Nonzero-sum risk-sensitive stochastic differential games: A multi-parameter eigenvalue problem approach

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 Hamilton–Jacobi–Bellman (HJB) equations. Exploiting the stochastic representation of the principal eigenfunctions we completely characterize Nash equilibrium points in the space of stationary Markov strategies. The complete characterization of Nash equilibrium points is established under an additive structural assumption on the running cost and the drift term.