Controlled Mean Field Stochastic Systems

The concept of weak solutions for stochastic differential equations (SDEs, for short) has been studied for about 70 years, since Skorohod [87], [88], the concept of weak solutions of SDEs has been widely studied, and has been playing an important role in the development of the theory of SDEs in recent decades. In the first half of the 70th, Japanese and Soviet probabilists made a great contribution to clarify the relation-ship between weak solution existence, strong existence, uniqueness in law and pathwise uniqueness.Mean-field SDEs, also known as McKean-Vlasov equations, the concept of weak solutions for McKean-Vlasov equations has been intensively studied for at least three decades by many authors. These works have shown that, as in the classical theory of SDEs (i.e., without mean-field term), also for the McKean-Vlasov equations the approach through a martingale problem turns out to be a powerful tool in the investigation of weak solutions.This paper, we study the existence and the uniqueness in law of the weak solutions of mean-field SDEs in two cases:The drift coefficient depends not only on the solution process but also on the law of solution process; The drift and diffusion coefficients are both depend on the law of the solution. Then we extend our work to the study of 2-person zero-sum stochastic differential games.Backward stochastic differential equations and forward-backward stochastic differ-ential equations (FBSDEs, for short) have been widely recognized as useful tools for applications in many fields, especially mathematical finance and the stochastic control theory, but also in partial differential equations (PDEs. for short) theory and in game theorv.Non-linear backward stochastic differential equations (BSDEs, for short) were first introduced by Pardoux and Peng [75] in 1990. Since then the theon of BSDEs has been studied by man)-researchers. The quick development of the theory of BSDEs has its origin in the mean field of its different applications, namely in economics and finance, for PDEs, in stochastic control and also in stochastic differential games. One of the objectives of this paper is to study the development and applications of BSDEs in the theories of stochastic control and stochastic differential games. There are many types of BSDEs. for example, mean-field BSDEs, decoupled FBSDEs, fully-coupled FBSDEs, BSDEs with jumps and reflected BSDEs, etc.This paper we study mean-field forward and backward SDEs with jumps, mean-field BSDEs with jumps involving the value function, and we give the probabilistic interpretation for the associated integro-partial differential equations.Let us introduce the main content and the organization of the thesis.In Chapter 1 the Introduction gives an overview of our topics in Chapter 2 to Chapter 5.Chapter 2 is devoted to the study of stochastic differential equations, whose diffusion coefficient σ(s.X.Λs) is Lipschitz continuous with respect to the path of the solution process X, while its drift coefficient b(s.X.Λs.Qxs) is only measurable with respect to X and depends continuously (in the sense of the 1-Wasserstein metric) on the law of the solution process. We first prove the existence and the uniqueness in law of the weak solution of the above stochastic differential equation. After we extend the results to the study of 2-person zero-sum stochastic differential games described by doubly controlled coupled mean-field forward-backward stochastic differential equations with dynamics whose drift coefficient is only measurable with respect to the state process. We obtain for them the existence of generalized saddle point controls under Isaacs’ condition and we discuss conditions under which we have generalized saddle point controls.The novelty of this chapter:The first to study the uniqueness in law of weak solution of the mean-field SDE. and get generalized saddle point controls under Isaacs’ condition.This chapter is mainly based on the paper:LI. J.,MIN, H., Weak