Sojourn times of Gaussian processes with trend

Abstract

We derive exact tail asymptotics of sojourn time above the level $u\ge 0$ $$\mathbb{P}\left(v(u){\int }_0^T\mathbb{I}(X(t)-ct>u)dt>x\right),x\ge 0$$ as $u\to \infty$, where $X$ is a Gaussian process with continuous sample paths, $c>0$, $v(u)$ is a positive function of $u$ and $T\in (0,\infty ]$. Additionally, we analyze asymptotic distributional properties of $${\tau }_u(x):=\inf \left\{t\ge 0:v(u){\int }_0^t\mathbb{I}(X(s)-cs>u)ds>x\right\},$$ as $u\to \infty$, $x\ge 0$, where $\inf \varnothing =\infty$. The findings of this contribution are illustrated by a detailed analysis of a class of Gaussian processes with stationary increments and a family of self-similar processes.

arXiv comment30 pages

References in corpus (3)

• Occupation times, drawdowns, and drawups for one-dimensional regular diffusions
• Generalized fractional Brownian motion
• Extremes of $\gamma$-reflected Gaussian process with stationary increments

Cited by in corpus (5)

• On the cumulative Parisian ruin of multi-dimensional Brownian motion models
• On the speed of convergence of discrete Pickands constants to continuous ones
• Sojourns of Stationary Gaussian Processes over a Random Interval
• Sojourn Ruin of a Two-Dimensional Fractional Brownian Motion Risk Process
• On generalized Piterbarg-Berman function