State and Control Path-Dependent Stochastic Optimal Control With Jumps

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 Skorohod 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.

Automated summary

This paper considers the path-dependent stochastic optimal control problem for jump-diffusion models. It proves the dynamic programming principle of the value functional and introduces the path-dependent integro-type Hamilton-Jacobi-Bellman equation. The verification theorem is provided to obtain the sufficient condition for optimality. The explicit optimal solutions are obtained for specific problems using the verification theorem and solving the corresponding PIHJB equation.