Sample path properties of the local time of multifractional Brownian motion

TL;DR

This paper investigates the fine path properties of the local time of multifractional Brownian motion, including continuity, law of iterated logarithm, and local self-similarity, providing new insights into its regularity and asymptotic behavior.

Contribution

It establishes new estimates for the local time's continuity, studies an analogue of Chung's law, and proves local asymptotic self-similarity for the local time of multifractional Brownian motion.

Findings

Established bounds for local and uniform moduli of continuity.

Derived an analogue of Chung's law of the iterated logarithm.

Proved local asymptotic self-similarity of the local time.

Abstract

We establish estimates for the local and uniform moduli of continuity of the local time of multifractional Brownian motion, $B^H=(B^{H(t)}(t),t\in\mathbb{R}^+)$. An analogue of Chung's law of the iterated logarithm is studied for $B^H$ and used to obtain the pointwise H\"{o}lder exponent of the local time. A kind of local asymptotic self-similarity is proved to be satisfied by the local time of $B^H$.