Zero-sum stochastic differential game with risk-sensitive cost

TL;DR

This paper analyzes zero-sum stochastic differential games with risk-sensitive costs, characterizing saddle point strategies and establishing the existence and uniqueness of the value function under stability conditions.

Contribution

It provides a complete characterization of saddle point strategies and proves the existence and uniqueness of the value function for such games.

Findings

Characterization of all saddle point strategies in stationary Markov controls

Existence and uniqueness of the value function for the Hamilton-Jacobi-Isaacs equation

Conditions under which the results hold, including geometric stability

Abstract

Zero sum games with risk-sensitive cost criterion are considered with underlying dynamics being given by controlled stochastic differential equations. Under the assumption of geometric stability on the dynamics , we completely characterize all possible saddle point strategies in the class of stationary Markov controls. In addition, we also establish existence-uniqueness result for the value function of the Hamilton-Jacobi-Isaacs equation.