On Gerber-Shiu functions and optimal dividend distribution for a L\'{e}vy risk process in the presence of a penalty function

TL;DR

This paper solves an optimal dividend distribution problem for a spectrally negative Lévy risk process, incorporating penalties at ruin and fixed costs, providing explicit conditions for optimal strategies and analyzing concrete examples.

Contribution

It offers a complete solution to the stochastic control problem, characterizes the value function, and establishes conditions for optimal dividend-band strategies using Gerber-Shiu functions.

Findings

Unique stochastic solution for the value function

Necessary and sufficient conditions for optimal dividend-band strategies

Explicit analysis of concrete examples

Abstract

This paper concerns an optimal dividend distribution problem for an insurance company whose risk process evolves as a spectrally negative L\'{e}vy process (in the absence of dividend payments). The management of the company is assumed to control timing and size of dividend payments. The objective is to maximize the sum of the expected cumulative discounted dividend payments received until the moment of ruin and a penalty payment at the moment of ruin, which is an increasing function of the size of the shortfall at ruin; in addition, there may be a fixed cost for taking out dividends. A complete solution is presented to the corresponding stochastic control problem. It is established that the value-function is the unique stochastic solution and the pointwise smallest stochastic supersolution of the associated HJB equation. Furthermore, a necessary and sufficient condition is identified for optimality of a single dividend-band strategy, in terms of a particular Gerber-Shiu function. A number of concrete examples are analyzed.