Non-zero Sum Stochastic Differential Games of Fully Coupled Forward-Backward Stochastic Systems

TL;DR

This paper investigates non-zero sum stochastic differential games involving fully coupled forward-backward systems, providing conditions for Nash equilibrium existence in complex multi-dimensional stochastic environments.

Contribution

It establishes both necessary and sufficient conditions for Nash equilibria in non-zero sum stochastic differential games with fully coupled forward-backward systems, expanding theoretical understanding.

Findings

Verification theorem for Nash equilibrium

Necessary conditions for equilibrium existence

Applicability to multi-dimensional systems

Abstract

In this paper, an open-loop two-person non-zero sum stochastic differential game is considered for forward-backward stochastic systems. More precisely, the controlled systems are described by a fully coupled nonlinear multi- dimensional forward-backward stochastic differential equation driven by a multi-dimensional Brownian motion. one sufficient (a verification theorem) and one necessary conditions for the existence of open-loop Nash equilibrium points for the corresponding two-person non-zero sum stochastic differential game are proved. The control domain need to be convex and the admissible controls for both players are allowed to appear in both the drift and diffusion of the state equations.