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Randomised Euler-Maruyama method for SDEs with Hölder continuous drift coefficient

Jianhai Bao, Yue Wu

TL;DR

This work develops and analyzes a randomised Euler–Maruyama scheme for additive SDEs with time-inhomogeneous drift that is $\alpha$-Hölder in time and $\beta$-Hölder in space. By introducing independent uniform random shifts $\tau_j$ in the time argument, the method gains improved convergence via stochastic sewing, and the authors prove a strong $L^p$-convergence rate of $\tfrac{1}{2}+\gamma-\varepsilon$ where $\gamma=\alpha\wedge(\beta/2)$. The analysis combines two complementary approaches: (i) sharp quadratic bounds for randomised quadrature errors and (ii) a PDE-based route using backward Kolmogorov equations to relate the numerical error to PDE regularity. The rate $\tfrac{1}{2}+\gamma-\varepsilon$ is near-optimal and surpasses the standard EM rate, highlighting the effectiveness of randomness and stochastic sewing in handling time irregularity for SDEs with irregular drift. The paper also contributes methodological advances by integrating stochastic sewing into randomised numerical schemes and offering an alternative PDE-based convergence proof.

Abstract

In this paper, we examine the performance of randomised Euler-Maruyama (EM) method for additive time-inhomogeneous SDEs with an irregular drift. In particular, the drift is assumed to be $α$-Hölder continuous in time and bounded $β$-Hölder continuous in space with $α,β\in (0,1]$. The strong order of convergence of the randomised EM in $L^p$-norm is shown to be $1/2+(α\wedge (β/2))-ε$ for an arbitrary $ε\in (0,1/2)$, higher than the one of standard EM, which is $α\wedge (1/2+β/2-ε)$. The proofs highly rely on the stochastic sewing lemma, where we also provide an alternative proof when handling time irregularity for a comparison.

Randomised Euler-Maruyama method for SDEs with Hölder continuous drift coefficient

TL;DR

This work develops and analyzes a randomised Euler–Maruyama scheme for additive SDEs with time-inhomogeneous drift that is -Hölder in time and -Hölder in space. By introducing independent uniform random shifts in the time argument, the method gains improved convergence via stochastic sewing, and the authors prove a strong -convergence rate of where . The analysis combines two complementary approaches: (i) sharp quadratic bounds for randomised quadrature errors and (ii) a PDE-based route using backward Kolmogorov equations to relate the numerical error to PDE regularity. The rate is near-optimal and surpasses the standard EM rate, highlighting the effectiveness of randomness and stochastic sewing in handling time irregularity for SDEs with irregular drift. The paper also contributes methodological advances by integrating stochastic sewing into randomised numerical schemes and offering an alternative PDE-based convergence proof.

Abstract

In this paper, we examine the performance of randomised Euler-Maruyama (EM) method for additive time-inhomogeneous SDEs with an irregular drift. In particular, the drift is assumed to be -Hölder continuous in time and bounded -Hölder continuous in space with . The strong order of convergence of the randomised EM in -norm is shown to be for an arbitrary , higher than the one of standard EM, which is . The proofs highly rely on the stochastic sewing lemma, where we also provide an alternative proof when handling time irregularity for a comparison.
Paper Structure (10 sections, 12 theorems, 94 equations)

This paper contains 10 sections, 12 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. The randomised Euler-Maruyama method
  3. Our contribution and organisation
  4. Notation and preliminaries
  5. Quadratic estimates
  6. Error analysis via a PDE approach
  7. Some PDE estimate
  8. The strong convergence of randomised EM
  9. Useful estimate from heat kernel Butkovsky2021
  10. An alternative proof of Lemma \ref{['lem:qb3_discrete']}

Key Result

Theorem 2.1

burkholder1966 For each $p \in (1,\infty)$ there exist positive constants $c_p$ and $C_p$ such that for every discrete-time martingale $(Y^n)_{n \in \mathbb{N}_0}$ and for every $n \in \mathbb{N}_0$ we have where $[Y]_n = |Y^{0}|^2 + \sum_{k=1}^{n} |Y^{k}-Y^{k-1}|^2$ is the quadratic variation of $(Y^n)_{n \in \mathbb{N}_0}$.

Theorems & Definitions (22)

  • Theorem 2.1
  • Theorem 2.2: Le2020
  • Lemma 3.1: Well-posedness
  • proof
  • Proposition 3.2: Quadratic bound 1
  • proof
  • Lemma 3.3: Quadratic bound 2
  • proof
  • Proposition 3.4: Quadratic bound 3
  • proof
  • ...and 12 more