Numerical approximation of SDEs with fractional noise and distributional drift

Ludovic Goudenège (1) , El Mehdi Haress (2) , Alexandre Richard (1, 2)

1 Fédération de Mathématiques de CentraleSupélec
2 MICS - Mathématiques et Informatique pour la Complexité et les Systèmes

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.

Mots clés: Numerical approximation; regularisation by noise; fractional Brownian motion

Domaines: Probabilités [math.PR]

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