Extremes of stationary Gasussian storage models

TL;DR

This paper derives precise asymptotic probabilities for the extremes of stationary Gaussian storage models, providing insights into their tail behavior and conditions for the strong Piterbarg property.

Contribution

It offers exact asymptotics for the supremum and infimum probabilities of stationary Gaussian storage processes, advancing understanding of their extreme value behavior.

Findings

Exact asymptotics for tail probabilities as u→∞

Conditions for the strong Piterbarg property

Characterization of extremal events in Gaussian storage models

Abstract

For the stationary storage process $\{Q(t), t\ge0\}$, with $ Q(t)=\sup_{ s \ge t}\left(X(s)-X(t)-c(s-t)^\beta\right), $ where $\{X(t),t\ge 0\}$ is a centered Gaussian process with stationary increments, $c>0$ and $\beta>0$ is chosen such that $Q(t)$ is finite a.s., we derive exact asymptotics of $\mathbb{P}\left(\sup_{t\in [0,T_u]} Q(t)>u\right)$ and $\mathbb{P}\left(\inf_{t\in [0,T_u]} Q(t)>u\right)$, as $u\rightarrow\infty$. As a by-product we find conditions under which strong Piterbarg property holds.