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An insurance risk process with a generalized income process: A solvency analysis
Insurance: Mathematics and Economics ( IF 1.9 ) Pub Date : 2021-03-06 , DOI: 10.1016/j.insmatheco.2021.02.005
Zijia Wang , David Landriault , Shu Li

In ruin theory, an insurer’s income process is usually assumed to grow at a deterministic rate of c>0 over time. For instance, both the well-known Cramér–Lundberg risk process and the Sparre Andersen risk model have this assumption built in the construction of their respective surplus processes. This assumption is mainly considered for purposes of mathematical tractability, but generally fails to accurately model an insurer’s income dynamics. To better characterize the variability and uncertainty of an insurer’s income process, several papers have studied insurance risk models with random incomes where the main emphasis is placed on carrying the related Gerber–Shiu analysis. However, a systematic and quantitative understanding of how the more volatile income processes impact an insurer’s solvency risk is still lacking. This paper aims to fill this gap in the literature by quantitatively assessing the impact of the choice of income process on some finite-time and infinite-time ruin quantities. To carry this analysis, we consider a generalized Sparre Andersen risk model with a random income process which renews at claim instants. For exponentially distributed claim sizes, we derive explicit expressions for some joint distributions involving the time to ruin and the number of claims until ruin. As special cases of the proposed insurance risk process, we consider income processes modelled by a subordinator or a particular varying premium rate model. Numerical examples are then carried to draw important risk management implications of a solvency nature for the insurer.



中文翻译:

具有广义收益过程的保险风险过程:偿付能力分析

在破产理论中,通常假定保险公司的收入过程以确定的增长率增长。 C>0随着时间的推移。例如,众所周知的Cramér-Lundberg风险过程和Sparre Andersen风险模型都在构建各自盈余过程时建立了这一假设。主要出于数学易处理性的目的考虑此假设,但通常无法准确地对保险公司的收入动态进行建模。为了更好地描述保险公司收入过程的可变性和不确定性,有几篇论文研究了随机收入的保险风险模型,其中主要着重于进行相关的Gerber-Shiu分析。但是,仍然缺乏对更加不稳定的收入过程如何影响保险公司偿付能力风险的系统和定量的了解。本文旨在通过定量评估收入过程的选择对某些有限时间和无限时间破产数量的影响来填补文献中的空白。为了进行此分析,我们考虑了带有随机收益过程的广义Sparre Andersen风险模型,该模型在索偿瞬间更新。对于呈指数分布的索赔额,我们导出了一些联合分布的显式表达式,其中涉及破坏时间和直到破产为止的索赔数量。作为拟议的保险风险过程的特殊情况,我们考虑由下属或特定的变动保费率模型建模的收入过程。然后通过数字示例得出保险公司具有偿付能力性质的重要风险管理含义。我们考虑具有随机收益过程的广义Sparre Andersen风险模型,该模型会在索赔时立即更新。对于呈指数分布的索赔额,我们导出了一些联合分布的显式表达式,其中涉及破坏时间和直到破产为止的索赔数量。作为拟议的保险风险过程的特殊情况,我们考虑由下属或特定的变动保费率模型建模的收入过程。然后通过数字示例得出保险公司具有偿付能力性质的重要风险管理含义。我们考虑具有随机收益过程的广义Sparre Andersen风险模型,该模型会在索赔时立即更新。对于呈指数分布的索赔额,我们导出了一些联合分布的显式表达式,其中涉及破坏时间和直到破产为止的索赔数量。作为拟议的保险风险过程的特殊情况,我们考虑由下属或特定的变动保费率模型建模的收入过程。然后通过数字示例得出保险公司具有偿付能力性质的重要风险管理含义。作为拟议的保险风险过程的特殊情况,我们考虑由下属或特定的变动保费率模型建模的收入过程。然后通过数字示例得出保险公司具有偿付能力性质的重要风险管理含义。作为拟议的保险风险过程的特殊情况,我们考虑由下属或特定的变动保费率模型建模的收入过程。然后通过数字示例得出保险公司具有偿付能力性质的重要风险管理含义。

更新日期:2021-03-29
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