Stochastic differential game of functional forward-backward stochastic system and related path-dependent HJBI equation

TL;DR

This paper investigates a stochastic differential game involving functional forward-backward stochastic differential equations, establishing the deterministic nature of value functions and their characterization as viscosity solutions to path-dependent HJBI equations.

Contribution

It introduces a framework for stochastic differential games with functional FBSDEs and extends HJBI equations to the path-dependent setting, proving the dynamic programming principle.

Findings

Value functions are shown to be deterministic.

Upper and lower value functions are viscosity solutions of path-dependent HJBI equations.

Generalization of HJBI equations to the path-dependent case.

Abstract

This paper is devoted to a stochastic differential game of functional forward-backward stochastic differential equation (FBSDE, for short). The associated upper and lower value functions of the stochastic differential game are defined by controlled functional backward stochastic differential equations (BSDEs, for short). Applying the Girsanov transformation method introduced by Buckdahn and Li [1], the upper and the lower value functions are shown to be deterministic. We also generalize the Hamilton-Jacobi-Bellman-Isaacs (HJBI, for short) equations to the path-dependent ones. By establishing the dynamic programming principal (DPP, for short), the upper and the lower value functions are shown to be the viscosity solutions of the corresponding upper and the lower path-dependent HJBI equations, respectively.