Jump Stochastic Optimal Control

The thesis is devoted to the study of optimal control problems for stochastic systems driven by both a Poisson random measure and a Brownian motion. It is divided into three parts.Part 1 is concerned with the dynamic programming principle of the optimal control and its associated stochastic Hamilton-Jacobi-Bellman (HJB) equation for finite dimensional non-Markov stochastic systems with jumps. The stochastic HJB equation is a fully nonlinear backward stochastic partial differential-integral equa-tions driven by a Poisson random measure, and its solution consists of a pair of random fields. Under some sufficiently regular assumptions, by dynamic program-ming principle and Ito-Ventzell formula, the value function of the stochastic optimal control problem is shown to be the first part of the solution to the stochastic HJB equation. Moreover, an existence and uniqueness result is presented for the weak solution of the stochastic HJB equation.Part 2 is concerned with the stochastic optimal control problem with jumps in infinite dimensional spaces. The second-order differential operator of the controlled stochastic system is featured to be random and time-varying. In terms of weak solutions, the stochastic maximum principle and the verification theorem for the optimal control are established for the convex admissible control set.Part 3 is concerned with the stochastic optimal control problem for fully coupled nonlinear forward-backward stochastic systems with jumps under partial informa-tion. Both sufficient (a verification theorem) and necessary conditions are proved for the optimal control.