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.

These papers were so far available only in journals, several of them only in French. Collected together here in English, they are accompanied by a foreword by H. Geman, professor of Finance at the Université Paris-Dauphine and ESSEC Includes supplementary material: sn.pub/extras

Inhalt

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), 202208.- 2. On Some Exponential Functionals of Brownian Motion Adv. Appl. Prob. 24 (1992), 509531.- 3. Some Relations between Bessel Processes, Asian Options and Confluent Hypergeometric Functions C.R. Acad. Sci., Paris, Sér. I 314 (1992), 417474 (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), 951956.- 5. Bessel Processes, Asian Options, and Perpetuities Mathematical Finance, Vol. 3, No. 4 (October 1993), 349375 (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), 2334.- 8. On Exponential Functionals of Certain Lévy Processes Stochastics and Stochastic Rep. 47 (1994), 71101 (with P. Carmona and F. Petit).- 9. On Some Exponential-integral Functionals of Bessel Processes Mathematical Finance, Vol. 3 No. 2 (April 1993), 231240.- 10. Exponential Functionals of Brownian Motion and Disordered Systems J. App. Prob. 35 (1998), 255271 (with A. Comtet and C. Monthus).