The study of the distributions of sums of dependent risks is a key topic in actuarial sciences, risk management, reliability and in many branches of applied and theoretical probability. However, there are few results where the distribution of the sum of dependent random variables is available in a closed form. In this paper, we obtain several analytical expressions for the distribution of the aggregated risks under dependence in terms of copulas. We provide several representations based on the underlying copula and the marginal distribution functions under general hypotheses and in any dimension. Then, we study stochastic comparisons between sums of dependent risks. Finally, we illustrate our theoretical results by studying some specific models obtained from Clayton, Ali-Mikhail-Haq and Farlie-Gumbel-Morgenstern copulas. Extensions to more general copulas are also included. Bounds and the limiting behavior of the hazard rate function for the aggregated distribution of some copulas are studied as well.

中文翻译：

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相关风险总和的 COPULA 表示：模型和比较

对相关风险总和分布的研究是精算科学、风险管理、可靠性以及应用和理论概率的许多分支的关键课题。但是，很少有结果可以以封闭形式获得因随机变量之和的分布。在本文中，我们根据 copulas 获得了依赖关系下的聚合风险分布的几个分析表达式。我们基于一般假设和任何维度下的基本 copula 和边际分布函数提供了几种表示。然后，我们研究相关风险总和之间的随机比较。最后，我们通过研究从 Clayton、Ali-Mikhail-Haq 和 Farlie-Gumbel-Morgenstern copulas 获得的一些特定模型来说明我们的理论结果。还包括对更一般的 copula 的扩展。还研究了一些 copula 的聚合分布的危险率函数的界限和限制行为。