  In this paper we consider the De Finetti's optimal dividend and capital
injection problem under a Markov additive model. We assume that the surplus
process before dividends and capital injections follows a spectrally positive
Markov additive process. Dividend payments are made only at the jump times of
an independent Poisson process. Capitals are required to be injected whenever
needed to ensure a non-negative surplus process to avoid bankruptcy. Our
purpose is to characterize the optimal periodic dividend and capital injection
strategy that maximizes the expected total discounted dividends subtracted by
the total discounted costs of capital injection. To this end, we first consider
an auxiliary optimal periodic dividend and capital injection problem with final
payoff under a single spectrally positive L\'evy process and conjecture that
the optimal strategy is a double barrier strategy. Using the fluctuation theory
and excursion-theoretical approach of the spectrally positive L\'evy process
and the Hamilton-Jacobi-Bellman inequality approach of the control theory, we
are able to verify the conjecture that some double barrier periodic dividend
and capital injection strategy solves the auxiliary problem. With the results
for the auxiliary control problem and a fixed point argument for recursive
iterations induced by the dynamic programming principle, the optimality of a
regime-modulated double barrier periodic dividend and capital injection
strategy is proved for our target control problem.
