Consider a multi-dimensional Brownian motion which models the surplus processes of multiple lines of business of an insurance company. Our main result gives exact asymptotics for the cumulative Parisian ruin probability as the initial capital tends to infinity. An asymptotic distribution for the conditional cumulative Parisian ruin time is also derived. The obtained results on the cumulative Parisian ruin can be seen as generalisations of some of the results derived in D bicki et al. [(2018). Extremal behavior of hitting a cone by correlated Brownian motion with drift. Stochastic Processes and Their Applications 128, 4171–4206]. As a particular interesting case, the two-dimensional Brownian motion risk model is discussed in detail.

中文翻译：

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多维布朗运动风险模型的累积巴黎废墟

考虑一个多维布朗运动，它对保险公司的多个业务线的盈余过程进行建模。我们的主要结果给出了累积巴黎破产概率的精确渐近线，因为初始资本趋于无穷大。还导出了条件累积巴黎废墟时间的渐近分布。在累积巴黎废墟上获得的结果可以看作是 D bicki 等人得出的一些结果的概括。[（2018）。通过与漂移相关的布朗运动击中锥体的极值行为。随机过程及其应用 128, 4171–4206]。作为一个特别有趣的案例，详细讨论了二维布朗运动风险模型。