solutions of mean-field stochastic differential equations and applications to zero-sum, stochastic differential games, SIAM Journal on Control and Optimization, accepted.Chapter 3 continues the studies in Chapter 2. Here we investigate the weak solutions of SDEs. whose drift and diffusion coefficients depend both on the law of the solution process. Under continuity and boundness conditions, and with the help of a generalized local martingale problem, we prove the existence and the uniqueness in law of the weak solution of the mean-field SDE.The novelty of this chapter:We extend the Ito formula associated with mean-field problems; Different from Stroock and Varadhan [89], we use a generalized local martingale problem to prove the existence and the uniqueness in law of weak solution of mean-field SDE.This chapter is mainly based on the paper:LI, J., MIN, H., The existence and the uniqueness in law of the weak solutions of mean-field stochastic differential equation, submitted.Chapter 4 investigates mean-field BSDEs with jumps. First we prove the existence and the uniqueness of solutions of mean-field SDEs under linear growth and Lipschitz conditions. Then we prove the existence and the uniqueness of the solution of mean-field BSDEs. continuous dependence of solutions on parameters and comparison theorem. In the following step we obtain the existence and the uniqueness of solutions of decoupled mean-field forward-backward SDEs. Finally, we prove that the value function given by the backward equation of the decoupled mean-field forward-backward SDE describes the unique viscosity solution of the associated PDE.The novelty of this chapter:We study mean-filed forward and backward SDEs with jumps, the solution of our mean-field BSDE with jumps provides a probabilistic formula for the unique viscosity solution of a parabolic PDE.This chapter is mainly based on the paper:LI,J.,MIN,H., Controlled mean-field backward stochastic differential equations with jumps involving the value function. Journal of Systems Science and Complexity, accepted.In Chapter 5 which is inspired by Chapter 4,we investigate a new type of controlled mean-field BSDEs with jumps, namely mean-field BSDEs with jumps which are strongly coupled with the value function of the associated control problem. We first prove the existence and the uniqueness as well as a comparison theorem for the above BSDEs. For this we suppose Lipschitz condition as well as linear growth and monotonicity as-sumptions. Then, with generalized stochastic backward semigroups, we get the dynamic programming principle for the value functions. Finally, we prove that the value func-tion is a viscosity solution of the associated nonlocal Hamilton-Jacobi-Bellman (HJB, for short) integro-partial differential equation, we also prove its uniqueness in an adequate space of continuous functions.The novelty of this chapter:We study a new type of controlled mean-field BSDEs with jumps. We prove the existence and the uniqueness of the solution of mean-field BS- DEs with jumps involving the value function, and provide a probabilistic interpretation for the associated HJB.This chapter is mainly based on the following paper:LI, J.,MIN, H., Controlled mean-field backward stochastic differential equations with jumps involving the value function, Journal of Systems Science and Complexity, accepted.This paper contains five chapters, we give an overview of the structure and the main results of this dissertation.Chapter 1 Introduction;Chapter 2 Weak solutions of mean-field SDEs and application to zero-sum stochas-tic differential games;Chapter 3 The existence and the uniqueness in law of weak solutions of mean-field SDEs;Chapter 4 Mean-field BSDEs with jumps;Chapter 5 Controlled mean-field BSDEs with jumps involving the value function.Chapter 2:We devote to study of mean-field SDEs whose diffusion coefficient a(s,X.As) is Lipschitz continuous with respect to the path of the solution process X, while drift coefficient b(s, X.As, Qxs) is only measurable with