Professor Terry Lyons
- Wallis Professor of Mathematics
Director, Oxford Man Institute of Quantitative Finance
President Designate, London Mathematical Society
Personal Web Page
Reception/Secretary: +44 1865 (6)16611
Direct: +44 1865 (6)16608
University of Oxford
Walton Well Road
24-29 St Giles'
I am the Director (and a founding member) of the Oxford Man Institute of Quantitative Finance, and the Wallis Professor of Mathematics; previously 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.
Recent Publications (from MathSciNet):
MR2920195 Reviewed Cass, Thomas; Litterer, Christian; Lyons, Terry Rough paths on manifolds.
New trends in stochastic analysis and related topics,
33–88, Interdiscip. Math. Sci., 12, World Sci. Publ., Hackensack, NJ, 2012. (Reviewer: Antoine J. Lejay) 60H30 (34F05 46N30 58J65)
MR2884615 Reviewed Litterer, C.; Lyons, T. Introducing cubature to filtering.
The Oxford handbook of nonlinear filtering,
768–796, Oxford Univ. Press, Oxford, 2011. (Reviewer: Ramon van Handel) 60H30 (60G35 60H10 65C30)
MR2857245 Reviewed Liang, Gechun; Lyons, Terry; Qian, Zhongmin Backward stochastic dynamics on a filtered probability space.
Ann. Probab. 39 (2011), no. 4, 1422–1448. (Reviewer: Anthony Réveillac) 60H10 (60H30 60J45)
MR2789080 Reviewed Gyurkó, Lajos Gergely; Lyons, Terry J. Efficient and practical implementations of cubature on Wiener
Stochastic analysis 2010,
73–111, Springer, Heidelberg, 2011. 65C05 (60H10 60H35 60J65 65M75)
MR2653219 Indexed Lyons, Terence J. Kiyoshi Itô (1915–2008).
Probab. Theory Related Fields 148 (2010), no. 1-2, 1–4. 62-03 (01A70)
MR2664768 Reviewed Gyurkó, Lajos Gergely; Lyons, Terry Rough paths based numerical algorithms in computational finance.
Mathematics in finance,
17–46, Contemp. Math., 515, Amer. Math. Soc., Providence, RI, 2010. 91-02 (60H99 65C30 91G60)
MR2630037 Reviewed Hambly, Ben; Lyons, Terry Uniqueness for the signature of a path of bounded variation and the
reduced path group.
Ann. of Math. (2) 171 (2010), no. 1, 109–167. (Reviewer: Isamu Dôku) 58J65 (60G17 60H05 60H30)
MR2590691 Reviewed Levin, Daniel; Lyons, Terry A signed measure on rough paths associated to a PDE of high order:
results and conjectures.
Rev. Mat. Iberoam. 25 (2009), no. 3, 971–994. 60H05 (60H07 60H15 60J65)
MR2681814 Reviewed Litterer, Christian; Lyons, Terry Cubature on Wiener space continued.
Stochastic processes and applications to mathematical finance,
197–217, World Sci. Publ., Hackensack, NJ, 2007. (Reviewer: Bo Zhang) 60H30 (60G35 93E11)
MR2681809 Indexed Hara, Keisuke; Lyons, Terry Smooth rough paths and the applications.
Stochastic processes and applications to mathematical finance,
115–125, World Sci. Publ., Hackensack, NJ, 2007. 42A20 (26A45 60G17)
MR2414505 Reviewed Hara, Keisuke; Lyons, Terry Smooth rough paths and applications for Fourier analysis.
Rev. Mat. Iberoam. 23 (2007), no. 3, 1125–1140. (Reviewer: Antoine J. Lejay) 42A20 (26A45)
MR2348055 Reviewed Lyons, Terry; Victoir, Nicolas An extension theorem to rough paths.
Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. 5, 835–847. (Reviewer: Antoine J. Lejay) 60H10 (60H07)
MR2314753 Reviewed Lyons, Terry J.; Caruana, Michael; Lévy, Thierry Differential equations driven by rough paths.
Lectures from the 34th Summer School on Probability Theory held in
Saint-Flour, July 6–24, 2004.
With an introduction concerning the Summer School by Jean Picard.
Lecture Notes in Mathematics, 1908. Springer, Berlin, 2007. xviii+109 pp. ISBN: 978-3-540-71284-8; 3-540-71284-4 (Reviewer: Bohdan Maslowski) 60H10 (60-02 60H07)
MR2290140 Reviewed Li, Xiang-Dong; Lyons, Terry J. Smoothness of Itô maps and diffusion processes on path spaces.
Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 4, 649–677. (Reviewer: Isamu Dôku) 60G17 (58J65 60J65)
MR2247845 Reviewed Lyons, Terry J.; Sidorova, Nadia On the radius of convergence of the logarithmic signature.
Illinois J. Math. 50 (2006), no. 1-4, 763–790 (electronic). (Reviewer: Habib Ouerdiane) 60H10 (34A34 34F05 93C35)
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.