Continuous-time zero-sum games for Markov chains with risk-sensitive finite-horizon cost criterion

In this paper, we study the two-person zero-sum stochastic games for controlled continuous time Markov chains with risk-sensitive finite-horizon cost criterion. The transition and cost rates are possibly unbounded. For the zero-sum risk-sensitive stochastic game, we prove the existence of the value of the game and a Markov saddle-point equilibrium in the class of all history-dependent multi-strategies under the suitable conditions. We achieve our results by studying the corresponding risk-sensitive finite-horizon optimality equations.