Mean-Field Forward-Backward Stochastic Differential Equations And Related Optimal Control, Differential Game Problems

Mean-field stochastic differential equations, namely McKean-Vlasov equations, have been widely used in many fields, such as, Statistical Mechanics, Physics, Quantum Me-chanics and Quantum Chemistry. In recent works, Lasry and Lions [57] extended the application fields of mean-field problems to Economics, Finance and game theory. Fol-lowing the development of the theory of mean-field stochastic differential equations (S-DEs), many researchers have been interested in partial differential equations (PDEs) of mean-field type, and tried to investigate this kind of equations with stochastic ap-proach. Inspired by the work of Lasry and Lions [57], Buckdahn. Djehiche, Li and Peng [17] investigated a special mean-field problem in a purely stochastic approach, and obtained a new kind of backward stochastic differential equations (BSDEs) under mean-field framework, namely mean-field BSDEs. Following their pioneering paper, Buckdahn, Li and Peng [20] proved in 2009 the existence and the uniqueness for mean-field BSDEs under Lipschitz condition, and they also gave a probabilistic interpretation for associ-ated nonlocal PDEs. Their both works inspired several researchers to study mean-field forward-backward stochastic differential equations (FBSDEs). A lot of works on the theory of mean-field FBSDEs and on its applications have been done recently.On the other hand, since Bellman [6] introduced the dynamic programming prin-ciple, his approach has become a main tool to investigate optimal control problems. It has established the relationship between controlled SDEs and Hamilton-Jacobi-Bellman PDEs, which has made possible to give a probabilistic interpretation for the associat-ed PDEs. The second and the third chapter in this paper investigate optimal control problems and stochastic differential games (SDGs) for decoupled mean-filed FBSDEs, whereas the forth and the five chapter are devoted to optimal control problems for ful-ly coupled FBSDEs in mean-field framework. A direct technical difficulty is that the dynamic programming principle does not hold true any more in the mean-field context. To overcome this difficulty, we have to freeze partially initial parameters, which leads to new kinds of FBSDEs as BSDEs coupled with the value function, BSDEs in games which are coupled with the upper and low value functions, but also to different kinds of fully coupled FBSDEs involving the value function.In 2013, the course given by P.L. Lions at College de France (or see the notes on this course written by Cardaliaguet [23]) has given a lot of impulses to the investigation of mean-field problems. In his course, Lions defined that a function f:P2(Rn)(?)R is called differentiable with respect to a measure, if the function f(ζ):=f(Pζ),ζ∈L2(Ω) is Frechet differentiable over the space of square integrable random variables L2(Ω). In-spired by this idea, Carmona and Delarue [27], [28] studied the master equation for large population equilibrium and FBSDEs, as well as controlled McKean Vlasov dynamics. Cardaliaguet [24] proved the existence and uniqueness of a weak solution for first or-der mean field game systems with local coupling by variational methods. In order to overcome the absence of the flow property in the classical mean-field SDE, Buckdah-n,Li, Peng and Rainer [21] split it into two decoupled mean-field SDEs, whose unique solution (Xt,ζ. Xt,x,Pζ) satisfies flow property. Moreover, for a sufficiently regular func-tion Φ, the value function involving the measure V(t,x,Pζ)= E[Φ(XT,t,x,Pζ, PXTt,ζ)] is the unique classical solution of the associated PDE. This approach made a great progress in overcoming partial freezing of initial data in order to obtain dynamic programming principle, see Buckdahn, Li and Peng [20]. Hao and Li [41], [42], [43], [44]. Based on the work of Buckdahn, Li, Peng and Rainer [21], mean-field SDEs with jumps are investigated and associated PDEs are interpreted stochastically in the sixth chapter.This paper mainly considers two topics:the first one is to investigate, under partially fixed initial data, optimal control problems and stochastic game problems for decoupled mean-field FBSDEs, as well as optimal control problems