发表机构
Jiangsu Normal University; Wuhan University（江苏师范大学; 武汉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对状态依赖快速切换的随机微分方程，提出三种基于异构多尺度方法的数值算法，证明其强L^p收敛性，并通过数值实验验证了计算优势。
AI 中文摘要
本文旨在为一类具有状态依赖快速切换过程的随机微分方程开发高效的数值逼近方法。当缩放参数较小时，直接使用Euler--Maruyama (EM) 格式会失效。基于\\(\cite{e2005analysis}\\)的异构多尺度方法，我们提出了三种算法，并证明了它们对任意\\(p\geq 2\\)具有显式收敛阶的强\\(L^p\\)收敛性。在第一种算法中，我们将平均原理与针对平均方程的EM格式相结合，其中马尔可夫链的不变测度可以通过求解一个线性系统显式获得。然而，随着切换状态数量的增加，计算该不变测度会产生三次方的成本。为避免求解大型线性系统，我们转而逼近不变测度。因此，第二种和第三种算法都将针对修正平均方程的宏观EM格式与估计平均漂移的微观求解器相结合。更具体地说，在第二种算法中，使用微观时间步长模拟离散时间马尔可夫链，并通过在有限次微观转移上取平均来获得平均漂移。然而，第一种和第二种算法仅在切换过程具有有限状态时有效。因此，我们引入了第三种算法，该算法允许切换过程具有可数无限状态，其中通过Gillespie算法生成精确的连续时间马尔可夫链，并通过在指定区间上进行精确时间平均来计算平均漂移。数值实验验证了理论结果，并展示了这三种算法的计算优势。
英文摘要
This paper aims to develop efficient numerical approximations for a class of stochastic differential equations with state-dependent fast switching processes. The direct Euler--Maruyama (EM) scheme fails when the scaling parameter is small. Based on the heterogeneous multiscale method of \cite{e2005analysis}, we propose three algorithms and prove their strong $L^p$-convergence with explicit rates for any $p\geq 2$. In the first algorithm, we combine the averaging principle with an EM scheme for the averaged equation, where the invariant measure of the Markov chain can be explicitly obtained by solving a linear system. However, computing this invariant measure incurs cubic cost as the number of switching states increases. To avoid solving large linear systems, we approximate the invariant measure instead. Thus the second and third algorithms both combine a macroscopic EM scheme for a modified averaged equation with micro-solvers that estimate the averaged drift. More precisely, in the second algorithm, a discrete-time Markov chain is simulated with a micro time step, and the averaged drift is obtained by averaging over finitely many micro transitions. However, both the first and second algorithms only work when the switching process has finite states. Therefore, we introduce a third algorithm that allows for switching processes with countably infinite states, in which exact continuous-time Markov chain is generated via the Gillespie algorithm, and the averaged drift is computed by exact time averaging over a specified interval. Numerical experiments verify the theoretical results and demonstrate the computational advantages of these three algorithms.
Comments31 pages, 11 figures
