Pathwise Stochastic Calculus with Local Times

TL;DR

This paper develops a pathwise stochastic calculus framework based on local times, extending existing definitions and establishing new formulae, with detailed analysis of how local times depend on partition choices.

Contribution

It introduces a general notion of pathwise local time, unifies existing concepts, and provides conditions for their existence and properties, including dependence on partitions.

Findings

Pathwise local time agrees with classical local time for semimartingales.

Pathwise Itô-Tanaka and change of variables formulas are established.

Dependence of quadratic variation and local time on partition choices is analyzed.

Abstract

We study a notion of local time for a continuous path, defined as a limit of suitable discrete quantities along a general sequence of partitions of the time interval. Our approach subsumes other existing definitions and agrees with the usual (stochastic) local times a.s. for paths of a continuous semimartingale. We establish pathwise version of the It\^o-Tanaka, change of variables and change of time formulae. We provide equivalent conditions for existence of pathwise local time. Finally, we study in detail how the limiting objects, the quadratic variation and the local time, depend on the choice of partitions. In particular, we show that an arbitrary given non-decreasing process can be achieved a.s. by the pathwise quadratic variation of a standard Brownian motion for a suitable sequence of (random) partitions; however, such degenerate behavior is excluded when the partitions are constructed from stopping times.