for fully coupled mean-field FBSDEs and general fully coupled mean-field FBSDEs; and the second one is to study mean-field SDEs with jumps, as well as associated nonlocal PDEs.In what follows we detail the content and structure of our paper in detail.Chapter I introduces the problems studied in Chapter II to Chapter VI.In Chapter II, for partially fixed initial values optimal control problems for con-trolled systems of mean-field BSDEs are considered. A new kind of BSDEs. named BSDEs coupled with the value function, is investigated. Making use of an iteration method, under the Lipschitz condition for the driving coefficient/the existence and the uniqueness for this kind of equations is obtained. When the coefficient f is non-decreasing in y’,a classical comparison theorem argument allows to show a comparison theorem of this new BSDEs. Thanks to a partial freezing of the initial data, the dynamic programming principle still holds true. With a new method which is different to BSDE approach introduced first by Peng in 1997, we prove the existence and uniqueness of viscosity solutions of the associated nonlocal HJB equations.More precisely, by fixing partially the initial data and controls, we overcome the dif-ficulty that the dynamic programming principle does not hold true in mean-field frame-work. Because this new kind of BSDEs involves the value function itself, we can not adopt the contraction mapping theory, which is used by Buckdahn, Li and Peng [20] in the case without control, in order to prove the existence and the uniqueness for our equations. A new iteration method is used to study this type of equations. On the other hand, once knowing the solution (Yt,x,v,Zt,x,v,W(t,x)) of the BSDE coupled with the value function, the optimal control problem for our controlled system can be treated like a classical optimal control problem by defining new coefficients. This allows to show with a more direct method the existence and uniqueness of viscosity solutions of related HJB equations.Chapter II is mainly based on the work:Hao, T., Li, J., Backward stochastic differential equations coupled with value func-tion and related optimal control problems, Abstract and Applied Analysis, Volume 2014, Article ID 262713,17 pages, http://dx.doi.org/10.1155/2014/262713.The main purpose of Chapter III is to study stochastic differential games for con-trolled systems of decoupled mean-field FBSDEs. Inspired by the work in Chapter II, we consider BSDEs coupled with the upper and the lower value functions, and we prove the existence and the uniqueness of this kind of equations, and we also show with the help of the Grisanov transformation that the value function is deterministic. The both nonlocal HJB-Isaacs equations associated with this kind of BSDEs are studied. As the coefficients contain expectation terms, the both HJB-Isaacs equations are coupled, which is different from the classical case. Last not least we introduce the Isaacs condition for this HJB-Isaacs system, under which the value function for this stochastic game exists.This Chapter is based on the work:Hao, T., Li, J., BSDEs in games, coupled with the value functions. Associated non- local Bellman-Isaacs Equations, submitted.Chapter IV is devoted to a probabilistic interpretation for nonlocal HJB equations associated with fully coupled FBSDEs involving the value function. We originally con-sider the optimal control problem of fully coupled FBSDEs in the mean-field framework. However, the same technical difficulty as that in Hao, Li [41] is met. So we adopt the approach introduced in [41]:We fix partially the initial parameters, and then by an iter-ation method we prove the existence and the uniqueness for this new equation. But the fact is that we have to deal with a fully coupled FBSDE:Any iteration procedure also involves the solution of the forward equation and its law. In order to get the convergence of the iteration sequence we have to suppose that the coefficients are Lipschitz in the law component with a sufficiently small Lipschitz constant, and we prove the convergence of the iterating sequence with a new method. The associated nonlocal HJB equations are also considered. Based on results by Li and Wei [63]. by using a short direct probabilistic argument, we prove that the value