Stress testing, and in particular, reverse stress testing, is a prominent exercise in risk management practice. Reverse stress testing, in contrast to (forward) stress testing, aims to find an alternative but plausible model such that under that alternative model, specific adverse stresses (i.e. constraints) are satisfied. Here, we propose a reverse stress testing framework for dynamic models. Specifically, we consider a compound Poisson process over a finite time horizon and stresses composed of expected values of functions applied to the process at the terminal time. We then define the stressed model as the probability measure under which the process satisfies the constraints and which minimizes the Kullback-Leibler divergence to the reference compound Poisson model.

We solve this optimization problem, prove existence and uniqueness of the stressed probability measure, and provide a characterization of the Radon-Nikodym derivative from the reference model to the stressed model. We find that under the stressed measure, the intensity and the severity distribution of the process depend on time and state, and hence the stressed model is not a compound Poisson process. We illustrate the dynamic stress testing by considering stresses on VaR and both VaR and CVaR jointly and provide illustrations of how the stochastic process is altered under these stresses. We generalize the framework to multivariate compound Poisson processes and stresses at times other than the terminal time. We illustrate the applicability of our framework by considering “what if” scenarios, where we answer the question: What is the severity of a stress on a portfolio component at an earlier time such that the aggregate portfolio exceeds a risk threshold at the terminal time? Furthermore, for general constraints, we propose an algorithm to simulate sample paths under the stressed measure, thus allowing to compare the effects of stresses on the dynamics of the process.

压力测试，特别是反向压力测试，是风险管理实践中的一项重要活动。与（正向）压力测试相比，反向压力测试旨在找到替代但合理的模型，以便在该替代模型下满足特定的不利应力（即约束）。在这里，我们提出了动态模型的反向压力测试框架。具体来说，我们考虑有限时间范围内的复合泊松过程以及由在最终时间应用于该过程的函数的期望值组成的应力。然后，我们将应力模型定义为概率测度，在该概率测度下，过程满足约束，并且最小化与参考复合泊松模型的 Kullback-Leibler 散度。

我们解决了这个优化问题，证明了应力概率测度的存在性和唯一性，并提供了从参考模型到应力模型的 Radon-Nikodym 导数的表征。我们发现，在压力测度下，过程的强度和严重性分布取决于时间和状态，因此压力模型不是复合泊松过程。我们通过联合考虑 VaR 以及 VaR 和 CVaR 上的压力来说明动态压力测试，并说明随机过程在这些压力下如何改变。我们将该框架推广到多元复合泊松过程和除最终时间之外的时间的应力。我们通过考虑“假设”场景来说明我们的框架的适用性，在这种情况下，我们回答了以下问题：投资组合组成部分在早期所承受的压力的严重程度是多少，以致总投资组合在最终时间超过了风险阈值？此外，对于一般约束，我们提出了一种算法来模拟压力测量下的样本路径，从而可以比较压力对过程动态的影响。