September 2014 Approximation of passage times of γ-reflected processes with FBM input
Enkelejd Hashorva, Lanpeng Ji
Author Affiliations +
J. Appl. Probab. 51(3): 713-726 (September 2014). DOI: 10.1239/jap/1409932669


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.


Download Citation

Enkelejd Hashorva. Lanpeng Ji. "Approximation of passage times of γ-reflected processes with FBM input." J. Appl. Probab. 51 (3) 713 - 726, September 2014.


Published: September 2014
First available in Project Euclid: 5 September 2014

zbMATH: 1303.60027
MathSciNet: MR3256222
Digital Object Identifier: 10.1239/jap/1409932669

Primary: 60G15
Secondary: 60G70

Keywords: fractional Brownian motion , Gaussian approximation , passage time , Pickands constant , Piterbarg constant , risk process with tax , workload process , γ-reflected process

Rights: Copyright © 2014 Applied Probability Trust


This article is only available to subscribers.
It is not available for individual sale.

Vol.51 • No. 3 • September 2014
Back to Top