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Let Z (t) = exp 2 B H (t) - t 2 H , t ∈ R with B H (t) , t ∈ R a standard fractional Brownian motion (fBm) with Hurst parameter H ∈ (0 , 1 ] and define for x non-negative the Berman function B Z (x) = E I { ϵ 0 (R Z) > x } ϵ 0 (R Z) ∈ (0 , ∞) , where the random variable R independent of Z has survival function 1 / x , x ⩾ 1 and ϵ 0 (R Z) = ∫ R I R Z (t) > 1 dt. 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. [ABSTRACT FROM AUTHOR]

We derive exact tail asymptotics of sojourn time above the level u ≥ 0 P v (u) ∫ 0 T I (X (t) - c t > u) d t > x , x ≥ 0 , as u → ∞ , where X is a Gaussian process with continuous sample paths, c is some constant, v(u) is a positive function of u and T ∈ (0 , ∞ ] . Additionally, we analyze asymptotic distributional properties of τ u (x) : = inf t ≥ 0 : v (u) ∫ 0 t I (X (s) - c s > u) d s > x , x ≥ 0 , as u → ∞ , where inf ∅ = ∞ . The findings of this contribution are illustrated by a detailed analysis of a class of Gaussian processes with stationary increments and a family of self-similar Gaussian processes. [ABSTRACT FROM AUTHOR]

We investigate the tail asymptotic behavior of the sojourn time for a large class of centered Gaussian processes X, in both continuous- and discrete-time framework. All results obtained here are new for the discrete-time case. In the continuous-time case, we complement the investigations of Berman (Commun Pure Appl Math 38(5):519–528, 1985a and Probab Theory Relat Fields 20(1):113–124, 1987) for non-stationary X. A by-product of our investigation is a new representation of Pickands constant which is important for Monte-Carlo simulations and yields a sharp lower bound for Pickands constant. [ABSTRACT FROM AUTHOR]

Abstract. In the paper we present a method of simulation of ruin probability over infinite horizon for fractional Brownian motion with parameter of self-similarity H > 1/2. We derive some theoretical results which show how fast the method works. As an application of our method we numerically compute the Pickands constant. [ABSTRACT FROM AUTHOR]