Ritsumeikan course - Abstract 

Lecture 1: Introduction to rough paths.

We first give an introduction and some motivation to rough path theory and its application to differential equations driven by fractional Brownian motion. Then we review stochastic calculus according to Ito, with a language which can be easily adapted to our future considerations. 

Lecture 2: Young integration and preliminary results.

We show that differential calculus with respect to fractional Brownian motion with Hurst parameter H different from 1/2 cannot be handled thanks to martingale type arguments. We then treat the case H>1/2 by means of Young integrals. This integration theory will be introduced with two different methods: (i) An elementary way based on Riemann sums. (ii) By means of algebraic integration techniques, which will be at the heart of our future discussions and can be introduced in an elementary manner in this context. 

Lecture 3: Algebraic integration formalism.

We treat the case of level 2 rough paths, an apply our results to differential equations driven by a fractional Brownian motion with Hurst parameter 1/3<H<1/2. We will try to give some insight on technical details. According to the time being left, we will then introduce Malliavin calculus tools applied to rough differential equations.