Zero-sum risk-sensitive stochastic differential games with reflecting diffusions in the orthant

We study zero-sum games with risk-sensitive cost criterion on the infinite horizon where the state is a controlled reflecting diffusion in the nonnegative orthant. We consider two cost evaluation criteria: discounted cost and ergodic cost. Under certain assumptions, we establish the existence of saddle point equilibria. We obtain our results by studying the corresponding Hamilton–Jacobi–Isaacs equations. For the ergodic cost criterion, exploiting the stochastic representation of the principal eigenfunction, we have completely characterized saddle point equilibrium in the space of stationary Markov strategies.