Abstract
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).
Highlights
In this paper we are interested in the probabilistic aspects of multiple simultaneous failures typically occurring due to pandemic-type events
As shown in Delsing et al (2020) such a risk model appears naturally in insurance applications.Since Brownian risk model (Brm) is a natural limiting model, it can be used as a benchmark for various complex models
Given the fundamental role of Brownian motion in applied probability and statistics, it is of theoretical interest to study failure events arising from this model
Summary
Let hereafter I denote a non-empty index set of {1, . Note in passing that the assumption in Theorem 2.1 that a has no more than k−1 non-positive components excludes the case that there exists a set I ⊂ {1, . For notational simplicity we consider the case I has d elements and avoid indexing by I. The index set I is unique with m = |I | ≥ 1 elements, see the lemma (or Debicki et al (2018)[Lem 2.1]). (a) has a unique solution agiven in (7) with I a unique non-empty index set with m ≤ d elements such that min x x≥a. Theorem 2.3 If a ∈ Rd has at least one positive component and is non-singular, for all S ∈ [0, 1).
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call