Volume 21, 2017
|Page(s)||495 - 535|
|Published online||08 January 2018|
Extremes of γ-reflected Gaussian processes with stationary increments
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.
Received: 1 April 2017
Revised: 31 August 2017
Accepted: 2 November 2017
For a given centered Gaussian process with stationary increments X(t),t ≥ 0 and c > 0, let Wγ(t) = X(t) − ct − γinf0 ≤ s ≤ t(X(s) − cs),t ≥ 0 denote the γ-reflected process, where γ ∈ (0,1). This process is important for both queueing and risk theory. In this contribution we are concerned with the asymptotics, as u → ∞, of ℙ(sup0≤t ≤ T Wγ(t)>u), t ∈ (o,∞]. 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.
Mathematics Subject Classification: 60G15 / 60G70
Key words: 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
© EDP Sciences, SMAI, 2017
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