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 study the well-posedness and numerical approximation of multidimensional stochastic differential equations (SDEs) with distributional drift, driven by a fractional Brownian motion. First, we prove weak existence for such SDEs. This holds under a condition that relates the Hurst parameter $H$ of the noise to the Besov regularity of the drift. Then under a stronger condition, we study the error between a solution $X$ of the SDE with drift $b$ and its tamed 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 result covers the critical case of the regime of strong existence and pathwise uniqueness. When the Besov regularity increases and the drift becomes a bounded measurable function, we recover the (almost) optimal rate of convergence $1/2-\varepsilon$. As a byproduct of this convergence, we deduce that pathwise uniqueness holds in a class of H\"older 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: Regularisation by noise; Numerical approximation; Fractional Brownian motion

Domaines: Probabilités [math.PR]

Ludovic Goudenège, El Mehdi Haress, Alexandre Richard. Numerical approximation of SDEs with fractional noise and distributional drift. 2023. ⟨hal-03715427v2⟩