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

Abstract

Define a γ-reflected processWγ(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 indexH∈ (0, 1) and a negative linear trend. In risk theoryRγ(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 reserveugoes 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 byYH, which we also investigate.

Gaussian risk models with financial constraints

“…This, somewhat surprising result, contrasts with the infinite-time case analyzed by Hüsler & Piterbarg (2008) and Hashorva & Ji (2014), where the limiting random variable is normally distributed.…”

“…For infinite-time horizon results in this direction are well known; see e.g. Asmussen & Albrecher (2010), Hüsler & Piterbarg (2008) and Hashorva & Ji (2014) for the normal approximation of the conditional distribution of the ruin time τ (u) given that τ (u) < ∞.…”

“…the seminal contribution Segerdahl (1955) and the monographs Embrechts et al (1997) and Asmussen & Albrecher (2010). Recent results for infinite-time Gaussian and Lévy risk models are derived in Hüsler (2006), Hüsler & Piterbarg (2008), Hüsler & Zhang (2008), Griffin & Maller (2012), Griffin (2013) and Hashorva & Ji (2014).…”