On the supremum of γ-reflected processes with fractional Brownian motion as input

Let {X-H(t), t >= 0} be a fractional Brownian motion with Hurst index H is an element of (0, 1] and define a gamma-reflected process W-gamma(t) = X-H(t) - ct - gamma inf(s is an element of[0,t])(X-H(s) - cs), t >= 0 with c > 0, gamma is an element of [0, 1] two given constants. In this paper we establish the exact tail asymptotic behaviour of M-gamma(T) = sup(t is an element of[0,T]) W-gamma(t) for any T is an element of (0, infinity]. Furthermore, we derive the exact tail asymptotic behaviour of the supremum of certain non-homogeneous mean-zero Gaussian random fields. (c) 2013 Elsevier B.V. All rights reserved.