Stochastic differential games for fully coupled FBSDEs with jumps

TL;DR

This paper studies stochastic differential games involving fully coupled forward-backward stochastic differential equations with jumps, establishing the deterministic nature of value functions, their characterization as viscosity solutions to HJBI equations, and conditions for game value existence.

Contribution

It introduces a new transformation to prove the determinism of value functions and establishes their characterization as viscosity solutions to associated HJBI equations in jump-diffusion SDGs.

Findings

Value functions are deterministic.

Value functions are viscosity solutions to HJBI equations.

Existence of game value under specific conditions.

Abstract

This paper is concerned with stochastic differential games (SDGs) defined through fully coupled forward-backward stochastic differential equations (FBSDEs) which are governed by Brownian motion and Poisson random measure. For SDGs, the upper and the lower value functions are defined by the controlled fully coupled FBSDEs with jumps. Using a new transformation introduced in [6], we prove that the upper and the lower value functions are deterministic. Then, after establishing the dynamic programming principle for the upper and the lower value functions of this SDGs, we prove that the upper and the lower value functions are the viscosity solutions to the associated upper and the lower Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations, respectively. Furthermore, for a special case (when $\sigma,\ h$ do not depend on $y,\ z,\ k$), under the Isaacs' condition, we get the existence of the value of the game.