The Stochastic Maximum Principle for Optimal Control Problem of Jump-Diffusion Systems with Fractional Brownian Motion

Abstract

This paper presents the stochastic maximum principle for optimal control problem of jump-diffusion type stochastic differential equations (SDEs) with fractional Brownian motion (fBm) of the Hurst parameter H is an element of 12,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H \in \left( \frac{1}{2},1\right) $$\end{document}. The control variable is involved in all diffusion coefficients of the SDE, and the control domain is not necessarily convex. We prove the maximum principle, represented by the variational inequality, through first- and second-order variational and duality analysis, together with the adjoint equations identified by jump-diffusion type backward SDEs with fBm. We obtain the precise estimates of first- and second-order variational equations. Then we have to apply appropriate Malliavin derivatives to deal with an additional second-order variation induced by stochastic integrals of fBm, which needs to be taken into account in the maximum principle.