  This paper studies the bailout optimal dividend problem with regime switching
under the constraint that dividend payments can be made only at the arrival
times of an independent Poisson process while capital can be injected
continuously in time. We show the optimality of the regime-modulated
Parisian-classical reflection strategy when the underlying risk model follows a
general spectrally negative Markov additive process. In order to verify the
optimality, first we study an auxiliary problem driven by a single spectrally
negative \lev process with a final payoff at an exponential terminal time and
characterise the optimal dividend strategy. Then, we use the dynamic
programming principle to transform the global regime-switching problem into an
equivalent local optimization problem with a final payoff up to the first
regime switching time. The optimality of the regime modulated
Parisian-classical barrier strategy can be proven by using the results from the
auxiliary problem and approximations via recursive iterations.
