Course 1: Essential of stochastic calculus - Mr ED DAHBI M'hamed
The objective of this course is to provide participants with the essential and necessary elements for developing stochastic differential calculus, including the theory of stochastic integration for Brownian motion (Wiener process). The most powerful tool in stochastic differential calculus is the change of variable formula, called Itô's formula, in this context. In particular, using this formula we construct other stochastic processes linked to the Brownian motion such as martingales. Ito's formula will be applied also to give probabilistic interpretations to partial differential equations of parabolic type. This introductory course will be presented in four chapters. The first one will be dedicated to general notions of stochastic processes in continuous time, including definitions, measurability, adaptations, regularities and definitions of martingales and their properties. The second chapter will be devoted to stochastic integration in the sense of Itô before a review on the Riemann-Stieltjes integral and then establish Itô's formula and apply it to evaluate some stochastic integrals. The third chapter deals with some applications of the Itô's formula to the resolution of some stochastic differential equations (SDEs), in particular existence and uniqueness of solutions under Lipschitz conditions on the coefficients. This course ends with chapter four which highlights the bilateral correspondence between solutions of partial differential equations (PDEs) and Markov processes. Applications in mathematical finance including pricing and hedging of financial derivatives will be presented using EDS and / or PDEs.
Course 2: Introduction to White Noise, Hida-Malliavin Calculus and Applications - Mme AGRAM Nacira
We first give a survey of classical Malliavin calculus based on the Wiener-Itô chaos expansion theorem for Brownian motion. Then we introduce the Hida white noise theory, and in this context we show that there is a natural extension of the Malliavin calculus from the classical domain D_{1,2} to all of L²(P). This extension is called the Hida-Malliavin calculus. The Hida-Malliavin calculus allows us to prove new results under weaker assumptions than could be obtained by the classical theory. In particular, we prove (i) a generalised fundamental theorem of stochastic calculus and (ii) a general solution representation theorem for backward stochastic differential equations with jumps, in terms of Hida-Malliavin derivatives. As an application of the above theory we consider the optimal control of stochastic Volterra integral equations.
Course 3: Probabilistic approximation by means of Malliavin calculus - Mr NOURDIN Ivan
Stein's method was invented in the sixties by the great statistician Charles Stein, mainly for teaching reasons. This latter was indeed looking for a simple and accessible proof to a student of a combinatorial central limit theorem due to Wald and Wolfowitz. In a completely disconnected way, Malliavin calculus was developed by the great analyst Paul Malliavin in the seventies, with his desire to provide a fully probabilist proof of the celebrated Hörmander criterion for hypoellipticity of partial differential equations. Although Stein's method and Malliavin calculus have been historically developed for completely different reasons, it turns out that they have in common to be both based on the use of genuine integration by parts formulas. This last fact was recently exploited by Ivan Nourdin and Giovanni Peccati in order to invent the so-called Malliavin-Stein approach, which represents nowadays a powerful theory for normal approximation of random variables measurable with respect to an underlying Gaussian process. The goal of this series of lectures is to introduce the students to this new technique, and to illustrate its power through selected examples coming from diverse areas of contemporary interests..
Course 4: Optimal Stop and US Options Applications - Mr OUKNINE Youssef
In this course we will give an overview of the general theory of processes by emphasizing the optional and predictable tribes as well as the classification of downtime, finally the famous section theorems and their applications. We introduce the optional martingales and their decompositions, a so-called Doob-Meyer-Mertens decomposition which is fundamental in the theory of the non-regular optimal stop. All these notions extend to the nonlinear case via the solutions of the retrograde differential equations (see the recent works of Miryana Grigorova (Peter Imkeller, Youssef Ouknine and Marie-Claire Quenez). Finally, we will end this course with applications to the valuation and hedging of US options.
Course 5: Poisson point process - Pr Xiaochuan Yang
Poisson point process is arguably the most fundamental stochastic object that is discrete in nature and is widely used in modeling. In this lecture series, I will present elements of Poisson point processes, Malliavin calculus on Poisson functionals, first and second order Poincare inequalities. The theory is somewhat parallel to Malliavin calculus on the Wiener space presented in Course 2. As application we show normal approximation for statistics of random geometric graphs based on Poisson points by the so-called Malliavin-Stein approach partly presented in Course 3.
Course 6: introduction to stochastic optimal switching - Mr HAMADANE Said
1) Introduction to the problem of stochastic optimal switching. Motivations and relation with systems of stochastic retrograde differential equations with interconnected obstacles. Existence of an optimal strategy. Link with Dynkin's game of zero sum. Some digital aspects.
2) Optimal switching and partial differential equation systems with interconnected obstacles of the associated HJB type. Existence and uniqueness of the system solution. Extensions.
3) Stochastic switching game of zero sum. Study of associated min-max and max-min PDEs. Existence of value at play. Extensions.
Practical work: Introduction to Python and applications to simulation of EDSs - Dr O. Kebiri, Pr A. Aibeche et Pr F.O.TEBBOUNE