Extremes of multifractional Brownian motion

TL;DR

This paper investigates the precise asymptotic behavior of the probability that the supremum of a standard multifractional Brownian motion exceeds a high threshold, depending on the structure of the Hurst function.

Contribution

It provides exact asymptotics for the tail probability of the supremum of multifractional Brownian motion under various H(t) structures.

Findings

Derived asymptotic formulas for different H(t) cases.

Identified key factors influencing tail behavior.

Extended understanding of extremes in multifractional processes.

Abstract

Let $B_{H}(t), t\geq [0,T], T\in(0,\infty)$ be the standard Multifractional Brownian Motion(mBm), in this contribution we are concerned with the exact asymptotics of \begin{eqnarray*} \mathbb{P}\left\{\sup_{t\in[0,T]}B_{H}(t)>u\right\} \end{eqnarray*} as $u\rightarrow\infty$. Mainly depended on the structures of $H(t)$, the results under several important cases are investigated.