Weak error analysis of time and particle discretization of nonlinear stochastic differential equations in the McKean sense.

Summary This thesis is devoted to the theoretical and numerical study of the weak error of time and particle discretization of nonlinear Stochastic Differential Equations in the McKean sense. In the first part, we analyze the weak convergence speed of the time discretization of standard SDEs. More specifically, we study the convergence in total variation of the Euler-Maruyama scheme applied to d-dimensional DEs with a measurable drift coefficient and additive noise. We obtain, assuming that the drift coefficient is bounded, a weak order of convergence 1/2. By adding more regularity on the drift, namely a spatial divergence in the sense of L[rho]-space-uniform distributions in time for some [rho] greater than or equal to d, we reach a convergence order equal to 1 (within a logarithmic factor) at terminal time. In dimension 1, this result is preserved when the spatial derivative of the drift is a measure in space with a total mass bounded uniformly in time. In the second part of the thesis, we analyze the weak discretization error in both time and particles of two classes of nonlinear DHSs in the McKean sense. The first class consists of multi-dimensional SDEs with regular drift and diffusion coefficients in which the law dependence occurs through moments. The second class consists of one-dimensional SDEs with a constant diffusion coefficient and a singular drift coefficient where the law dependence occurs through the distribution function. We approximate the EDS by the Euler-Maruyama schemes of the associated particle systems and we obtain for both classes a weak order of convergence equal to 1 in time and in particles. In the second class, we also prove a result of chaos propagation of optimal order 1/2 in particles and a strong order of convergence equal to 1 in time and 1/2 in particles. All our theoretical results are illustrated by numerical simulations.