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# Convergence in total variation of the Euler-Maruyama scheme applied to diffusion processes with measurable drift coefficient and additive noise

2 MATHRISK - Mathematical Risk Handling
Inria de Paris, ENPC - École des Ponts ParisTech, UPEM - Université Paris-Est Marne-la-Vallée
Abstract : We are interested in the Euler-Maruyama discretization of a stochastic differential equation in dimension $d$ with constant diffusion coefficient and bounded measurable drift coefficient. In the scheme, a randomization of the time variable is used to get rid of any regularity assumption of the drift in this variable. We prove weak convergence with order $1/2$ in total variation distance. When 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 $\rho \ge d$, 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. We confirm our theoretical analysis by numerical experiments.
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Pré-publication, Document de travail
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https://hal-enpc.archives-ouvertes.fr/hal-02613774
Contributeur : Oumaima Bencheikh Connectez-vous pour contacter le contributeur
Soumis le : mercredi 20 mai 2020 - 13:01:41
Dernière modification le : vendredi 4 février 2022 - 03:13:41

### Identifiants

• HAL Id : hal-02613774, version 1
• ARXIV : 2005.09354

### Citation

Oumaima Bencheikh, Benjamin Jourdain. Convergence in total variation of the Euler-Maruyama scheme applied to diffusion processes with measurable drift coefficient and additive noise. 2020. ⟨hal-02613774⟩

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