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

Abstract

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 $W_\Ga(t)=X_H(t)-ct-\gammainf_{s\in[0,t]}\left(X_H(s)-cs \right)$, $t\ge0$ 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 $T\in (0,\IF]$. Furthermore, we derive the exact tail asymptotic behaviour of the supremum of certain non-homogeneous mean-zero Gaussian random fields.