| In 1990,Peng[74]first introduced the second-order term in the Taylor expansion of the variation and obtained the global maximum principle for the classical stochas-tic optimal control problem with control-dependent diffusion case.Since then,many researchers investigate this kind of optimal control problems for various stochastic sys-tems.In 1993,Peng[76]first established a local stochastic maximum principle for the classical stochastic recursive optimal control problem.But when the control domain is nonconvex,one encounters an essential difficulty when trying to derive the first-order and second-order expansions for BSDE and it is proposed as an open problem in Peng[78].Recently,Hu[39]studied this open problem and obtained a completely novel global maximum principle.As for the relationship between the MP and the DP-P,within the framework of viscosity solution,Nie,Shi and Wu[68]studied the local case;Nie,Shi and Wu[69]studied the general case with the help of the first-order and second-order adjoint equations which are introduced in Hu[39].In this thesis,we mainly study a stochastic optimal control problem for fully coupled forward-backward stochastic control systems with a nonempty control domain,dynamic programming principle and the existence and uniqueness for the related HJB equation,the relationship between the maximum principle and dynamic programming principle.There are six chapters in this thesis.The first chapter is the introduction,and the rest parts are organized as follows.Chapter 2:A global stochastic maximum principle for fully coupled forward-backward stochastic systemsIn this Chapter,we study a stochastic optimal control problem for fully coupled forward-backward stochastic control systems with a nonempty control domain.For our problem,the first-order and second-order variational equations are fully coupled linear FBSDEs.Inspired by Hu[39],we develop a new decoupling approach by introducing an adjoint equation which is a quadratic BSDE.By revealing the relations among the terms of the first-order Taylor's expansions,we estimate the orders of them and derive a global stochastic maximum principle which includes a completely new term.Applications to stochastic linear quadratic control problems are investigated.We first give an Lp-estimate for the following fully coupled forward-backward stochastic differential equation(FBSDEs):For p>1,set?p:=Cp2p+1(1+Tp)c1p,(0.47)where c1 ? max{L2,L3},Cp is defined in Lemma A.1 in appendix.Theorem 0.1.Suppose Assumption 2.1 holds and ?p<1 for some p>1.Then the equation(0.46)admits a unique solution(X(·),Y(·),Z(·))? LFp(?;C([0,T],Rn))×where C depends on T,p,L1,c1.Consider the following fully coupled stochastic control system:Our optimal control problem is to minimize the cost functional J(u(·))=Y(0)over U[0,T]:In order to obtain stochastic maximum principle,we need to introduce the first-order and second-order adjoint equation as follows whereK1(t)=(1-p(t)?z(t))-1[?x(t)p(t)+?y(t)+?y(t)p2(t)+q(t)],(0.51)where H(t,x,y,z,u,p,q)+g(t,x,y,z,u,)+pb(t,x,y,z,u)+q?(t,x,y,z,u),K2(t)=(1-p(t)?z(t))-1{p(t)?y(t)+2[?x(t)+?y(t)p(t)+?z(t)K1(t)]}P(t)+(1-p(t)?z(t))-1{Q(t)+p(t)[1,p(t),K1(t)]D2?(t)[1,p(t),K1(t)]T}.Define H(t,x,y,z,u,p,q,P)=pb(t,x,y,z+?(t),u)+q?(t,x,y,z+?(t),u)(0.53)+1/2P(?(t,x,y,z,+?(t),u-?(t,X(t),Y(t),Z(t),u(t)))2+g(t,x,y,z+?(t),u),where ?(t)is defined by,for t ?[0,T]?(t)=p(t)(?(t,X(t),Y(t),Z(t)+?(t),u)-?(t,X(t),Y(t),Z(t),u(t))).