An Euler–Maruyama method for Caputo–Hadamard fractional stochastic differential equations on exponential meshes and its fast approximation

IF 1.7 3区 数学 Q2 MATHEMATICS, APPLIED
Min Li, Shangjiang Guo, Peng Hu, Haiyan Song
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Abstract

This paper studies the numerical solutions of Caputo–Hadamard fractional stochastic differential equations. Firstly, we construct an Euler–Maruyama (EM) scheme for the equations, and the corresponding convergence rate is investigated. Secondly, we propose a fast EM scheme based on the sum-of-exponentials approximation to decrease the computational cost of the EM scheme. More concretely, the fast EM scheme reduces the computational cost from \(O(N^2)\) to \(O(N\log ^2 N)\) and the storage from O(N) to \(O(\log ^2 N)\) when the final time \(T\approx e\), where N is the total number of time steps. Moreover, considering the statistical errors from Monte Carlo path approximations, multilevel Monte Carlo (MLMC) techniques are utilized to reduce computational complexity. In particular, for a prescribed accuracy \(\varepsilon >0\), the EM scheme and the fast EM scheme, integrated with the MLMC technique, respectively reduce the standard EM scheme’s computational complexity from \(O(\varepsilon ^{-2-\frac{2}{\widetilde{\alpha }}})\) to \(O(\varepsilon ^{-\frac{2}{\widetilde{\alpha }}})\) and the fast EM scheme’s complexity to \(O(\varepsilon ^{-\frac{1}{\widetilde{\alpha }}}\left|\log \varepsilon \right|^3)\), where \(0<\widetilde{\alpha }=\alpha -\frac{1}{2}<\frac{1}{2}\). Finally, numerical examples are included to verify the theoretical results and demonstrate the performance of our methods.

Abstract Image

指数网格上的卡普托-哈达玛德分数随机微分方程的欧拉-丸山方法及其快速近似值
本文研究 Caputo-Hadamard 分数随机微分方程的数值解法。首先,我们构建了方程的 Euler-Maruyama (EM) 方案,并研究了相应的收敛速率。其次,我们提出了一种基于指数和近似的快速 EM 方案,以降低 EM 方案的计算成本。更具体地说,当最终时间为(T/approx e/),其中N为总时间步数时,快速EM方案将计算成本从(O(N^2)\)降低到(O(N/log ^2 N)\),存储成本从(O(N))降低到(O(\log ^2 N)\)。此外,考虑到蒙特卡罗路径近似的统计误差,多级蒙特卡罗(MLMC)技术被用来降低计算复杂度。特别是,对于规定精度 \(\varepsilon >;0),EM方案和快速EM方案与MLMC技术相结合、分别将标准EM方案的计算复杂度从(O(\varepsilon ^{-2-\frac{2}{widetilde{\alpha }})降低到(O(\varepsilon ^{-\frac{2}{widetilde{\alpha }})。\而快速 EM 方案的复杂度为 O(^{-\frac{1}{widetilde\alpha }}left|\log \varepsilon \right|^3)\)、其中 \(0<;\widetilde{alpha }=\alpha -\frac{1}{2}<\frac{1}{2}\).最后,我们还列举了一些数值示例来验证理论结果,并展示我们的方法的性能。
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来源期刊
Numerical Algorithms
Numerical Algorithms 数学-应用数学
CiteScore
4.00
自引率
9.50%
发文量
201
审稿时长
9 months
期刊介绍: The journal Numerical Algorithms is devoted to numerical algorithms. It publishes original and review papers on all the aspects of numerical algorithms: new algorithms, theoretical results, implementation, numerical stability, complexity, parallel computing, subroutines, and applications. Papers on computer algebra related to obtaining numerical results will also be considered. It is intended to publish only high quality papers containing material not published elsewhere.
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