Abstract:

There will be three lectures, the first will consider the signature of a path; the second will look at the general theory of rough paths, explaining the most basic theorems; the third will look at how the signature of a path, together with a re-combination technique, can be used to produce high order substitute methods for Monte Carlo. Putting these ideas together one is able to create a family of novel and very accurate numerical methods.

1. Ben Hambly and Terry Lyons Uniqueness for the signature of a path of bounded variation and the reduced path group, (to appear Ann. Math) http://people.maths.ox.ac.uk/~hambly/PDF/Papers/signew3.pdf

2. Lyons, Terry J., Caruana, Michael, Lévy, Thierry Differential Equations Driven by Rough Paths Ecole d'Eté de Probabilités de Saint-Flour XXXIV-2004 Series: Lecture Notes in Mathematics Vol. 1908

3. Lyons, T.J. & Victoir, N. 2004 Cubature on Wiener Space, Proc. R. Soc. Lond. A 460,. 169-198.

4. Christian Litterer and Terry Lyons, High order recombination and an application to cubature on Wiener space, manuscript to appear.