Nonzero-sum risk-sensitive continuous-time stochastic games with ergodic costs

TL;DR

This paper investigates nonzero-sum risk-sensitive stochastic games in continuous time, establishing existence and characterization of Nash equilibria under unbounded transition and cost rates using eigenvalue methods.

Contribution

It introduces a novel approach using principal eigenvalues to prove existence and characterization of Nash equilibria in risk-sensitive stochastic games with unbounded rates.

Findings

Existence of Nash equilibrium in stationary strategies.

Characterization of Nash equilibria via principal eigenfunctions.

Applicable to systems with unbounded transition and cost rates.

Abstract

We study nonzero-sum stochastic games for continuous time Markov decision processes on a denumerable state space with risk-sensitive ergodic cost criterion. Transition rates and cost rates are allowed to be unbounded. Under a Lyapunov type stability assumption, we show that the corresponding system of coupled HJB equations admits a solution which leads to the existence of a Nash equilibrium in stationary strategies. We establish this using an approach involving principal eigenvalues associated with the HJB equations. Furthermore, exploiting appropriate stochastic representation of principal eigenfunctions, we completely characterize Nash equilibria in the space of stationary Markov strategies.