The field of risk theory has traditionally focused on ruin-related quantities. In particular, the socalled Expected Discounted Penalty Function has been the object of a thorough study over the years. Although interesting in their own right, ruin related quantities do not seem to capture path-dependent properties of the reserve. In this article we aim at presenting the probabilistic properties of drawdowns and the speed at which an insurance reserve depletes as a consequence of the risk exposure of the company. These new quantities are not ruin related yet they capture important features of an insurance position and we believe it can lead to the design of a meaningful risk measures. Studying drawdowns and speed of depletion for Lévy insurance risk processes represent a novel and challenging concept in insurance mathematics. In this paper, all these concepts are formally introduced in an insurance setting. Moreover, using recent results in fluctuation theory for Lévy processes, we derive expressions for the distribution of several quantities related to the depletion problem. Of particular interest are the distribution of drawdowns and the Laplace transform for the speed of depletion. These expressions are given for some examples of Lévy insurance risk processes for which they can be calculated, in particular for the classical Cramer-Lundberg model.
Ruin probabilities in tough times - Part 2 - Heavy-traffic approximation for fractionally differentiated random walks in the domain of attraction of a nonGaussian stable distribution
Ruin probabilities in tough times - Part 1 - Heavy-traffic approximation for fractionally integrated random walks in the domain of attraction of a nonGaussian stable distribution
On the time spent in the red by a refracted Lévy risk process
本文引入了一个具有自适应保险费率的保险破产模型，简称重组/折射模型，其中区分了经典破产和破产。在这个模型中，一旦财富过程落入红色区域，保险费率就会增加，当财富过程恢复时，保险费率就会恢复到正常水平。分析主要集中在折射的Lévy风险过程在红色区域的时间（类似于负盈余的持续时间）。基于Kyprianou and Loeffen（2010）和Loeffen et al.（2012）的结果，我们确定了与折射谱负Lévy过程的占据时间相关的各种泛函的分布。例如，这些结果被用来计算破产概率和巴黎破产的概率，在这个模型与重组。
Risk Modelling on Liquidations with Lévy Processes
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。
On Fair Reinsurance Premiums; Capital Injections in a Perturbed Risk Model
我们考虑一个风险模型，破产后的赤字是由一种新的再保险合同，提供资本注入。为了保证保险公司在破产后的生存，再保险人只在破产时注入资本，而破产时的跳跃大于所选择的自留水平。否则，必须从股东那里筹集资金以弥补小额亏损。这里的问题是确定足够的再保险保费。将再保险净保费建立在任何未来注资的贴现预期价值的基础上似乎是公平的。受Huzak et al.（2004）和Ben Salah（2014）关于连续破产事件的结果的启发，我们证明了在总索赔由从属子和布朗扰动建模的情况下，这些再保险保费的显式公式存在。在这里，破产事件是由于布朗振荡或跳跃和再保险资本注入只适用于后一种情况。两个特定的风险模型，并在一些数值例子的结果明确说明。
