Marc Yor

Exponential Functionals of Brownian Motion and Related Processes

• Broschiertes Buch

Produktbeschreibung

This volume collects papers about the laws of geometric Brownian motions and their time-integrals, written by the author and coauthors between 1988 and 1998. These functionals play an important role in Mathematical Finance, as well as in (probabilistic) studies related to hyperbolic geometry, and also to random media. Throughout the volume, connections with more recent studies involving exponential functionals of Lévy processes are indicated. Some papers originally published in French are made available in English for the first time.

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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.

Produktdetails

• Springer Finance
• Verlag: Springer / Springer Berlin Heidelberg / Springer, Berlin
• Artikelnr. des Verlages: 978-3-540-65943-3
• Softcover reprint of the original 1st ed. 2001
• Seitenzahl: 216
• Erscheinungstermin: 14. August 2001
• Englisch
• Abmessung: 235mm x 155mm x 12mm
• Gewicht: 336g
• ISBN-13: 9783540659433
• ISBN-10: 3540659439
• Artikelnr.: 08937607

Inhaltsangabe

0. Functionals of Brownian Motion in Finance and in Insurance.- 1. On Certain Exponential Functionals of Real-Valued Brownian Motion J Appl. Prob. 29 (1992), 202-208.- 2. On Some Exponential Functionals of Brownian Motion Adv. Appl. Prob. 24 (1992), 509-531.- 3. Some Relations between Bessel Processes, Asian Options and Confluent Hypergeometric Functions C.R. Acad. Sci., Paris, Sér. I 314 (1992), 417-474 (with Hélyette Geman).- 4. The Laws of Exponential Functionals of Brownian Motion, Taken at Various Random Times C.R. Acad. Sci., Paris, Sér. I 314 (1992), 951-956.- 5. Bessel Processes, Asian Options, and Perpetuities Mathematical Finance, Vol. 3, No. 4 (October 1993), 349-375 (with Hélyette Geman).- 6. Further Results on Exponential Functionals of Brownian Motion.- 7. From Planar Brownian Windings to Asian Options Insurance: Mathematics and Economics 13 (1993), 23-34.- 8. On Exponential Functionals of Certain Lévy Processes Stochastics and Stochastic Rep. 47 (1994), 71-101 (with P. Carmona and F. Petit).- 9. On Some Exponential-integral Functionals of Bessel Processes Mathematical Finance, Vol. 3 No. 2 (April 1993), 231-240.- 10. Exponential Functionals of Brownian Motion and Disordered Systems J. App. Prob. 35 (1998), 255-271 (with A. Comtet and C. Monthus).

Rezensionen

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 ihreAnwendungen, Vol. 21 (1), 2002)