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We study a De Finetti's optimal dividend and capital injection problem undera Markov additive model. The surplus process before dividend and capitalinjection is assumed to follow a spectrally positive Markov additive process(MAP). Dividend payments are made only at the jump times of an independentPoisson process and capitals are injected to avoid bankruptcy. The aim of thepaper is to characterize an optimal periodic dividend and capital injectionstrategy that maximizes the expected total discounted dividends subtracted bythe total discounted costs of capital injection. Applying the fluctuation andexcursion theory for Levy processes and the stochastic control theory, we firstaddress an auxiliary periodic dividend and capital injection control problemwith a terminal payoff under the spectrally positive Levy process. Usingresults obtained for this auxiliary problem and a fixed point argument foriterations induced by dynamic program, we characterize the optimal strategy ofour prime control problem as a regime-modulated double-barrierperiodic-continuous-reflection dividend and capital injection strategy.

De Finetti's Control Problem with Poisson Observations under Spectrally Positive Markov Additive Process