Optimal Dividend Problem for a Compound Poisson Risk Model

Abstract

In this note we study the optimal dividend problem for a company whose surplus process, in the absence of dividend payments, evolves as a generalized compound Poisson model in which the counting process is a generalized Poisson process. This model including the classical risk model and the Pólya-Aeppli risk model as special cases. The objective is to find a dividend policy so as to maximize the expected discounted value of dividends which are paid to the shareholders until the company is ruined. We show that under some conditions the optimal dividend strategy is formed by a barrier strategy.Moreover, two conjectures are proposed.

Branching particle systems and compound Poisson processes related to Pólya-Aeppli distributions

“…They were considered as early as in [5, p. 96]. More recently, their realizations in specific stochastic models, characterizations, properties and applications were discussed in [3], [4], [15], [16], [22], [32].…”

“…(Note in passing that the subclass of Hougaard processes which corresponds to the value of the power parameter p = 3/2 comprises the entire family of compound Poisson-exponential processes.) Some applications in Risk Theory for more general classes of Lévy processes, but which all contain Pólya-Aeppli processes as their components associated with the structure of the jumps of these processes, were addressed in [15], [22], [32].…”