Some Optimal Control And Differential Game Problems In Forward-Backward Stochastic Systems

General nonlinear backward stochastic differential equations (BSDEs, for short) were first introduced by Pardoux, Peng [65] in1990. Since that pioneering paper from1990, the theory of BSDEs has been intensively studied by a lot of researchers attracted by its various applications, namely in stochastic control, stochastic differential game, finance, and the theory of partial differential equations (PDEs, for short). By now, the theory of BSDEs has been a significant branch of probability theory and stochastic analysis. The aim of this paper is to study the development and applications of BSDE in the fields of stochastic control and differential game theory.One core of the modern control theory is stochastic optimal control theory. The stochastic optimal control theory has been widely applied since early1960s. The s-tochastic optimal control we studied is the one that optimizes some cost functional of the stochastic controlled system. There are two main approaches of solving stochastic optimal control problems:Pontryagin’s maximum principle and the Bellman’s dynamic programming principle. The two approaches characterize the optimal control from two aspects:the maximum principle presents the necessary condition of the optimal con-trol; the dynamic programming principle establishes the relationship between a family of optimal control problems with different initial times and initial states and the second-order partial differential equations (HJB equations), the optimal control is derived by optimizing the Hamilton function.From these two aspects, this doctoral thesis studies the stochastic control and d-ifferential game problems of fully coupled FBSDE, fully coupled FBSDE with jumps, functional FBSDE and forward-backward stochastic Volterra integral equations system-atically.Let us introduce the main content and the organization of the thesis. In Chapter1, the Introduction gives an overview of our topics in Chapter2to Chapter5.Chapter2investigates the optimal control problem of fully coupled FBSDEs.This chapter studies the optimal control problem of fully coupled FBSDE from the angle of PDE theory. We carry out the discussion in two cases:the diffusion coefficient a of the forward equation depends on z, but doesn’t depend on the control u;σ depends on the control u. but doesn’t depend on z. For the two cases, we present the probabilistic representation of the viscosity solution of the corresponding HJB equation. For these. we also discuss some useful estimates for fully coupled FBSDE under the monotonicity condition, and the existence, uniqueness theorem, comparison theorem, LP estimates for fully coupled FBSDE on the small interval when assuming the Lipschitz condition and the linear growth condition.This chapter is mainly based on the paper:LI, J.WEI, Q., Optimal control problems of fully coupled FBSDEs and viscosity solutions of Hamilton-Jacobi-Bellman equation. SIAM J. Control Optim., submitted.In Chapter3, inspired by Chapter2, we research the zero-sum stochastic differential game problems of fully coupled FBSDEs with jumps. First we prove the LP estimates for the solution of fully coupled FBSDE with jumps under the monotonicity condition, and the existence, uniqueness theorem, comparison theorem, LP estimates for fully coupled FBSDE with jumps on the small interval under the Lipschitz and linear growth assumptions. Based on these, we investigate the differential game problems. We deal with the problem when σ depends on z and the controls u, v at the same time which improves the method in the second chapter. We derive that the upper and lower value functions are the viscosity solutions of the associated integral-differential operator Isaacs’ equations, respectively. In a spacial case (when σ, h don’t depend on z, k), we get the uniqueness of the viscosity solution, i.e., under the Isaacs’condition, the stochastic differential game has a value.This chapter is mainly based on the following papers:LI, J., WEI, Q., LP estimates for fully coupled FBSDEs with jumps, submitted.LI, J., WEI, Q., Stochastic differential games for fully coupled FBSDEs with jumps, submitted.In