Strong rate of convergence for the Euler--Maruyama scheme of SDEs with unbounded Hölder continuous drift coefficient
Tsukasa Moritoki, Dai Taguchi
TL;DR
The paper addresses the problem of obtaining a strong convergence rate for the Euler–Maruyama scheme applied to multi‑dimensional SDEs with a drift $b$ that is uniformly locally Hölder continuous and unbounded, in the presence of multiplicative noise. It combines the Itô--Tanaka (Zvonkin) transformation to remove the Hölder drift with stochastic sewing lemma techniques and heat‑kernel density estimates to control the driftless EM scheme, enabling a rigorous $L^p$ error analysis via a Girsanov change of measure. The main contribution is a sharp strong convergence rate: for any $p\ge1$ and $\varepsilon \in (0,(1+\alpha)/2)$, $\sup_{t\in[0,T]} \mathbb{E}|X_t - X_t^n|^p]^{1/p}$ scales as $n^{-1/2}$ when $\sigma$ is nonconstant and smooth ($$\sigma \in C_b^{3}$$, uniformly elliptic), and as $n^{-(1+\alpha)/2+\varepsilon}$ when $\sigma = I_d$, thereby extending results from bounded drift to unbounded Hölder drift with sub‑linear growth. This work delivers near‑optimal convergence rates for a broad class of SDEs and provides a robust methodological framework for stochastic numerical analysis with unbounded coefficients.
Abstract
In this paper, we provide the strong rate of convergence for the Euler--Maruyama scheme for multi-dimensional stochastic differential equations with uniformly locally (unbounded) Hölder continuous drift and multiplicative noise. Our technique is based on Itô--Tanaka trick (Zvonkin transformation) for unbounded drift. Moreover, in order to apply the stochastic sewing lemma, we use the heat kernel estimate for the density function of the Euler--Maruyama scheme.