We consider settings in which the distribution of a multivariate random variable is partly ambiguous. We assume the ambiguity lies on the level of the dependence structure, and that the marginal distributions are known. Furthermore, a current best guess for the distribution, called reference measure, is available. We work with the set of distributions that are both close to the given reference measure in a transportation distance (e.g., the Wasserstein distance), and additionally have the correct marginal structure. The goal is to find upper and lower bounds for integrals of interest with respect to distributions in this set. The described problem appears naturally in the context of risk aggregation. When aggregating different risks, the marginal distributions of these risks are known and the task is to quantify their joint effect on a given system. This is typically done by applying a meaningful risk measure to the sum of the individual risks. For this purpose, the stochastic interdependencies between the risks need to be specified. In practice, the models of this dependence structure are however subject to relatively high model ambiguity. The contribution of this paper is twofold: First, we derive a dual representation of the considered problem and prove that strong duality holds. Second, we propose a generally applicable and computationally feasible method, which relies on neural networks, in order to numerically solve the derived dual problem. The latter method is tested on a number of toy examples, before it is finally applied to perform robust risk aggregation in a real‐world instance.

中文翻译：

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通过神经网络进行稳健的风险聚合


我们考虑多元随机变量的分布部分不明确的设置。我们假设模糊性在于依赖结构的水平，并且边际分布是已知的。此外，当前对分布的最佳猜测（称为参考测量）是可用的。我们使用一组分布，这些分布既接近运输距离中给定的参考度量（例如 Wasserstein 距离），又具有正确的边际结构。目标是找到关于该集合中的分布的感兴趣积分的上限和下限。所描述的问题自然出现在风险聚合的背景下。当汇总不同的风险时，这些风险的边际分布是已知的，任务是量化它们对给定系统的联合影响。这通常是通过对各个风险的总和应用有意义的风险度量来完成的。为此，需要指定风险之间的随机相互依赖性。然而，在实践中，这种依赖结构的模型会受到相对较高的模型模糊性的影响。本文的贡献是双重的：首先，我们推导了所考虑问题的对偶表示，并证明强对偶性成立。其次，我们提出了一种普遍适用且计算上可行的方法，该方法依赖于神经网络，以便数值解决导出的对偶问题。后一种方法在许多玩具示例上进行了测试，然后最终应用于现实世界实例中执行稳健的风险聚合。