全耦合随机微分方程的扩散近似

M. Rockner, Longjie Xie
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引用次数: 32

摘要

我们考虑了$\mathbb R^d$中对应于遍历扩散过程的椭圆算子的泊松方程。在系数较温和的条件下,得到了参数的最优正则性和光滑性。然后将结果应用于仅含Holder连续系数的全耦合多时间尺度随机微分方程的一般扩散近似。得到了四种不同的平均方程及其收敛速率。此外,证明了收敛性仅依赖于系数相对于慢变量的规律性,而不依赖于它们相对于快分量的规律性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Diffusion approximation for fully coupled stochastic differential equations
We consider a Poisson equation in $\mathbb R^d$ for the elliptic operator corresponding to an ergodic diffusion process. Optimal regularity and smoothness with respect to the parameter are obtained under mild conditions on the coefficients. The result is then applied to establish a general diffusion approximation for fully coupled multi-time-scales stochastic differential equations with only Holder continuous coefficients. Four different averaged equations as well as rates of convergence are obtained. Moreover, the convergence is shown to rely only on the regularities of the coefficients with respect to the slow variable, and does not depend on their regularities with respect to the fast component.
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