Let $(X_t)_{t \ge 0}$ be the solution of the stochastic differential equation $$dX_t = b(X_t) dt+A dZ_t, \quad X_{0}=x,$$ where $b: \mathbb{R}^d \rightarrow \mathbb R^d$ is a Lipschitz function, $A \in \mathbb R^{d \times d}$ is a positive definite matrix, $(Z_t)_{t\geq 0}$ is a $d$-dimensional rotationally invariant $\alpha$-stable L\'evy process with $\alpha \in (1,2)$ and $x\in\mathbb{R}^{d}$. We use two Euler-Maruyama schemes with decreasing step sizes $\Gamma = (\gamma_n)_{n\in \mathbb{N}}$ to approximate the invariant measure of $(X_t)_{t \ge 0}$: one with i.i.d. $\alpha$-stable distributed random variables as its innovations and the other with i.i.d. Pareto distributed random variables as its innovations. We study the convergence rate of these two approximation schemes in the Wasserstein-1 distance. For the first scheme, when the function $b$ is Lipschitz and satisfies a certain dissipation condition, we show that the convergence rate is $\gamma^{1/\alpha}_n$. Under an additional assumption on the second order directional derivatives of $b$, this convergence rate can be improved to $\gamma^{1+\frac 1 {\alpha}-\frac{1}{\kappa}}_n$ for any $\kappa \in [1,\alpha)$. For the second scheme, when the function $b$ is twice continuously differentiable, we obtain a convergence rate of $\gamma^{\frac{2-\alpha}{\alpha}}_n$. We show that the rate $\gamma^{\frac{2-\alpha}{\alpha}}_n$ is optimal for the one dimensional stable Ornstein-Uhlenbeck process. Our theorems indicate that the recent remarkable result about the unadjusted Langevin algorithm with additive innovations can be extended to the SDEs driven by an $\alpha$-stable L\'evy process and the corresponding convergence rate has a similar behaviour. Compared with the previous result, we have relaxed the second order differentiability condition to the Lipschitz condition for the first scheme.

翻译：设$(X_t)_{t \ge 0}$为随机微分方程$$dX_t = b(X_t) dt+A dZ_t, \quad X_{0}=x$$的解，其中$b: \mathbb{R}^d \rightarrow \mathbb R^d$为Lipschitz函数，$A \in \mathbb R^{d \times d}$为正定矩阵，$(Z_t)_{t\geq 0}$为$d$维旋转不变$\alpha$-稳定Lévy过程（$\alpha \in (1,2)$），且$x\in\mathbb{R}^{d}$。本文采用两种递减步长$\Gamma = (\gamma_n)_{n\in \mathbb{N}}$的Euler-Maruyama格式逼近$(X_t)_{t \ge 0}$的不变测度：一种以独立同分布的$\alpha$-稳定随机变量作为增量，另一种以独立同分布的Pareto分布随机变量作为增量。我们研究了这两种逼近格式在Wasserstein-1距离下的收敛速率。对于第一种格式，当函数$b$满足Lipschitz条件及特定的耗散条件时，证明收敛速率为$\gamma^{1/\alpha}_n$；若进一步假设$b$具有二阶方向导数，则该速率可改进为对任意$\kappa \in [1,\alpha)$的$\gamma^{1+\frac 1 {\alpha}-\frac{1}{\kappa}}_n$。对于第二种格式，当$b$二阶连续可微时，我们获得收敛速率$\gamma^{\frac{2-\alpha}{\alpha}}_n$，并证明该速率在一维稳定Ornstein-Uhlenbeck过程中是最优的。定理表明：近期关于加性增量未调整Langevin算法的显著结果可推广至$\alpha$-稳定Lévy过程驱动的随机微分方程，且相应收敛速率呈现类似特性。与以往结果相比，我们已将第一种格式的二阶可微性条件放松为Lipschitz条件。