A maximum principle for Markov regime-switching forward‐backward stochastic differential games and applications

In this paper, we present an optimal control problem for stochastic differential games under Markov regime‐switching forward–backward stochastic differential equations with jumps. First, we prove a sufficient maximum principle for nonzero‐sum stochastic differential games problems and obtain equilibrium point for such games. Second, we prove an equivalent maximum principle for nonzero‐sum stochastic differential games. The zero‐sum stochastic differential games equivalent maximum principle is then obtained as a corollary. We apply the obtained results to study a problem of robust utility maximization under a relative entropy penalty and to find optimal investment of an insurance firm under model uncertainty.