Optimal Dividend Strategy in the Compound Poisson Model with Constant Interest

Abstract

In this paper, we study optimal dividend problems in a compound Poisson risk process with constant interest on investments. Throughout the paper, we assume that the rate of dividend pay out is bounded by some positive constant. It is shown that the optimal value function of the dividends can be characterized by the Hamilton–Jacobi–Bellman equation. For the case of an exponential claim amount distribution, it is shown that the optimal dividend strategy is a threshold strategy. Then the Laplace transform of the time of ruin, under a threshold strategy, can be calculated. Kummer's confluent hypergeometric equation and confluent hypergeometric functions play a key role in this paper.

Keywords:

• Compound Poisson risk model
• Confluent hypergeometric function
• Constant interest force
• Kummer's confluent hypergeometric equation
• Threshold strategy

Mathematics Subject Classification:

• 60J75
• 91B30

ACKNOWLEDGMENTS

The authors would like to thank the referees for valuable suggestions to improve the earlier version of the paper. The work is supported by NNSF (grant No. 10571092) of China and the Research Fund for the Doctorial Program of Higher Education.