BSDE Approach for $\alpha$-Potential Stochastic Differential Games

TL;DR

This paper introduces a BSDE-based method to analyze $oldsymbol{oldsymbol{ ext{ extalpha}}}$-potential stochastic differential games, providing new insights into derivatives of the objective function and their dependence on game features.

Contribution

It develops a novel BSDE approach to characterize derivatives in $ extalpha$-potential stochastic games, including linear-quadratic cases with common noise.

Findings

Derivatives of the objective function are expressed via adjoint equations.

Rigorous estimates for $ extalpha$ are established.

Dependence of $ extalpha$ on game parameters is analyzed.

Abstract

In this paper, we examine a class of $\alpha$-potential stochastic differential games with random coefficients via the backward stochastic differential equations (BSDEs) approach. Specifically, we show that the first and second order linear derivatives of the objective function for each player can be expressed through the corresponding first and second-order adjoint equations, which leads to rigorous estimates for $\alpha$. We illustrate the dependence of $\alpha$ on game characteristics through detailed analysis of linear-quadratic games, and with common noise.