Nonzero-Sum Risk-Sensitive Stochastic Differential Games: A Multi-parameter Eigenvalue Problem Approach

TL;DR

This paper investigates nonzero-sum risk-sensitive stochastic differential games, establishing Nash equilibria via a multi-parameter eigenvalue approach and coupled HJB equations, providing a complete characterization of equilibrium strategies.

Contribution

It introduces a novel multi-parameter eigenvalue method to prove the existence and characterization of Nash equilibria in risk-sensitive stochastic differential games.

Findings

Existence of Nash equilibrium in stationary Markov strategies.

Characterization of equilibrium points through coupled HJB equations.

Use of eigenvalue approach for solving complex stochastic games.

Abstract

We study nonzero-sum stochastic differential games with risk-sensitive ergodic cost criterion. Under certain conditions, using multi-parameter eigenvalue approach, we establish the existence of a Nash equilibrium in the space of stationary Markov strategies. We achieve our results by studying the relevant systems of coupled HJB equations. Exploiting the stochastic representation of the principal eigenfunctions we completely characterize Nash equilibrium points in the space of stationary Markov strategies.