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 A, we mean the limit [GRAPHICS] where H-eta(T) = E exp(max(tis an element of[0,T]) (root2eta(t) - sigma(eta)(2)(t))) and eta(t) is a centered Gaussian process with stationary increments and variance function sigma(eta)(2)(t). Under some mild conditions on sigma(eta)(2)(t) we prove that H-eta 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 psi(u) = P(sup(tgreater than or equal to0) zeta(t) - ct > u) as u --> infinity where zeta(x) = f(0)(x) Z(s) ds and Z(s) is a stationary centered Gaussian process with covariance function R(t) fulfilling some integrability conditions. (C) 2001 Elsevier Science B.V. All rights reserved.