Optimal dividend policies with transaction costs for a class of jump-diffusion processes

This paper addresses the problem of finding an optimal dividend policy for a class of jump-diffusion processes. The jump component is a compound Poisson process with negative jumps, and the drift and diffusion components are assumed to satisfy some regularity and growth restrictions. Each dividend payment is changed by a fixed and a proportional cost, meaning that if ξ is paid out by the company, the shareholders receive kξ−K, where k and K are positive. The aim is to maximize expected discounted dividends until ruin. It is proved that when the jumps belong to a certain class of light-tailed distributions, the optimal policy is a simple lump sum policy, that is, when assets are equal to or larger than an upper barrier <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\bar{u}^{*}$</EquationSource> </InlineEquation>, they are immediately reduced to a lower barrier <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\underline{u}^{*}$</EquationSource> </InlineEquation> through a dividend payment. The case with K=0 is also investigated briefly, and the optimal policy is shown to be a reflecting barrier policy for the same light-tailed class. Methods to numerically verify whether a simple lump sum barrier strategy is optimal for any jump distribution are provided at the end of the paper, and some numerical examples are given. Copyright Springer-Verlag 2013