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

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