(0.54)Then we obtain the following maximum principle.Theorem 0.2.Suppose Assumptions 2.4,2.7 and 2.14 hold.Let u(·)? U[0,T]be optimal and(X(·),Y(·),Z(·))be the corresponding state processes of(0.48).Then the following stochastic maximum principle holds:H(t,X(t),Y(t),Z(t),u,p(t),q(t),P(t))(0.55)?H(t,X(t),Y(t),Z(t),u(t),p(t),q(t)P(t)),(?)u?U,a.e.,a.s.,where(p(·),q(·)),(P(·),Q(·))satisfy(0.50),(0.52)respectively,and ?(·)satisfies(0.54).We also considered the case without assumption of q(·)ia bounded.However,we just prove the maximum principle under ?(t,x,y,z,u)= A(t)z +? 1(t,x,y,u)and||A(·)||? is small enough.In this case,the first-order adjoint equation becomes where K1(t)=(1-p(t)A(t))-1[?x(t)p(t)+?y(t)p2(t)+q(t)].And the second-order adjoint equation becomes:where H(t,x,y,z,u,p,q)=g(t,x,y,z,u)+pb(t,x,y,z,u)+q?(t,x,y,z,u)?K2(t)=(1-p(t)A(t))-1{p(t)?y(t)+2[?x(t)+?y(t)p(t)+A(t)K1(t)]}P(t)+(1-p(t)A(t))-1{Q(t)+p(t)[1,p(t)]D2?1(t)[1,p(t)]T}.By the same analysis as in Theorem 0.2,we obtain the following maximum principle.Theorem 0.3.Suppose Assumptions 2.4,2.7 and 2.23 hold.Let u(·)?U[0,T]be optimal and(X(·),Y(·),Z(·))be the corresponding state processes of(0.48).Then the following stochastic maximum principle holds:H(t,X(t),Y(t),Z(t),u,p(t),q(t),P(t))?H(t,X(t),Y(t),Z(t),u(t),p(t),q(t),P(t)),(?)u?Ua.e.,a.s..By similar analysis as for 1-dimensional case,we can obtain the case of d-dimensional Brownian motion in(0.48).Moreover,we study a linear quadratic control problem by the above results.For simplicity of presentation,we suppose all the processes are one dimensional.Consider the following linear forward-backward stochastic control system and minimizing the following cost functional J(u(·))=E[?t0(A4(t)X(t)2+B4(t)Y(t)2+C4(t)Z(t)2+D4(t)u(t)2)dt+GX(T)2+Y(0)2],(0.59)where Ai,Bi,Ci,Di,i = 1,2,3,4 are deterministic R-valued functions,F,G are deterministic constants and J is FT-measurable bounded random variable.We obtain the following stochastic maximum principle for problem(0.58)-(0.59).Theorem 0.4.Suppose Assumptions 2.4 and.2.7 hold.Let u(·)? U[0,T]be optimal and(X(·),Y(·),Z(·))be the corresponding state processes of(0.58).Then the following stochastic maximum principle holds:(D1(t)m(t)+D2(t)+D3(t)h(t)+2D4(t)u(t))(u-u(t))+[C4p(t)D2(t)2/(1-p(t)C2(t))2+D4(t)+P(t)D2(t)2](u-u(t))2?0,(?)u?U,a.e.,a.s.,where p(·)satisfies the following ODE with K1(t)=(1-p(t)C2(t))-1[A2(t)p(t)+B2(t)p2(t)],P(·)satisfies the following ODE,where R1(t)=2(A1(t)+B1(t)p(t)+C1(t)K1(t))+(A2(t)+B2(t)p(t)+C2(t)k1(t))2.Chapter 3:The existence and uniqueness of viscosity solution to a kind of generalized Hamilton-Jacobi-Bellman equationIn this Chapter,we study the existence and uniqueness of viscosity solutions to generalized Hamilton-Jacobi-Bellman(HJB)equations combined with algebra equa-tions.This generalized HJB equation is related to a stochastic optimal control problem for which the state equation is described by a fully coupled FBSDE.With the help of the uniqueness of the solution to FBSDEs,we propose a novel probabilistic approach to study the uniqueness of the solution to this generalized HJB equation.We obtain that the value function is the minimum viscosity solution to this generalized HJB equation.Especially,when the coefficients are independent of the control variable or the solution is smooth,the value function is the unique viscosity solution.We study the existence and uniqueness of viscosity solution to the following gen-eralized HJB equation combined with an algebra equation Here H(·)is defined as follows H(t,x,v,p,A,u)=1/2tr[??T(t,x,v,V(t,x,v,p,u),u)A]+pTb(t,x,u,V(t,x,v,p,u),u)+g(t,x,u,V(t,x,v,p,u),u),(0.63)(t,x,v,p,A,u)?