Risk-sensitive continuous-time stochastic games with the average criterion and a compact state space

This paper attempts to study the risk-sensitive average continuous-time stochastic game with compact state and action spaces. We derive an equivalent Shapley equation for the risk-sensitive average criterion. By building a novel parametric operator and analyzing the properties of an eigenvalue of the operator, we prove the equivalent Shapley equation admits a solution, and then establish the existence of the value and a Nash equilibrium over the class of history-dependent policies. Moreover, we design an iterative algorithm for computing the value of the game and prove the convergence of the algorithm. Finally, two examples are given to verify our results.