论文标题
通过Euler-Maruyama方案对稳定SDE的不变度度量的近似
Approximation of the invariant measure of stable SDEs by an Euler--Maruyama scheme
论文作者
论文摘要
我们提出了两个Euler-Maruyama（EM）类型的数值方案，以近似于由$α$稳定的Lévy流程驱动的随机微分方程（SDE）的不变度量（$ 1 <α<2 $）：一种与Pareto-distribibibibibibibibibibibibibibibibibibibs的近似方案（近似方案）。使用Duhamel原理的离散版本和Malliavin Colculus中的Bismut公式，我们证明，Wasserstein- $ 1 $距离的错误范围为$η^{1-ε} $和$η^{\frac2α-1} $，分别是$ IS $ IS $ IS $ IS $ IS $ IS $ IS $ IS $ IS $ IS $ is（0,1,1）。方案。对于帕累托驱动的方案，Ornstein-uhlenbeck $α$稳定过程的显式计算表明，价格$η^{\frac2α-1} $无法改善。
We propose two Euler-Maruyama (EM) type numerical schemes in order to approximate the invariant measure of a stochastic differential equation (SDE) driven by an $α$-stable Lévy process ($1<α<2$): an approximation scheme with the $α$-stable distributed noise and a further scheme with Pareto-distributed noise. Using a discrete version of Duhamel's principle and Bismut's formula in Malliavin calculus, we prove that the error bounds in Wasserstein-$1$ distance are in the order of $η^{1-ε}$ and $η^{\frac2α-1}$, respectively, where $ε\in (0,1)$ is arbitrary and $η$ is the step size of the approximation schemes. For the Pareto-driven scheme, an explicit calculation for Ornstein--Uhlenbeck $α$-stable process shows that the rate $η^{\frac2α-1}$ cannot be improved.
