Discounted stochastic games for continuous-time jump processes with an uncountable state space

In this paper we study the nonzero-sum stochastic games for continuous-time jump processes under the expected discounted payoff criterion. The state space and action spaces for players are Borel spaces and the reward and transition rates are allowed to be unbounded. Under certain reasonable conditions, we first introduce new auxiliary static games and obtain the corresponding properties of these static games. Then by an approximation approach, we show the existence of a stationary almost Markov Nash equilibrium in the class of randomized history-dependent strategy profiles. Moreover, we use two examples to illustrate our main results.