On Exponential Functionals of Brownian Motion and Related Processes

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.

From the reviews:

"This book is a collection of papers that deal with the laws of Geometric Brownian Motion and their time-integrals with an emphasis on Asian Options. Each paper is self-contained and presents the topics at a high level. ... Thus, this book provides a valuable reference for people investigating and applying this mathematics to the study of Asian Options." (Moreno Fasolo, www.quantnotes.com, November, 2001)

"This book is a collection of ten papers on the law of certain functionals of geometric Brownian motion. ... The volume combines a great variety of different techniques, especially from Stochastic Analysis, and wonderfully illustrates their applicability. ... Some of these papers are made available in English for the first time. They are supplemented by an updated list of references and a short review of further progress made since publication of the presented results." (Peter Bank, Zentralblatt MATH, Vol. 999 (24), 2002)

"Most of the papers are motivated by financial considerations, in particular Asian options ... . Each paper is appended with a postscript, which, in most cases, indicates the current context of the article by commenting on recent developments and including additional references. An index has also been provided. ... this book gathers together a collection of interesting papers, some of which contain some quite elegant results." (W. P. Wood, The Australian Mathematical Society Gazette, Vol. 29 (2), 2002)

"The present book is of great importance to mathematical finance. That is the reason why it is published in the new series Springer Finance. ... The present volume is a collection of papers written by the author and 5 co-authors between 1988 and 1998, partly translated from French originals. ... it is welcome to reprint interesting papers in mathematical finance which are spread over different journals not easily available to the reader." (H.-J. Girlich, Zeitschrift für Analysis und ihre Anwendungen, Vol. 21 (1), 2002)