ON DIVIDENDS AND GERBER-SHIU ANALYSIS WITH CONSTANT INTEREST AND A PERIODIC-THRESHOLD MIXED STRATEGY

doi:10.1007/s10473-024-0606-0

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (6): 2139-2164.doi: 10.1007/s10473-024-0606-0

ON DIVIDENDS AND GERBER-SHIU ANALYSIS WITH CONSTANT INTEREST AND A PERIODIC-THRESHOLD MIXED STRATEGY

Zhang LIU^1,†, Ping CHEN²

1. School of Computer and Information Engineering, Jiangxi Agricultural University, Nanchang 330045, China;
2. Department of Economics, The University of Melbourne, Victoria 3010, Australia

收稿日期:2023-09-07 修回日期:2024-03-12 发布日期:2024-12-06
通讯作者: † Zhang LIU, E-mail: liuzhang1006@163.com
作者简介:Ping CHEN, E-mail: pche@unimelb.edu.au
基金资助:
Liu's research was supported by the Nation-al Natural Science Foundation of China (12361095) and the Jiangxi Provincial Natural Science Foundation (20232BAB201028).

ON DIVIDENDS AND GERBER-SHIU ANALYSIS WITH CONSTANT INTEREST AND A PERIODIC-THRESHOLD MIXED STRATEGY

Zhang LIU^1,†, Ping CHEN²

1. School of Computer and Information Engineering, Jiangxi Agricultural University, Nanchang 330045, China;
2. Department of Economics, The University of Melbourne, Victoria 3010, Australia

Received:2023-09-07 Revised:2024-03-12 Published:2024-12-06
Contact: † Zhang LIU, E-mail: liuzhang1006@163.com
About author:Ping CHEN, E-mail: pche@unimelb.edu.au
Supported by:
Liu's research was supported by the Nation-al Natural Science Foundation of China (12361095) and the Jiangxi Provincial Natural Science Foundation (20232BAB201028).

摘要/Abstract

摘要： In recent years, the research focus in insurance risk theory has shifted towards multi-type mixed dividend strategies. However, the practical factors and constraints in financial market transactions, such as interest rates, tax rates, and transaction fees, inevitably impact these strategies. By incorporating appropriate constraints, a multi-type mixed strategy can better simulate real-world transactions. Following the approach of Liu et al. [28], we examine a classical compound Poisson risk model that incorporates the constraints of constant interest rates and a periodic-threshold mixed dividend strategy. In this model, the surplus process of insurance companies is influenced by several factors. These factors include constant interest rates, continuously distributed dividends within intervals (threshold dividend strategy), and dividends at discrete time points (periodic dividend strategy). We derive the piecewise integro-differential equations (IDEs) that describe the expected present value of dividends (EPVDs) until ruin time and the Gerber-Shiu expected discounted penalty function. Furthermore, we provide explicit solutions to these IDEs using an alternative method based on the inverse Laplace transform combined with the Dickson-Hipp operator. This enables us to obtain explicit expressions for the dividend and Gerber-Shiu functions. Additionally, we present examples to illustrate the application of our results.

关键词: Cramér-Lundberg model, mixed dividend strategy, interest, EPVDs, Gerber-Shiu function

Abstract: In recent years, the research focus in insurance risk theory has shifted towards multi-type mixed dividend strategies. However, the practical factors and constraints in financial market transactions, such as interest rates, tax rates, and transaction fees, inevitably impact these strategies. By incorporating appropriate constraints, a multi-type mixed strategy can better simulate real-world transactions. Following the approach of Liu et al. [28], we examine a classical compound Poisson risk model that incorporates the constraints of constant interest rates and a periodic-threshold mixed dividend strategy. In this model, the surplus process of insurance companies is influenced by several factors. These factors include constant interest rates, continuously distributed dividends within intervals (threshold dividend strategy), and dividends at discrete time points (periodic dividend strategy). We derive the piecewise integro-differential equations (IDEs) that describe the expected present value of dividends (EPVDs) until ruin time and the Gerber-Shiu expected discounted penalty function. Furthermore, we provide explicit solutions to these IDEs using an alternative method based on the inverse Laplace transform combined with the Dickson-Hipp operator. This enables us to obtain explicit expressions for the dividend and Gerber-Shiu functions. Additionally, we present examples to illustrate the application of our results.

Key words: Cramér-Lundberg model, mixed dividend strategy, interest, EPVDs, Gerber-Shiu function

中图分类号:

35B10

Zhang LIU, Ping CHEN. ON DIVIDENDS AND GERBER-SHIU ANALYSIS WITH CONSTANT INTEREST AND A PERIODIC-THRESHOLD MIXED STRATEGY[J]. 数学物理学报(英文版), 2024, 44(6): 2139-2164.

Zhang LIU, Ping CHEN. ON DIVIDENDS AND GERBER-SHIU ANALYSIS WITH CONSTANT INTEREST AND A PERIODIC-THRESHOLD MIXED STRATEGY[J]. Acta mathematica scientia,Series B, 2024, 44(6): 2139-2164.

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ON DIVIDENDS AND GERBER-SHIU ANALYSIS WITH CONSTANT INTEREST AND A PERIODIC-THRESHOLD MIXED STRATEGY

ON DIVIDENDS AND GERBER-SHIU ANALYSIS WITH CONSTANT INTEREST AND A PERIODIC-THRESHOLD MIXED STRATEGY

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