Chapter4, we consider the stochastic differential games of functional FBSDE, where the cost functional is defined by the solution Y of the functional FBSDE. We prove that the upper and lower functions are the viscosity solutions of the associat-ed integral-differential operator Isaacs’equation, and generalize the Hamilton-Jacobi-Bellman-Isaacs (HJBI, for short) equation to the path-dependent case.This chapter is mainly based on the paper:JI, S., WEI, Q., The dynamic programming method of stochastic differential game for functional forward-backward stochastic system, Mathematical Problems in Engineer-ing, Volume2013, Article ID958920,14pages, http://dx.doi.org/10.1155/2013/958920In Chapter5, we consider the stochastic maximum principle of forward-backward stochastic Volterra integral equation. In view of the particularity of stochastic Volterra integral equation, we introduce a kind of new control u(·,·) which depends on two time variables t and s. In addition, we regard the terminal process ψ(·) as the control. Assuming the convex control domain and using the duality principle, we characterize the necessary condition of the optimal control. At last, we give the examples to explain the feasibility of the new kind controls.This chapter is mainly based on the paper:JI, S., WEI, Q.,An optimal control problem of forward-backward stochastic Volter-ra integral equations with state constraints, submitted.This paper contains five chapters, we give an overview of the structure and the main results of this dissertation.Chapter1Introduction;Chapter2Optimal control problems of fully coupled FBSDEs and viscosity so-lutions of HJB equations;Chapter3Stochastic differential game problems of fully coupled FBSDE with jumps;Chapter4Stochastic differential game of functional FBSDE and the related path-dependent HJBI equation;Chapter5An optimal control problem of forward-backward stochastic Volterra integral equation with state constriants.Chapter2:we prove that the value function W(t,x) defined through fully coupled FBSDE is deterministic, satisfies DPP and is the viscosity solution of the associated Hamilton-Jacobi-Bellman (HJB, for short) equation; we get some useful estimates of fully coupled FBSDE under monotonicity con-dition, derive the wellposedness and regularity properties of fully coupled FBSDE on the small interval when assuming the Lipschitz and linear growth conditions.We consider the following fully coupled forward-backward stochastic controlled system: For a given process u(·)∈ut,T, we define the associated cost functional as follows: The value function is defined asIn this chapter, we mainly adopt the method introduced by Buckdahn, Li [13]. without assuming the coefficients to be Holder-continuous with respect to the con-trol variable, we prove that the value function is deterministic. Also combined with the notion of backward stochastic semigroup introduced by Peng, we get the dynamic programming principle, and derive the continuity property of the value function with respect to the state and time variables.Proposition2.2.3.(Deterministic) Under (H2.1.1),(H2.1.2), for any (t, x)￡[0, T]×Rn, W(t,x) is a deterministic function in the sense that W(t,x)=E[W(t,x)], P-a.s.Lemma2.2.6.(Monotonicity property) Under the assumptions (H2.1.1),(H2.1.2), for any u∈ut,T, the cost functional J(t,x;u). and the value function W(t,x) are monotonic in the following sense: for each x,x∈Rn, t∈[0, T],Theorem2.2.9.(DPP) Under the assumptions (H2.1.2),(H2.2.1) and (H2.2.2), the value function W(t,x) satisfies the following DPP: there exists sufficiently small δ0>0, such that for any0≤δ≤δ0, t∈[0,T-δ], x∈Rn, where (Xst,x;u,Yst,x;u,Zst,x;u)t≤s≤t+δ is the solution of the following fully coupled FBSDE:Theorem2.2.11.