Path integral solutions for n-dimensional stochastic differential equations under α-stable Lévy excitation

IF 3.2 3区工程技术 Q2 MECHANICS Theoretical and Applied Mechanics Letters Pub Date : 2023-03-01 DOI:10.1016/j.taml.2023.100430

Wanrong Zan , Yong Xu , Jürgen Kurths

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引用次数: 2

Abstract

In this paper, the path integral solutions for a general n-dimensional stochastic differential equations (SDEs) with $α$ -stable Lévy noise are derived and verified. Firstly, the governing equations for the solutions of n-dimensional SDEs under the excitation of $α$ -stable Lévy noise are obtained through the characteristic function of stochastic processes. Then, the short-time transition probability density function of the path integral solution is derived based on the Chapman-Kolmogorov-Smoluchowski (CKS) equation and the characteristic function, and its correctness is demonstrated by proving that it satisfies the governing equation of the solution of the SDE, which is also called the Fokker-Planck-Kolmogorov equation. Besides, illustrative examples are numerically considered for highlighting the feasibility of the proposed path integral method, and the pertinent Monte Carlo solution is also calculated to show its correctness and effectiveness.

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α-稳定l激励下n维随机微分方程的路径积分解

本文推导并验证了具有α-稳定Lévy噪声的一般n维随机微分方程的路径积分解。首先，通过随机过程的特征函数，得到了在α-稳定Lévy噪声激励下n维SDE解的控制方程。然后，基于Chapman-Kolmogorov-Smoluchowski（CKS）方程和特征函数，导出了路径积分解的短时转移概率密度函数，并通过证明它满足SDE解的控制方程，即Fokker-Planck-Kolmokorov方程，证明了它的正确性。此外，为了突出所提出的路径积分方法的可行性，还对相应的蒙特卡罗解进行了数值计算，以表明其正确性和有效性。

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来源期刊

Theoretical and Applied Mechanics Letters MECHANICS-

CiteScore

6.20

自引率

2.90%

发文量

545

审稿时长

12 weeks

期刊介绍： An international journal devoted to rapid communications on novel and original research in the field of mechanics. TAML aims at publishing novel, cutting edge researches in theoretical, computational, and experimental mechanics. The journal provides fast publication of letter-sized articles and invited reviews within 3 months. We emphasize highlighting advances in science, engineering, and technology with originality and rapidity. Contributions include, but are not limited to, a variety of topics such as: • Aerospace and Aeronautical Engineering • Coastal and Ocean Engineering • Environment and Energy Engineering • Material and Structure Engineering • Biomedical Engineering • Mechanical and Transportation Engineering • Civil and Hydraulic Engineering Theoretical and Applied Mechanics Letters (TAML) was launched in 2011 and sponsored by Institute of Mechanics, Chinese Academy of Sciences (IMCAS) and The Chinese Society of Theoretical and Applied Mechanics (CSTAM). It is the official publication the Beijing International Center for Theoretical and Applied Mechanics (BICTAM).