Tel: 01865 273544
Web Link: http://sag.maths.ox.ac.uk/samath/
I lead the Stochastic Analysis Group in Maths.
BA and Part III, Trinity College, Cambridge; DPhil, Oxford; Doctor, honoris causa, UPS, Toulouse; Junior Research Fellow, Jesus College, Oxford; Hedrick Assistant Professor, University of California, Los Angeles; Lecturer, Imperial College, London; Colin MacLaurin Professor of Mathematics, Edinburgh; Professor of Mathematics, Imperial College, London; SRC/EPSRC Senior Fellow; Wallis Professor of Mathematics, Oxford; Member of the Oxford-Man Institute; Director of the Wales Institute for Mathematical and Computational Sciences (WIMCS); joined St Annes 2000.
Analysis, Probability, Martingales, Stochastic Calculus and Differential Equations. Applications in finance and in other areas of pure mathematics.
Stochastic Analysis in all shapes and forms. Computing for mathematics (at a high level in C++). I run a weekly seminar for graduate students in Stochastic Analysis throughout the year as well as a weekly Stochastic Analysis seminar (typically on Tuesdays in the morning) with two outside speakers every Monday afternoon in term. Many of the Stochastic Analysis Graduate students are at St Anne’s and come to this so I get to know most of them.
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.
Terry J. Lyons, ‘Differential equations driven by rough signals’, Rev. Mat. Iberoamericana 14 (1998), no. 2, 215-310
Terry Lyons and Zhongmin Qian, System control and rough paths, Oxford Mathematical Monographs. Oxford Science Publications (2002)
T.J. Lyons and T.S. Zhang, ‘Decomposition of Dirichlet processes and its application’, Ann. Probab. 22 (1994), no. 1, 494-524
Terence J. Lyons and Wei Zheng, ‘An A crossing estimate for the canonical process on a Dirichlet space and a tightness result’, Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987), Astérisque No. 157-58 (1988), 249-71
Terry Lyons, ‘Differential equations driven by rough signals. I. An extension of an inequality of L.C. Young’, Math. Res. Lett. 1 (1994), no. 4, 451-64
Terry Lyons and Dennis Sullivan, ‘Function theory, random paths and covering spaces’, J. Differential Geom. 19 (1984), no. 2, 299-323
Dan Crisan and Terry Lyons, ‘Nonlinear filtering and measure-valued processes’, Probab. Theory Related Fields 109 (1997), no. 2, 217-44
T.J. Lyons and W.A. Zheng, ‘On conditional diffusion processes’, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), no. 3-4, 243-55
M. Ledoux, T. Lyons and Z. Qian, ‘Lévy area of Wiener processes in Banach spaces’, Ann. Probab. 30 (2002), no. 2, 546-78
J.G. Gaines and T.J. Lyons, ‘Variable step size control in the numerical solution of stochastic differential equations’, SIAM J. Appl. Math. 57 (1997), no. 5, 1455-84