On Berman Functions.

Let Z(t)=exp2BH(t)-t2H,t∈R with BH(t),t∈R a standard fractional Brownian motion (fBm) with Hurst parameter H∈(0,1] and define for x non-negative the Berman function BZ(x)=EI{ϵ0(RZ)>x}ϵ0(RZ)∈(0,∞),where the random variable R independent of Z has survival function 1/x,x⩾1 and ϵ0(RZ)=∫RIRZ(t)>1dt.In this paper we consider a general random field (rf) Z that is a spectral rf of some stationary max-stable rf X and derive the properties of the corresponding Berman functions. In particular, we show that Berman functions can be approximated by the corresponding discrete ones and derive interesting representations of those functions which are of interest for Monte Carlo simulations presented in this article.

Pandemic-type failures in multivariate Brownian risk models.

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 DÈ©bicki et al. (J. Appl. Probab. 57(2), 597-612 2020) and DÈ©bicki et al. (Stoch. Proc. Appl. 128(12), 4171-4206 2018).