论文标题
随机微分方程的统计数据和分解的近似值
Statistics for stochastic differential equations and approximations of resolvent
论文作者
论文摘要
统计数据的数值评估在统计物理及其应用领域中起着至关重要的作用。可以通过相应的后退kolmogorov方程来评估具有高斯白噪声的随机微分方程的统计数据。重要的通知是，无需在整个域上获得向后的kolmogorov方程的解。在某个点对应于随机微分方程的初始坐标的一定点上，评估溶液的值就足够了。为此，最近已经开发了基于Compinatorics的算法。在本文中，我们讨论了分解的高阶近似，并提出了基于二阶近似的算法。提出的算法显示二阶收敛。此外，天真算法的收敛性自然会导致推外方法。它们可以很好地计算出更准确的价值，而计算成本则更少。通过Ornstein-Uhlenbeck过程和嘈杂的Van der Pol System证明了所提出的方法。
The numerical evaluation of statistics plays a crucial role in statistical physics and its applied fields. It is possible to evaluate the statistics for a stochastic differential equation with Gaussian white noise via the corresponding backward Kolmogorov equation. The important notice is that there is no need to obtain the solution of the backward Kolmogorov equation on the whole domain; it is enough to evaluate a value of the solution at a certain point that corresponds to the initial coordinate for the stochastic differential equation. For this aim, an algorithm based on combinatorics has recently been developed. In this paper, we discuss a higher-order approximation of resolvent, and an algorithm based on a second-order approximation is proposed. The proposed algorithm shows a second-order convergence. Furthermore, the convergence property of the naive algorithms naturally leads to extrapolation methods; they work well to calculate a more accurate value with fewer computational costs. The proposed method is demonstrated with the Ornstein-Uhlenbeck process and the noisy van der Pol system.
