Risk-sensitive zero-sum stochastic differential game for jump–diffusions

Abstract

This article is concerned with the infinite horizon risk-sensitive zero-sum stochastic differential game problem for a class of jump–diffusions controlled through the drift, and driven by a compensated Poisson process and a Wiener process. Under certain geometric stability assumption on the dynamics, we completely characterize all possible saddle point strategies in the class of stationary Markov strategies. We obtain our result by exploiting the stochastic representation of the principal eigenfunction of the associated Hamilton–Jacobi–Isaac (HJI) equation, which is a semilinear integro-partial differential equation.