Professor Paul Malliavin began his long career as a pure mathematician, specialising in mathematical analysis, the branch of pure mathematics that grew out of calculus. He ended it by being best known for his work on probability theory, and in particular as the discoverer of the type of calculus that bears his name.
Ordinary differential calculus, going back to Newton and Leibniz in the 17th century, provides a general method for finding extreme points — the points where some function takes its maximum or minimum values. This leads on to the calculus of variations, developed in the 18th century, in which one finds maxima and minima, not as points where a function takes an extreme value, but as functions where some integral takes an extreme value. Such problems are all around us in physics. For example, a stick appears bent in water because light travels more slowly in water than in air, and the light rays we perceive are those for which the time of travel is minimised.
We live in a random world, and one of the most important developments of calculus ideas in the last century was the introduction of random, or stochastic, functions. This dates from the work of the great Japanese probabilist Kiyosi Itô (obituary, November 20, 2008) in 1944; the resulting stochastic, or Itô, calculus is now well established, both in academic mathematics and as a working tool used in the City by quantitative analysts (“quants”). Malliavin’s contribution was to develop a stochastic calculus of variations; his breakthrough paper appeared in 1976.
The resulting theory has been extensively developed, by Malliavin himself and many others, and is now known as the Malliavin calculus. This is an essentially infinite-dimensional theory, and so makes use of functional analysis, the infinite-dimensional extension of ordinary analysis. There are close links with some areas of theoretical physics, where some of the relevant mathematics originated; it is no accident that many quants have a physics background.
An important conference took place in 2001 on numerical and computer implementation of Malliavin calculus, with particular emphasis on its application by quants to calibration of models in mathematical finance. Following this, Malliavin’s last book (with Anton Thalmaier, in 2006) was devoted to its extensive applications in mathematical finance.
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Paul Malliavin was born in 1925 at Neuilly-sur-Seine. He took his first degree (agrégé de mathématiques) in 1946; his doctorate followed in 1954, from the University of Paris. Malliavin had an excellent education in the best traditions of French mathematics: his doctoral adviser was the prominent analyst Szolem Mandelbrojt, himself a pupil of Jacques Hadamard, one of the great figures of the 19th century. His thesis was on complex analysis.
During a postdoctoral year in 1954-55 at the Institute of Advanced Study at Princeton, Malliavin met the Swedish analyst Arne Beurling, with whom he worked on harmonic analysis. This area, which dates back to Fourier’s work on heat in the early 19th century, deals with the decomposition of functions into periodic components, and with the concept of the spectrum in physics. Going in the other direction leads to the problem of “spectral synthesis”. The work that made Malliavin’s name was a series of papers on the impossibility of spectral synthesis, in various contexts, in 1959. Also important from this period was his collaboration with Beurling in 1962. Malliavin taught at the University of Caen from 1955 to 1962, and then spent the rest of his career at the University of Paris (1962-93). He was elected a corresponding member of the Academy of Sciences in 1977 and a full member in 1979.
Malliavin travelled widely. He made several visits to Princeton, teaching there and at Stanford, Chicago, Jerusalem, Madrid and Stockholm. He was a familiar figure at conferences in Britain, with his amiable smile, pork-pie hat and typically French speaking style. He collaborated extensively, with his wife Marie-Paul Malliavin-Brameret, with Mandelbrojt, Beurling, Jean-Pierre Kahane (his fellow-academician, who also took his PhD in 1954 under Mandelbrojt), Katznelson (another pupil of Mandelbrojt), with David Nualart of Barcelona (the author of the standard work on Malliavin calculus), and many others, including Nualart’s daughter Eulàlia. He was noted for his editorial work, for example as a long-serving editor (from 1967) of the Journal of Functional Analysis.
His mathematical output was impressive for its sheer extent (some 180 papers and several books), its depth, and its exceptionally wide mathematical range. His papers cover real, complex, harmonic and functional analysis, approximation theory, differential geometry and probability theory. He was widely honoured, receiving many prizes; he was a foreign member of the Swedish Academy of Sciences.
He is survived by his wife and two daughters.
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Professor Paul Malliavin, mathematician, was born on September 11, 1925. He died on June 3, 2010, aged 84