Ruin probability for Gaussian integrated processes

Pickands constants play an important role in the exact asymptotic of extreme values for Gaussian stochastic processes. By the generalized Pickands constant Hη we mean the limitHη=limT→∞ Hη(T)T,where Hη(T)=E exp(maxt∈[0,T] (2η(t)−ση2(t))) and η(t) is a centered Gaussian process with stationary increments and variance function ση2(t). some mild conditions on ση2(t) we prove that Hη 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. application we obtain the exact asymptotic behavior of ψ(u)=P(supt⩾0 ζ(t)−ct>u) as u→∞, where ζ(x)=∫0xZ(s) ds and Z(s) is a stationary centered Gaussian process with covariance function R(t) fulfilling some integrability conditions.