Extremes of γ-reflected Gaussian processes with stationary increments

Dȩbicki, Krzysztof; Hashorva, Enkelejd; Liu, Peng

doi:10.1051/ps/2017019

Dȩbicki, Krzysztof ¹ ; Hashorva, Enkelejd ² ; Liu, Peng ³

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 $X (t), t \geq 0$ and $c > 0$ , let $W_{γ} (t) = X (t) - c t - γ i n f_{0 \leq s \leq t} (X (s) - c s), t \geq 0$ denote the $γ$ -reflected process, where $γ \in (0, 1)$ . This process is important for both queueing and risk theory. In this contribution we are concerned with the asymptotics, as $u \to \infty$ , of $ℙ (s u p_{0 \leq t \leq T} W_{γ} (t) > u), t \in (o, \infty]$ . 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.

Reçu le : 2017-04-01
Accepté le : 2017-11-02

MR Zbl

DOI : 10.1051/ps/2017019

Classification : 60G15, 60G70
Mots-clés : 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

Affiliations des auteurs :

Dȩbicki, Krzysztof ¹ ; Hashorva, Enkelejd ² ; Liu, Peng ³

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     title = {Extremes of \ensuremath{\gamma}-reflected {Gaussian} processes with stationary increments},
     journal = {ESAIM: Probability and Statistics},
     pages = {495--535},
     publisher = {EDP-Sciences},
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Dȩbicki, Krzysztof; Hashorva, Enkelejd; Liu, Peng. Extremes of γ-reflected Gaussian processes with stationary increments. ESAIM: Probability and Statistics, Tome 21 (2017), pp. 495-535. doi : 10.1051/ps/2017019. http://www.numdam.org/articles/10.1051/ps/2017019/

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