Numerical approximation of SDEs with fractional noise and distributional drift
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910719077777408 |
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| author | Goudenège, Ludovic Haress, El Mehdi Richard, Alexandre |
| author_facet | Goudenège, Ludovic Haress, El Mehdi Richard, Alexandre |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_11455 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Numerical approximation of SDEs with fractional noise and distributional drift Goudenège, Ludovic Haress, El Mehdi Richard, Alexandre Probability Numerical Analysis 60H10, 65C30, 60G22, 60H50, 34A06 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. |
| title | Numerical approximation of SDEs with fractional noise and distributional drift |
| topic | Probability Numerical Analysis 60H10, 65C30, 60G22, 60H50, 34A06 |
| url | https://arxiv.org/abs/2302.11455 |