Optimal control of coupled forward-backward stochastic system with jumps and related Hamilton-Jacobi-Bellman equations

TL;DR

This paper studies optimal control problems for coupled forward-backward stochastic systems with jumps, establishing the regularity of solutions, the deterministic nature of the value function, and its characterization as a viscosity solution to Hamilton-Jacobi-Bellman equations.

Contribution

It introduces a framework for analyzing coupled stochastic systems with jumps, proving the value function's regularity, determinism, and its role as a viscosity solution to related HJB equations.

Findings

Value function is deterministic and satisfies dynamic programming principle.

Value function is a viscosity solution of Hamilton-Jacobi-Bellman equations with integral operators.

Regularity results for coupled forward-backward stochastic differential equations.

Abstract

In this paper we investigate a kind of optimal control problem of coupled forward-backward stochastic system with jumps whose cost functional is defined through a coupled forward-backward stochastic differential equation with Brownian motion and Poisson random measure. For this end, we first study the regularity of solutions for this kind of forward-backward stochastic differential equations. We obtain that the value function is a deterministic function and satisfies the dynamic programming principle for this kind of optimal control problem. Moreover, we prove that the value functions is a viscosity solutions of the associated Hamilton-Jacobi-Bellman equations with integral-differential operators.