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In this paper we consider the De Finetti's optimal dividend and capitalinjection problem under a Markov additive model. We assume that the surplusprocess before dividends and capital injections follows a spectrally positiveMarkov additive process. Dividend payments are made only at the jump times ofan independent Poisson process. Capitals are required to be injected wheneverneeded to ensure a non-negative surplus process to avoid bankruptcy. Ourpurpose is to characterize the optimal periodic dividend and capital injectionstrategy that maximizes the expected total discounted dividends subtracted bythe total discounted costs of capital injection. To this end, we first consideran auxiliary optimal periodic dividend and capital injection problem with finalpayoff under a single spectrally positive L\'evy process and conjecture thatthe optimal strategy is a double barrier strategy. Using the fluctuation theoryand excursion-theoretical approach of the spectrally positive L\'evy processand the Hamilton-Jacobi-Bellman inequality approach of the control theory, weare able to verify the conjecture that some double barrier periodic dividendand capital injection strategy solves the auxiliary problem. With the resultsfor the auxiliary control problem and a fixed point argument for recursiveiterations induced by the dynamic programming principle, the optimality of aregime-modulated double barrier periodic dividend and capital injectionstrategy is proved for our target control problem.

On De Finetti's control under Poisson observations: optimality of a double barrier strategy in a Markov additive model