This is with deep sadness that I learnt that my former Phd advisor Marc Yor passed away on Thursday, January 9. I remember him as an extraordinary kind and gentle person with an unlimited amount of patience and energy for his students and collaborators. His knowledge of all aspects of the Brownian motion and of the associated literature was phenomenal. His book, Continuous martingales and Brownian motion, jointly written with Daniel Revuz, formed a generation of probabilists and is certainly one of the most complete expositions of the theory of continuous stochastic processes in continuous time.
His mathematical legacy is very large and includes about 400 research articles written with more than 150 different co-authors.
The first part of his career in the 1970’s and the beginning of the 1980’s was devoted to the general theory of stochastic processes in the spirit of the French school in probability that was led by Paul-Andre Meyer at that time. In this period he already obtained several groundbreaking results. I wish to mention his improvements and extensions of the Burkholder-Davis-Gundy inequalities, and his a simple approach to the Skorokhod embedding problem. Together with Thierry Jeulin, he also founded the theory of enlargement of filtrations which nowadays is widely used in mathematical finance.
In the 1980’s and thereafter, several of his most influential works are related to the study of functionals of the Brownian motion.
- The study of planar Brownian motion. His 1980 paper on windings of Brownian motions was a milestone in the fine study of the planar Brownian motion and somehow inspired several later developments in the study of conformally invariant processes, like the SLE process.
- The study of quadratic functionals. He wrote numerous influential papers devoted to the computation of exact distributions of quadratic functionals of the Brownian motion and about Bessel processes and their connections with excursion theory.
- The study of exponential functionals. In the 1990’s he developed a deep interest for the exponential functionals of the Brownian motion. In particular, motivated by the problem of finding explicitly, as much as possible, a formula for the price of Asian options, he obtains a formula for the Laplace transform in time of the distribution of an exponential functional. Several of his beautiful papers on this topic may be found in his book: Exponential functionals of Brownian motion and related processes. Though motivated by mathematical finance, it is interesting that exponential functionals have been found to have connections with representation theory of Lie groups and integrable systems (see also this paper).
- Webpage of the Paris 6 mathematics department
- Jim Pitman’s webpage about Marc Yor
- Zhan Shi webpage about Marc Yor
- IMS bulletin obituary
- Une excursion avec Marc Yor
More recent interests and important results include the study of penalisation of Brownian motion paths.
The following video taken at the University of Bristol in December 2008, shows him as he was: humble, expert and passionate on his subject.
The following video (in French, January 2008) shows him explaining a piece of history of modern probability theory.