Fokker-Planck equation driven by asymmetric Levy motion
Fokker-Planck equation driven by asymmetric Levy motion
复制标题
由非对称 Levy 运动驱动的 Fokker-Planck 方程
DOI：
10.1007/s10444-018-9642-4
复制
发表时间：
2019
影响因子：
1.7
通讯作者：
Huang Yanghong
中科院分区：
文献类型：
--
作者：
Wang Xiao;Shang Wenpeng;Li Xiaofan;Duan Jinqiao;Huang Yanghong
非高斯 Lévy 噪声存在于许多模型中，用于理解物理学、金融、生物学等的基本原理。在这项工作中，我们考虑由一维非对称 Lévy 运动产生的 Fokker-Planck 方程 (FPE)，它是一个非局部偏微分方程。我们提出了非局部 FPE 中奇异积分的精确数值求积，并开发了一种快速求和方法，以将复杂度从 O(J2) 降低到一个时间步长，其中 J 是未知数的数量。我们还提供了数值格式满足极大值原理的条件。我们的数值方法通过与特殊情况的精确解进行比较而得到验证。我们还讨论了概率密度函数的性质以及各种因素对解的影响，包括稳定性指数、偏度参数、漂移项、高斯和非高斯噪声以及域大小。
Non-Gaussian Lévy noises are present in many models for understanding underlining principles of physics, finance, biology, and more. In this work, we consider the Fokker-Planck equation (FPE) due to one-dimensional asymmetric Lévy motion, which is a non-local partial differential equation. We present an accurate numerical quadrature for the singular integrals in the non-local FPE and develop a fast summation method to reduce the order of the complexity fromO(J2) toin one time step, whereJis the number of unknowns. We also provide conditions under which the numerical schemes satisfy maximum principle. Our numerical method is validated by comparing with exact solutions for special cases. We also discuss the properties of the probability density functions and the effects of various factors on the solutions, including the stability index, the skewness parameter, the drift term, the Gaussian and non-Gaussian noises, and the domain size.