Zero-Sum Risk-Sensitive Stochastic Differential Games

We study zero-sum risk-sensitive stochastic differential games on the infinite horizon with discounted and ergodic payoff criteria. Under certain assumptions, we establish the existence of values and saddle-point equilibria. We obtain our results by studying the corresponding Hamilton--Jacobi--Isaacs equations. Finally, we show that the value of the ergodic payoff criterion is a constant multiple of the maximal eigenvalue of the generators of the associated nonlinear semigroups.

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