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

Let {XH(t),t≥0}{XH(t),t≥0} be a fractional Brownian motion with Hurst index H∈(0,1]H∈(0,1] and define aγγ-reflected process Wγ(t)=XH(t)−ct−γinfs∈[0,t](XH(s)−cs)Wγ(t)=XH(t)−ct−γinfs∈[0,t](XH(s)−cs), t≥0t≥0 with c>0,γ∈[0,1]c>0,γ∈[0,1] two given constants. In this paper we establish the exact tail asymptotic behaviour of Mγ(T)=supt∈[0,T]Wγ(t)Mγ(T)=supt∈[0,T]Wγ(t) for any T∈(0,∞]T∈(0,∞]. Furthermore, we derive the exact tail asymptotic behaviour of the supremum of certain non-homogeneous mean-zero Gaussian random fields.