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.
Krzysztof.Debicki@math.uni.wroc.pl
² Enkelejd Hashorva, Department of Actuarial Science, University of Lausanne, UNIL-Dorigny 1015 Lausanne, Switzerland.
enkelejd.hashorva@unil.ch
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.
peng.liu@unil.ch

Received: 1 April 2017
Revised: 31 August 2017
Accepted: 2 November 2017

Abstract

For a given centered Gaussian process with stationary increments X(t),t ≥ 0 and c > 0, let W_γ(t) = X(t) − ct − γinf_{0 ≤ 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(sup_{0≤t ≤ T} W_γ(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.