Article Dans Une Revue Stochastic Processes and their Applications Année : 2025

Numerical approximation of SDEs with fractional noise and distributional drift

Résumé

We study the numerical approximation of SDEs with singular drifts (including distributions) driven by a fractional Brownian motion. Under the Catellier-Gubinelli condition that imposes the regularity of the drift to be strictly greater than $1-1/(2H)$, we obtain an explicit rate of convergence of a tamed Euler scheme towards the SDE, extending results for bounded drifts. Beyond this regime, when the regularity of the drift is $1-1/(2H)$, we derive a non-explicit rate. As a byproduct, strong well-posedness for these equations is recovered. Proofs use new regularising properties of discrete-time fBm and a new critical Gr\"onwall-type lemma. We present examples and simulations.
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Dates et versions

hal-03715427 , version 1 (06-07-2022)
hal-03715427 , version 2 (23-02-2023)
hal-03715427 , version 3 (30-01-2025)

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Ludovic Goudenège, El Mehdi Haress, Alexandre Richard. Numerical approximation of SDEs with fractional noise and distributional drift. Stochastic Processes and their Applications, 2025, 181, ⟨10.1016/j.spa.2024.104533⟩. ⟨hal-03715427v3⟩
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