Risk-Sensitive Stochastic Control and Differential Games

Abstract

This thesis studies risk-sensitive stochastic optimal control and differential game problems. First, we study risk-sensitive stochastic differential games for controlled reflecting diffusion processes in a smooth bounded domain in Rd . We consider both nonzero-sum and zero-sum cases. We treat two cost evaluation criteria namely discounted cost and ergodic cost. Under certain assumptions, we establish the existence of a Nash/saddle-point equilibria for relevant cases. For ergodic cost criterion, we use principal eigenvalue approach to study the game problems. This approach enables us to obtain a complete characterization of Nash/saddle point equilibrium in the space of stationaryMarkov strategies. Subsequently, we study risk-sensitive ergodic control problem for controlled reflecting diffusion processes in the non-negative orthant. Under a certain Lyapunov type stability assumption and some other technical assumptions, we first establish the existence of a solution to the multiplicative Poisson equation for each stationary Markov control. Using this result, we establish the existence of a unique solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation. This, in turn, leads to the complete characterization of optimal control in the space of stationary Markov controls. Then we study risk-sensitive zero-sum/nonzero-sumstochastic differential games 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 a saddle point/Nash equilibria, for relevant cases. We obtain our results by studying the corresponding Hamilton-Jacobi-Isaacs (HJI)/coupled HJB equations. For the ergodic cost criterion, we completely characterize a saddle point/Nash equilibria in the space of stationary strategies. Finally, we study nonzero-sum stochastic differential games with risk-sensitive ergodic cost criteria, where the state space is a controlled diffusion process in Rd . Under certain conditions, we establish the existence of a Nash equilibriumin stationary strategies. We achieve our results by studying the relevant systems of coupled HJB equations. Also, we completely characterize a Nash equilibrium in the space of stationary strategies.