[0,T]×Rn×R×Rn×Sn×U,V(t,x,v,p,u)is the solution to the following algebra equation V(t,x,v,p,u)=PT?(t,x,v,V(t,x,v,p,u),u).This kind of problem has the following stochastic optimal control interpretation.Let t?[0,T],??L2(Ft;Rn)and an admissible control u(·)?U[t,T].Consider the following controlled fully coupled FBSDE:For each given(t,x)?[0,T]× Rn,define the value functionWe prove that this value function satisfies the following dynamic programming principle.Theorem 0.5.Suppose Assumption 3.1 holds.Then for each(t,x)?[0,T)×Rn and??(0,T-t],we haveIn addition,we show that the value function W(t,x)defined in(0.65)is a viscosity solution to the HJB equation(0.62)in the following Theorem.Theorem 0.6.Suppose Assumptions 3.1 and 3.13 hold.Then the value function W(t,x)is the viscosity solution to the HJB equation(0.62).As to the uniqueness of the viscosity solution to the HJB equation(0.62).We consider three cases.(?)? independent of y and z,that is the following Theorem.Theorem 0.7.Suppose Assumption 3.1(i)holds.Then there exists at most one viscosity solution to(0.62)without variable y and z in the class of continuous functions which are Lipschitz continuous with respect to x.(?)? depends on y and z,we separate this case into three subcases.The first case is that ? independent of z.Theorem 0.8.Suppose ? is independent of z,W is a viscosity solution to HJB equation(0.62);and one of the following two conditions holds true:(i)Assumption 3.1 holds;(ii)Assumptions 3.1(i)and 3.20 hold.Moreover,satisfies Assumption 3.20.Let W be the value function.Furthermore,we assume that W is Lipschitz continuous in x.Then W ? W.The second case is that ? dependent on y and z.Theorem 0.9.Suppose one of the following two conditions holds true:(i)Assumptions 3.1 and 3.13 hold;(ii)Assumptions 3.1(i),3.13 and 3.20 hold.Let W be the value function and W be a viscosity solution to HJB equation(0.62).Furthermore,we assume that W is Lipschitz continuous in(t,x),DW is Lipschitz continuous in x and ||DW||?L3<1.Then W<W.The third case is that the coefficients of the controlled system b,? and g are independent of control variable u.It is obviously that for this case the corresponding HJB equation(0.62)degenerates to a semilinear parabolic equation with an algebra equation.Theorem 0.10.Suppose b,a and g are independent of control variable u,? is inde-pendent ofz,W is a viscosity solution to HJB equation(0.62);and one of the following two conditions holds true:(i)Assumption 3.1 holds;(ii)Assumptions 3.1(i)and 3.20 hold.Moreover,satisfies Assumption 3.20.Let W be the value function.Furthermore,we assume that W is Lipschitz continuous in x.Then W = W.Similarly,we have the following theorem.Theorem 0.11.Suppose b,? and g are independent of control variable u;and one of the following two conditions holds true:(i)Assumptions 3.1 and 3.13 hold;(?)Assumptions 3.1(i),3.13 and 3.20 hold.Let W be the value function and W be a viscosity solution to HJB equation(0.62).Furthermore,we assume that W is Lipschitz continuous in(t,x),DW is Lipschitz continuous in x and ||DW||?L3<1.Then W = W.(?)The smooth caseIn this case,we assume that the solution of the HJB equation W ? C1,2([0,T]×Rn).Then,we have the following theorem.Theorem 0.12.Suppose one of the following two conditions holds true:(i)Assumptions 3.1 and 3.13 hold;(ii)Assumptions 3.1(i),3.13 and 3.20 hold.Let W be the value function and W ?C1,2([0,T]×Rn)be a solution to the HJB equation(0.62).Furthermore,we assume ||?||?<?,||DW||?L3<1 and ||D2W||?