Stochastic control with dividend payments and capital injections for Markov additive processes

TL;DR

This paper investigates optimal dividend and capital injection strategies for Markov additive processes, extending previous models to more general modulating processes and providing new optimality conditions and strategies.

Contribution

It introduces a framework for stochastic control of MAPs with general modulating processes, deriving optimality conditions for barrier strategies and proposing new proof techniques.

Findings

Optimality of Markov-modulated barrier strategies established.

Necessary and sufficient conditions for dividend strategy optimality derived.

Insight into strategies when dividends and injections occur at arbitrary times obtained.

Abstract

Motivated by de Finetti's optimal dividend problem with capital injections, we study a stochastic control problem for the additive component of a Markov additive process (MAP). In contrast to previous studies, the modulating component is allowed to be a general right process on a Radon space, so the model is not restricted to finite-state regime switching and cannot in general be reduced to a finite collection of L\'evy process control problems. Capital injections are allowed at arbitrary times. We first consider the case in which dividend payments are allowed only at prescribed discrete times and establish necessary and sufficient conditions for the optimality of a strategy. These conditions then yield the optimality of a class of Markov-modulated periodic--classical barrier strategies. Combining this optimality result with an approximation argument, we obtain insight into the possible form of optimal strategies in the case where dividend payments, like capital injections, may be made at arbitrary times. Because of the generality of the MAPs considered here, the proof techniques used in previous studies of similar problems are not directly applicable. We therefore develop an alternative argument based on the additive structure of MAPs and dynamic programming between dividend opportunities. The argument also suggests a possible approach to other stochastic control problems involving general MAPs.