I am a PhD candidate in Probability at the CERMICS, Ecole des Ponts ParisTech. With my advisor Benjamin Jourdain, we study the weak error for time and particle discretizations of some Stochastic Differential Equations non linear in the sense of McKean.
This PhD is dedicated to the theoretical and numerical study of the weak error for time and particle discretizations of some SDEs non linear in the sense of McKean.
In the first part of this thesis, we address the weak error analysis for the time discretization of standard SDEs. More specifically, we study the convergence in total variation of the Euler-Maruyama scheme applied to multi-dimensional SDEs with additive noise and a measurable drift coefficient.
In the second part of this thesis, we analyze the weak error for both time and particle discretizations of two classes of nonlinear SDEs in the sense of McKean. The first class consists in multi-dimensional SDEs with regular drift and diffusion coefficients in which the dependence in law intervenes through moments. The second class consists in one-dimensional SDEs with a constant diffusion coefficient and a singular drift coefficient where the dependence in law intervenes through the cumulative distribution function. We approximate the SDEs by the Euler-Maruyama schemes of the associated particle systems. We also study, for the second class, the trajectorial propagation of chaos as well as strong convergence in time and in particles.
All our theoretical results are illustrated by numerical experiments.
I am a former student of the National School of Computer Science and Applied Mathematics of Grenoble (Ensimag) , IAE Grenoble School of Management and DEA Lamberton. A detailed resume can be found here.
CERMICS, École Nationale des Ponts et Chaussées
6 et 8, av. Blaise Pascal
Cité Descartes - Champs-sur-Marne
77455 Marne-la-Vallée cedex 2
E-mail 1: oumaima[dot]bencheikh[at]enpc[dot]fr
E-mail 2: oumaima[dot]bencheikh[at]hotmail[dot]com