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We study a finite-horizon two-person zero-sum risk-sensitive stochastic gamefor continuous-time Markov chains and Borel state and action spaces, in whichpayoff rates, transition rates and terminal reward functions are allowed to beunbounded from below and from above and the policies can be history-dependent.Under suitable conditions, we establish the existence of a solution to thecorresponding Shapley equation (SE) by an approximation technique. Then, by theSE and the extension of the Dynkin's formula, we prove the existence of a Nashequilibrium and verify that the value of the stochastic game is the uniquesolution to the SE. Moreover, we develop a value iteration-type algorithm forapproaching to the value of the stochastic game. The convergence of thealgorithm is proved by a special contraction operator in our risk-sensitivestochastic game. Finally, we demonstrate our main results by two examples.

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