Perturbation Problems For Ordinary Differential Equation And Optimal Control For FBSDEs

The thesis consists in two main topics:the first one concerns random perturbation for multi-dimensional ordinary differential equation (ODE in short) and large deviation for forward-backward stochastic differential equations (FBSDEs in short). The second topic deals with the investigation of the optimal control of FBSDEs.In Chapter1, we consider the following ordinary differential equation where b:Rd→Rd. There is a general local existence theory if b is only supposed to be continuous (Peano’s Theorem), even though uniqueness may fail in this case. However, the perturbed stochastic differential equation (SDE in short) where W is a d-dimensional standard Brownian motion, has a unique strong solution when b is assumed to be continuous and bounded. Moreover, when ε→0+, the solutions to the perturbed SDEs converge, in a suitable sense, to the solutions of the ODE. This phenomenon has been extensively studied for one-dimensional case in literature. The goal of this chapter is to analyze some multi-dimensional cases (which need slightly different techniques that in dimension one). When b has an isolated zero and is non Lipschitz continuous at zero, the ODE may have infinitely many solutions. Our main result shows which solutions of the ODE can be the limits of the solutions of the SDEs (as ε→0+). Such a result was only known in the one dimensional case and was obtained by techniques which cannot be extended to the higher dimensional case. The main novelty consists in the treatment of multi-dimensional case.In Chapter2we consider a kind of coupled forward-backward stochastic differential equations (FBSDEs in short) with parameter ε>0, The convergence of distributions of (Xε,t,x, Yε,t,x) is studied as ε→0+, and the Freidlin-Wentzell’s large deviation principle is proved as well.In particular, this work extends already obtained results to the case where b and σ can depend on the variable Y.In Chapter3, we study the near-optimal control for a kind of linear stochastic control systems which is governed by FBSDEs, where both the drift and diffusion terms are allowed to depend on controls and the control domain non-convex. New necessary and sufficient conditions of the near-optimality are established in the form of Pontryagin stochastic maximum principle.The main contribution lies on the fact of that it is possible to consider non-convex domain and diffusion terms containing control variable.In Chapter4, we study quasilinear stochastic partial differential equations with random coefficients as follows: where The existence and uniqueness of solutions to above stochastic Hamilton-Jacobi-Bellman (H-J-B in short) equations are obtained. An optimal control interpretation is given as well.In Chapter5, we investigate the controlled systems described by forward-backward stochastic differential equations with the control contained in drift, diffusion and gen-erator of BSDEs. A new verification theorem is derived within the framework of vis-cosity solutions without involving any derivatives of the value functions. It is worth to pointing out that this theorem has wider applicability than the classical verification theorems. Furthermore, it is used to build the optimal stochastic feedback controls for forward-backward systems.