Relationship between maximum principle and dynamic programming for systems driven by normal martingales

Abstract

The aim of this paper is to prove a sufficient stochastic maximum principle for the optimal control of systems driven by normal martingales. 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.