Approximation of Passage Times of γ-Reflected Processes with FBM Input. (September 2014)

• Record Type: Journal Article
• Authors: Hashorva, Enkelejd; Ji, Lanpeng
• Is Part Of: Journal of applied probability. Volume 51:Number 3(2014)
• Page Start: 713
• Page End: 726
• Publication Date: 2014-09
• Subjects: Gaussian approximation, -- passage time, -- γ-reflected process, -- workload process, -- risk process with tax, -- fractional Brownian motion, -- Piterbarg constant, -- Pickands constant; 60G15, -- 60G70; 519.2
• DOI: 10.1017/S0021900200011621
• Languages: English
• ISSNs: 0021-9002

Abstract

Abstract : Define a γ-reflected process W γ ( t ) = Y H ( t ) - γinf s ∈[0, t ] Y H ( s ), t ≥ 0, with input process { Y H ( t ), t ≥ 0}, which is a fractional Brownian motion with Hurst index H ∈ (0, 1) and a negative linear trend. In risk theory R γ ( u ) = u - W γ ( 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 Y H, which we also investigate.