具有不同尺度的分数阶随机微分方程的均质化原理与数值分析
Homogenization principle and numerical analysis for fractional stochastic differential equations with different scales
AI总结：
本文研究了不同尺度的分数阶随机微分方程，在非Lipschitz条件下证明了解的存在唯一性，基于时间均质化思想在均方意义下证明了均质化原理，并构造Euler-Maruyama格式分析数值误差，展示了均质化系统的计算优势。
摘要：
本文研究具有不同尺度的分数阶随机微分方程。我们在非Lipschitz条件下，建立了Caputo分数阶随机微分系统解的存在性与唯一性。基于时间均质化的思想，我们在一种新的均质化假设下，证明了均质化原理（平均化原理）在均方（$L^2$范数）收敛意义下成立。此外，我们构造了非自治系统的Euler-Maruyama格式，并分析了其数值误差。最后，给出了两个数值算例以验证理论结果。与现有文献不同，我们从数值角度展示了均质化自治系统的计算优势。
英文摘要：
This work is concerned with fractional stochastic differential equations with different scales. We establish the existence and uniqueness of solutions for Caputo fractional stochastic differential systems under the non-Lipschitz condition. Based on the idea of temporal homogenization, we prove that the homogenization principle (averaging principle) holds in the sense of mean square ($L^2$ norm) convergence under a novel homogenization assumption. Furthermore, an Euler-Maruyama scheme for the non-autonomous system is constructed and its numerical error is analyzed. Finally, two numerical examples are presented to verify the theoretical results. Different from the existing literature, we demonstrate the computational advantages of the homogenized autonomous system from a numerical perspective.
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