(The continuity property with respect to t) Under (H2.1.2),(H2.2.1) and (H2.2.2), the value function W(t,x) is1/2-Holder continuous in t.Furthermore, we prove the value function to be the viscosity solution of the corre-sponding HJB equation in the following two cases:Case1: We suppose that σ does not depend on z, but depends on u. Theorem2.3.2. Under the assumptions (H2.1.2) and (H2.2.1), the value function W(t,x) is the viscosity solution of the following HJB equation: where t∈[0, T], x∈R,y∈R,p∈R,X∈R,Case2: We suppose that σ depends on z, and does not depend on u.Theorem2.3.10. Under the assumptions (H2.1.2),(H2.2.1),(H2.2.2),(H2.3.1) and (H2.3.2), the value function W(t,x) is the viscosity solution of the following HJB equation: where t∈[0,T], x∈Rn, The Case2is more complicate, the associated HJB equation is combined with an algebraic equation, which is inspired by Wu and Yu [91],[92], we use a new method-the continuation method combined with the fixed point theorem in order to prove for the first time that the algebraic equation has a unique solution, and give the representation for the solution. The following is the important Representation Theorem for the solution of the algebraic equation.Proposition2.3.12. For any s∈[0,T], ζ∈Rd,y∈R,x∈Rn, there exists a unique z such that z=ζ+Dψ(s, x).σ(s, x, y+φ(s, x), z). That means, the solution z can be written as z=h(s, x, y, ζ), where the function h is Lipschitz with respect to y, ζ, and|h(s,x,y, ζ|≤C(1+|x|+|y|+|ζ|). The constant C is independent of s,x,y,ζ And z=h(s, x,y, ζ) is continuous with respect to s.In the proofs of the above results, the Lp estimates of solution of fully coupled FBSDE for arbitrary time intervals under monotonicity condition and the wellposed-ness, regularity properties of solution of FBSDE on the small time interval under the Lipschitz and linear growth conditions play the key role. We present the main useful results as follows:Consider the following fully coupled FBSDE with initial condition(t, ζ)∈[0, T]×L2(Ω,Ft,P;Rn):Proposition2.5.1. Under the assumptions (H2.1.1),(H2.1.2),(H2.5.1),(H2.5.2), for any01≤t≤T and the associated initial states ζ, ζ’∈L2(Ω, Ft,P;Rn), we have the following estimates, P-a,s.: If σ also satisfies:(H2.5.3) for any t∈[0, T] and any (x, y, z)∈Rn×R×Rd, P-a.s.,|σ(t, x, y,z)|≤L(1+|x|+|y|), then we can get The following result is about the continuous dependence of the fully coupled FB-SDE on the terminal condition.Proposition2.5.4. Under the assumptions (H2.1.1),(H2.1.2),(H2.5.1) and (H2.5.2), for any0≤t≤T, the associated initial state ζ∈L2(Ω,Ft,P; Rn) and ξ∈L2(Ω,FT,P;R), we let (Xst,ζ,Yst,ζ,Zst,ζ)s∈[t,T] be the solution of FBSDE (2.5.1) associated with (6, σ,f,ζ,Φ), and (Xst,ζ,Vst,ζ,Zst,ζ)s∈[t,T]be the solution of FBSDE associated with (6, σ,f,ζ,Φ+ξ). Then we haveProposition2.5.8. Under the assumptions (H2.1.1),(H2.1.2),(H2.5.1),(H2.5.2) and (H2.5.3), for anyp≥2,0≤t≤T and the associated initial state ζ∈Lp(Ω,Ft,P;Rn), there exists δ0≥0, which depends on p and Lipschitz constant K and the linear growth constant L, such that where (Xst,ζ,Yst,ζ,Zst,ζ)s∈[t,T] is the solution of FBSDE associated with (6, σ,f,ζ,Φ).In the proofs of the existence of the viscosity solutions, the Lp estimates of fully coupled FBSDE on the small integral are the key points. First we present the existence and uniqueness theorem, comparison theorem for fully coupled FBSDE on the small integral.Proposition2.5.9. We suppose the assumptions (H2.5.1),(H2.5.2),(H2.5.4) hold true, where the assumption (H2.5.4) is the following hypothesis:(H2.5.4) the Lipschitz constant Lσ≥0of σ with respect to z is sufficiently small, i.e., there exists Lσ≥0small enough such that, for all t∈[0,T], u∈U, x1,x1∈Rn,y1,y2∈R,z1,z2∈Rd,P-a.s.,Then, there exists a constant0≤δ0, only depending on the Lipschitz constant K, such that for every0≤δ≤δ0and ζ∈L2(Ω,Ft,P;Rn), FBSDE (2.5.1) has a unique solution (Xst,ζ,Yst,ζ,Zst,ζ)s∈[t,t+δ] on the time interval [t, t+δ].Theorem2.5.11.(Generalized Comparison Theorem) We suppose the assumptions (H2.5.1),(H2.5.2),(H2.5.4) are satisfied. Let δ0>0be a constant, only depending on the Lipschitz constant K, such that for every0≤δ≤δ0and ζ∈L2(Ω,Ft,P;Rn), FBSDE (2.5.1) has a unique solution {Xsi,Ysi,Zsi)s∈[t,t+δ] associated with (b,σ,f,ζ,Φi) on the time interval [t,t+δ], respectively. Then, if for any0≤δ≤δ0we have Φ1(Xt+δ2≥Φ2(Xt+δ2), P-a.s.,(resp., Φ1(Xt+δ1)≥Φ2(Xt+δ1) P-a-s.), we also have Yt1≥Yt2, P-a.s.Proposition2.5.12. Let Φ be deterministic. We suppose the assumptions (H2.5.1).