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## Numerical approximation of SDEs with fractional noise and distributional drift

Ludovic Goudenège
El Mehdi Haress
• Function : Author
• PersonId : 1147316
Alexandre Richard

#### 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.

#### Domains

Mathematics [math] Probability [math.PR]

### Dates and versions

hal-03715427 , version 1 (06-07-2022)
hal-03715427 , version 2 (23-02-2023)

### Identifiers

• HAL Id : hal-03715427 , version 1

### Cite

Ludovic Goudenège, El Mehdi Haress, Alexandre Richard. Numerical approximation of SDEs with fractional noise and distributional drift. 2022. ⟨hal-03715427v1⟩

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