Hostname: page-component-5c6d5d7d68-tdptf Total loading time: 0 Render date: 2024-09-01T06:10:38.761Z Has data issue: false hasContentIssue false
  • English
  • Français

The stationary moments of Poisson-driven non-linear dynamical systems

Published online by Cambridge University Press:  14 July 2016

Charles E. Smith  and
Loren Cobb
Charles E. Smith*
Affiliation:
Medical University of South Carolina
Loren Cobb*
Affiliation:
Medical University of South Carolina
*
∗Postal address: Department of Biometry, College of Medicine, Medical University of South Carolina, Charleston, SC 29425, U.S.A.
∗Postal address: Department of Biometry, College of Medicine, Medical University of South Carolina, Charleston, SC 29425, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

Moment recursion relations have previously been derived for the stationary probability density functions of continuous-time stochastic systems with Wiener (white noise) input. These results are extended in this paper to the case of Poisson (shot noise) input. The non-linear dynamical systems are expressed, in general, as stochastic differential equations, with an independent increment input. The transition probability density function evolves according to the appropriate Kolmogorov equation. Moments of the stationary density are obtained from the Fourier transform of the stationary density. The moment relations can be used to estimate the parameters of linear and non-linear stochastic systems from empirical moments, given either Wiener or Poisson input.

Keywords

MOMENT RECURSION RELATIONSNON-LINEAR STOCHASTIC DIFFERENTIAL EQUATIONSFILTERED POINT PROCESSES
Type
Short Communications
Information
Journal of Applied Probability , Volume 19 , Issue 3 , September 1982 , pp. 702 - 706
Copyright
Copyright © Applied Probability Trust 1982 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This research was supported by the National Science Foundation, Grant #80–11451, and the South Carolina State Appropriation for Biomedical Research.

References

Cobb, L. (1978) Stochastic catastrophe models and multimodal distributions. Behavioral Sci. 23, 360–374.Google Scholar
Cobb, L., Koppstein, P. and Chen, N. H. (1982) Estimation and moment recursion relations for multimodal distributions of the exponential family. J. Amer. Statist. Assoc. To appear.Google Scholar
Gihman, I. I. and Skorohod, A. V. (1972) Stochastic Differential Equations. Springer-Verlag, New York.Google Scholar
Hanson, F. B. and Tuckwell, H. C. (1978) Persistence times of populations with large random fluctuations. Theoret. Popn Biol. 14, 46–61.Google Scholar
Hirsch, M. W. and Smale, S. (1974) Differential Equations, Dynamic Systems, and Linear Algebra. Academic Press, New York.Google Scholar
Mcgarty, T. P. (1974) Stochastic Systems and State Estimation. Wiley, New York.Google Scholar
Ricciardi, L. M. (1977) Diffusion Processes and Related Topics in Biology. Springer-Verlag, New York.Google Scholar
Snyder, D. L. (1975) Random Point Processes. Wiley, New York.Google Scholar