Optimal dividend of compound poisson process under a stochastic interest rate

Linlin Tian; Xiaoyi Zhang; Yizhou Bai

doi:10.3934/jimo.2019047

Article Contents

2020, Volume 16, Issue 5: 2141-2157. Doi: 10.3934/jimo.2019047

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Optimal dividend of compound poisson process under a stochastic interest rate

1.
School of Mathematical Sciences, Nankai University, Tianjin 300071, China
2.
School of Economics and Management, Hebei University of Technology, Tianjin 300401, China

^* Corresponding author: Xiaoyi Zhang
^* Corresponding author: Xiaoyi Zhang

Received: January 2018

Revised: November 2018

Early access: May 2019

Published: August 2020

Research is supported by Chinese NSF Grants No.11471171 and No.11571189

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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.

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Mathematics Subject Classification: Primary: 93E20; Secondary: 91B30.

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Figure 1. The shape of the value function

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Figure 2. Left picture: The sensitivity of $ V $ about parameter $ \beta $. Right picture: The sensitivity of $ V $ about parameter $ \lambda $

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Figure 3. the realization of $ \exp\{-U_s^r\} $, $ r = 1, a = 1, {\hat{\delta }} = 1 $ for $ \hat{b} = 2 $ and $ \hat{b} = -2 $

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