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Exponential Functionals of Brownian Motion and Related Processes

  • Marc Yor

Part of the Springer Finance book series (FINANCE)

About this book

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 Springer Book Archive
  • 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