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Strong rate of convergence of the Euler scheme for SDEs with irregular drift driven by Levy noise

Oleg Butkovsky, Konstantinos Dareiotis, Máté Gerencsér

TL;DR

This work studies the strong convergence of the Euler scheme for SDEs of the form $dX_t=b(X_t)\,dt+dL_t$ with irregular Hölder drift and Lévy noise of index $\alpha\in(0,2]$. By developing a novel extension of stochastic sewing together with Le’s quantitative John–Nirenberg inequality, the authors obtain strong $L_p$ and almost sure convergence rates that are independent of $p$ over the full admissible range $\alpha\in[2/3,2]$ and $\beta>1-\alpha/2$, improving prior results both in the $\alpha$-range and the moment dependence. The main rate is $\|X-X^n\|_{\mathcal{C}^0([0,1]);L_p} \lesssim n^{-(\tfrac{1}{2}+\tfrac{\beta}{\alpha}\wedge \tfrac{1}{2})+\varepsilon}$ (plus mesh-size in the initial condition), and almost sure convergence follows with an explicit rate. The paper also yields strong $L_p$ bounds for approximations of nonsmooth additive functionals of Lévy processes and demonstrates the approach on a broad class of Lévy processes satisfying natural semigroup and generator conditions, thereby widening the applicability of Euler-type schemes under irregular noise.

Abstract

We study the strong rate of convergence of the Euler--Maruyama scheme for a multidimensional stochastic differential equation (SDE) $$ dX_t = b(X_t) \, dt + dL_t, $$ with irregular $β$-Hölder drift, $β> 0$, driven by a Lévy process with exponent $α\in (0, 2]$. For $α\in [2/3, 2]$, we obtain strong $L_p$ and almost sure convergence rates in the entire range $β> 1 - α/2$, where the SDE is known to be strongly well-posed. This significantly improves the current state of the art, both in terms of convergence rate and the range of $α$. Notably, the obtained convergence rate does not depend on $p$, which is a novelty even in the case of smooth drifts. As a corollary of the obtained moment-independent error rate, we show that the Euler--Maruyama scheme for such SDEs converges almost surely and obtain an explicit convergence rate. Additionally, as a byproduct of our results, we derive strong $L_p$ convergence rates for approximations of nonsmooth additive functionals of a Lévy process. Our technique is based on a new extension of stochastic sewing arguments and Lê's quantitative John-Nirenberg inequality.

Strong rate of convergence of the Euler scheme for SDEs with irregular drift driven by Levy noise

TL;DR

This work studies the strong convergence of the Euler scheme for SDEs of the form with irregular Hölder drift and Lévy noise of index . By developing a novel extension of stochastic sewing together with Le’s quantitative John–Nirenberg inequality, the authors obtain strong and almost sure convergence rates that are independent of over the full admissible range and , improving prior results both in the -range and the moment dependence. The main rate is (plus mesh-size in the initial condition), and almost sure convergence follows with an explicit rate. The paper also yields strong bounds for approximations of nonsmooth additive functionals of Lévy processes and demonstrates the approach on a broad class of Lévy processes satisfying natural semigroup and generator conditions, thereby widening the applicability of Euler-type schemes under irregular noise.

Abstract

We study the strong rate of convergence of the Euler--Maruyama scheme for a multidimensional stochastic differential equation (SDE) with irregular -Hölder drift, , driven by a Lévy process with exponent . For , we obtain strong and almost sure convergence rates in the entire range , where the SDE is known to be strongly well-posed. This significantly improves the current state of the art, both in terms of convergence rate and the range of . Notably, the obtained convergence rate does not depend on , which is a novelty even in the case of smooth drifts. As a corollary of the obtained moment-independent error rate, we show that the Euler--Maruyama scheme for such SDEs converges almost surely and obtain an explicit convergence rate. Additionally, as a byproduct of our results, we derive strong convergence rates for approximations of nonsmooth additive functionals of a Lévy process. Our technique is based on a new extension of stochastic sewing arguments and Lê's quantitative John-Nirenberg inequality.
Paper Structure (14 sections, 19 theorems, 227 equations, 1 figure)

This paper contains 14 sections, 19 theorems, 227 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Examples of Lévy processes satisfying the main assumptions
  4. Preliminaries
  5. Conditional shifted stochastic sewing lemma
  6. Weighted John-Nirenberg inequality
  7. Heat kernel and related bounds
  8. Proof of the main results
  9. A priori bounds
  10. General bounds
  11. Bounds on $\mathcal{E}^{f,n,1}$, $\mathcal{E}^{f,n,2}$, $\mathcal{E}^{f,n,3}$
  12. Main proofs
  13. Proofs of the well-posedness of SDE \ref{['eq:main-SDE']}
  14. Proofs of the auxiliary statements of the article

Key Result

Theorem 2.1

Suppose that $L$ satisfies A:1--A:3. Let $\eta$ be a $\mathcal{F}_0$--measurable random vector taking values in $\mathbb{R}^d$. Suppose additionally that and let $b\in \mathcal{C}^\beta(\mathbb{R}^d,\mathbb{R}^d)$. Then equation eq:main-SDE with the initial condition $X_0=\eta$ has a unique strong solution. Furthermore, this solution $X$ is the limit of the Picard iterations, namely

Figures (1)

  • Figure 1: Convergence rates. is the required lower bound for $\beta$; is the classical condition \ref{['eq:exponents-WP']}; shading indicates rate of $L_p$-convergence from $1/2$ () to $1$ ().

Theorems & Definitions (57)

  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • Remark 2.7
  • Proposition 2.8
  • Proposition 2.9: Schmoments
  • Example 2.10: General non-degenerate $\alpha$--stable process, $\alpha\in(0,2)$
  • ...and 47 more