The Euler-Maruyama method for S(F)DEs with Hölder drift and α-stable noise

Approximation of the ergodic measure of SDEs with singular drift by Euler-Maruyama scheme

“…If the drift b is locally Lipschitz, Mattingly et al [16] obtained the convergence rate of its EM approximation for invariant measure under a certain distance. [13,24] studied the strong convergence of EM scheme in a finite time interval when b is Höler continuous, in which they used the Zvonkin's transform.…”

A robust and efficient component-wise WENO scheme for Euler equations

“…When the driven noise is an α-stable process with α ∈ (0, 2), and the drift coefficient is only β-Hölder continuous with β > 1 − α/2, the strong well-posedness of (1.1) has been studied in [TTW74], [Pri12], [Pri15], [CSZ18], and [CZZ21]. However, only for α ∈ [1, 2), the rate of strong convergence for the Euler-Maruyama approximation of SDE (1.1) has been studied in [MPT17], [MX18], [HL18], and [KS19]. Notably, all of the convergence rates obtained in these works depend on the regularity of the drift coefficient, and they become increasingly worse as β approaches 1 − α/2.…”

New regularity of kolmogorov equation and application on approximation of semi-linear spdes with Hölder continuous drifts

“…There are many results on this topic, see [4,7,8,9,10,13,20,28,29,30,31,32], and references therein. Encouraged by this idea, some researchers have adopted Zvonkin's transform to study the strong convergence rate of the Euler Maruyama (EM) method for SDEs with singular drift, for instance, [14,24,25,26] and the SDEs with Jumps [17]. However, so far, there are no results on the numerical method for the semi-linear SPDEs with singular drift.…”

“…Thus it is very hard to obtain (17). We will leave the multiplicative noise case in the future research.…”