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

Matteo Cagnotti

TL;DR

This work tackles numerical approximation of one-dimensional SDEs with drift in a distributional Besov space, driven by Brownian motion. By combining a Zvonkin-type transformation with mollification and the Yamada–Watanabe approximation, it reduces the problem to solving a regularized SDE and analyzes the error via a stochastic Grönwall framework, yielding $L^p$-sup convergence for $p\ge2$ with explicit rates and an $L^1$-sup convergence rate. The results hold under $b \in C_T^{1/2}\mathcal{C}^{(-\beta)+}$ with $\beta\in(0,1/2)$ and provide precise rate formulas involving $r(\beta,\varepsilon)$, improving understanding of numerical schemes for SDEs with highly singular drifts. The methods integrate virtual solutions, mollified Euler schemes, and local-time techniques, offering a robust approach to strong convergence in Besov-drift settings with potential implications for rough forcing models and singular interactions.

Abstract

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.

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

TL;DR

This work tackles numerical approximation of one-dimensional SDEs with drift in a distributional Besov space, driven by Brownian motion. By combining a Zvonkin-type transformation with mollification and the Yamada–Watanabe approximation, it reduces the problem to solving a regularized SDE and analyzes the error via a stochastic Grönwall framework, yielding -sup convergence for with explicit rates and an -sup convergence rate. The results hold under with and provide precise rate formulas involving , improving understanding of numerical schemes for SDEs with highly singular drifts. The methods integrate virtual solutions, mollified Euler schemes, and local-time techniques, offering a robust approach to strong convergence in Besov-drift settings with potential implications for rough forcing models and singular interactions.

Abstract

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 , for all , 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 - norm.
Paper Structure (9 sections, 15 theorems, 84 equations)

This paper contains 9 sections, 15 theorems, 84 equations.

Table of Contents

  1. Introduction and Main Results
  2. Framework and Preliminaries
  3. Notation, Spaces, and Other Useful Estimates
  4. Virtual solutions
  5. Auxiliary Estimates
  6. Proof of Theorem \ref{['rate:Lp']}: $L^p$-sup Convergence
  7. A Yamada-Watanabe Approximation Based Proof
  8. An Equivalent Local Time-Based Approach
  9. Proof of Theorem \ref{['rate:L1']}: $L^1$-sup Convergence

Key Result

Theorem 1.1

Let Assumption hypothesis:b hold. Then, for any choice of $p\geq 2$ and $\varepsilon\in(0,\frac{1}{2} - \beta)$ there exists a constant $C>0$ such that where $r(\beta,\varepsilon)$ was defined in eq:rate.

Theorems & Definitions (23)

  • Theorem 1.1: $L^p$-$\sup$ Convergence
  • Remark 1.2
  • Theorem 1.3: $L^1$-$\sup$ Convergence
  • Remark 1.4
  • Lemma 2.1: Bernstein inequality
  • Lemma 2.2: Schauder estimates
  • Lemma 2.3: gyongy_note_2011
  • Lemma 2.4: de_angelis_numerical_2022
  • Lemma 2.5: chaparro_jaquez_convergence_2025
  • Lemma 2.6: chaparro_jaquez_convergence_2025
  • ...and 13 more