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.