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

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.