function is a viscosity solution of our nonlocal HJB equation.The novelty of this chapter is that a new method is employed to prove the con-vergence of the iteration sequence. The existence for fully coupled FBSDEs involving the value function is proved and a probabilistic interpretation for the associated HJB equations of mean-field type is provided.This Chapter is based on:Hao, T., Li, J., Fully coupled forward-backward SDEs involving the value function. Nonlocal Hamilton-Jacobi-Bellman equations, ESAIM:COCV, Control, Optimisation and Calculus of Variations.22(2) (2016), pp.519-538.The main purpose of Chapter V is to extend the work of Chapter IV to the general case. More precisely, in Chapter IV the coefficient σ of forward SDEs depends on control v, but does not depend on z, while in this Chapter a depends not only on control v hut-also on z. We call this new type of fully coupled FBSDEs,’’general fully coupled FBSDEs involving the value function". Under suitable assumptions we prove the existence and the uniqueness for such equations. The work in this chapter extends the results of the previous chapter. The related HJB equations are nonlocal and they are combined with an algebraic equation. With the help of an existence and uniqueness theorem for the general fully coupled FBSDEs involving the value function, the existence of viscosity solutions of the associated nonlocal HJB equations combined with algebraic equations is obtained.This chapter is based on:Hao, T., Zhao, N., General fully coupled FBSDEs involving the value function and related nonlocal Hamilton-Jacobi-Bellman equations combined with algebraic equations, submitted to Chinese Annals of Mathematics Series B, under the second round of review.In Chapter Ⅵ we generalize the work of Buckdahn, Li, Peng and Rainer [21] (with-out jumps) to the case of mean-field SDEs with jumps, and moreover, we interpret the related mean-field PDEs stochastically. In order to realize the extension, we need to prove several new results, in particular a new Ito formula (see Theorem 6.6.2), whose proof is far from being a direct extension of the Ito formula for mean-field processes without jumps in [21]. Indeed, the key for the proof of the Ito formula consists in the study of the derivative of f(PXst,ξ) with respect to s (see Theorem 6.6.1). While in the case without jumps in [21] this derivative was obtained as a direct consequence of a kind of second order Taylor expansion of f(Pξ0+η) at ξ0∈ L2(P), as E[|η|3]→0, this approach is here in the case of jumps not possible anymore:Indeed, while in the case without jumps the solution of a mean-field SDE satisfies E[|Xs+ht,ξ-Xst,ξ|3]= O(h3/2), as 0<h→0, in the case with jumps (6.2.1) we only have E[|Xs+ht,ξ=Xst,ζ|3]= O(h). To overcome this difficulty which has its origin in the jump part, we have to consider the filtration generated by the Poisson random measure and enlarged by the full informa-tion on the underlying Brownian motion, in order to apply the Ito’s formula just to the jump part of the Ito process. This, combined with a series of subtle estimates, allows to compute the derivative of f(PXst,ξ) in Theorem 6.6.1 under the standard assumption that f∈ Cb2,1 (P2(Rd)), and thus later to prove the Ito’s formula in Theorem 6.6.2. Using this new Ito formula, we prove that, for sufficiently regular function Φ:R×P2 (Rd)→R, the value function V(t,x,Pξ):= E[<Φ(XTt,x,Pξ,PXTt,ξ)] is the unique classical solution of the associated nonlocal PDE.The novelties of this chapter lie in two points:The first one is the achievement of a Ito formula for jump processes in the mean-field context, the second one is the proba-bilistic interpretation of the associated nonlocal integral-PDEs.This chapter is based on:Hao, T., Li, J., Mean-field SDEs with jumps and nonlocal integral-PDEs, Nonlinear Differential Equations and Applications,23(2) (2016), pp.1-51.This dissertation is composed of the six chapters mentioned above. Let us introduce the outline and the main conclusions.Chapter I Introduction;Chapter II BSDEs coupled with value function and related optimal control prob-lems;Chapter III BSDEs coupled with the upper and lower value functions and re-lated stochastic differential game problems;Chapter IV Fully coupled FBSDEs involving the value function and viscosity