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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