In this paper, we consider the following stochastic differential equation for (Xt)t⩾0 on ℝd and its Euler-Maruyama (EM) approximation \((Y_{t_{n}})_{n\in \mathbb{Z}^{+}}\) : \(\eqalign{& d{X_t} = b({X_t})dt + \sigma ({X_t})d{B_t}, \cr & {Y_{{t_{n + 1}}}} = {Y_{{t_n}}} + {\eta _{n + 1}}b({Y_{{t_n}}}) + \sigma ({Y_{{t_n}}})({B_{{t_{n + 1}}}} - {B_{{t_n}}}),}\) where b: ℝd ↦ ℝd, σ: ℝd ↦ ℝd×d are measurable, Bt is the d-dimensional Brownian motion, t0:= 0, and \({t_n}: = \sum\nolimits_{k = 1}^n {{\eta _k}}\) for constants ηk > 0 satisfying limk→∞ηk = 0 and \(\sum\nolimits_{k = 1}^\infty {\;{\eta _k}} = \infty\) . We investigate the convergence rates of \(Y_{t_{n}}\) under both additive and multiplicative noise settings for different smoothness levels of b. When the noise is additive and partial dissipation conditions hold, we obtain explicit convergence rates of \(\mathbb{W}_{p}(ℒ(Y_{{t_n}}), ℒ(X_{{t_n}}))+\mathbb{W}_{p}(ℒ(Y_{{t_n}}),\mu)\rightarrow 0\) as n → ∞, where \(\mathbb{W}_{p}\) is the Lp-Wasserstein distance for p ∈ [0, 1], ℒ(ξ) denotes the distribution of ξ, and μ is the unique invariant probability measure of (Xt)t⩾0. When the noise is multiplicative and global dissipation conditions hold, the convergence rate of \(\mathbb{W}_{p}(ℒ)(Y_{{t_n}}), ℒ(X_{{t_n}}))\) for p ⩾ 2 is studied. Compared with the existing results where b is usually C1 or C2 smooth, our estimates apply to Hölder continuous drift and clearly demonstrate the dependence of the convergence rate on the smoothness of b.
