Extremes of γ-reflected Gaussian processes with stationary increments

Dȩbicki, Krzysztof¹ ; Hashorva, Enkelejd²  ; Liu, Peng³

¹ Krzysztof Dȩbicki, Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
² Enkelejd Hashorva, Department of Actuarial Science, University of Lausanne, UNIL-Dorigny 1015 Lausanne, Switzerland.
³ Department of Actuarial Science, University of Lausanne, UNIL-Dorigny 1015 Lausanne, Switzerland and Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.

ESAIM: Probability and Statistics, Tome 21 (2017), pp. 495-535

Résumé

For a given centered Gaussian process with stationary increments $X\left(t\right),t\ge 0$ and $c>0$, let $W_{\gamma }\left(t\right)=X\left(t\right)-ct-\gamma in{f}_{0\le s\le t}(X\left(s\right)-cs),t\ge 0$ denote the $\gamma$-reflected process, where $\gamma \in (0,1)$. This process is important for both queueing and risk theory. In this contribution we are concerned with the asymptotics, as $u\to \infty$, of $\mathbb{P}(su{p}_{0\le t\le T}{W}_{\gamma }\left(t\right)>u),t\in (o,\infty ]$. Moreover, we investigate the approximations of first and last passage times for given large threshold $u$. We apply our findings to the cases with $X$ being the multiplex fractional Brownian motion and the Gaussian integrated process. As a by-product we derive an extension of Piterbarg inequality for threshold-dependent random fields.

Détail

Reçu le : 2017-04-01
Accepté le : 2017-11-02
DOI : 10.1051/ps/2017019

Classification : 60G15, 60G70
Keywords: γ-reflected Gaussian process, uniform double-sum method, first passage time, last passage time, fractional brownian motion, gaussian integrated process, pickands constant, piterbarg constant, piterbarg inequality