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

TL;DR

This paper characterizes the optimal control strategy for dividends and capital injections in a Markov additive model with Poisson observation times, proving the optimality of a double barrier strategy using advanced stochastic analysis.

Contribution

It introduces a novel optimal control framework under a Markov additive process with Poisson observation, establishing the optimality of a double barrier strategy through rigorous fluctuation and control theory methods.

Findings

Optimal double barrier strategy is proven to be optimal.

The auxiliary problem solution supports the main conjecture.

The approach combines fluctuation theory and dynamic programming.

Abstract

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.