This page contains some lecture notes about courses I had the
opportunity to teach in the past.
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 will
be 6 classes of 45 minutes
. The course
will be at graduate level.
The purpose of the course will be 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, where I
was invited by Patrick Cheridito. 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, 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.