solutions of associated HJB equations;Chapter V General fully coupled FBSDEs involving the value function and re-lated nonlocal Hamilton-Jacobi-Bellman equations combined with algebraic equations;Chapter VI Mean-field SDEs with jumps and nonlocal integral-PDEs; Chapter II:We propose a new type of BSDEs, namely BSDEs coupled with value function; we prove that the BSDEs coupled with value function exist unique adopted solutions; the value function defined through the solution of a BSDE coupled with value function, is deterministic, and moreover, is the unique viscosity solution of the associated nonlocal PDE.Frozen partially initial value (x0, v) ∈Rn× V0,T,for given (t, x) ∈ [0, T] ×Rn, making use of an iteration method, we prove that the following equation possesses a unique solution (Yt,x,v,Zt,x,v,W(t,x)).Theorem 2.2.2 Under assumptions (H2.2.1) and (H2.2.2), the above BSDE coupled with value function has a unique solution (Yt,x;v Zt,x;v W), (t, x, v)∈[0, T]×Rn×Vt,T, with (Yt,x;v,Zt,x;v)∈SF2(t,T;]R)×HF2(t,T;Rd) and W:[0, T]×Ω×Rn→R a random field such that satisfies(i) W(t,x) is Ft-measurable, for all (t,x)∈ [0,T] ×]Rn;(ⅱ) |W(t,x)-W(t,x))|≤ C|x-x, P-a.s., (t,x), (t,x) ∈ [0, T]×Rn;(ⅲ) |W(t,x)|≤C(1+|x|), B-a.s., (t,x)∈ [0, T]× Rn, for some constant C > 0.With the help of the extended notion of backward semigroup given by Peng in 1997, we prove that the value function W(t, x) satisfies dynamic programming principle. Theorem 2.3.1 (DPP) Under assumptions (H2.2.1) and (H2.2.2) we have, for all (t,x) ∈[0,T]×Rn, 0<t<T-δ, P-a.s.,Using a new approach which is different to the backward semigroup method intro-duced by Peng, we show that the value function W(t, x) is the unique viscosity solution of the following nonlocal PDE:Theorem 2.4.1 (Existence) Under assumptions (H2.2.1) and (H2.2.2), the value func-tion W ∈Cp([0, T]×Rn) in Theorem 2.2.2 is a viscosity solution of the above PDE.Theorem 2.4,2 (Uniqueness) The function W is the unique viscosity solution of the above PDE in the class 8.Chapter Ⅲ: We consider stochastic differential games for controlled system-s of decoupled FBSDEs in mean-field framework; we establish BSDEs in games, which are coupled with the upper and the lower value functions; both the upper and the lower value functions are deterministic, and dynamic pro-gramming principle holds true for them; associated HJB-Isaacs equations are coupled; the pair (W, U) is the unique viscosity solution for system composed of the two coupled HJB-Isaacs equations.Fixed (x0; u,v)∈Rn×U0,T×V0,T, the following BSDE coupled with the upper and the lower value functions: exists a unique adopt solution (Yt,x,u,v, Zt,x,u,v, W (t,x), U(t,x)).Theorem 3.1.2 Under assumptions (H3.1.1) 和 (H3.1.2), the above equation possesses a unique solution (Yt,x;u,v, Zt,x;u,v, W, U). Moreover, W and U satisfy: There exists a constant C depending only on L, such that for all x,x∈Rn,t∈[0, T], P-a.s., (H3.1.3) For all (s, x’, x, y, z, u, v) ∈ [0, T]×Rn×]Rn×Rd×U×V, function f(s, x’, y’, y", x, y, z, u, v) is nondecreasing with respect to y’ and y".Thereom 3.1.3 (Comparison Theorem) Let fi = fi(t, x’, y’, y", x, y, z, u, v) and Φi, i = 1, 2, satisfy (H3.1.2), (H3.1.3). Let (Yi,t,x;u,v, Zi,t,x;u,v, Wi, Ui) be the solution of a BS-DE coupled with value function with coefficient (fi,Φi). Then, if f1≥f2 and Φ1≥Φ2, P-a.s., we have Ys,1,t,x;u,v≥Ys2,t,x;u,v,P-a.s∈[t, T], (t,x)∈[0, T]×Rn, u∈Ut,T, v ∈Vt,T. Moreover, W1(t,x)≥W2(t,x), U1(t,x)≥U2(t,x), P-a.s., (t,x) ∈[0, T]×Rn.The related two HJB-Isaacs equations are coupled, and the pair (W, U) is the unique viscosity solution for this system.For (t,x) ∈[0,T] x Rn, the following two theorems show that the equation exists a unique viscosity solution.Theorem 3.2.1 (Existence) Under assumptions (H3.1.1) and (H3.1.2), the pair (W, U) ∈ Cp([0, T] ×Rn; R2) in Theorem 3.1.1 is a viscosity solution of the above PDE.Theorem 3.2.2 (Uniqueness) The pair (W, U) in Theorem 3.1.1 is the unique viscosity solution of the above PDE in (?) ×(?).Chapter IV:Using a new iteration method, we prove that fully coupled F-BSDEs involving the value function exist unique solutions; we defined the value function by the solution of a fully coupled FBSDEs involving the value function; the value function satisfying dynamic programming principle and regular condition, is