<?.Then W=W.Chapter 4:Stochastic maximum principle,dynamic programming prin-ciple,and their relationship for fully coupled forward-backward stochastic control systemsWithin the framework of viscosity solution,we study the relationship between the maximum principle(MP)in Chapter 2 and the dynamic programming principle(DPP)in Chapter 3 for a fully coupled forward-backward stochastic controlled system(FBSCS)with a nonconvex control domain.For a fully coupled FBSCS,both the corresponding MP and the corresponding Hamilton-Jacobi-Bellman(HJB)equation combine an algebra equation respectively.So this relationship becomes more com-plicated and almost no work involves this issue.With the help of a new decoupling technique,we obtain the desirable estimates for the fully coupled forward-backward variational equations and establish the relationship.Furthermore,for the smooth case,we discover the connection between the derivatives of the solution to the algebra e-quation and some terms in the first-and second-order adjoint equations.Finally,we study the local case under the monotonicity conditions as in[55,87]and obtain the relationship between the MP in[87]and the DPP in[55].Consider the following controlled fully coupled FBSDE:for s?[t,T],For each given(t,x)?[0,T]×R,define the cost functional J(t,x;u(.))=Yt,x;u(t),(0.67)and the value function as follows:We introduce the following generalized HJB equation combined with an algebra equa-tion for W(·,·):where G(t,x;W(t,x),V(t,x,u),u)=Wx(t,x)·b(t,x,W(t,x),V(t,x,u),u)+1/2Wxx(t,x)(?(t,x,W(t,x),V(t,x,u),u))2(0.70)+g(t,x,W(t,x),V(t,x,u),u).The corresponding stochastic maximum principle as shown in last part,The viscosity solution to(0,69)can be equivalently defined by sub-jets and super-jets.The notions of second-order super-and sub-jets in the spatial variable x are defined as follows.For w?C([0,T]×R and(t,x)?[0,T]×R,defineTheorem 0.13.Let Assumptions 4.1,4.8 and 4.10 hold.Let u(.)be optimal for problem(0.68),and let(p(·),q(·))and(P(·),Q(·))?(0,T;R)×L2?1F([0,T];R)be the solution to equation(0.50)and(0.52)respectively.Furthermore,suppose that q(·)is bounded.ThenSimilarly,the notions of right super-and sub-jets in the time variable t are defined in what follows.For w ? C([0,T]× R)and(t,x)?[0,T)×R,defineTheorem 0.14.Suppose the same assumptions as in Theorem 0.13.Then,for each s ?[t,T],where H1(s,Xt,x;u(s),Yt,x;u(s),Zt,x;u(s))=-H(s,Xt,x;u(s),Yt,x;u(s),Zt,x;u(s),u(s),p(s),q(s),P(s))+P(s)?(s)2.We consider three special cases.(I)the value function W is supposed to be smooth.(?)the diffusion term a of the forward stochastic differential equation in(0.66)is linear in z.(?)the local case in which the control domain is convex and compact.(?)The value function W is supposed to be smooth.Theorem 0.15.Let Assumptions 4.1 4.8 and 4.10 hold.Letw(t,x)? C1,2([0,T]×R)be a solution of the HJB equation(0.69).If||?||?<? and ||Wx||?||?z||?<1,then w(t,x)?J(t,x;u(·)),(?)u(·)?Uw[t,T],(t,x)?[0,T]×R.Furthermore,if u(·)?Uw[t,T]such that G(s,Xt,x;u(s),w(s,Xt,x;u(s)),v(s,Xt,x;u(s),u(s)),u(s))+ws(s,Xt,x;u(s))= 0,where(Xt,x;u(·),Yt,x;u(·),Zt,x;u(·))is the solution to FBSDE(0.66)corresponding to u(·)?nd v(s,x,u)= wx(t,x)?(s,x,w(s,x),v(s,x,u),u),(?)(s,x)?[t,?]?R,u?U,then u(·)is an optimal control.We study the relationship between the derivatives of the value function W and the adjoint processes.Theorem 0.16.Let Assumptions 4.1,4.8 and 4.10 hold.Suppose that u(·)? Uw[t,T]is an optimal Control,and(Xt,x;u(·),Y,x;u(·),Zt,x;u(·))is the corresponding optimal s-tate.Let(p(·),q(·))be the solution to(0.50).If the value function W(·,·)?C1,2([t,T]x R),then Yt,x;u(s)= W(s,Xt,x;u(s)),Zt,x;u(s)=V(s,Xt,x;u(s),u(s)),s?