Convergence of Numerical Time-Averaging and Stationary Measures via Poisson Equations
Convergence of Numerical Time-Averaging and Stationary Measures via Poisson Equations
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DOI：
10.1137/090770527
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发表时间：
2009-08
期刊：
影响因子：
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通讯作者：
Jonathan C. Mattingly;A. Stuart;M. Tretyakov
中科院分区：
文献类型：
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作者：
Jonathan C. Mattingly;A. Stuart;M. Tretyakov
考虑随机微分方程 (SDE) 长期行为的数值近似。获得时间平均估计器的误差估计，然后用它来表明数值方法的平稳行为收敛于 SDE 的平稳行为。误差分析基于使用底层 SDE 的相关泊松方程。这种方法的主要优点是其简单性和通用性。它同样适用于一系列显式和隐式方案，包括那些具有随机变量简单模拟的方案以及亚椭圆 SDE。为了简化说明，我们只考虑SDE的状态空间是环面的情况，并且只研究平滑测试函数。然而，我们预计该方法可以得到更广泛的应用。我们的方法与斯坦因的方法之间存在类比。讨论了结果的一些实际意义。
Numerical approximation of the long time behavior of a stochastic differential equation (SDE) is considered. Error estimates for time-averaging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the SDE. The error analysis is based on using an associated Poisson equation for the underlying SDE. The main advantages of this approach are its simplicity and universality. It works equally well for a range of explicit and implicit schemes, including those with simple simulation of random variables, and for hypoelliptic SDEs. To simplify the exposition, we consider only the case where the state space of the SDE is a torus, and we study only smooth test functions. However, we anticipate that the approach can be applied more widely. An analogy between our approach and Stein's method is indicated. Some practical implications of the results are discussed.