scholarly journals Comment on “Fokker-Planck equations for nonlinear dynamical systems driven by non-Gaussian Lévy processes” [J. Math. Phys. 53, 072701 (2012)]

2016 ◽  
Vol 57 (3) ◽  
pp. 034101 ◽  
Author(s):  
Marcin Magdziarz ◽  
Tomasz Zorawik
Keyword(s):  
Dynamical Systems ◽  
Lévy Processes ◽  
Nonlinear Dynamical Systems ◽  
Levy Processes ◽  
Nonlinear Dynamical ◽  
Non Gaussian ◽  
Fokker Planck Equations ◽  
Math Phys ◽  
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.

A sequential importance sampling filter with a new proposal distribution for state and parameter estimation of nonlinear dynamical systems

2007 ◽  
Vol 464 (2089) ◽  
pp. 25-47 ◽  
Author(s):  
Shuva J Ghosh ◽  
C.S Manohar ◽  
D Roy
Keyword(s):  
Dynamical Systems ◽  
Parameter Identification ◽  
Importance Sampling ◽  
Nonlinear Dynamical Systems ◽  
Gaussian White Noise ◽  
Sequential Importance Sampling ◽  
State Variables ◽  
Nonlinear Dynamical ◽  
Taylor Expansions ◽  
Non Gaussian

The problem of estimating parameters of nonlinear dynamical systems based on incomplete noisy measurements is considered within the framework of Bayesian filtering using Monte Carlo simulations. The measurement noise and unmodelled dynamics are represented through additive and/or multiplicative Gaussian white noise processes. Truncated Ito–Taylor expansions are used to discretize these equations leading to discrete maps containing a set of multiple stochastic integrals. These integrals, in general, constitute a set of non-Gaussian random variables. The system parameters to be determined are declared as additional state variables. The parameter identification problem is solved through a new sequential importance sampling filter. This involves Ito–Taylor expansions of nonlinear terms in the measurement equation and the development of an ideal proposal density function while accounting for the non-Gaussian terms appearing in the governing equations. Numerical illustrations on parameter identification of a few nonlinear oscillators and a geometrically nonlinear Euler–Bernoulli beam reveal a remarkably improved performance of the proposed methods over one of the best known algorithms, i.e. the unscented particle filter.

Sign in / Sign up

Export Citation Format

Share Document