Approximation of the invariant measure for stable stochastic differential equations by the Euler–Maruyama scheme with decreasing step sizes

Abstract

Let $(X_t)_{t \geqslant 0}$ be the solution of the stochastic differential equation \[\mathrm{d} X_t = b(X_t) \,\mathrm{d} t+A\,\mathrm{d} Z_t, \quad X_{0}=x,\] where $b\,:\, \mathbb{R}^d \rightarrow \mathbb{R}^d$ is a Lipschitz-continuous function, $A \in \mathbb{R}^{d \times d}$ is a positive-definite matrix, $(Z_t)_{t\geqslant 0}$ is a d -dimensional rotationally symmetric $\alpha$ -stable Lévy 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 $(X_t)_{t \geqslant 0}$ : one uses independent and identically distributed $\alpha$ -stable random variables as innovations, and the other employs independent and identically distributed Pareto random variables. We study the convergence rates of these two approximation schemes in the Wasserstein-1 distance. For the first scheme, under the assumption that the function b is Lipschitz and satisfies a certain dissipation condition, we demonstrate a convergence rate of $\gamma^{\frac{1}{\alpha}}_n$ . This convergence rate can be improved to $\gamma^{1+\frac {1}{\alpha}-\frac{1}{\kappa}}_n$ for any $\kappa \in [1,\alpha)$ , provided b has the additional regularity of bounded second-order directional derivatives. For the second scheme, where the function b is assumed to be twice continuously differentiable, we establish a convergence rate of $\gamma^{\frac{2-\alpha}{\alpha}}_n$ ; moreover, we show that this rate is optimal for the one-dimensional stable Ornstein–Uhlenbeck process. Our theorems indicate that the recent significant result of [34] concerning the unadjusted Langevin algorithm with additive innovations can be extended to stochastic differential equations driven by an $\alpha$ -stable Lévy process and that the corresponding convergence rate exhibits similar behaviour. Compared with the result in [6], our assumptions have relaxed the second-order differentiability condition, requiring only a Lipschitz condition for the first scheme, which broadens the applicability of our approach.