Author Archives: Fabrice Baudoin

Based on the results of the previous Lecture, it should come as no surprise that differential equations driven by the Brownian rough path should correspond to Stratonovitch differential equations. In this Lecture, we prove that it is indeed the case. … Continue reading →

Since a -dimensional Brownian motion is a -rough path for , we know how to give a sense to the signature of the Brownian motion. In particular, the iterated integrals at any order of the Brownian motion are well defined … Continue reading →

Our goal in the next two lectures will be to prove that rough differential equations driven by a Brownian motion seen as a -rough path, are nothing else but stochastic differential equations understood in the Stratonovitch sense. The proof of … Continue reading →

We now turn to the proof of Lyons’ continuity theorem. Theorem: Let . Assume that are -Lipschitz vector fields in . Let such that with . Let be the solutions of the equations There exists a constant depending only on … Continue reading →

We now turn to the proof of the continuity theorem. We start with several lemmas, which are not difficult but a little technical. The first one is geometrically very intuitive. Lemma: Let such that with and with . Then, there … Continue reading →

We are now ready to state the main result of rough paths theory: the continuity of solutions of differential equations with respect to the driving path. Theorem: Let . Assume that are -Lipschitz vector fields in . Let such that … Continue reading →

We now turn to the proof of Davie’s estimate. We follow the approach by Friz-Victoir who smartly use interpolations by geodesics in Carnot groups. Theorem: Let . Assume that are -Lipschitz vector fields in . Let . Let be the … Continue reading →

In this Lecture, we prove one of the fundamental estimates of rough paths theory. This estimate is due to Davie. It provides a basic estimate for the solution of the differential equation in terms of the -variation of the lift … Continue reading →

In the previous Lecture we proved that Brownian motion paths almost surely have a bounded -variation for every . In this lecture, we are going to prove that they even almost surely are -rough paths for . To prove this, … Continue reading →