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Session I.7 - Stochastic Computation

Monday, June 12, 17:00 ~ 17:30

Numerical analysis of Euler scheme for SDE driven by fractional Brownian motion

Ludovic Goudenège

CNRS, France   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In a first part, we will present how to use stochastic sewing lemma from [Le20] for building solutions of SDE driven by additive fractional Brownian motion. When the drift is regular or bounded, we can build these solutions as the limit of Euler schemes and obtain a strong rate of convergence, as in [BDG21,DGL21,DAGI19].

Moreover, using roughness of fractional Brownian motion, we can define solutions to SDEs with distributional drift as limit of the previous solutions build with bounded regular drifts. In a second part, we will present numerical simulations of these singular SDEs involving Dirac measure or indicator functions in dimension 1 or 2.

Finally, we will present how to obtain a strong rate of convergence by combining the speed of approximation of the distributional drift and the time-step size in the Euler scheme. However it will force the noise to be ``rough enough'', essentially by adding constraints on the H\"older regularity of the noise [GHR22].


[BDG21] O. Butkovsky, K. Dareiotis, and M. Gerencs\'er. Approximation of SDEs: a stochastic sewing approach. {\em Probab. Theory Related Fields}, 181(4), 975-1034, 2021.

[GHR22] L. Gouden\`ege, E. M. Haress et A. Richard. Numerical approximation of SDEs with fractional noise and distributional drift. {\em hal-03715427v1}, 2022.

[Le20] K. L\^e. A stochastic sewing lemma and applications. {\em Electronic Journal of Probability}, 25, 1-55, 2020.

[DGL21] K. Dareiotis, M. Gerencs\'er and K. L\^e. Quantifying a convergence theorem of Gyongy and Krylov. {\em arXiv preprint arXiv:2101.12185}, 2021.

[DAGI19] T. De Angelis, M. Germain and E. Issoglio. A numerical scheme for stochastic differential equations with distributional drift. {\em arXiv preprint arXiv:1906.11026}, 2019.

Joint work with El Mehdi Haress (MICS - CentraleSupélec - Paris-Saclay University, France) and Alexandre Richard (MICS - CentraleSupélec - Paris-Saclay University, France).

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