(H2.5.2),(H2.5.4) hold true. Then, for every p≥2, there exists sufficiently small constant5≥0, only depending on the Lipschitz constant K, and some constant Cp.K, only depending on p, the Lipschitz constant K and the linear growth constant L, such that for every0≤δ≤δ and ζ∈Lp(Ω,Ft,P;Rn), where (Xst,ζ,Yst,ζ,Zst,ζ)s∈[t,t+δ] is the solution of FBSDE associated with (b, σ, f,ζΦ) and with the time horizon t+δ.Chapter3, we obtain the Lp estimates for fully coupled FBSDEs with jumps under monotonicity condition, and derive the wellposedness and regularity results of solutions on the small interval when assuming the Lipschitz, lin-ear growth conditions; we prove the lower and upper value functions W(t, x), U(t,x) are the viscosity solutions of the associated HJBI equations, respec-tively.Inspired by Chapter2. we study the stochastic differential game problems of fully coupled FBSDE with jumps. First we still need the Lp estimates of solution of fully coupled FBSDE with jumps under the monotonicity condition and on the small time interval.We consider the following fully coupled FBSDE with jumps: where Πst,ζ=(Xst,ζ,Yst,ζ,Zst,ζ),Πs-t,ζ=(Xs-t,ζ,Yst,ζ,Ζst,ζ). The main estimates are as follows:Proposition3.2.1. Under the assumptions (H3.1.2),(H3.1.3),(H3.2.1), for any0<t <T and any associated initial states ζ,ζ’∈L2(Ω,Ft,P;Rn), we have the following estimates, P-a.s.: If σ, h also satisfy:(H3.2.2)for any t∈[0,T] and any (x,y, z, k)∈Rn×M×Rd×R, P-a.s.,|σ(t,x,y,z,k)|≤L(1+|x|+|y|),|h(t,x,y,z,k,e)|≤ρ(e)(1+|x|+|y|), then we can getProposition3.2.6. Under the assumptions (H3.1.2),(H3.1.3),(H3.2.1),(H3.2.2), for any p≥2,0≤t≤T and the associated initial states ζ,ζ’∈Lp(Ω, Ft,P;Rn), there exists δ0>0which depends on p and the Lipschitz constant and the linear growth constant L, such thatIn order to investigate the existence of the viscosity solution, we still need the existence and uniqueness of the solution of fully coupled FBSDE with jumps, as well as the comparison theorem, Lp estimates as follows:Theorem3.2.9. We suppose the assumptions (H3.1.2),(H3.2.1),(H3.2.3) hold true, where assumption (H3.2.3) is the following:(H3.2.3) The Lipschitz constant Lσ≥0of σ with respect to z, k is sufficiently small, i.e., there exists some Lσ≥0small enough such that, for all t∈[0,T], x1,x2∈Rn,y1,y2∈R,z1,z2∈Rd,k1,k2∈R, Also the Lipschitz coefficient Lh(·) of h with respect to z, k is sufficiently small, i.e. there exists a function Lh: E�'R+with Ch:=max(supLh2(e),∫ELh2(e)λ(de))<e∈E+∞sufficiently small, and for all t∈[0, T], x1,x2∈Rn,y1,y2∈R,z1,z2∈Rd,k1,k2∈R,e∈EThen, there exists a constant δ0>0only depending on the Lipschitz constants K and Lσ,Ch such that, for every0≤δ≤δ0, and ζ∈L2(Ω,Ft,P;Rn), FBSDE (3.2.1) has a unique solution (Πst,ζ,Kst,ζ)s∈[t,t+δ] on the time interval [t, t+δ].Theorem3.2.11.(Generalized Comparison Theorem) We suppose that the assump-tions (H3.1.2),(H3.2.1),(H3.2.3) are satisfied. Let δ0>0be a constant, only depending on the Lipschitz constants K, Lσ and Lh(·), such that for every0≤δ≤δ0and ζ∈L2(Ω,Ft,P;Rn), FBSDE (3.2.22) has a unique solution (Xsi,Ysi,Zsi,Ksi)s∈[t,t+δ] associated with (b,σ, g,ζ,Φi) on the time interval [t. t+δ], respectively. Then, if for any0≤δ≤δ0it holds Φ1(Xt+δ2)≥Φ2(Xt+δ2), P-a.s.(resp.,Φ1(Xt+δ1)≥Φ2(Xt+δ1), P-a.s.), we also get Yt1≥Yt2, P-a.s.Proposition3.2.12. Let Φ be deterministic, and suppose the assumptions (H3.1.2),(H3.2.1),(H3.2.3),(H3.2.4) hold true. Then, for every p>2, there exists a sufficient-ly small constant5>0, only depending on the Lipschitz constants K and Lσ, Lh(·), and some constant Cp,k, only depending on p, the Lipschitz constants K, Lσ, Lh(·) and the linear growth constant L, such that for every0<5<5and ζ∈Lp(Ω,Ft,P;Rn), where (Xst,ζ,Ystζ,Zst,ζ, Kst,ζ)s∈[t,t+δ] is the solution of FBSDE (3.2.22) associated with (6,σ, g,ζ,Φ) and the time horizon t+δ.The following results are about the stochastic differential game.Suppose that the control state spaces U, V are compact metric spaces. By u (resp., V) we denote the admissible control set of all U (resp., V)-valued Ft-predictable processes for the first (resp., second) player. If u∈u (resp., v∈V), we call u (resp., v) an admissible control.Inspired by Li, Wei [54], we research stochastic differential game of fully coupled FBSDE with jumps as follows: wheres∈[t,T], Πst,ζ;u,v=(Xst,ζ;u,v,Yst,ζ;u,v, Zst,ζ;u,v) and Πs-t,ζ;u,v=(Xs-t,ζu,v;Ys-t,ζ;u,v,Zst,ζ;u,v)For given processes u(·)∈Ut,T,v(·)∈Vt,T, the