Level crossings of fractional Brownian motion

TL;DR

This paper extends the classical characterization of Brownian motion's local time via level crossings to fractional Brownian motion, providing a path-by-path construction and analyzing the (1/H)-th variation along level crossing partitions.

Contribution

It offers the first fractional analogue of the local time characterization through level crossings, employing novel methods suited for non-Markovian processes.

Findings

Path-by-path construction of fractional Brownian local time.

Almost-sure convergence of (1/H)-th variation along Lebesgue partitions.

Raises conjecture on the limit capturing non-Markovian features.

Abstract

Since the classical work of L\'evy, it is known that the local time of Brownian motion can be characterized through the limit of level crossings. While subsequent extensions of this characterization have primarily focused on Markovian or martingale settings, this work presents a highly anticipated extension to fractional Brownian motion -- a prominent non-Markovian and non-martingale process. Our result is viewed as a fractional analogue of Chacon et al. (1981). Consequently, it provides a global path-by-path construction of fractional Brownian local time. Due to the absence of conventional probabilistic tools in the fractional setting, our approach utilizes completely different argument with a flavor of the subadditive ergodic theorem, combined with the shifted stochastic sewing lemma recently obtained in Matsuda and Perkowski (22, arXiv:2206.01686). Furthermore, we prove an almost-sure convergence of the (1/H)-th variation of fractional Brownian motion with the Hurst parameter H, along random partitions defined by level crossings, called Lebesgue partitions. This result raises an interesting conjecture on the limit, which seems to capture non-Markovian nature of fractional Brownian motion.