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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 apositive definite matrix, $(Z_t)_{t\geq 0}$ is a $d$-dimensional rotationallyinvariant $\alpha$-stable L\'evy process with $\alpha \in (1,2)$ and$x\in\mathbb{R}^{d}$. We use two Euler-Maruyama schemes with decreasing stepsizes $\Gamma = (\gamma_n)_{n\in \mathbb{N}}$ to approximate the invariantmeasure of $(X_t)_{t \ge 0}$: one with i.i.d. $\alpha$-stable distributedrandom variables as its innovations and the other with i.i.d. Paretodistributed random variables as its innovations. We study the convergence rateof these two approximation schemes in the Wasserstein-1 distance. For the firstscheme, when the function $b$ is Lipschitz and satisfies a certain dissipationcondition, we show that the convergence rate is $\gamma^{1/\alpha}_n$. Under anadditional assumption on the second order directional derivatives of $b$, thisconvergence rate can be improved to $\gamma^{1+\frac 1{\alpha}-\frac{1}{\kappa}}_n$ for any $\kappa \in [1,\alpha)$. For the secondscheme, when the function $b$ is twice continuously differentiable, we obtain aconvergence 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 stableOrnstein-Uhlenbeck process. Our theorems indicate that the recent remarkableresult about the unadjusted Langevin algorithm with additive innovations can beextended to the SDEs driven by an $\alpha$-stable L\'evy process and thecorresponding convergence rate has a similar behaviour. Compared with theprevious result, we have relaxed the second order differentiability conditionto the Lipschitz condition for the first scheme.

Approximation of the invariant measure for stable SDE by the Euler-Maruyama scheme with decreasing step-sizes