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