On the Optimal Dividend Problem for Insurance Risk Models with Surplus-Dependent Premiums

TL;DR

This paper solves an optimal dividend distribution problem for insurance models with surplus-dependent premiums, providing a complete solution and conditions for optimal strategies using Hamilton-Jacobi-Bellman equations.

Contribution

It introduces a comprehensive solution to the stochastic control problem for surplus-dependent premiums, including conditions for optimal dividend-band strategies.

Findings

Derived Hamilton-Jacobi-Bellman equation for the model

Established necessary and sufficient conditions for optimality

Analyzed concrete examples demonstrating the results

Abstract

This paper concerns an optimal dividend distribution problem for an insurance company with surplus-dependent premium. In the absence of dividend payments, such a risk process is a particular case of so-called piecewise deterministic Markov processes. The control mechanism chooses the size of dividend payments. The objective consists in maximazing the sum of the expected cumulative discounted dividend payments received until the time of ruin and a penalty payment at the time of ruin, which is an increasing function of the size of the shortfall at ruin. A complete solution is presented to the corresponding stochastic control problem. We identify the associated Hamilton-Jacobi-Bellman equation and find necessary and sufficient conditions for optimality of a single dividend-band strategy, in terms of particular Gerber-Shiu functions. A number of concrete examples are analyzed.