall only point out a key estimate. For any given e i 2L 2 F W ([0;T ] ;R 4 ), obviously we have a unique solution ( i ; i ) of (6.19), i = 1; 2, respectively, i.e., 8 > < > : d i t =fA t +E 0 [G t e i t g(t;y)] y=U u t +H t e i t h t gdt ( e i )(t;dt) 0 (t;dt) + i t dW t ; i T = : We dene b = 1 2 , i = i ; i , i = 1; 2, respectively. b e = e 1 e 2 . Noting the linearity of BSDE (6.19) we see thatb satises: b t = Z T t n E 0 [G s b e s g(s;y)] y=U u s +H s b e s h s o ds + Z T t ( b e )(s;ds)M T t ; (6.22) where M T t 4 = R T t b s dW s . Therefore, jb t +M T t j 2 2 n Z T t n E 0 [G s b e s g(s;y)] y=U u s +H s b e s h s o ds 2 + Z T t ( b e )(s;ds) 2 o : 107 Taking expectation on both sides above and noting thatE 0 [b t M T t ] = 0 and E 0 n Z T t n E 0 [G s b e s g(s;y)] y=U u s +H s b e s h s o ds 2 o C(Tt)E 0 h Z T t j b e s j 2 ds i ; we have E 0 [jb t j 2 ] +E 0 h Z T t j b s j 2 ds i C(Tt)E 0 h Z T t j b e s j 2 ds i +E 0 n Z T t ( b e )(s;ds) 2 o : (6.23) To estimate the term involving ( b e ) we note that (recall (6.20)) if a square-integrable processV is increasing and continuous, then so is its dual predictable projection p V . Thus, by the denition of p V we have E 0 h Z T t d[ p V s ] 2 i = 2E 0 h Z T t ( p V s p V t )d[ p V s ] i = 2E 0 h Z T t ( p V s p V t )dV s i 2E 0 [( p V T p V t )(V T V t )] 2 E 0 h Z T t d[ p V s ] 2 i 1=2 E 0 h Z T t dV s 2 i 1=2 : That is, E 0 h Z T t d[ p V s ] 2 i 4E 0 h Z T t dV s 2 i : (6.24) Applying this to V t 4 = R T 0 R t^r 0 j b e r jj 1 r (ds)jdr, t2 [0;T ], we have E 0 h Z T t ( b e )(s;ds) 2 i E 0 h Z T t p Z T s j b e r jj 1 r (ds)jdr 2 i 4E 0 h Z T t Z T s j b e r jj 1 r (ds)jdr 2 i 4E 0 h Z T t Z T s j b e r j`(r;ds)dr 2 o CE 0 h Z T t j b e r jdr 2 o C(Tt)E 0 h Z T 0 j b e s j 2 ds i ; 108 and therefore (6.23) becomes E 0 [jb t j 2 ] +E 0 h Z T t j b s j 2 ds i C(Tt)E 0 h Z T t j b e s j 2 ds i : (6.25) With this estimate, and following the standard argument one shows that BSDE (6.18) is well-posed on [T;T ] for some (uniform) > 0. Iterating the argument one can then obtain the well-posedness on [0;T ]. We leave the details to the interested reader. We are now ready to prove the main result of this paper. Let us dene the Hamiltonian: for (';)2 C T P(C T ), and k : [0;T ] ! R adapted process, (t;!;z)2 [0;T ] R, H (t;!;' ^t ;;z;k) 4 =k t (!)(t;' ^t ;;z) +f(t;' ^t ;;z): (6.26) We have the following theorem. Theorem 6.3 (Stochastic Maximum Principle). Assume that the Assumptions 2.3 and 2.2 hold. Assume further that the mapping z7!H (t;' ^t ;;z) is convex. Let u = u 2U ad be an optimal control and X u the corresponding trajectory. Then, for dtdQ 0 -a.e. (t;!)2 [0;T ] it holds that H (t;!;X u ^t ; u t ;u t ;q t ) = inf v2U H (t;!;X u ^t ; u t ;v;q t ); (6.27) where (p;q; ~ q) and (Q;M; ~ M) constitute the unique solution of the BSDE (6.18). 109 Proof. We rst recall from (4.14) that C u;v f (t) = E 0 [L u t @ z f(t;' ^t ;U u t ;z)]j '=X u ;z=ut =@ z f(t;X u ^t ; u t ;u t ); C u;v (t) = E 0 n L u t @ z (t;' ^t ;U u t ;z) io '=X u ;z=ut =@ z (t;X u ^t ; u t ;u t ): Then (6.16) implies that 0 E 0 h Z T 0 [q t C u;v (t) +C u;v f (t)](v t u t )dt i (6.28) = E 0 h Z T 0 @ z H (t;!;X u ^t ; u t ;u t ;q t )(v t u t )dt i : Therefore for dtdQ 0 -a.e. (t;!)2 [0;T ] , and any v2U, it holds that @ z H (t;!;X u ^t ; u t ;u t ;q t )(vu t ) 0: (6.29) Now, for any v2U, one has, dtdQ 0 -a.e. on [0;T ] , H (t;!;X u ^t ; u t ;v;q t )H (t;!;X u ^t ; u t ;u t ;q t ) = Z 1 0 @ z H (t;!;X u ^t ; u t ;u t +(vu t );q t )(vu t )d = Z 1 0 h @ z H (t;!;X u ^t ; u t ;u t +(vu t );q t )@ z H (t;!;X u ^t ; u t ;u t ;q t ) i (vu t )d +@ z H (t;!;X u ^t ; u t ;u t ;q t )(vu t ) 0;: Here the rst integral on the right hand side above is nonnegative due to the convexity ofH in variablez, and the last term is non-negative because of (6.29). The identity (6.27) now follows immediately. 110 Remark 6.4. In stochastic control literature the inequality (6.28) is sometimes referred to as Stochastic Maximum Principle in integral form, which in many appli- cations is useful, as it does not require the convexity assumption on the Hamiltonian H . 111 Bibliography [1] Aase, K.K., Bjuland, T., and Oksendal, B.. Strategic insider trading equilibrium: a lter theory approach. Afr. Mat. 23, 145-162 (2012). [2] Back, K., Insider trading in continuous time. Rev. Financ. Stud. 5, 387-409 (1992). [3] Back, K. and Pedersen, H., Long-lived information and intrady patterns. Journal of Financial Market, 1, 3-4, 385-402 (1998). [4] Benssussan, A., Stochastic Control of Partially Observable Systems, Cambridge University Press (1992). [5] Biagini, F., Hu, Y., Myer-Brandis, T., and Oksendal, B., Insider trading equilib- rium in a market with memory. Math Finan Econ 6, 229-247 (2012). [6] Bj ork, T., Arbitrage Theory in Continuous Time. 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Abstract (if available)