[t,T]and-Ws(s,Xt,x;u(s))=G(s,Xt,x;u(s),W(s,Xt,x;u(s)),V(s,Xt,x;u(s),u(s)),u(s))=min G(s,Xt,x;u(s),W(s,Xt,x;u(s)),V(s,Xt,x;u(s),u),u),s?[t,T].Moreover,if W(·,·)? C1,3([t,T]x R)and Wsx(·,·)is continuous,then,for s ?[t,T],p(s)=Wx(s,Xt,x,u(S)),q(s)=Wxx(s,Xt,x,u(s))?(s,Xt,x;u(s),Yt,x;u(s),Zt,x;u(s),u(s)).Furthermore,if W(·,·)?C1,4([t,T]×R)and Wsxx(·,·)is continuous,then P(s)?Wxx(s,Xt,x;u(s)),s?[t,T],where(P(·),Q(·))satisfies(0.52).If the value function is smooth enough,we can use the DPP to derive the MP in the following theorem.Theorem 0.17.Let Assum,ptions 4.1,4.8 and 4.10 hold.Suppose that u(·)?uw[w,T]is an optimal control,and(Xt,x;u(·),Yt,x;u(·),Zt,x;u(·))is the corresponding optimal state.Let(p(·),q(·))and(P(·),Q(·))be the solutions to(0.50)and(0.52)respectively.IfW(·,·)? C1,4([t,T]× R)and Wsxx(·,·)is continuous,then H(s,Xt,x;u(s),Yt,x;u(s),Zt,x;u(s),u,p(s),q(s),P(s))?H(s,Xt,x;u(s),Yt,x;u(s),Zt,x;u(s),u(s),p(s),q(s),P(s)),(0.72)(?)u?U,a.e.,a.s..(?)The diffusion term ? of the forward stochastic differential equation in(0.66)is linear in z.Under this case,we do not need the assumption that q(·)is bounded.Theorem 0.18.Suppose Assumptions 4.1,4.8,4.10 atnd 4.21 hold.Let u(·)be optimal for our problem(0.68),and let(p(·),q(·))?LF?(0,T;R)×LF2,2(0,T;R)and(P(·),Q(·))?LF2(?;C(0,T],R))x LF2,2(0,T;R)be the solution to equation(0.50)and(0.52)respectively.ThenTheorem 0.19.Suppose the same assumptions as in Theorem 0.18.Then,for each s?[t,T],where H1(s,Xt,x;u(s);Yt,x;u(s),Zt,x;u(s))=-H(s,Xt,x;u(s),Yt,x;u(s)Zt,x;u(s),u(s),p(s),q(s),P(s))+P(s)?(s)2.(?)The local case.The control domain is assumed to be a convex and compact set.Under monotonicity condition,we prove the relationship between MP and DPP.Theorem 0.20.Suppose Assumptions 4.1(i)and 4.24 hold.Let u(·)be optimal for our problem(0.68)and(h(·),m(·),n(·))be the solution to FBSDE(4.83).If L3 is small enough,then Dx1,-W(s,Xt,x;u(s))(?){m(s)h-1(s)}(?)D1x+W(s,Xt,x;u(s)),(?)s?[t,T],P-a.s..Theorem 0.21.Suppose Assu,mptions 4.1(i)and 4.24 hold.Let u(·)be optimal for problem(0.68)and(h(·),m(·),n(·))be the solution to FBSDE(4.83).If L3 is small enough and the value function W(·,·)? C1,2([t,T]× R),then,Yt,x;u(s)=W(s,Xt,x;u(s)),(0.73)Zt,x;u(s)=V(s,Xt,x;u(s),W(s,Xt,x;u(s)),Wx(s,Xt,x;u(s)),u(s)),s?[t,T]and for any s?[t,T],-Ws(s,Xt,x;u(s))=G(s,Xt,x;u(s),W(s,Xt,x;u(s)),Wx(s,Xt,x;u(s)),Wxx(s,Xt,x;u(s)),u(s))(0.74)=min u?U G(s,Xt,x;u(s),W(s,Xt,x;u(s)),Wx(s,Xt,x;u(s)),Wxx(s,Xt,x;u(s)),u).Moreover,if W(·,·)?C1,3([t,T]×R)and Wsx(·,·)is continuous,then,for s ?[t,T],m(s)=Wx(s,Xt,x;u(s))h(s),n(s)=(1-Wx(s,Xt,x;u(s))?z(s))-1bz(s),(Wx(s,Xt,x;u(s)))2(0.75)+gz(s)Wx(s,Xt,x;u(s))?(s)h(s),and(?)u?U,a.e.s?[t,T],P-a.s.<Htu(s,Xt,x;u(s),Yt,x;u(s),Zt,x;u(s),h(s),m(s);n(s)),u-u(s))?0.(0.76)Chapter 5.A Stochastic Maximum Principle for Linear Quadratic Problem with Nonconvex Control DomainWe consider the stochastic linear quadratic optimal control problem in which the control domain is nonconvex.By the functional analysis and convex perturbation methods,we establish a novel maximum principle.And finally we apply the obtained stochastic maximum principle to a concrete example.For a given x ? Rn,consider the following linear stochastic differential equation:where A,B,C,D,b,? are deterministic matrix-valued functions of suitable sizes.In addition,the quadratic cost functional is given by J(u(·))=E{1/2?to[?Q(t)X(t),X(t)?+2?S(t)X(t),u(t)?+?R(t)u(t),u(t)?]dt(0.78)+1/2?GX(T),X(T)?