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Sharp lower error bounds for strong approximation of SDEs with piecewise Lipschitz continuous drift coefficient

Simon Ellinger

Abstract

We study pathwise approximation of strong solutions of scalar stochastic differential equations (SDEs) at a single time in the presence of discontinuities of the drift coefficient. Recently, it has been shown by Müller-Gronbach and Yaroslavtseva (2022) that for all $p \in [1, \infty)$ a transformed Milstein-type scheme reaches an $L^p$-error rate of at least $3 / 4$ when the drift coefficient is a piecewise Lipschitz-continuous function with a piecewise Lipschitz-continuous derivative and the diffusion coefficient is constant. It has been proven by Müller-Gronbach and Yaroslavtseva (2023) that this rate $3 / 4$ is optimal if one additionally assumes that the drift coefficient is bounded, increasing and has a point of discontinuity. While boundedness and monotonicity of the drift coefficient are crucial for the proof of the matching lower bound of Müller-Gronbach and Yaroslavtseva (2023), we show that both conditions can be dropped. For the proof we apply a transformation technique which was so far only used to obtain upper bounds.

Sharp lower error bounds for strong approximation of SDEs with piecewise Lipschitz continuous drift coefficient

Abstract

We study pathwise approximation of strong solutions of scalar stochastic differential equations (SDEs) at a single time in the presence of discontinuities of the drift coefficient. Recently, it has been shown by Müller-Gronbach and Yaroslavtseva (2022) that for all a transformed Milstein-type scheme reaches an -error rate of at least when the drift coefficient is a piecewise Lipschitz-continuous function with a piecewise Lipschitz-continuous derivative and the diffusion coefficient is constant. It has been proven by Müller-Gronbach and Yaroslavtseva (2023) that this rate is optimal if one additionally assumes that the drift coefficient is bounded, increasing and has a point of discontinuity. While boundedness and monotonicity of the drift coefficient are crucial for the proof of the matching lower bound of Müller-Gronbach and Yaroslavtseva (2023), we show that both conditions can be dropped. For the proof we apply a transformation technique which was so far only used to obtain upper bounds.
Paper Structure (9 sections, 19 theorems, 167 equations)

This paper contains 9 sections, 19 theorems, 167 equations.

Table of Contents

  1. INTRODUCTION
  2. PROOF OF THE MAIN RESULT
  3. Idea of the proof
  4. The transformation and further tools
  5. General results for $(\mu 1)$-functions and SDEs
  6. On the transformation
  7. On the coupling of noise
  8. Proof of Theorem \ref{['theorem:lowerBound:SDEs:discDrift']}
  9. Proof of Corollary \ref{['cor:GeneralLowerBound']}

Key Result

Theorem 1

Let $\mu \colon \mathbb{R} \rightarrow \mathbb{R}$ satisfy $(\mu 1), (\mu 2)$ and $(\mu 3)$. Let $x_0 \in \mathbb{R}$ and let $X: [0,1] \times \Omega \rightarrow \mathbb{R}$ be a strong solution of the SDE on the time interval $[0,1]$ with initial value $x_0$ and driving Brownian motion $W$. Then there exists a constant $c > 0$ such that for all $n \in \mathbb{N}$,

Theorems & Definitions (32)

  • Theorem 1
  • Corollary 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • Lemma 5
  • ...and 22 more