Optimal dividend of compound poisson process under a stochastic interest rate

,School Of Mathematical Sciences, Nankai University, Tianjin 300071, China ,Linlin Tian,Xiaoyi Zhang, ,Yizhou Bai

doi:10.3934/jimo.2019047

,School Of Mathematical Sciences, Nankai University, Tianjin 300071, China , Linlin Tian + Show 3 more

Open Access

https://doi.org/10.3934/jimo.2019047

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Abstract

In this paper we assume the insurance wealth process is driven by the compound Poisson process. The discounting factor is modelled as a geometric Brownian motion at first and then as an exponential function of an integrated Ornstein-Uhlenbeck process. The objective is to maximize the cumulated value of expected discounted dividends up to the time of ruin. We give an explicit expression of the value function and the optimal strategy in the case of interest rate following a geometric Brownian motion. For the case of the Vasicek model, we explore some properties of the value function. Since we can not find an explicit expression for the value function in the second case, we prove that the value function is the viscosity solution of the corresponding HJB equation.

Highlights

The optimal dividend problem has been discussed for a long time in the literature
In the case of a surplus process following a compound Poisson process, Gerber and Shiu [12] showed that the optimal strategy is a threshold strategy when claim size are exponentially distributed for restricted dividend rates
They found an explicit expression for the value function of the optimal strategy for both restricted and unrestricted dividends in the case of geometric Brownian motion

Summary

Introduction

The optimal dividend problem has been discussed for a long time in the literature. In 1957 De Finetti [9] proposed that an insurance company should allow cash leakages and measure their performance during its life time instead of only focussing on ruin probability. In the setting of constant interest rate, Asmussen and Taksar [5] solved the optimal dividend problem for the special case of Brownian motion. The discounting factor is modelled as a stochastic process: at first as a geometric Brownian motion, as an exponential function of an integrated Ornstein-Uhlenbeck process They found an explicit expression for the value function of the optimal strategy for both restricted and unrestricted dividends in the case of geometric Brownian motion. It is natural to consider the problem in the framework of viscosity solutions

Problem Formulation

Geometric Brownian motion as a discounting factor

Solving HJB Equation

Ornstein-Uhlenbeck Process as a Interest Rate

Properties of the Value Function

Concluding Remarks