ABSTRACT This paper considers a Cramér–Lundberg risk setting, where the components of the underlying model change over time. We allow the more general setting of the cumulative claim process being modeled as a spectrally positive Lévy process. We provide an intuitively appealing mechanism to create such parameter uncertainty: at Poisson epochs, we resample the model components from a finite number of d settings. It results in a setup that is particularly suited to describe situations in which the risk reserve dynamics are affected by external processes. We extend the classical Cramér–Lundberg approximation (asymptotically characterizing the all-time ruin probability in a light-tailed setting) to this more general setup. In addition, for the situation that the driving Lévy processes are sums of Brownian motions and compound Poisson processes, we find an explicit uniform bound on the ruin probability. In passing we propose an importance-sampling algorithm facilitating efficient estimation, and prove it has bounded relative error. In a series of numerical experiments we assess the accuracy of the asymptotics and bounds, and illustrate that neglecting the resampling can lead to substantial underestimation of the risk.

中文翻译：

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具有重新采样环境的废墟模型

摘要 本文考虑了 Cramér-Lundberg 风险设置，其中基础模型的组件随时间变化。我们允许将累积索赔过程的更一般设置建模为谱正 Lévy 过程。我们提供了一种直观吸引人的机制来创建这种参数不确定性：在 Poisson epochs，我们从有限数量的 d 设置中重新采样模型组件。它产生的设置特别适合描述风险准备金动态受外部过程影响的情况。我们将经典的 Cramér-Lundberg 近似（渐近表征轻尾设置中的所有时间破产概率）扩展到这个更一般的设置。此外，对于驱动 Lévy 过程是布朗运动和复合泊松过程的总和的情况，我们在破产概率上找到了一个明确的统一界限。顺便说一下，我们提出了一种促进有效估计的重要性采样算法，并证明它具有有界的相对误差。在一系列数值实验中，我们评估了渐近线和边界的准确性，并说明忽略重采样会导致严重低估风险。