The Pedestrian's Guide to Local Time

TL;DR

This paper provides an accessible introduction to local time in stochastic processes, covering theoretical foundations, intuitive explanations, and formal proofs, with applications to Brownian motion and stochastic differential equations.

Contribution

It offers a comprehensive, pedestrian-oriented overview of local time theory, combining intuitive arguments with formal proofs and applications in stochastic analysis.

Findings

Clarifies the relationship between local time and the Tanaka formula

Provides formal proofs of key results in local time theory

Includes applications to regulated stochastic differential equations

Abstract

These notes contains an introduction to the theory of Brownian and diffusion local time, as well as its relations to the Tanaka Formula, the extended Ito-Tanaka formula for convex functions, the running maximum process, and the theory of regulated stochastic differential equations. The main part of the exposition is very pedestrian in the sense that there is a considerable number of intuitive arguments, including the use of the Dirac delta function, rather than formal proofs. For completeness sake we have, however, also added a section where we present the formal theory and give full proofs of the most important results. In the appendices we briefly review the necessary stochastic analysis for continuous semimartingales.