Sojourn Times of Gaussian Processes with Trend

We derive exact tail asymptotics of sojourn time above the level u≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\ge 0$$\end{document}Pv(u)∫0TI(X(t)-ct>u)dt>x,x≥0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathbb {P}} \left( v(u)\int _0^T {\mathbb {I}}(X(t)-ct>u)\text {d}t>x \right) , \quad x\ge 0, \end{aligned}$$\end{document}as u→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\rightarrow \infty $$\end{document}, where X is a Gaussian process with continuous sample paths, c is some constant, v(u) is a positive function of u and T∈(0,∞]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T\in (0,\infty ]$$\end{document}. Additionally, we analyze asymptotic distributional properties of τu(x):=inft≥0:v(u)∫0tI(X(s)-cs>u)ds>x,x≥0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \tau _u(x):=\inf \left\{ t\ge 0: {v(u)} \int _0^t {\mathbb {I}}(X(s)-cs>u)\text {d}s>x\right\} , \quad x \ge 0, \end{aligned}$$\end{document}as u→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\rightarrow \infty $$\end{document}, where inf∅=∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\inf \emptyset =\infty $$\end{document}. 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 Gaussian processes.