On the Bail-Out Optimal Dividend Problem

This paper studies the optimal dividend problem with capital injection under the constraint that the cumulative dividend strategy is absolutely continuous. We consider an open problem of the general spectrally negative case and derive the optimal solution explicitly using the fluctuation identities of the refracted-reflected L\'evy process. The optimal strategy as well as the value function are concisely written in terms of the scale function. Numerical results are also provided to confirm the analytical conclusions.Comment: To appear in Journal of Optimization Theory and Applications. Keywords: stochastic control, scale functions, refracted-reflected L\'evy processes, bail-out dividend proble

Optimal dividend and capital injection under spectrally positive Markov additive models

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