Optimal dividend and capital injection under spectrally positive Markov additive models

TL;DR

This paper addresses an optimal dividend and capital injection problem within spectrally positive Markov additive models, demonstrating the optimality of a regime-modulated double barrier strategy through analytical and recursive methods.

Contribution

It introduces a novel approach to solve the optimal dividend and capital injection problem in Markov additive models, characterizing optimal barriers analytically and confirming strategy optimality.

Findings

Double barrier strategy is optimal under the model.

Optimal barriers are explicitly characterized using fluctuation identities.

Regime-modulated strategies are effective for complex stochastic models.

Abstract

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.