respect to X and depends continuously (in the sense of the 1-Wasserstein metric) on the law of the solution process. We prove the existence and the uniqueness in law of the weak solution of the above SDEs. Moreover, we extend the results to the study of 2-person zero-sum stochastic differential games.We want to study the weak solution of the following mean-field SDE, whose drift coefficient depends on the law of the solution process: where (Bt)t∈[o,T] is a d-dimensional Brownian motion under the probability measure Q, and X0 is independent of B under Q and obeys a given law probability μ0.Under the drift coefficient is bounded and measurable, and modulus of continuity with respect to measure conditions, we study the existence and the uniqueness in law of the weak solution of the above equation.Theorem 2.2.1 Under assumption (H2.2.1), for any μ0 ∈P1(Rd), mean-field SDE (0.0.24) with QX0= μ0 has a weak solution (Ω,F, F, Q, B, X).Theorem 2.3.1 Suppose that assumption (H2.3.1) holds, and let (Ωi,Fi, Qi, Bi, Xi), i = 1, 2, be two weak solutions of mean-field SDE (0.0.24) with QX011 = QX022 ∈ P1(Rd). Then (B1, X1) and (B2, X2) have the same law under their respective probability measures, i.e., Q(B1,X1)1=Q2(B2,X2).Since we have the existence and the uniqueness in law of the weak solution, we extend the result to 2-person zero-sum stochastic differential games. Given an arbitrary μ0∈P1 (Rd) and (u, v)∈u×V, we consider the following coupled mean-field forward-backward SDE: where Qu,v is a probability measure with respect to which Bu,v is a Brownian motion.As cost functional of the game we consider the first player wants to minimize it through his control u ∈u, and the second player, for whom it is a pay-off functional, wants to maximize it using the control v∈v.In order to rewrite (0.0.25) in a different form, we suppose temporarily that (0.0.25) admits a weak solution with (Ω,F,F, P,X) = (Ω,F, F, P, X). For (u, v)∈u×v, putting ×U× V, we have from (0.0.25) that From the Girsanov Theorem and the definition of F and the filtration F it follows that Denoting by Lsu,v the density of Qu,v restricted to Fs with respect to P restricted to Fs, we see that the above discussed Girsanov transformation allows to give to (0.0.25) the following form, a system considered under the probability P: where for (s,φ·Λs,y,z,v,u,v)∈[0, T] ×C([0,T];Rd)×R×R×P1(Rd+1) ×U ×V,Then we have the following results.Theorem 2.4.1 Under assumption (H2.4.1), on (Ω,F,F, P, W, X), for any (u,v)∈ u×v, system (0.0.27) has a solution (Bu,v,Qu,v), (Ysuv,Zsu,v)s∈[0,T]∈(S(F,P)2(0,T;R)× h(f,p)2(0,T;Rd))∩(S(F,Qu,v)2(0,T;R)×H2(F,Qu,v)(0,T;Rd)), where Bu,v is an (F,Qu,v)-Brownian motion and Lsu,v the density of Qu,v/Fs with respect to P/Fs,s∈ [0, T].For (t,ψ,y,z,v,u,v)∈[0,T] ×C([0,T];Rd)×R×Rd×P1(Rd+1)×U×V, we introduce the Hamiltonian associated with the gameg(t,φ·Λt,y,z,v,u,v) = <b(t,φ·Λt, y,z,v,u,v),z>+ f(t,φ.Λt,y,z,v,u,v). (0.0.28) We suppose now throughout this section that the Isaacs’ condition holds, that is, for any (t,φ,y, z,v)∈ [0,T]× C([0,T]; Rd)×R×Rd×P1(Rd+1),Theorem 2.4.2 Suppose that assumptions (H2.4.1) and (H2.4.2) hold. Then the system has a solution (Y*, Z*) ∈(S(F,p)2(0, T;R)×H2(F,P))(0,T;Rd))∩ (S(F,Q*)2)(0, T;R)×H(F,Q.)2 (0,T;Rd)), (B*,Q*), where B* is an (F, Q*)-Brownian motion and Ls* is the density of Q*/F, with respect to P/F, s E [0, T].Thus, we can get an important result of the application to zero-sum stochastic differential games.Theorem 2.4.3 Let (H2.4.1) and (H2.4.3) be satisfied. Then, for all (u, v) E/4 x 1;, the solution of (0.0.27) (Bu,v,Qu,v),(Yu,v,Zu,v) is unique on (Ω,F,F,,P,W,X) in the sense of pathwise uniqueness for the processes (Bu,v, Lu,v,Yu,v, Zu,v). Moreover, supposing that there is (u*, v*) ∈u×V such that for all (u, v)∈u×v, dtdP-a.e., we have, for all (u, v)∈u×, i.e., (u*, v*) is a couple of generalized saddle point controls. The constant C depends only on the coefficients b and f.Chapter 3: We study weak solution of mean-field SDEs, whose dift b(s, X,, Qxs) and diffusion