Exponential Functionals of Brownian Motion and Related Processes [Paperback]

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.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, S?r. I 314 (1992), 417474 (with H?lyette Geman).- 4. The Laws of Exponential Functionals of Brownian Motion, Taken at Various Random Times C.R. Acad. Sci., Paris, S?r. I 314 (1992), 951956.- 5. Bessel Processes, Asian Optionl“0