Stochastic differential games with controlled regime-switching

In this article, we consider a two-player zero-sum stochastic differential game with regime-switching. Different from the results in existing literature on stochastic differential games with regime-switching, we consider a game between a Markov chain and a state process which are two fully coupled stochastic processes. The payoff function is given by an integral with random terminal horizon. We first study the continuity of the lower and upper value functions under some additional conditions, based on which we establish the dynamic programming principle. We further prove that the lower and upper value functions are unique viscosity solutions of the associated lower and upper Hamilton–Jacobi–Bellman–Isaacs equations with regime-switching, respectively. These two value functions coincide under the Isaacs condition, which implies that the game admits a value. We finally apply our results to an example.