# Exponential Functionals of Brownian Motion and Related Processes

• Marc Yor
Book

Part of the Springer Finance book series (FINANCE)

1. Front Matter
Pages i-ix
2. Hélyette Geman
Pages 1-13
3. Marc Yor
Pages 14-22
4. Marc Yor
Pages 23-48
5. Marc Yor
Pages 63-92
6. Marc Yor
Pages 93-122
7. Marc Yor
Pages 123-138
8. Marc Yor
Pages 139-171
9. Marc Yor
Pages 172-181
10. Marc Yor
Pages 182-203
11. Back Matter
Pages 200-205

### Introduction

This monograph contains: - ten papers written by the author, and co-authors, between December 1988 and October 1998 about certain exponential functionals of Brownian motion and related processes, which have been, and still are, of interest, during at least the last decade, to researchers in Mathematical finance; - an introduction to the subject from the view point of Mathematical Finance by H. Geman. The origin of my interest in the study of exponentials of Brownian motion in relation with mathematical finance is the question, first asked to me by S. Jacka in Warwick in December 1988, and later by M. Chesney in Geneva, and H. Geman in Paris, to compute the price of Asian options, i. e. : to give, as much as possible, an explicit expression for: (1) where A~v) = I~ dsexp2(Bs + liS), with (Bs,s::::: 0) a real-valued Brownian motion. Since the exponential process of Brownian motion with drift, usually called: geometric Brownian motion, may be represented as: t ::::: 0, (2) where (Rt), u ::::: 0) denotes a 15-dimensional Bessel process, with 5 = 2(1I+1), it seemed clear that, starting from (2) [which is analogous to Feller's repre­ sentation of a linear diffusion X in terms of Brownian motion, via the scale function and the speed measure of X], it should be possible to compute quan­ tities related to (1), in particular: in hinging on former computations for Bessel processes.

### Keywords

Asian options Bessel functions Bessel process Bessel processes Brownian motion Lévy process beta-gamma variables geometric Brownian motion

#### Authors and affiliations

• Marc Yor
• 1
1. 1.Laboratoire de Probabilités et Modèles AléatoiresUniversité de Paris VIParisFrance

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-642-56634-9
• Copyright Information Springer-Verlag Berlin Heidelberg 2001
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-540-65943-3
• Online ISBN 978-3-642-56634-9
• Series Print ISSN 1616-0533
• Buy this book on publisher's site
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