On the Supremum of gamma-reflected Processes with Fractional Brownian Motion as Input

Let $X_H(t),t\ge 0$ be a fractional Brownian motion with Hurst index H\in(0,1} and define a gamma-reflected process $W_{\text{Ga}}(t)=X_H(t)-ct-{\text{gammainf}}_{s\in [0,t]}(X_H(s)-cs)$, $t\ge 0$ with $c>0,\gamma \in [0,1]$ two given constants. In this paper we establish the exact tail asymptotic behaviour of ${\sup }_{t\in [0,T]}{W}_{\gamma }(t)$ for any $T\in (0,\text{IF}]$. Furthermore, we derive the exact tail asymptotic behaviour of the supremum of certain non-homogeneous mean-zero Gaussian random fields.