  This paper studies the bail-out optimal dividend problem with regime
switching under the constraint that the cumulative dividend strategy is
absolutely continuous. We confirm the optimality of the regime-modulated
refraction-reflection strategy when the underlying risk model follows a general
spectrally negative Markov additive process. To verify the conjecture of a
barrier type optimal control, we first introduce and study an auxiliary problem
with the final payoff at an exponential terminal time and characterize the
optimal threshold explicitly using fluctuation identities of the
refracted-reflected Levy process. Second, we transform the problem with
regime-switching into an equivalent local optimization problem with a final
payoff up to the first regime switching time. The refraction-reflection
strategy with regime-modulated thresholds can be shown as optimal by using
results in the first step and some fixed point arguments for auxiliary
recursive iterations.
