The Milstein scheme for singular SDEs with Hölder continuous drift

TL;DR

The paper proves that the Milstein scheme attains a strong convergence rate of $\frac{1+\alpha}{2}$ for SDEs with Hölder continuous drift $b\in\mathcal{C}^\alpha$ and elliptic diffusion $\sigma\sigma^*$, under $\sigma\in\mathcal{C}^3$. The authors develop sharp density estimates for Milstein-type processes via Malliavin calculus, employ stochastic sewing to bound additive functionals, and use a Zvonkin transformation (solving a PDE with large $\theta$) to regularise the drift, with a Girsanov transform bridging driftless and drifted dynamics. The main result follows from a detailed error decomposition that combines these tools, yielding an $L^p$-bound of the form $\|\sup_{t\in[0,1]}|X_t-X_t^n|\|_{L^p} \le N|x_0-x_0^n|+N n^{-(1+\alpha)/2+\varepsilon}$. This advances the understanding of higher-order numerical schemes for SDEs with irregular coefficients and highlights density-estimation techniques as a key ingredient in strong approximation analysis.

Abstract

We study the $L^p$ rate of convergence of the Milstein scheme for SDEs when the drift coefficients possess only Hölder regularity. If the diffusion is elliptic and sufficiently regular, we obtain rates consistent with the additive case. The proof relies on regularisation by noise techniques, particularly stochastic sewing, which in turn requires (at least asymptotically) sharp estimates on the law of the Milstein scheme, which may be of independent interest.

Theorems & Definitions (29)

• Theorem 1.2
• Remark 1.3
• Theorem 2.1
• Theorem 2.2
• proof
• Remark 2.3
• Proposition 2.4
• Remark 2.5
• proof
• Proposition 2.6