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Analyse de l'erreur faible de discrétisation en temps et en particules d'équations différentielles stochastiques non linéaires au sens de McKean

Abstract : This thesis is dedicated to the theoretical and numerical study of the weak error for time and particle discretizations of some Stochastic Differential Equations non linear in the sense of McKean. In the first part, 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 d-dimensional SDEs with additive noise and a measurable drift coefficient. We prove weak convergence with order 1/2 when assuming boundedness on the drift coefficient. By adding more regularity to the drift, namely the drift has a spatial divergence in the sense of distributions with [rho]-th power integrable with respect to the Lebesgue measure in space uniformly in time for some [rho] superior or egal to d, the order of convergence at the terminal time improves to 1 up to some logarithmic factor. In dimension d=1, this result is preserved when the spatial derivative of the drift is a measure in space with total mass bounded uniformly in time. 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 and obtain for both classes a weak order of convergence equal to 1 in time and particles. We also prove, for the second class, a trajectorial propagation of chaos result with optimal order 1/2 in particles as well as a strong order of convergence equal to 1 in time and 1/2 in particles. All our theoretical results are illustrated by numerical experiments.
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Oumaima Bencheikh. Analyse de l'erreur faible de discrétisation en temps et en particules d'équations différentielles stochastiques non linéaires au sens de McKean. Analyse numérique [math.NA]. Université Paris-Est, 2020. Français. ⟨NNT : 2020PESC1030⟩. ⟨tel-03129074⟩

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