Ruin probability for Gaussian integrated processes

Abstract

Pickands constants play an important role in the exact asymptotic of extreme values for Gaussian stochastic processes. By the generalized Pickands constant H η we mean the limit H η= lim T→∞ H η(T) T , where H η(T)= E exp( max t∈[0,T] ( 2 η(t)−σ η 2(t))) and η( t) is a centered Gaussian process with stationary increments and variance function σ η 2( t). Under some mild conditions on σ η 2( t) we prove that H η is well defined and we give a comparison criterion for the generalized Pickands constants. Moreover we prove a theorem that extends result of Pickands for certain stationary Gaussian processes. As an application we obtain the exact asymptotic behavior of ψ(u)= P( sup t⩾0 ζ(t)−ct>u) as u→∞, where ζ(x)= ∫ 0 x Z(s) ds and Z( s) is a stationary centered Gaussian process with covariance function R( t) fulfilling some integrability conditions.