Differential game with discrete stopping time

We consider a two agents zero-sum differential game with discrete stopping times. The payoff function may depend both on the stopping time, on the state of the system at this time, and on the player who constitutes this stopping time. To formalize strategies, non-anticipating càdlàg stochastic processes are used. Under the Isaacs condition, the existence of the game value is established. The corresponding near-optimal strategy is constructed with stochastic guide based on an auxiliary model game governed by continuous-time Markov chain.