σ(s,X,,QXs) depend not only on the state process X,, but also on its law. We suppose that b and a are bounded and continuous in the state as well as the probability law; the continuity with respect to theprobability law is understood in the sense of the 2-Wasserstein metric. Using the ap-proach through a local martingale problem,then we prove the existence and the uniqueness in law of the weak solution of mean-field SDEs.Inspired by Chapter 2, in this chapter we study the following mean-field SDE, whose diffusion and drift coefficient are both depend on the law of the solution process: where ζ∈L2(Ω,F0, P;Rd) with given distribution v = Qζ, and (Bt)te[O,T] is a d-dimensional Brownian motion under the probability measure Q.Using the approach through a generalized local martingale problem, we prove the existence and the uniqueness in law of weak solution of mean-field SDE. At first we extend the It5 formula associated with mean-field problems given by Buckdahn et al. [18].Theorem 3.1.1 Let σ = (σs), γ = (%) be Rd×d-valued and b = (bs),β = (βs) be Rd-valued adapted stochastic processes, such that (i) There exists a constant q > 6 such that E[(f0T|σs|q+|bs|q)ds)3.4]<+∞ (ii) ∫0T(|s|2+|s|2)ds<+∞,P-a.s.F∈ Cb1,21,([0, T] ×Rd×P(Rd). Then, for the It5 processes we haveThen we introduce the generalized local martingale problem, before we study the existence of the weak solution, we first consider the equivalence, that isProposition 3.2.1 The existence of a weak solution (Ω, F,F, Q, B, X) to equation (0.0.32) with initial distribution v, on B(Rd) is equivalent to the existence of a solution to the local martingale problem associated with A, with Py(0) = v.Then we get the first main result of this chapter.Theorem 3.2.1 Under assumption (H3.1.1) mean-field SDE (0.0.32) has a weak so-lution (Ω,F, F, Q, X, B).Now, we study the uniqueness in law of the weak solution. Firstly, we have:Theorem 3.3.1 For given f∈C0∞(Rd), consider the Cauchy problem where (t, x, v)∈[0, ∞)× Rd×P2{Rd). We assume that (0.0.36) has a solution vf ∈Cb([0,∞)× Rd ×P2(Rd))∩Cb1,2,1((0,∞)×Rd×P2(Rd)), for all f∈ C0∞(Rd). Then, for every P satisfying Py(0)=δx,x∈Rd, the local martingale problem associated with A has at most one solution.Corollary 3.3.1 Under the assumption of Theorem 3.3.1, we have for the mean-field S-DE (0.0.32) the weak solutions have the uniqueness in law, that is, for any weak solutions (Ωi,Fi,FiQi,Bi,Xi), i= 1,2, of (0.0.32), we QX11,= Q2X2.Chapter 4:We prove the existence and the uniqueness as well as the compari-son theorem for mean-field BSDEs under Lipschitz condition. Then we prove the existence and the uniqueness of the solutions of decoupled mean-field forward-backward SDEs, and the associated dynamic programming principle (DPP, for short). Finally, we prove the value function given by the backward equation of the decoupled mean-field forward-backward SDE is the unique viscosity solution of the associated PDE.We consider three types of SDEs, i.e., mean-field SDEs with jumps, mean-field BSDEs with jumps and decoupled mean-field forward-backward SDEs with jumps.We study the following mean-field BSDE with jumps: where 0< t< T. Under Lipschitz condition, with the help of fixed point theorem, we prove equation (0.0.37) has a unique solution.Theorem 4.3.1 Under the assumption (H4.3.2), for any random variable ζ∈L2(Ω, P), equation (0.0.37) has a unique adapted solutionTheorem 4.3.2 (Comparison Theorem) Let h : Ω×[0, T]×R×Rd×R such that, h(.,., y, z, k) is F-predictable, for all (y, z, k), and(i) There exists a constant C > 0 such that, for all t ∈[0, T], yl, y: ∈R, z1, z2 ∈Rd k1, k2 ∈, a.s.,(ii) h(.,0,0,0)∈HF2(0, T;R);(iii) kâ†' h(t, y, z, k) is non-decreasing, for all (t, y, z)∈ [0, T]×R×Rd; Moreover, let l :Ω×[0, T]×E be measurable, l(.,-, e) F-predictable, for all e∈E, and let I satisfyLet fi = fi(ω,t,y’,z’,k’,y,z,k), i = 1,2, be two drivers, and f2 satisfy (H4.3.2), λ; R). Furthermore, we assume:(i) One of the both coefficients is independent of z’;(ii) One of the both coefficients is independent