Optimal Control And Differential Game Of Partial Information Forward-Backward Stochastic Systems

A backward stochastic differential equation (BSDE, in short) is an Ito's stochas-tic differential equation (SDE, in short) in which the terminal rather than the initial condition is specified. The BSDEs were introduced by Bismut [7] in the linear case and independently by Pardoux and Peng [39] and Duffie and Epstein [13] in the non-linear case. A BSDE coupled with a forward SDE formulates a forward-backward stochastic differential equation (FBSDE, in short). Since their introduction. FBSDEs have received considerable research attention in number of different areas, especially in stochastic control and financial mathematics. For instance, the classical Hamilto-nian system arising from necessary conditions for stochastic optimal control problems belongs to one of such kind of equations; the celebrated Black-Scholes formula for options pricing can be recovered via an FBSDE. For more details, especially refer to the monographs by Ma and Yong [34] and Yong and Zhou [70]. Since FBSDEs are well-defined dynamic systems, it is natural to consider optimal control and differential game problems of FBSDEs. This thesis is dedicated to studying stochastic filtering, optimal control and differential game of FBSDEs with complete or partial information.Wang and Wu [54] originally studied the filtering theory of forward-backward stochastic systems, where the state and observation equations are driven by standard Brownian motions. They proposed a kind of backward separation techniques, which is more convenient to solve the partially observable optimal control problem than that of Wonham [59]. Inspired by Wang and Wu [54], we study a more general case where the state and observation equations are driven by both Brownian motions and Poisson processes. Due to the property of random jumps from Poisson processes, we obtain some new and interesting results which are distinguished from that of Wang and Wu [54].Shi and Wu [47] investigated a kind of optimal control problems of forward-backward stochastic differential equations with random jumps (FBSDEPs. in short), and Wu [61] studied optimal control of partially observable FBSDEs, both in the case of the convex control domain and the diffusion term allowing to contain the control variable. Wang and Wu [55] studied a kind of stochastic recursive optimal control problem in the case where the control domain is not necessarily convex and the dif-fusion term of forward equation does not contain the control variable. Based on their works, we consider the optimal control of partially observable FBSDEPs and establish a necessary and a sufficient conditions for an optimal control. The results extend those of Shi and Wu [47] and Wu [61] to the cases of partial observation and random jumps respectively, and partly generalize those of Liptser and Shiryayev [33], Bensoussan [6], Tang [50] and Wang and Wu [54,55] to the cases of forward-backward systems or random jumps.However, the works mentioned above do not deal with the correlation between the states and observations. To my best knowledge, there is only one paper about this topic (see Tang [50]), but Tang only considered the forward system dynamics driven by Brownian motion and proved a general stochastic maximum principle. Here, we study the optimal control problems of FBSDEPs with correlated noisy observations. In the case of convex control domain, a local maximum principle and a verification theorem are proved. The present results are a partial extension to Shi and Wu [48], Tang and Hou [51], Tang [50], Wang and Wu [55], Xiao [63] and Meng [35] for Brownian motion case only, or Poisson point processes only, or forward SDEs only, or uncorrelated noisy observations. Full information control problem can be considered a special case of partial information control problem. From this point of view, the present results represent partial extension to the relevant ones in Peng [41], Shi and Wu [47] and Xu [66].Up till now, there are only two papers about differential games of BSDEs:one is Yu and Ji [72], where an linear quadratic (LQ, in short) nonzero-sum game was studied by a standard completion of squares techniques and the explicit form of a Nash equi-librium point was obtained:the other one is Wang and Yu [56], where the game system was a nonlinear BSDE, and a necessary and a sufficient conditions in the form of maxi-mum principle were established. The game problems mentioned above are restricted to backward stochastic systems. To my best knowledge, there are only two papers about the differential games of forward-backward stochastic systems (see Buckdahn and Li [8] and Yu [71]). In Buckdahn and Li [8], the game system is described by a decoupled FBSDE, and the performance criterion is defined by the solution