Maximum principle for optimal control problems of forward–backward regime-switching system and applications

In this paper, we derive the stochastic maximum principle for optimal control problems of the forward–backward Markovian regime-switching system. The control system is described by forward–backward SDEs and modulated by continuous-time, finite-state Markov chains. We first obtain the necessary and sufficient conditions for the optimal control. Thereafter, we apply the maximum principle to recursive utility investment–consumption problems and LQ problems with Markovian regime-switching.