of k’;(iii) One of the both coefficients is nondecreasing in y’;Let ζ1,ζ2∈ L2(Ω,FT,P) and denote by (Y1,Z1,K1) and (Y2, Z2,K2) the solution of the mean-field BSDE with jumps (0.0.37) with data (ζ1, f1) and (ζ2, f2), respectively. Then, if ζ1≤ζ2, P-a.s., and f1≤f2,P-a.s., we have Y5I≤Yt2, t∈ [0, T], P-a.s.Now let us introduce the random field: where Yt,x is the solution of backward equation.We consider the following PDE: where Here the functions b, a,γf and Φ are supposed to satisfy the assumptions (H4.2.1), (H4.4.1) and (H4.4.2).Now we give the main results of this subsection.Theorem 4.5.1 (Existence) Under (H4.2.1), (H4.4.1) and (H4.4.2) the value func-tion u(t,x) defined by (0.0.38) is a viscosity solution of equation (0.0.39).Theorem 4.5.2 (Uniqueness) Under assumptions (H4.2.1). (H4.4.1) and (H4.4.2) the value function u(t.x) denned by (0.0.38) is the unique viscosity solution of PDE (0.0.39) in the 6.Chapter 5:Using an approximation method, under Lipschitz, linear growth and boundness condition, we prove the mean-field BSDE with jumps strong coupled with the value function has a unique solution and we also get com-parison theorem. With the help of the generalized stochastic backward semi-groups, we get the DPP for the value functions. Furthermore, we prove the value function is a unique viscosity solution of the associated nonlocal HJB integro-partial differential equation, in an adequate space of continuous functionsWe consider the following mean-field BSDE with jumps, coupled with the value function of the associated control problem:In order to prove the existence of the solution, we choose an iterative approach, first to prove the existence of the solution of the following BSDE: Putting (Yt,x;v,0,Zt,x;v,0,Kt,x;v,0) ≡(0,0,0), for t∈[0,T], x ∈Rn, v∈ Vt,T. Then we haveLemma 5.2.1 For all m≥1, equation (0.0.41) has a unique solution (yt,x;v,m,Zt,x;v,m, gt,x;v,m)∈SF2(0,T;R)×HF2(0,T;Rd)×KF2,λ(0,T;R), (t,x)∈ [0,T]×Rn. Moreover, Wm :Ω× [0, T]×Rnâ†'R is a measurable random field such that(i) Wm(t,x) is F’t-measurable, (t,x)∈ [0, T] × Rn;(ii) There exists a constant C > 0 independent of m such that, P-a.s., for all t ∈ [0, T], X, x∈Rn, (1) |Wm(t,x) -Wm(t,x)|≤C|-x|; (2) |Wm(t,x)|≤C(1+|x|).With the help of the above lemma, we can state now one of our main results: Theorem 5.2.1 Under the assumptions (H5.1.1) and (H5.2.1) equation (0.0.40) coupled with the associated value function has a unique solution (Yt,x;v,Zt,x;v,Kt,x;v,W), (t,x,v)∈ [0, T]×Rn× Vt,T, with (Yt,x;v,Zt,x;v, Kt,x;v)∈ SF2(0, T;R)×HF2(0, T;Rd)×~κF,λ2(0, T;R) and W : [0, T] ×Ω×Rnâ†'R being the random field Moreover, W satisfies(i) W(t,x)is Ft-measurable, (t,x)∈[0,T]×Rn; (0.0.42)(ii) |W(t,x) - W(t,x)|≤C|x -x|, P-a.s., (t,x), (t,x)∈ [0,T]×Rn; (0.0.43)(iii) ]W(t,x)≤< C(1 +|x|), P-a.s., (t,x)∈ [0, T]×Rn, (0.0.44_for some constant C > 0.Theorem 5.3.1 (Comparison Theorem) Let fi =fi(t,x’,x,y’,y,z,k,v), be two gener-ators which satisfy the assumptions (H5.2.1) and (H5.3.1) and let ζi∈ L2(Ω, FT, P), i=1,2. For i = 1,2, we denote by (Yi,t,x;,v,Zi,t,x;v,Ki,t,xp;v,W),∈Vt,T, (t,x)∈ [0, T]×Rn, the solution of the following BSDE with jumps, involving the value function, where Xt,x,v,is defined by the following mean-field SDE (0.0.46) with the coefficients b, 7 and a satisfying (H5.1.1): Then, if ζ1≥ζ2, P-a.s., and f1≥f2, we have that Moreover, in this case also W1(t,x)≥W2(t,x), P-a.s., (t,x)∈[0,T] ×Rn.Theorem 5.4.1 Under the assumptions (H5.1.1) and (H5.2.1) we have, for all (t, x) ∈ [0, T) x Rn,0≤ t≤T-δ, P-a.s.,At last we consider the following non-local HJB integro-partial differential equation: where DW and D2W are the gradient and the Hessian matrix of W with respect to a;, respectively.Then we have the following results.Theorem 5.5.1 (Existence) Under the assumptions (H5.1.1) and (H5.2.1), the value function W e Cp([0,T] x Rn) given by equation (0.0.40) in Theorem 5.2.1 is a viscosity solution of PDE (0.0.47).Theorem 5.5.2 (Uniqueness) In the class O the function W is the unique viscosity solution of PDE (0.0.47).