variable of BSDE, at the value at time 0. Buckdahn and Li proved a dynamic programming principle for both the upper and the lower value functions of the game, and showed that these two functions are the unique viscosity solutions to the upper and the lower Hamilton-Jacobi-Bellman-Isaacs equations. Recently, Yu [71] studied a linear-quadratic case of nonzero-sum game problem for forward-backward stochastic systems, where the FB-SDE method was employed to obtain an explicit Nash equilibrium point. However, in the present paper, we shall study the problem in more general situation. Com-bining FBSDE theory with certain classical convex variational techniques, we prove a necessary and a sufficient conditions for a Nash equilibrium point of nonzero-sum differential game of FBSDEs in the form of maximum principle, as well for a saddle point of zero-sum differential game of FBSDEs. Meanwhile, an example of a nonzero-sum differential game is worked out to illustrate theoretical applications. In terms of maximum principle, the explicit form of an equilibrium point is obtained.Inspired by finding an equilibrium point of an LQ nonzero-sum differential game of partial information backward doubly stochastic differential equations (BDSDEs, in short) and well describing so-called informal trading phenomena such as "insider trad-ing" in the market, we are concerned with a new type of differential game problem of partial information forward-backward doubly stochastic differential equations (FBDS-DEs, in short). This problem possesses fine generality. Firstly, the FBDSDE game system covers many systems as its particular case. For example, if we drop the terms on backward Ito's integral or forward equation or both them, then the FBDSDE can be reduced to FBSDE or BDSDE or BSDE. Secondly, all the results can be reduced to the case of full information. Finally, if the present zero-sum stochastic differential game has only one player, the game problem is reduced to some related optimal control. In detail, our results are a partial extension to optimal control of partial information BSDEs and FBSDEs (see Huang, Wang and Xiong [20] and Xiao and Wang [64]) and full information BDSDEs (see Han, Peng and Wu [18]), and to differential games of full information BSDEs and partial information BSDEs (see Wu and Yu [56], Yu and Ji [72], Zhang [73] and Wu and Yu [57]).The thesis consists of four chapters. We list the main results as follows.Chapter 1:We give a brief introduction on the problems investigated from Chap- ter 2 to Chapter 4.Chapter 2:We study stochastic filtering of linear FBSDEPs. By applying the filtering equation established, we solve a partially observable LQ control problem, where an explicit observable optimal control is determined by the optimal filtering estimation.Theorem 2.1 Let (H2.1) and (H2.2) hold, and assume that there exists a solution to (2.14). Then the filtering estimation (πt(x),πt(y),πt(z1),πt(z2),πt(ri),πt(r2)) of the state (x, y, z1,z2, r1, r2) are given by (2.14), (2.22) and (2.23), and the conditional mean square error of the filtering estimationπt(x) is given by (2.21).Corollary 2.1 If we assume that a6(·)=a10(·)=0, that is, x(·) and N1(·) have no common jump times, then (2.14) becomes and (2.22), (2.23) and (2.21) still hold where the corresponding a6(·) and aw(·) inΣ(·),Λ(·) and (2.21) are replaced by 0.Corollary 2.2 If c5(·)=0, that is, the observation process Z(·) has no jumps, (2.24) is also the corresponding filtering equation, and (2.22), (2.23) and (2.21) still hold.Theorem 2.2 Let (H2.1)-(H2.3) hold. For any v(·)∈Uad, the state variable xv(·), which is the solution to (2.30), has the filtering estimation and where we adopt the notationγ(t)=E[(x"(t)-πt(xv))2|FtZ].Theorem 2.3 If(H2.1)-(H2.4) hold, then u(·) in (2.44)is an indeed optimal control for the aforesaid partially observable optimal control problem. Theorem 2.4 Let (H2.1)-(H2.4) hold. Then the optimal control u(·) and the corre-sponding performance criterion J(u(·)) are given by (2.44) and (2.54), respectively.Chapter 3:We firstly study optimal control of partially observable FBSDEPs when the states and observations are non-correlated. Based on this, we further extend it to the case where the states and observations are correlated. For these two cases, we establish the corresponding necessary and the sufficient conditions in the form of maximum principle. We also work out two examples to illustrate the theoretical applications.Lemma 3.1 Let assumption (H3.1) hold. ThenLemma 3.2 Let assumption (H3.1) hold. ThenLemma 3.3 Let assumption (H3.1) hold. Then the following variational inequality holds:Theorem 3.1 Let (H3.1) hold. Let u(·) be an optimal control for our stochastic optimal control problem, (x(·),y(·), z(·), r(·,·)) be the corresponding optimal trajectory, and (p(·),q(·), k(·),L(·,·))be the