Research Topic

The PhD was 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 the 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 the 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.

Publications & Preprints

• O.Bencheikh, B. Jourdain. Bias behaviour and antithetic sampling in mean-field particle approximations of SDEs nonlinear in the sense of McKean ESAIM: PROCEEDINGS AND SURVEYS, Vol.65, p. 219-235,2019.
• O. Bencheikh, B. Jourdain. Weak and strong error analysis for mean-field rank based particle approximations of one dimensional viscous scalar conservation laws Preprint arXiv:1910.11237, 2019.
• O. Bencheikh, B. Jourdain. Convergence in total variation of the Euler-Maruyama scheme applied to diffusion processes with measurable drift coefficient and additive noise Preprint arXiv:2005.09354, 2020.
• O. Bencheikh, B. Jourdain. Approximation rate in Wasserstein distance of probability measures on the real line by deterministic empirical measures Preprint arXiv:2012.09729, 2020.

Conferences and Talks

• Groupe de Travail Méthodes Stochastiques et Finance - CERMICS, 05/11/20
• Young Researchers Seminar - CERMICS, 19/10/20
• Séminaire de Probabilités de Rennes - IRMAR, Rennes, France, 03/02/20
• Advances in Financial Mathematics, Paris, France, 14/01/20-17/01/20
• 12th European Summer School in Financial Mathematics, Padova, Italy, 02/09/19-06/09/19
• Nonlinear Processes and their Applications, Saint-Etienne, France, 02/07/19-05/07/19
• 9th General AMaMeF Conference, Paris, France, 11/06/19-14/06/19
• Perturbation Techniques in Stochastic Analysis and Its Applications (CIRM), Marseille, France, 11/03/19-15/03/19
• Semaine d'études Maths-Entreprises (SEME), Orsay, France, 14/01/19-18/01/19
• 11th European Summer School in Financial Mathematics, Paris, France, 27/08/18-31/08/18
• Groupe de Travail Méthodes Stochastiques et Finance - CERMICS, 14/06/18
• Journée des Doctorants, Université Paris-Est, 04/06/18
• Young Researchers Seminar - CERMICS, 22/05/18

Teaching

• Numerical statistics and data analysis, Second year of the Engineering program at École des Ponts ParisTech (Lectures & Tutorials)
• Decision in Uncertainty, First year of the Engineering program at École des Ponts ParisTech (Tutorials)

Curriculum

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.

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