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Numerical approximation of SDEs with fractional noise and distributional drift

Abstract : We prove weak existence for multi-dimensional SDEs with distributional drift driven by a fractional Brownian motion. This holds under a condition that relates the Besov regularity of the drift to the Hurst parameter $H$ of the noise. Then under a stronger condition, we study the numerical error between a solution $X$ of the SDE with drift $b$ and its Euler scheme with mollified drift $b^n$. We obtain a rate of convergence in $L^m(\Omega)$ for this error, which depends on the Besov regularity of the drift. This rate holds for any Hurst parameter smaller than the critical value imposed by the ``strong'' condition. Close to the critical value, the rate is $H-\varepsilon$. When the Besov regularity increases and the drift becomes a bounded measurable function, we recover the optimal rate of convergence $1/2-\varepsilon$. As a byproduct of this convergence, we deduce that pathwise uniqueness holds in a class of regular Hölder continuous solutions and that any such solution is strong. The proofs rely on stochastic sewing techniques, especially to deduce new regularising properties of the discrete-time fractional Brownian motion. We also present several examples and numerical simulations that illustrate our results.
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https://hal-centralesupelec.archives-ouvertes.fr/hal-03715427
Contributor : Alexandre Richard Connect in order to contact the contributor
Submitted on : Wednesday, July 6, 2022 - 1:53:55 PM
Last modification on : Saturday, July 9, 2022 - 3:32:47 AM

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  • HAL Id : hal-03715427, version 1

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Ludovic Goudenège, El Mehdi Haress, Alexandre Richard. Numerical approximation of SDEs with fractional noise and distributional drift. 2022. ⟨hal-03715427⟩

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