Optimal control for stochastic differential equations governed by normal martingales

Abstract

This thesis presents two research topics, the first one being divided into two parts. In the first part, we study an optimal control problem where the state equation is driven by a normal martingale. We prove a sufficient stochastic maximum and we also show the relationship between stochastic maximum principle and dynamic programming in which the control of the jump size is essential and the corresponding Hamilton-Jacobi-Bellman (HJB) equation in this case is a mixed second order partial differential-difference equation. As an application, we solve explicitly a mean-variance portfolio selection problem. In the second part, we study a non smooth version of the relationship between MP and DPP for systems driven by normal martingales in the situation where the control domain is convex. The second topic, is to characterize sub-game perfect equilibrium strategy of a partially observed optimal control problems for mean-field stochastic differential equations (SDEs) with correlated noises between systems and observations, which is time-inconsistent in the sense that it does not admit the Bellman optimality principle.