This paper studies De Finetti's optimal dividend problem with capital
injection under spectrally positive Markov additive models. Based on dynamic
programming principle, we first study an auxiliary singular control problem
with a final payoff at an exponential random time. The double barrier strategy
is shown to be optimal and the optimal barriers are characterized in analytical
form using fluctuation identities of spectrally positive Levy processes. We
then transform the original problem under spectrally positive Markov additive
models into an equivalent series of local optimization problems with the final
payoff at the regime-switching time. The optimality of the regime-modulated
double barrier strategy can be confirmed for the original problem using results
from the auxiliary problem and the fixed point argument for recursive
iterations.Comment: Keywords: Spectrally positive Levy process, regime switching, De
Finetti's optimal dividend, capital injection, double barrier strategy,
singular contro