Numerical analysis and applications of Fokker-Planck equations for stochastic dynamical systems with multiplicative α-stable noises

2020 ◽  
Vol 87 ◽  
pp. 711-730
Author(s):  
Yanjie Zhang ◽  
Xiao Wang ◽  
Qiao Huang ◽  
Jinqiao Duan ◽  
Tingting Li
Keyword(s):  
Numerical Analysis ◽  
Dynamical Systems ◽  
Stochastic Dynamical Systems ◽  
Fokker Planck Equations ◽  
Fokker Planck

Governing equations for probability densities of Marcus stochastic differential equations with Lévy noise

2016 ◽  
Vol 17 (05) ◽  
pp. 1750033 ◽  
Author(s):  
Xu Sun ◽  
Xiaofan Li ◽  
Yayun Zheng
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Probability Density ◽  
Lévy Processes ◽  
Planck Equation ◽  
Levy Processes ◽  
Stochastic Dynamical Systems ◽  
Non Gaussian ◽  
Fokker Planck Equations ◽  
Fokker Planck

Marcus stochastic differential equations (SDEs) often are appropriate models for stochastic dynamical systems driven by non-Gaussian Lévy processes and have wide applications in engineering and physical sciences. The probability density of the solution to an SDE offers complete statistical information on the underlying stochastic process. Explicit formula for the Fokker–Planck equation, the governing equation for the probability density, is well-known when the SDE is driven by a Brownian motion. In this paper, we address the open question of finding the Fokker–Planck equations for Marcus SDEs in arbitrary dimensions driven by non-Gaussian Lévy processes. The equations are given in a simple form that facilitates theoretical analysis and numerical computation. Several examples are presented to illustrate how the theoretical results can be applied to obtain Fokker–Planck equations for Marcus SDEs driven by Lévy processes.

THE NUMERICAL SOLUTION OF NONLINEAR STOCHASTIC DYNAMICAL SYSTEMS: A BRIEF INTRODUCTION

1991 ◽  
Vol 01 (02) ◽  
pp. 277-286 ◽  
Author(s):  
P. E. KLOEDEN ◽  
E. PLATEN ◽  
H. SCHURZ
Keyword(s):  
Numerical Analysis ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Numerical Solution ◽  
Stochastic Differential Equations ◽  
Taylor Expansion ◽  
Rapid Development ◽  
Stochastic Calculus ◽  
Stochastic Dynamical Systems ◽  
The Subject

The numerical analysis of stochastic differential equations, currently undergoing rapid development, differs significantly from its deterministic counterpart due to the peculiarities of stochastic calculus. This article presents a brief, pedagogical introduction to the subject from the perspective of stochastic dynamical systems. The key tool is the stochastic Taylor expansion. Strong, pathwise approximations are distinguished from weak, functional approximations, and their role in stability with Lyapunov exponents and stiffness is discussed.

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