arXiv 2007.014262020-07-02q-fin.RM

基于Lévy过程的清算风险模型

Risk Modelling on Liquidations with Lévy Processes

Aili Zhang, Ping Chen, Shuanming Li, Wenyuan Wang

备注：: 4 figures

摘要：

破产理论的学术界将保险公司的破产定义为其盈余降至零以下已经有几十年了。然而，这种简化需要仔细调整，以模仿现实世界的清算过程。受Broadie et al.（2007）和Li et al.（2020）的启发，本文使用三障碍模型来描述保险公司破产的财务压力。将保险公司的财务状况分为偿付能力、破产和清算三种状态，其中保险公司在偿付能力和破产状态下的盈余过程由两个谱负的Lévy过程来模拟，这两个谱负的Lévy过程被认为是模拟保险风险的好的候选过程.在这个三障碍模型中，我们给出了破产清算时间的严格定义。利用波动理论中的偏移技术，研究了清算时间、清算时盈余和清算前盈余的历史最高值的联合分布，推广了Gerber和Shiu（1998）关于经典的期望折现罚金函数的已知结果.所得结果用尺度函数和与两个Lévy过程相关的Lévy三重态表示。研究了两个Lévy过程重合的特殊情形，得到了仅用尺度函数表示的完备结果.相应的结果与Landriault et al.（2014），Baurdoux et al.（2016）和Frostig and Keren-Pinhasik（2019）中关于具有（或不具有）较低障碍的巴黎破产的现有文献具有良好的一致性。此外，数值例子说明了清算破产的基本特征。

英文摘要：

It has been decades since the academic world of ruin theory defined the insolvency of an insurance company as the time when its surplus falls below zero. This simplification, however, needs careful adaptions to imitate the real-world liquidation process. Inspired by Broadie et al. (2007) and Li et al. (2020), this paper uses a three-barrier model to describe the financial stress towards bankruptcy of an insurance company. The financial status of the insurer is divided into solvent, insolvent and liquidated three states, where the insurer's surplus process at the state of solvent and insolvent is modelled by two spectrally negative Lévy processes, which have been taken as good candidates to model insurance risks. We provide a rigorous definition of the time of liquidation ruin in this three-barrier model. By adopting the techniques of excursions in the fluctuation theory, we study the joint distribution of the time of liquidation, the surplus at liquidation and the historical high of the surplus until liquidation, which generalizes the known results on the classical expected discounted penalty function in Gerber and Shiu (1998). The results have semi-explicit expressions in terms of the scale functions and the Lévy triplets associated with the two underlying Lévy processes. The special case when the two underlying Lévy processes coincide with each other is also studied, where our results are expressed compactly via only the scale functions. The corresponding results have good consistency with the existing literatures on Parisian ruin with (or without) a lower barrier in Landriault et al. (2014), Baurdoux et al. (2016) and Frostig and Keren-Pinhasik (2019). Besides, numerical examples are provided to illustrate the underlying features of liquidation ruin.

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摘要：

英文摘要：

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