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Approximation of Passage Times of γ-Reflected Processes with FBM Input

Part of: Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Enkelejd Hashorva  and
Lanpeng Ji
Enkelejd Hashorva*
Affiliation:
University of Lausanne
Lanpeng Ji*
Affiliation:
University of Lausanne
*
∗ Postal address: Department of Actuarial Science, University of Lausanne, UNIL-Dorigny 1015 Lausanne, Switzerland.
∗ Postal address: Department of Actuarial Science, University of Lausanne, UNIL-Dorigny 1015 Lausanne, Switzerland.
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Abstract

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Define a γ-reflected process Wγ(t) = YH(t) - γinfs∈[0,t]YH(s), t ≥ 0, with input process {YH(t), t ≥ 0}, which is a fractional Brownian motion with Hurst index H ∈ (0, 1) and a negative linear trend. In risk theory Rγ(u) = u - Wγ(t), t ≥ 0, is referred to as the risk process with tax payments of a loss-carry-forward type. For various risk processes, numerous results are known for the approximation of the first and last passage times to 0 (ruin times) when the initial reserve u goes to ∞. In this paper we show that, for the γ-reflected process, the conditional (standardized) first and last passage times are jointly asymptotically Gaussian and completely dependent. An important contribution of this paper is that it links ruin problems with extremes of nonhomogeneous Gaussian random fields defined by YH, which we also investigate.

Keywords

Gaussian approximationpassage timeγ-reflected processworkload processrisk process with taxfractional Brownian motionPiterbarg constantPickands constant

MSC classification

Primary: 60G15: Gaussian processes
Secondary: 60G70: Extreme value theory; extremal processes
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 3 , September 2014 , pp. 713 - 726
Copyright
© Applied Probability Trust 

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