State and Control Path-Dependent Stochastic Optimal Control with Jumps

Abstract

We consider the state and control path-dependent stochastic optimal control problem for jump-diffusion models, where the dynamics and the objective functional are dependent on (current and past) paths of state and control processes. We prove the dynamic programming principle of the value functional, for which unlike the existing literature, the Skoro-hod metric is necessary to maintain the separability of cadlag (state and control) spaces. We introduce the state and control path-dependent integro-type Hamilton-Jacobi-Bellman (PIHJB) equation, which includes the Levy measure in the corresponding nonlocal path-dependent integral operator. Then by using the functional Ito calculus of a cadlag path, we show the verification theorem, which constitutes the sufficient condition for optimality in terms of the solution to the PIHJB equation. We finally apply our verification theorem to the linear-quadratic optimal control problem of jump-diffusion models with delay and the control path-dependent problem, for which the explicit optimal solutions are obtained by solving the corresponding PIHJB equation.