Modelling of multiple simultaneous failures in insurance, finance and other areas of applied probability is important especially from the point of view of pandemic-type events. A benchmark limiting model for the analysis of multiple failures is the classical d-dimensional Brownian risk model (Brm), see Delsing et al. (Methodol. Comput. Appl. Probab. 22(3), 927–948 2020). From both theoretical and practical point of view, of interest is the calculation of the probability of multiple simultaneous failures in a given time horizon. The main findings of this contribution concern the approximation of the probability that at least k out of d components of Brm fail simultaneously. We derive both sharp bounds and asymptotic approximations of the probability of interest for the finite and the infinite time horizon. Our results extend previous findings of Dȩbicki et al. (J. Appl. Probab. 57(2), 597–612 2020) and Dȩbicki et al. (Stoch. Proc. Appl. 128(12), 4171–4206 2018).

对保险、金融和其他应用概率领域中的多个同时失败进行建模非常重要，特别是从大流行类型事件的角度来看。用于分析多重故障的基准限制模型是经典的d维布朗风险模型 (Brm)，参见 Delsing 等人。（Methodol. Comput. Appl. Probab. 22 (3), 927–948 2020）。从理论和实践的角度来看，感兴趣的是计算给定时间范围内多个同时失效的概率。这一贡献的主要发现涉及到d中至少k个概率的近似值Brm 的组件同时发生故障。我们推导出有限和无限时间范围内感兴趣概率的锐界和渐近近似。我们的结果扩展了 Dȩbicki 等人先前的研究结果。（J. Appl. Probab. 57 (2), 597–612 2020）和 Dȩbicki 等人。（Stoch. Proc. Appl. 128 (12), 4171–4206 2018）。