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Lectures on stochastic sewing with applications

Oleg Butkovsky

TL;DR

The notes develop and extend the stochastic sewing lemma (SSL) to analyze stochastic perturbations across SDEs driven by Brownian, fractional Brownian, and Lévy noises, yielding Krylov-type bounds for integrals with rough drifts. A central advancement is the shifted stochastic sewing lemma and a generalized coupling framework, which together enable strong well-posedness for Hölder and distributional drifts and weak well-posedness via ergodic-type arguments. The work further provides sharp convergence rates for Euler schemes in the irregular-drift regime and extends the toolkit to Sobolev drifts with taming, leveraging the John–Nirenberg inequality to handle higher moments under Lévy noise. Overall, these methods broaden the applicability of SSL in stochastic analysis and numerical analysis, offering robust tools for SDEs and related stochastic systems with highly irregular inputs.

Abstract

These are lecture notes for a mini-course on stochastic sewing, taught at the University of Edinburgh and Beijing Institute of Technology in Spring/Summer 2025. The aim is to introduce the reader to stochastic sewing techniques and to show how they can be successfully applied to study various problems in stochastic analysis, including: regularization by noise for stochastic differential equations driven by Brownian motion or fractional Brownian motion, well-posedness of stochastic PDEs with irregular drift, the study of averaging operators and local times, and the analysis of numerical algorithms.

Lectures on stochastic sewing with applications

TL;DR

The notes develop and extend the stochastic sewing lemma (SSL) to analyze stochastic perturbations across SDEs driven by Brownian, fractional Brownian, and Lévy noises, yielding Krylov-type bounds for integrals with rough drifts. A central advancement is the shifted stochastic sewing lemma and a generalized coupling framework, which together enable strong well-posedness for Hölder and distributional drifts and weak well-posedness via ergodic-type arguments. The work further provides sharp convergence rates for Euler schemes in the irregular-drift regime and extends the toolkit to Sobolev drifts with taming, leveraging the John–Nirenberg inequality to handle higher moments under Lévy noise. Overall, these methods broaden the applicability of SSL in stochastic analysis and numerical analysis, offering robust tools for SDEs and related stochastic systems with highly irregular inputs.

Abstract

These are lecture notes for a mini-course on stochastic sewing, taught at the University of Edinburgh and Beijing Institute of Technology in Spring/Summer 2025. The aim is to introduce the reader to stochastic sewing techniques and to show how they can be successfully applied to study various problems in stochastic analysis, including: regularization by noise for stochastic differential equations driven by Brownian motion or fractional Brownian motion, well-posedness of stochastic PDEs with irregular drift, the study of averaging operators and local times, and the analysis of numerical algorithms.
Paper Structure (26 sections, 44 theorems, 381 equations, 1 figure)

This paper contains 26 sections, 44 theorems, 381 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Example 1: Regularization by noise for SDEs
  3. Example 2: Numerical methods
  4. Sewing and stochastic sewing
  5. Exercises
  6. Regularization by noise for SDEs driven by Brownian motion
  7. Step 1: Bounding integral functionals of a Brownian motion
  8. Step 2: Bounding integrals along a perturbed Brownian path
  9. Proof of strong uniqueness
  10. Exercises
  11. Weak well-posedness of SDEs driven by Brownian motion
  12. SDEs with distributional drifts
  13. Challenges and proof strategy
  14. Key integral bounds
  15. Weak existence
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $d\in \mathbb{N}$, $x\in\mathbb{R}^d$, $b\colon\mathbb{R}^d\to\mathbb{R}^d$ be a bounded measurable function. Then SDE mainSDE has a unique strong solution.

Figures (1)

  • Figure 5.1: Time points for bounding $\mathsf{E} ^{s-(t-s)}\delta A_{s,u,t}$

Theorems & Definitions (100)

  • Theorem 1.1: ver80zvonkin74
  • Theorem 1.2: bib:zz17FIR17
  • Theorem 1.3: Pr12chen2017wellbib:csz15
  • Theorem 1.4: BDG
  • Theorem 1.5: BDGLevy
  • Theorem 2.1: Sewing lemma, PressFGubi
  • Remark 2.2
  • proof : Proof of \ref{['t:SL']}
  • Theorem 2.3: Stochastic sewing lemma, LeSSL
  • Remark 2.4
  • ...and 90 more