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.