The Global Maximum Principle for Progressive Optimal Control of Partially Observed Forward-Backward Stochastic Systems with Random Jumps

TL;DR

This paper develops a global maximum principle for partially observed forward-backward stochastic systems with jumps, providing theoretical foundations and applications in optimal control with non-convex domains and correlated noises.

Contribution

It introduces a novel maximum principle for partially observed stochastic systems with jumps, including existence, uniqueness, and explicit control representation.

Findings

Established existence and uniqueness of solutions with jumps.

Derived a global maximum principle for partially observed systems.

Provided explicit control feedback via differential equations.

Abstract

IIn this paper, we study a partially observed progressive optimal control problem of forward-backward stochastic differential equations with random jumps, where the control domain is not necessarily convex, and the control variable enter into all the coefficients. In our model, the observation equation is not only driven by a Brownian motion but also a Poisson random measure, which also have correlated noises with the state equation. For preparation, we first derive the existence and uniqueness of the solutions to the fully coupled forward-backward stochastic system with random jumps and its estimation in $L^\beta(\beta\geq2)$-space under some assumptions, and the non-linear filtering equation of partially observed stochastic system with random jumps. Then we derive the partially observed global maximum principle with random jumps. To show its applications, a partially observed linear quadratic progressive optimal control problem with random jumps is investigated, by the maximum principle and stochastic filtering. State estimate feedback representation of the optimal control is given in a more explicit form by introducing some ordinary differential equations.