Local times for continuous paths of arbitrary regularity

TL;DR

This paper introduces a new concept of local time for continuous paths with finite p-th variation, extending classical stochastic calculus results to paths of arbitrary regularity, including fractional Brownian motions.

Contribution

It defines a pathwise local time of order p, establishes change of variable formulas, and connects this to fractional Brownian motion and collision local times.

Findings

Generalizes Tanaka and Meyer formulas to arbitrary regularity paths

Provides identities extending semimartingale theory

Connects local times with fractional Brownian motion

Abstract

We study a continuous pathwise local time of order p for continuous functions with finite p-th variation along a sequence of time partitions, for even integers p >= 2. With this notion, we establish a Tanaka-type change of variable formula, as well as Tanaka-Meyer formulae. We also derive some identities involving this high-order pathwise local time, each of which generalizes a corresponding identity from semimartingale theory. We then use collision local times between multiple functions of arbitrary regularity, to study the dynamics of ranked continuous functions of arbitrary regularity. We present also another definition of pathwise local time which is more natural for fractional Brownian Motions, and give a connection with the previous notion of local time.