A note on transient Gaussian fluid models
Let \(\xi=(\xi(t))_{0\leq t\leq T}\) be a centered Gaussian process with stationary increments, \(\xi(0)=0\) and with a.s.~continuous paths. Given a shift function \(d:[0,T]\to\mathbb R\) the authors investigate the behaviour of \[ \Psi_T(u):=\mathbb P\left(\sup_{0\leq t\leq T}[\xi(t)-d(t)]\geq u\right) \] as \(u\to\infty\). Under some regularity conditions about the shift \(d\) and the covariance function of \(\xi\) the authors are able to describe the asymptotic of \(\Psi_T\) completely (with exact constants). As examples they apply these results to the fractional Brownian motion (fBm) as well as to so-called integrated Gaussian processes. In the case of fBm \(B_H\), \(0<H<1\), the behaviour of \(\Psi_T\) turns out to be very different for \(0<H<1/2\), \(H=1/2\) and \(1/2<H<1\), respectively. For \(B_H\) on the whole positive real line and \(d(t)=c\cdot t\) similar results (with non-specified constants) were proved by \textit{J. Hüsler} and \textit{V. I. Piterbarg} [Stochastic Processes Appl. 83, No. 2, 257-271 (1999; Zbl 0997.60057)] and by \textit{O. Narayan} [Adv. Performance Anal. 1, 39-63 (1999)].
scientific article; zbMATH DE number 1790159