Two-sided jumps risk model with proportional investment and random observation periods

Chunwei Wang; Jiaen Xu; Naidan Deng; Shujing Wang; Chunwei Wang; Jiaen Xu; Naidan Deng; Shujing Wang

doi:10.3934/math.20231137

AIMS Mathematics

2023, Volume 8, Issue 9: 22301-22318. doi: 10.3934/math.20231137

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Research article

Two-sided jumps risk model with proportional investment and random observation periods

School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471023, Henan, China
$ \dagger $ These authors contributed equally to this work.

Received: 07 May 2023 Revised: 29 June 2023 Accepted: 03 July 2023 Published: 13 July 2023
MSC : 65C30, 91B05, 91G05

In this paper, we consider a two-sided jumps risk model with proportional investments and random observation periods. The downward jumps represent the claim while the upward jumps represent the random returns. Suppose an insurance company invests all of their surplus in risk-free and risky investments in proportion. In real life, corporate boards regularly review their accounts rather than continuously monitoring them. Therefore, we assume that insurers regularly observe surplus levels to determine whether they will ruin and that the random observation periods are exponentially distributed. Our goal is to study the Gerber-Shiu function (i.e., the expected discounted penalty function) of the two-sided jumps risk model under random observation. First, we derive the integral differential equations (IDEs) satisfied by the Gerber-Shiu function. Due to the difficulty in obtaining explicit solutions for the IDEs, we utilize the sinc approximation method to obtain the approximate solution. Second, we analyze the error between the approximate and explicit solutions and find the upper bound of the error. Finally, we discuss examples of sensitivity analysis.
- two-sided jumps,
- Gerber-Shiu function,
- proportional investment,
- random observation,
- sinc numerical method
Citation: Chunwei Wang, Jiaen Xu, Naidan Deng, Shujing Wang. Two-sided jumps risk model with proportional investment and random observation periods[J]. AIMS Mathematics, 2023, 8(9): 22301-22318. doi: 10.3934/math.20231137

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Abstract

In this paper, we consider a two-sided jumps risk model with proportional investments and random observation periods. The downward jumps represent the claim while the upward jumps represent the random returns. Suppose an insurance company invests all of their surplus in risk-free and risky investments in proportion. In real life, corporate boards regularly review their accounts rather than continuously monitoring them. Therefore, we assume that insurers regularly observe surplus levels to determine whether they will ruin and that the random observation periods are exponentially distributed. Our goal is to study the Gerber-Shiu function (i.e., the expected discounted penalty function) of the two-sided jumps risk model under random observation. First, we derive the integral differential equations (IDEs) satisfied by the Gerber-Shiu function. Due to the difficulty in obtaining explicit solutions for the IDEs, we utilize the sinc approximation method to obtain the approximate solution. Second, we analyze the error between the approximate and explicit solutions and find the upper bound of the error. Finally, we discuss examples of sensitivity analysis.

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