solution of (3.19). Then we haveTheorem 3.2 Let (H3.1) and (H3.2) hold. Let Zv(·) be FtY-adapted, u(·)∈Uad be an admissible control, and (x(·),y(·), z(·),r(·,·)) be the corresponding trajectories. Let β(·) and (p(·),q(·),k(·), L(·,·)) satisfy (3.17) and (3.19), respectively. Moreover the Hamiltonian H is convex in (x,y,z.r,v), and Then u(·) is an optimal control.Theorem 3.3 Minimizing the cost functional (3.29) over v(·)∈Uad in (3.32), subject to (3.30) and (3.33), formulates a partially observed optim,al control problem. Then the candidate optimal control u(·) in (3.35) is the desired unique optimal control, and its explicit expression is denoted by (3.57).Theorem 3.4 Assume that the hypothesis (H3.1) holds. Let u(·) be an optimal con-trol and{p, (Q, K. K, R), (q, k, k, r)} be the corresponding Ft-adapted square-integrable solution of FBSDEP (3.68). Then the necessary maximum principle is true for any v(·)∈Uad defined by (3.1).Theorem 3.5 Let (H3.1) and (H3.2) hold,ρv(·) be FtY-adapted and u(·)∈Uad be an admissible control with be the corresponding trajectories (x(·),y(·), z(·),z(·), r(·,·)). Further, we suppose that{p,(Q,K,K,R).(q,k,k,r)} satisfies equation (3.68), the Hamiltonian H(t, u(t)) is convex in (x,y, z, z, r, v), and Then u(}) is an optimal control.Chapter 4:We firstly study the differential games of terminal coupled FBSDEs. One of the motivations of this study is the problem of finding a saddle point in an LQ zero-sum differential game with generalized expectation. We give a necessary and a sufficient optimality conditions for the foregoing games. Inspired by finding an equilib-rium point of an LQ nonzero-sum differential game of partial information BDSDEs and well describing so-called informal trading phenomena such as "insider trading" in the market, we also further investigate differential games of partial information FBDSDEs. A necessary and a sufficient conditions for a Nash equilibrium point of nonzero-sum game are given, as well for a saddle point of zero-sum game. Lemma 4.1 Let assumption(H4.1)hold. Then it yields,for i=1,2,Lemma 4.2 Let assumpti.ons (H4.1)and (H4.2) hold. Then the following variational inequality holds for i=1,2:Theorem 4.1 Le (H4.1) and (H4.2) hold. Let(u1(·),u2(·))6e an equilibrium point of ProblemⅠwith the corresponding solutions(X(·),y(·),z(·))and (pi(·),qi(·),ki(·)) of (4.10)and(4.23).Then it follows that and are true for any(v1(·),v2(.))∈u1×u2,a.e.,a.s..Theorem 4.2 Let(H4.1),(H4.2)and(H4.3)hold. Let(u1(·),u2(·))∈u1×u2 with the corresponding solutions(x,y,z)and(pi,qi,ki)of equations(4.10)and(4.23).Suppose exist for all(t,a,b,c)∈[0,T]×Rn×Rm×Rm×d,and are concave in(a,b,c)for all t∈[0,T](the Arrow condition. Moreover Then (u1(·),u2(·)) is an equilibrium point of Problem I.Theorem 4.3 Let the assumptions (H4.1) and (H4.2) hold. Let (u1(·),u2(·))∈u1xu2 be a saddle point of ProblemⅡwith corresponding solutions (x, y, z) and (p. q, k) of equations (4.10) and (4.23) where the Hamiltonian functions H1 and H2 are defined by (4.37) and (4.38) respectively. Then it follows that and are true for any (v1(·), v2(·))∈u1×u2, a.e., a.s..Theorem 4.4 Let (H4.1), (H4.2) and (H4.3) hold. Let(u1(·),u2(·))∈u1×u2 with the corresponding solutions (x,y.z) and (p,q,k) of equations (4.10) and (4.41). Sup-pose that the Hamiltonian function H satisfies the following conditional mini-maximum principle: (i)Assum,e that bothφandγare concave, and exists for all (t, a, b, c)∈[0..T]×Rn×Rm×Rm×d, and is concave in (a,b,c). Then we have and (ⅱ)Assume that bothφandγare convex, and exists for all (t,a,b, c)∈[0,T]×Rn×Rm×Rm×d, and is convex in (a,b,c). Then we have. and (ⅲ)If both (ⅰ) and (ⅱ)are true, then (u1(·),u2(·)) is a saddle point which impliesTheorem 4.5 Let (H4.4) hold and(u1(·),u2(·))be an equilibrium point Problem (NZSG).Further,(y(·),z(·),Y(·),Z(·)) and (pi(·),pi(·),qi(·),qi(·)) are the solutions of(4.62)and(4.63) corresponding to the control(u1(·),u2(·)),respectively.Then it follows that and are true for any(v1(·),v2(·))∈u1×u2,a.e.a.s.Corollary 4.1 Suppose thatε1=F1 for allt.Let (H4.4) hold,and (u1(·),u2(·))be an eguilibrium point of Problem(NZSG).Moreove,(y(·),z(·),Y(·),Z(·)) and (pi(·),pi(·) qi(v),qi(·)) are the solutions of (4.62) and (4.63) corresponding to the control(u1(·), u2(·)),respectively. Then it follows that and are true for any(v1(·),v2(·))∈u1×u2,a.e.a.s.Theorem 4.6 Let.(H4.4) and (H4.5) hold. Let(y,z,Y,Z) and (pi.pi,qi,qi) be the solu-tions of equations (4.62)and(4.63)corresponding to the admissible control(u1(·),u2(·)), respectively. Suppose thatφi andγi are concave Y and y(i=1,2)respectively, and that for all(t,y,z,Y,Z,)∈[0,T]×Rn×Rn×l×Rm×Rm×d, (y,z.Y,v1)→,(t,y,z,Y,Z,v1,u2(t),p1(t),p1(t),q1(t),q1(t)), (y,z,Y,Z,v2)→H2(t,y,z,Y,Z,u1(t):v2,p2(t):p2(t),q2(t),q2(t)) are concave. Moreover, Then (u1(·),u2(·)) is an equilibrium point of Problem (NZSG).