Abstract

Rough path analysis has provided new insights in the analysis of stochastic dierential equations and stochastic partial dierential equations. When applied to stochastic systems, rough path analysis provides a means to construct a pathwise solution theory which, in many respects, behaves much like the theory of deterministic dierential equations and provides a clean break between analytical and probabilistic arguments. It provides a toolbox allowing to recover many classical results without using specic probabilistic properties such as predictability or the martingale property. The study of stochastic PDEs has recently led to a signicant extension, Hairer’s theory of regularity structures, and the second half of this course is devoted to a gentle introduction.

Pre-requisite Mathematics

• Upper undergraduate analysis, interest (and a little maturity) in stochastic analysis.

Literature

1. Terry J. Lyons, Michael Caruana, and Thierry Levy, Dierential equations driven by rough paths, Lecture Notes in Mathematics, vol. 1908, Springer, Berlin, 2007

2. Peter K. Friz, Nicolas Victoir, Multidimensional stochastic processes as rough paths, Cambridge Studies in Advanced Mathematics, vol. 120, Cambridge University Press, Cambridge, 2010

3. Peter K. Friz and Martin Hairer, A course on rough paths: With an introduction to regularity structures, Springer Universitext, 2014.

4. Martin Hairer, A theory of regularity structures, Inventiones mathematicae (2014), 1-236