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Numerical approximation of SDEs with fractional noise and distributional drift

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  • Goudenège, Ludovic
  • Haress, El Mehdi
  • Richard, Alexandre

Abstract

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 Grönwall-type lemma. We present examples and simulations.

Suggested Citation

  • Goudenège, Ludovic & Haress, El Mehdi & Richard, Alexandre, 2025. "Numerical approximation of SDEs with fractional noise and distributional drift," Stochastic Processes and their Applications, Elsevier, vol. 181(C).
    • Handle: RePEc:eee:spapps:v:181:y:2025:i:c:s0304414924002412
    DOI: 10.1016/j.spa.2024.104533
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    References listed on IDEAS

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    1. Andreas Neuenkirch & Ivan Nourdin, 2007. "Exact Rate of Convergence of Some Approximation Schemes Associated to SDEs Driven by a Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 20(4), pages 871-899, December.
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    8. Catellier, R. & Gubinelli, M., 2016. "Averaging along irregular curves and regularisation of ODEs," Stochastic Processes and their Applications, Elsevier, vol. 126(8), pages 2323-2366.
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