Optimal Dividend and Capital Injection Strategies for a Risk Model under Force of Interest

Ying Fang,Zhongfeng Qu

doi:10.1155/2013/750547

Abstract

As a generalization of the classical Cramér-Lundberg risk model, we consider a risk model including a constant force of interest in the present paper. Most optimal dividend strategies which only consider the processes modeling the surplus of a risk business are absorbed at 0. However, in many cases, negative surplus does not necessarily mean that the business has to stop. Therefore, we assume that negative surplus is not allowed and the beneficiary of the dividends is required to inject capital into the insurance company to ensure that its risk process stays nonnegative. For this risk model, we show that the optimal dividend strategy which maximizes the discounted dividend payments minus the penalized discounted capital injections is a threshold strategy for the case of the dividend payout rate which is bounded by some positive constant and the optimal injection strategy is to inject capitals immediately to make the company's assets back to zero when the surplus of the company becomes negative.

Highlights

In the mathematical finance and the actuarial literature, the optimal dividend problem has attracted much attention
[7] considered the optimal dividend problem when the risk process is modeled by a spectrally negative Levy process. They drew on the fluctuation theory of spectrally negative Levy processes and gave sufficient conditions under which the optimal strategy is of barrier type
We show that the optimal dividend strategy is a threshold strategy

Summary

Introduction

In the mathematical finance and the actuarial literature, the optimal dividend problem has attracted much attention. Albrecher and Thonhauser [5] studied the optimal dividend strategy by viscosity theory in the constant force of interest model They pointed out that the optimal dividend strategy in the general case is again of band type and for exponential claim sizes collapses to a barrier strategy. If the surplus process is a Brownian motion with drift, they found that the optimal injection policy is to invest the minimum such that the controlled surplus remains positive They show that the optimal dividend policy is a barrier strategy. In the more general framework of spectrally negative Levy processes, Avram et al [6] studied the optimality of barrier strategies with capital injections.

The Maximal Value Function

Optimal Dividend Strategy

Concluding Remarks