## Journal of Applied Probability

### Approximation of passage times of γ-reflected processes with FBM input

#### Abstract

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.

#### Article information

Source
J. Appl. Probab., Volume 51, Number 3 (2014), 713-726.

Dates
First available in Project Euclid: 5 September 2014

https://projecteuclid.org/euclid.jap/1409932669

Digital Object Identifier
doi:10.1239/jap/1409932669

Mathematical Reviews number (MathSciNet)
MR3256222

Zentralblatt MATH identifier
1303.60027

Subjects
Primary: 60G15: Gaussian processes
Secondary: 60G70: Extreme value theory; extremal processes

#### Citation

Hashorva, Enkelejd; Ji, Lanpeng. Approximation of passage times of γ-reflected processes with FBM input. J. Appl. Probab. 51 (2014), no. 3, 713--726. doi:10.1239/jap/1409932669. https://projecteuclid.org/euclid.jap/1409932669

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