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Strong solutions and sharp Euler--Maruyama approximations for SDEs with Lebesgue--Dini drift

Jinlong Wei, Junhao Hu, Guangying Lv, Chenggui Yuan

TL;DR

This work addresses strong approximation for SDEs with drift in $L^2([0,1];\mathcal{D}_b^\rho)$, i.e., square-integrable in time and Dini continuous in space, while the diffusion is non-constant and uniformly elliptic. The authors deploy a refined It\ô--Tanaka trick and parabolic regularity to prove strong well-posedness and the stochastic flow property, and, under a Lipschitz condition on the diffusion matrix, establish a sharp strong convergence rate for a polygonal Euler--Maruyama scheme of $n^{-1/2}$ up to a $(\log n)^{3/2}$ factor. They also demonstrate that the rate $1/2$ is optimal, even for smooth diffusion with vanishing drift, underscoring the sharpness of the results within the Lebesgue--Dini drift framework. These contributions provide the first sharp quantitative strong convergence estimates in this borderline setting, linking regularity theory for degenerate/drifting SDEs with precise numerical error bounds and stochastic-flow behavior.

Abstract

We investigate the strong approximation of stochastic differential equations whose drift is square-integrable in time and Dini continuous in space, while the diffusion coefficient is non-constant and uniformly elliptic. Using a refined Itô--Tanaka trick combined with parabolic regularity estimates, we first establish strong well-posedness and the stochastic flow property. Under additional Lipschitz regularity of the diffusion matrix, we then analyze a polygonal-type Euler--Maruyama scheme and prove the strong error estimate \[ \Big\|\sup_{0\le t\le1}|X_t-X_t^n|\Big\|_{L^p(Ω)} \le C n^{-\frac12}\log(n)^{\frac32}, \quad p\ge2. \] We further show that this rate is sharp: even under smooth and uniformly elliptic diffusion coefficients with vanishing drift, the convergence order $1/2$ cannot be improved. These results provide the first sharp quantitative strong convergence estimates in a Lebesgue--Dini drift framework.

Strong solutions and sharp Euler--Maruyama approximations for SDEs with Lebesgue--Dini drift

TL;DR

This work addresses strong approximation for SDEs with drift in , i.e., square-integrable in time and Dini continuous in space, while the diffusion is non-constant and uniformly elliptic. The authors deploy a refined It\ô--Tanaka trick and parabolic regularity to prove strong well-posedness and the stochastic flow property, and, under a Lipschitz condition on the diffusion matrix, establish a sharp strong convergence rate for a polygonal Euler--Maruyama scheme of up to a factor. They also demonstrate that the rate is optimal, even for smooth diffusion with vanishing drift, underscoring the sharpness of the results within the Lebesgue--Dini drift framework. These contributions provide the first sharp quantitative strong convergence estimates in this borderline setting, linking regularity theory for degenerate/drifting SDEs with precise numerical error bounds and stochastic-flow behavior.

Abstract

We investigate the strong approximation of stochastic differential equations whose drift is square-integrable in time and Dini continuous in space, while the diffusion coefficient is non-constant and uniformly elliptic. Using a refined Itô--Tanaka trick combined with parabolic regularity estimates, we first establish strong well-posedness and the stochastic flow property. Under additional Lipschitz regularity of the diffusion matrix, we then analyze a polygonal-type Euler--Maruyama scheme and prove the strong error estimate We further show that this rate is sharp: even under smooth and uniformly elliptic diffusion coefficients with vanishing drift, the convergence order cannot be improved. These results provide the first sharp quantitative strong convergence estimates in a Lebesgue--Dini drift framework.
Paper Structure (9 sections, 8 theorems, 173 equations, 3 figures, 2 tables)

This paper contains 9 sections, 8 theorems, 173 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Setup and notation
  3. Main results
  4. Analytic and stochastic preliminaries
  5. Proof of Theorem \ref{['the1.2']}
  6. Proof of Theorem \ref{['the1.4']}
  7. Numerical experiments
  8. Example A: one-dimensional Lebesgue--Dini drift
  9. Example B: two-dimensional coupled drift and diffusion

Key Result

Theorem 1.2

Let $b\in L^2([0,1];\mathcal{D}_b^\rho(\mathbb{R}^d;\mathbb{R}^d))$ such that $\rho$ is slowly varying at zero and $\rho^{1/2}$ is a Dini function. Suppose that $\sigma\in L^2([0,1];W^{1,\infty}(\mathbb{R}^d;\mathbb{R}^{d\times d}))$ and that $a(t,x)=\sigma(t,x)\sigma^{T}(t,x)$ satisfies: $\bullet$ $\bullet$ There exists $\alpha\in(0,1]$ such that Then there is a unique stochastic flow of homeom

Figures (3)

  • Figure 1: Spatial profiles of $g_K(x)=\sum_{k=1}^{K} a_k\varphi(2^k x)$ for different truncation levels $K$.
  • Figure 2: Example A. Log--log plots of strong errors versus $n$ (endpoint and pathwise supremum).
  • Figure 3: Example B. Log--log plot of strong errors versus $n$ (endpoint and supremum).

Theorems & Definitions (13)

  • Definition 1.1: Kun90, p. 114
  • Theorem 1.2
  • Example 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • ...and 3 more