  This paper presents three versions of maximum principle for a stochastic
optimal control problem of Markov regime-switching forward-backward stochastic
differential equations with jumps (FBSDEJs). A general sufficient maximum
principle for optimal control for a system driven by a Markov regime-switching
forward and backward jump-diffusion model is developed. After, an equivalent
maximum principle is proved. Malliavin calculus is also employed to derive a
general stochastic maximum principle. The latter does not require concavity of
Hamiltonian. Applications of the stochastic maximum principle to non-concave
Hamiltonian and recursive utility maximization is also discussed.
