The Pedestrian’s Guide to Local Time

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. I am very grateful to Mariana Khapko for valuable comments, and for giving me the necessary motivation to write this paper. Many thanks are also due to Boualem Djehiche for valuable comments and suggestions.