cost functional is defined as follows: where the process Yt,x;u,v is defined by the above FBSDE.For ζ=x∈R, we define the lower value function of our stochastic differential games and its upper value functionSimilar to the first section, we also get that the lower value function W and the upper value function U are deterministic, monotonic, continuous and satisfy DPP. The main results are as follows: Theorem3.4.5. The lower value function W, the upper value function Uare the viscosity solution of the following second-order Isaacs’ type integral-partial differential equation: where where t∈[0,T], x∈R,φ:[0,T]×R�'R, ψ:[0,T]×R×U×V�'Rd.Compared with Chapter2, the research of the viscosity solution in Chapter3has been improved inspired by [52]. Unlike the discussion in Chapter2, we consider the case when σ, h depend on z and the controls u, v at the same time.Also, when σ, h do not depend on y, z, k, we get the uniqueness of the viscosity solution.Chapter4: we prove the lower and upper value functions defined by the decoupled functional FBSDE are the viscosity solutions of the corresponding HJBI equation, respectively.There really exist some systems which are modeled only by stochastic systems whose evolutions depend on the past history of the states. Bases on this phenomenon, It is necessary to study the functional controlled system.The dynamic of the stochastic differential game is described by the following func-tional SDE:The cost functional J(γt;u,v) is denned as Yγt;u,v(t), which is the solution of the fol-lowing functional BSDE: where γt is a path on [0, t]. The two equations compose a decoupled functional FBSDE.For γt∈A, the lower value function and the upper value function are defined as follows:As we known, the essential infimum and essential supremum of a family random variables are still random variables. But by the Girsanov transformation method intro-duced by Buckdahn, Li [13], we get W(γt), U(γt) is deterministic.Proposition4.2.4. For any t∈[0, T], γt∈∧, W(γt) is a deterministic function, i.e., W(γt)=E[W(γt)], P-a.s.By introducing the notion of stochastic backward semigroup, we have the following DPP:Theorem4.2.8. Suppose (H4.2.1), the lower value function W(γt) obeys the follow-ing DPP: for any t∈[0, T],γt∈∧, δ>0,After getting the determinacy, continuity and DPP of the value functions, we have the following main result:Theorem4.3.2. The lower value function and the upper value function are the viscosity solution of the following path-dependent HJBI equations, respectively:where (γt, y,p, X)∈∧×M×Rd×Sd (Sd denotes the set of d×d symmetric matrices). Chapter5: we research the optimal control problem of forward-backward stochastic Volterra integral equations with state constraints; using the Eke-land’s variational principle, we get one kind of variational inequality; by the duality principle, we derive the maximum principle of the optimal control.In this chapter, we study the optimal control problem of the following equation: where f(·)∈LF2(0,T), b: Ω×Δ[0, T]×Rm×K�'Rm, σ:Ω×Δ[0, T]×Rm×K�'Rm×d, g:Ω×Δc[0,T]×Rm×Rm×Rm×d×K�'Rm.The control pair (ψ(·), u(·,·))∈u, where K is the nonempty convex subset of Rm.For each pair (ψ(·), u(·,·))∈u, we consider the following cost functional: where l1: Δ[0, T]×Rm×K�'R,l2: Δc[0, T]×Rm×Km×Em×d×K�'R,q:Rm�'R,h: Rm�'R,k:Rm�'R.Our control problem is where ρ:[0, T]�'Rm is continuous and satisfies∫0T|ρ(t)|2dt <∞.We introduce a new type of controls u(·,·) which depend on both time variables t and s. In our model, as time goes by, the controlled system is changing, therefore it is reasonable to keep changing the controls to get more better objectives. Now we haven’t found the finance applications for such kind of controls.Using the Ekeland’s variational principle, we get one kind of variational inequality; and by the duality principle, we derive the following maximum principle of the optimal control: Theorem5.2.7. Assume that (A1)-(A4) hold and l1, l2=0. Let {ψ*(·), u*(·,·)) be the optimal control pair;(X*(·), Y*(·), Z*(·,·)) be the corresponding optimal trajectory. Then there exist a deterministic function h0(·)∈Rm, h0∈Rm, h1,h1,h2≤0such that (?)(ψ(·),u(·,·))∈u, where (m(·), n(·,·),p(·)) is the solution of the adjoint equation.