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

TL;DR

This paper investigates the precise tail behavior of the supremum of gamma-reflected fractional Brownian motion over finite and infinite intervals, providing exact asymptotics for these stochastic processes.

Contribution

It establishes the exact tail asymptotics for the supremum of gamma-reflected fractional Brownian motion and related Gaussian fields, extending understanding of their extreme value behavior.

Findings

Exact tail asymptotics for supremum over finite interval

Exact tail asymptotics for supremum over infinite interval

Extension to non-homogeneous Gaussian fields

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.