The stochastic maximum principle for relaxed control problem with regime-switching

We study a stochastic relaxed control problem with regime-switching, in which the control enters both the drift and the diffusion coefficients. The goal is to establish the stochastic maximum principle and verification theorem for this problem with the convex variational method. A backward stochastic differential equation (BSDE) with regime-switching is given as the adjoint equation whose martingale terms are generated by both the Brownian motion and the Markov chain. A linear quadratic example in relaxed formulation is also solved explicitly.