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Quantifying a convergence theorem of Gyöngy and Krylov

Konstantinos Dareiotis, Máté Gerencsér, Khoa Lê

Abstract

We derive sharp strong convergence rates for the Euler-Maruyama scheme approximating multidimensional SDEs with multiplicative noise without imposing any regularity condition on the drift coefficient. In case the noise is additive, we show that Sobolev regularity can be leveraged to obtain improved rate: drifts with regularity of order $α\in (0,1)$ lead to rate $(1+α)/2$.

Quantifying a convergence theorem of Gyöngy and Krylov

Abstract

We derive sharp strong convergence rates for the Euler-Maruyama scheme approximating multidimensional SDEs with multiplicative noise without imposing any regularity condition on the drift coefficient. In case the noise is additive, we show that Sobolev regularity can be leveraged to obtain improved rate: drifts with regularity of order lead to rate .

Paper Structure

This paper contains 16 sections, 22 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Literature
  3. On the proof
  4. Main results
  5. Localising irregularities
  6. Preliminaries
  7. Notation
  8. Heat kernel bounds
  9. Sewing
  10. PDE estimates
  11. A Gronwall-type lemma
  12. Integral estimates
  13. Multiplicative noise
  14. Additive noise
  15. An intermediate stability estimate
  16. ...and 1 more sections

Key Result

Theorem 1.2

Given asn:multiplicative-basic, let $\varepsilon\in(0,1)$, $p\in(0,\infty)$. Then for all $n\in \mathbb{N}$ the following bound holds with some constant $N$ depending only on $d,p,\varepsilon,\lambda$ and $\sup(|b|+|\sigma|+|\nabla\sigma|+|\nabla^2\sigma|)$.

Theorems & Definitions (45)

  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Corollary 1.8
  • proof
  • Remark 1.9
  • Lemma 2.1
  • proof
  • ...and 35 more