Ju Y. Local Times and Excursion Theory for Brownian Motion...2013

Textbook in PDF format

This monograph discusses the existence and regularity properties of local times associated to a continuous semimartingale, as well as excursion theory for Brownian paths. Realizations of Brownian excursion processes may be translated in terms of the realizations of a Wiener process under certain conditions. With this aim in mind, the monograph presents applications to topics which are not usually treated with the same tools, e.g.: arc sine law, laws of functionals of Brownian motion, and the Feynman-Kac formula.
Front Matter
Prerequisites
Front Matter
The Existence and Regularity of Semimartingale Local Times
Lévy’s Representation of Reflecting BM and Pitman’s Representation of BES(3)
Paul Lévy’s Arcsine Laws
Front Matter
Brownian Excursion Theory: A First Approach
Two Descriptions of n: Itô’s and Williams’
A Simple Path Decomposition of Brownian Motion Around Time t = 1
The Laws of, and Conditioning with Respect to, Last Passage Times
Integral Representations Relating W and n
Front Matter
The Feynman–Kac Formula and Excursion Theory
Some Identities in Law
Back Matter