Abstract This paper contains two parts. In the first part, we extend Kyle-Back model of insider trading to the setting in which the insider has a dynamic information on an asset, under a linear conditional mean-field-type dynamic model. Such dynamics contain many existing models as special cases, but it is put into a rigorous mathematical framework for the first time. We shall first prove a general well-posedness result for a class of linear conditional mean-field SDEs, which will be the foundation for the underlying dynamics of the optimization problem. We give a general necessary condition for the existence of optimal intensity of trading strategy in such a case, and when the dynamics is actually Vasicek, we find a closed form of optimal intensity of trading strategy including the form of dynamic pricing rule set market makers and point out that in the equilibrium, all information is released by the insider and the intensity goes to infinity at the end of auction time. In the second part, we study the optimal control problem for a class of general mean-field stochastic differential equations (SDEs), in which the coefficients depend, nonlinearly, on both the state process as well as of its law. In particular, we assume that the control set is a general open set that is not necessary convex, and the coefficients are only continuous on the control variable without any further regularity or convexity. We extend the Stochastic Maximum to this general case.

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Creator Sun, Rentao (author)

Core Title Conditional mean-fields stochastic differential equation and their application

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Applied Mathematics

Publication Date 09/29/2017

Defense Date 09/29/2017

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag conditional mean-field SDEs,Kyle-Back equilibrium,maximum principle,McKean-Vlasov equation,mean-field SDE,OAI-PMH Harvest,optimal closed-loop system,reference measures,stochastic control,strategic insider trading

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Ma, Jin (committee chair), Lv, Jinchi (committee member), Zhang, Jianfeng (committee member)

Creator Email rentaosu@usc.edu,sunrentao@gmail.com

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