$L^p$-sup Convergence of the Euler-Maruyama Scheme for SDEs with Distributional Besov Drift

In this paper we extend existing results on the numerical approximation of one-dimensional SDEs with drift in a negative order Besov space and driven by Brownian motion. Using the Yamada-Watanabe approximation technique, we prove rates in $L^p$, for all $p\geq 2$, applying a Gronwall-type lemma previously used in the literature for SDEs with Hölder continuous coefficients. Additionally, we obtain an explicit convergence rate in the $L^1$-$\sup$ norm.

💡 Research Summary

The paper investigates strong convergence of the Euler–Maruyama discretisation for one‑dimensional stochastic differential equations (SDEs) whose drift belongs to a negative‑order Besov space (C^{-\beta+}) with (\beta\in(0,\tfrac12)). Classical theory fails for such singular drifts because the integral (\int_0^t b(s,X_s),ds) is not defined in the usual sense. The authors therefore employ a Zvonkin‑type transformation: they solve the backward Kolmogorov PDE \