Tail Asymptotics of Supremum of Certain Gaussian Processes over Threshold Dependent Random Intervals

TL;DR

This paper investigates the tail behavior of the maximum of Gaussian processes over random intervals that depend on a threshold, with applications to ruin probabilities in risk processes involving fractional Brownian motion.

Contribution

It derives asymptotic formulas for the supremum of Gaussian processes over threshold-dependent random intervals, extending existing results to more complex stochastic settings.

Findings

Asymptotic expressions for supremum probabilities over random intervals.

Application to ruin probabilities in fractional Brownian motion risk models.

Extension of Gaussian process tail asymptotics to threshold-dependent intervals.

Abstract

Let $\{X(t),t\ge0\}$ be a centered Gaussian process and let $\gamma$ be a non-negative constant. In this paper we study the asymptotics of $P\{\underset{t\in [0,\mathcal{T}/u^\gamma]}\sup X(t)>u\}$ as $u\to\infty$, with $\mathcal{T}$ an independent of $X$ non-negative random variable. As an application, we derive the asymptotics of finite-time ruin probability of time-changed fractional Brownian motion risk processes.