},where G ?Sn,Q,and R are Sn,Rk×n-and Sk-evalued functions,respectively.Let B:={0,1}k,C(?)Rk be a cloesd and convex set,U=C?B??,e=(1,1,…,1)T?Rk,E={u?Rk:0?u?e}and U=E?B.We set uad:={u(·)?L2F(0,T;Rk)|u(t)?U,a.e.,a.s.}.An element of Uad is called an admissible control.Our stochastic linear quadratic control problem is to find an admissible control u(·)such that By functional analysis,we can obtain another form of the cost functional:J(u(·))=1/2{<Nu(·),u(·)>+2<H(x),u(·)>+M(x)},(0.80)whereWe first transform the original linear quadratic problem into a quadratic optimiza-tion problem in a Hilbert space by functional analysis approach.Then through the introduced parameter,we can turn the original problem into a concave control problem with convex control domain.Then we can apply the stochastic maximum principle,which can deal with stochastic control problem with convex control domain,to the transformed concave problem,and obtain the new stochastic maximum principle.Theorem 0.22.Suppose Assumption 5.1 hold and let(X(·),u(·))be an optimal pair of the stochastic linear quadratic control problem(0.77)-(0.79).Then there exists an adapted solution(p(·),q(·))?LF2(0,T;Rn)×LF2(0,T;Rn)to the following backward stochastic differential equation such that where the Hamiltonian function H?is defined by H?(t,x,u,p,q)=<p,A(t)x+B(t)u+b(t)>+<q,C(t)x+D(t)u+?(t)>(0.82)-1/2{[<Q(t)x,x>+2<S(t)x-?I,u>+<(R(t)+?I)u(t),u(t)>]},the parameter ? is defined as;-? is the largest eigenvalue of N in(0.81),the operator I is identity operator.Chapter 6.The stochastic maximum principle in singular optimal con-trol with recursive utilitiesIn this chapter,we consider stochastic recursive optimal control problem,in which the control variable has two components with the first absolutely continuous and the second singular.The control domain of the first component needs not to be convex.By using a spike variation on the absolutely continuous part of the control and a convex perturbation on the singular one respectively,we obtain a stochastic maximum principle of the optimal control.Also,we give the relationship of the backward variational equation,the adjoint equation and forward variational equation.Consider the following state equation:The performance functional is defined by the solution of a backward stochastic differ-ential equation at time 0.More precisely,J(u?)= Y(0),(0.84)where Y(t)=(?)(X(T))+?Tt f(s,X(s),Y(s),Z(s),u(s))ds-?Tt Z(s)dB(s),(0.85)The stochastic recursive control problem is to minimize J(u,?)subject to(0.83)over U.In order to obtain the variational inequality,we introduce the following adjoint equation:And the second order adjoint equation(P(t),Q(t))satisfies the following equation:Define the Hamiltonian as follows:H(t,X,Y,Z,u,p,q,P)=<p,b(t,X,u)>+<q,?(t,X,u)>+1/2<P(?(t,X,u)-?(t,X,u)),(?(t,X,u)-?(t,X,u))>+f(t,X,Y,Z+<p,(?(t,X,u)-?(t,X,u))>,u)where(p,q,P)is given in equation(0.86)and(0.87).Then we have the following theorem.Theorem 0.23.Suppose Assumptions 6.2 and 6.3 hold.Let(u,?)be an optimal control minimizing the cost J over u and(X,Y,Z)denote the corresponding solution,then there exist two unique couples of adapted processes(p(·),q(·))?LF2([0,T];Rn)×LF2([0,T];Rn),(P(·),Q(·))?LF2([0,T];Rn×n)×LF2([0,T];Rn×n),which are respectively solutions of backward stochastic differential equations(0.86)and(0.87)such that H(t,X(t),Y(t),Z(t),u(t),p(t),q(t),P(t))(0.88)?H(t,X(t),Y(t),Z(t),v,p(t),q(t),P(t)),(?)u?U1,a.e.,a.s.,P{(?)t?[0,T],(?):GiT(t)p(t)?0}1,(0.89)where GiT(t)is the transpose of the i-th column of G(t),i = 1,2,...m. |
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