Lecture notes

More lecture notes are posted on my blog.

Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations, Institut Henri Poincare, September 2014

These notes are the basis of a course given at the Institut Henri Poincare in September 2014. We survey some recent results related to the geometric analysis of hypoelliptic diffusion operators on totally geodesic Riemannian foliations. We also give new applications to the study of hypocoercive estimates for Kolmogorov type operators.

The lectures are also posted on my blog.

Rough Paths Theory 

There have been 30 classes of 45 minutes. The lectures are posted on my blog

Stochastic differential equations driven by fractional Brownian motions, 34th Finnish summer school on Probability theory and Statistics, Päivölän Kansanopisto from June 4th to June 8th, 2012

There have been  6 classes of 45 minutes . The course is at a graduate level.

The purpose of the course is to provide an introduction to the study of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H>1/2.

Lecture 1: Fractional Brownian motion
Lecture 2: Young's integrals and basic estimates
Lecture 3: Stochastic differential equations driven fractional Brownian motions: Existence and uniqueness (1)
Lecture 4: Stochastic differential equations driven fractional Brownian motions: Existence and uniqueness (2)
Lecture 5: Malliavin calculus
Lecture 6: Existence of a density for the solution

The lecture notes may be downloaded here.

Modelling anticipations on a financial market, Princeton University, 2003

Download pdf file

This course was given in Princeton University in 2003. It is intended to graduate, post graduate students. These notes were published (in a different form) by Springer: In Paris-Princeton Lectures on Mathematical Finance, LNM 1814, (2003).

Financial markets obviously have asymmetry of information. That is, there are different type of traders whose behavior is induced by different types of information that they possess. Let us consider a "small" investor who trades in a arbitrage free financial market so as to maximize the expected utility of his wealth at a given time horizon. We assume that he possesses extra information about the future price of a stock. Our basic question is: What is the value of this information ?

Basic probability theory, Ho Chi Minh city, Vietnam, 2006

Download pdf file

This course was given in Vietnam in January 2006. It is a first course in probability theory. The notes are a bit rough but were useful to the students.

Stochastic calculus, In French, Toulouse University, 2004-2007

Le mouvement brownien est un processus stochastique omniprésent en théorie des probabilités. Il fut étudié au début du siècle par Bachelier, Einstein et Wiener. Dans les années quarante, Ito s'en sert pour développer un calcul stochastique permettant de résoudre des équations différentielles perturbées aléatoirement.

Le calcul stochastique est un mariage de la théorie des probabilités et du calcul différentiel et intégral, qui a trouvé depuis beaucoup d'applications (équations aux dérivées partielles, géométrie différentielle, mathématiques financières, télécommunications, etc...). Dans ce cours, nous présentons le mouvement brownien et le calcul stochastique qui lui est associé. L'accent est mis sur la théorie des diffusions.

Chapitre 0: Quelques rappels de théorie des probabilités

Chapitre 1: Processus stochastiques

Chapitre 2: Martingales

Chapitre 3: Mouvement brownien

Chapitre 4: Calcul d'Itô

Stochastic Taylor expansions and heat kernel asymptotics, Spring School of Mons, June 2009

Download pdf file

These notes focus on the applications of the stochastic Taylor expansion of solutions of stochastic differential equations to the study of heat kernels in small times. As an illustration of these methods we provide a new heat kernel proof of the Chern-Gauss-Bonnet theorem.