Wallis Professor of Mathematics

+44 1865 616611

## Address

University of Oxford

Andrew Wiles Building

Radcliffe Observatory Quarter

Woodstock Road

Oxford

OX2 6GG

## Recent Books:

System control and rough paths

ISBN-10: 0-19-850648-1
(2002)

## Highlighted Publications:

High order recombination and an application to cubature on Wiener space

Annals of Applied Probability
volume 22
page 1301-1327
(2012)

Backward stochastic dynamics on a filtered probability space

Annals of Probability
issue 4
volume 39
page 1422-1448
(1 July 2011)

Integrability Estimates for Gaussian Rough Differential Equations

(2011)

Expected signature of two dimensional Brownian Motion up to the first exit time of the domain

(2011)

Uniqueness for the signature of a path of bounded variation and the reduced path group

Annals of Mathematics. Second Series
volume 171
page 109-167
(2010)

## Recent Publications:

Hyperbolic development and inversion of signature

Journal of Functional Analysis
issue 7
volume 272
page 2933-2955
(1 April 2017)

The signature of a rough path: Uniqueness

ADVANCES IN MATHEMATICS
volume 293
page 720-737
(30 April 2016)
Full text available

THE ADAPTIVE PATCHED CUBATURE FILTER AND ITS IMPLEMENTATION

COMMUNICATIONS IN MATHEMATICAL SCIENCES
issue 3
volume 14
page 799-829
(2016)
Full text available

The theory of rough paths via one-forms and the extension of an argument of Schwartz to rough differential equations

JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN
issue 4
volume 67
page 1681-1703
(October 2015)
Full text available

EXPECTED SIGNATURE OF BROWNIAN MOTION UP TO THE FIRST EXIT TIME FROM A BOUNDED DOMAIN

ANNALS OF PROBABILITY
issue 5
volume 43
page 2729-2762
(September 2015)
Full text available

## Research Interests:

I am the Wallis Professor of Mathematics; I was a founding member (2007) of, and then Director (2011-2015) of, the Oxford Man Institute of Quantitative Finance; I was the Director of the Wales Institute of Mathematical and Computational Sciences (WIMCS; 2008-2011). I came to Oxford in 2000 having previously been Professor of Mathematics at Imperial College London (1993-2000), and before that I held the Colin Maclaurin Chair at Edinburgh (1985-93).

My long-term research interests are all focused on Rough Paths, Stochastic Analysis, and applications - particularly to Finance and more generally to the summarsing of large complex data. That is to say I am interested in developing mathematical tools that can be used to effectively model and describe high dimensional systems that exhibit randomness. This involves me in a wide range of problems from pure mathematical ones to questions of efficient numerical calculation.

## Further Details:

Stochastic analysis. This is the area of mathematics relating to the rigorous description of high-dimensional systems that have randomness. It is an area of wide-reaching importance. Virtually all areas of applied mathematics today involve considerations of randomness, and a mobile phone would not work without taking advantage of it. Those who provide fixed-rate mortgages have to take full account of it. My interests are in identifying the fundamental language and the basic results that are required to model the interaction between highly oscillatory systems where the usual calculus is inappropriate. If you google ‘Rough Paths’ and ‘Lyons’ you will find further information. My St Flour Lecture notes provide a straightforward technical introduction with all the details put as simply as possible. A more general introduction can be found in my talk/paper to the European Mathematical Society in Stockholm in 2002.

My approach is that of a pure mathematician, but my research has consequences for numerical methods, finance, sound compression and filtering. At the moment I am (speculatively) exploring their usefulness in understanding sudden shocks on dynamical systems, and also trying to understand the implications for geometric measure theory. The focus of my research directed to ‘Rough paths’ can be viewed as a successful approach to understanding certain types of non-rectifiable currents.

I actively look for applications in the mathematics I do, but my experience has led me to believe strongly in the importance of being rigorous in the development of the core mathematical ideas. For me, the word proof is synonymous with the more palatable ‘precise, convincing and detailed explanation’, and I believe it is important, even essential, to find rigorous proofs of the key mathematical intuitions so that mathematics can reliably grow and ideas can be passed on to the next generation.