a unique viscosity solution of the related nonlocal PDE; when coefficient a does not depend on (y, z), this value function is also the unique viscosity solution of this PDE.We consider the following fully coupled FBSDE involving the value function:Theorem 4.2.1 Suppose the assumptions (H4.2.1), (H4.2.2L0), (H4.2.3) hold true. Then the above fully coupled FBSDE involving the value function has a unique solution {(Xst,x;v,Yst,x;v,Zst,x;v)s∈[t,T]∈SF2(t,T;Rn)×HF2(t,T;Rd)∈[0,T]×Rn, W∈W}.Theorem 4.2.2 (DPP) Let the assumptions (H4.2.1), (H4.2.2L0), (H4.2.3) hold true. Then there is a positive δ0 depending on L such that, for all 0≤t≤T-δ with 0<δ≤δ0,Let us consider the related PDE:Theorem 4.3.1 Assume that the assumptions (H4.2.1), (H4.2.2L0), (H4.2.3) hold true. Then the value function W(t,x) ∈C([0,T] x Rn) given in Theorem 4.2.1 is a viscosity solution of the above HJB equation.Theorem 4.3.2 When a also does not depend on y, under assumptions (H4.2.1), (H4.2.2L0), (H4.2.3) the value function W(t, x) defined in Theorem 4.2.1 is the unique viscosity solution of the above HJB equation in class 0.Chapter V:We investigate general fully coupled FBSDEs involving the value function, i.e., coefficient a depends on z and v at the same time; making use of an iteration approach, we prove the existence and the uniqueness theorem for this kind of equations; when all coefficients are deterministic, the value function is deterministic, and it is also a viscosity solution of the related non-local PDE combined with an algebraic equation.We now consider the general fully coupled FBSDEs involving the value function:Theorem 5.1.1 Under assumptions (H5.1.1)-(H5.1.4) and (H5.1.5)’, (H5.1.6), the above general fully coupled FBSDE involving the value function possesses a unique so-lution{(Xt,x;v,Yt,x;v)∈ SF2(t, T; Rn) ×SF2(t, T; R) ×HF2(t, T; Rd),W E WLo}.The viscosity solution of the following HJB equation combined with an algebraic equation exists.Theorem 5.3.1 Let (H5.1.1)-(H5.1.4) and (H5.1.5)’, (H5.1.6) hold true, the value function W in Theorem 5.1.1 is a viscosity solution of the above equation.Theroem 5.3.2 Suppose a does not depend on (y, z) and the assumptions (H5.1.1)-(H5.1.4) and (H5.1.5)’, (H5.1.6) hold true, the value function W(t, x) given in Theorem 5.1.1 is the unique viscosity solution of the above equation in 0.Chapter VI:We study mean-field SDEs with jumps, and prove the existence and the uniqueness for systems of general mean-field SDEs with jumps; the pair (Xt,ζ, Xt,x,Pζ) satisfies flow property; we obtain the general Ito’s formula for more suitably dealing with jump terms, which allows to show that the value function involving the measure V(t,x,Pζ)= E[Φ(XTt,x,Pζ,PXTt,ζ)] is the u-nique classical solution of related PDE.The following equations: exist unique solutions.Theroem 6.2.1 Under assumption (H6.2.1), the above both equations admit unique solutions Xt,ζ= (Xst,ζ)s∈~[t,T] and Xt,x,ζ=(Xst,x,ζ)s[t,T]in SF2(t,T;Rd).The solution Xt,xζ is independent of Ft.Theorem 6.a.1 Under (H6.a.1), the L2--derivative (:)xXt,x,ζ= ((?)xXt,x,Pζ,j)1≤j≤d of Xt,x,Pζ, exists.Theorem 6.a.a Suppose the same assumptions as in Theorem 6.3.2. Then, for all 0≤t≤T, x∈Rd, the lifted mapping Xt,x,:L2(Ft;Rd)→L2(Fs;Rd) is Frechet differcntiable and its Frechet derivative is just Gateaux derivative, i.e. where forRemark 6.3.2 From Theorem 6.3.3 we know that Xt,x,ζ is Frechet differentiable with respect to ζ. In extension of the definition of the derivative of function f:P2(Rd)→R over Pg(Rd), t.his allows to consider Xt,x.Pζas differentiable with respect to the law, and the derivative is just Nt,x,Pζ(y), i.e.,(?)μXst,x,Pζ(y)=Nst,x,Pζ(y),s∈[t,T],y∈Rd, 0≤t≤T,x∈Rd,ζ∈L2(Ft;Rd).The following two theorems play crucial role in this chapter. Theorem 6.61. Suppose fCb2.1(P2∈Rd)).Then. under assumption (H6.4.1), for all 0≤t≤s≤T,ζ∈L2(Ft;Rd). we haveTheorem 6.6.2 Let ψ∈Cb1.(2.1)([0, T]×Rd×P2(Rd)). Under assumption (H6.4.2), we have the following Itos formula: for 0≤t≤s≤T. x∈Rd.ζ∈L2(Ft:Rd). The related nonlocal integral-PDE: possesses a unique classical solution.Thereom 6.6.3 Let　Φ∈Cb2.1(Rd×P2(Rd))and let assumption(H6.4.2)hold true Then the function V(t,x,Pζ):=E[Φ(XT,t,x,Pζ,PXTt,ζ)],(t,x,ζ)∈[0.T]×Rd×L2(Ft;Rd), is the unique classical solution in Cb1.(2.1)([0,T]×Rd×P2(Rd))of the above PDE.