Zero-sum risk-sensitive continuous-time stochastic games with unbounded reward and transition rates in Borel spaces

Abstract

This paper investigates a finite-horizon two-player zero-sum risk-sensitive stochastic game in continuous-time Markov chains with Borel state and action spaces. The model accommodates unbounded reward rates, transition rates, and terminal reward functions, while permitting history-dependent policies. The risk metric is the exponential utility function. Under appropriate conditions, we establish the existence of a solution to the corresponding Shapley equation (SE) through an approximation technique. Using the SE and an extension of Dynkin’s formula, we prove the existence of saddle-point equilibrium and demonstrate that the stochastic game’s value is the unique solution to the SE. Furthermore, we develop a value iteration algorithm for approximating the stochastic game’s value, with convergence guaranteed by a specialized contraction operator within our risk-sensitive stochastic game framework. Finally, we illustrate our main findings through an example.