Approximation of passage times of gamma-reflected processes with fBm input

Define a gamma-reflected process W-gamma(t) = Y-H(t) - gamma inf(s is an element of[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 is an element of (0, 1) and a negative linear trend. In risk theory R-gamma(u) = u - W-gamma(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 infinity. In this paper we show that, for the gamma-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