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

Abstract

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.