Optimal Dividend Payments for the Piecewise-Deterministic Poisson Risk Model

TL;DR

This paper analyzes optimal dividend strategies in piecewise-deterministic Poisson risk models, deriving threshold and barrier policies for restricted and unrestricted schemes, with explicit solutions under exponential claims and practical examples.

Contribution

It provides a unified framework for both restricted and unrestricted dividend schemes, deriving explicit optimal strategies and conditions for exponential claim sizes, extending prior separate analyses.

Findings

Threshold strategy is optimal for restricted schemes.

Barrier strategy is optimal for unrestricted schemes.

Explicit conditions for strategy optimality with exponential claims.

Abstract

This paper considers the optimal dividend payment problem in piecewise-deterministic compound Poisson risk models. The objective is to maximize the expected discounted dividend payout up to the time of ruin. We provide a comparative study in this general framework of both restricted and unrestricted payment schemes, which were only previously treated separately in certain special cases of risk models in the literature. In the case of restricted payment scheme, the value function is shown to be a classical solution of the corresponding HJB equation, which in turn leads to an optimal restricted payment policy known as the threshold strategy. In the case of unrestricted payment scheme, by solving the associated integro-differential quasi-variational inequality, we obtain the value function as well as an optimal unrestricted dividend payment scheme known as the barrier strategy. When claim sizes are exponentially distributed, we provide easily verifiable conditions under which the threshold and barrier strategies are optimal restricted and unrestricted dividend payment policies, respectively. The main results are illustrated with several examples, including a new example concerning regressive growth rates.