Skip to main content

Mathematical problems of modeling stochastic nonlinear dynamic systems


The purpose of this report is to introduce the engineer to the area of stochastic differential equations, and to point out the mathematical techniques and pitfalls in this area. Topics discussed include continuous-time Markov processes, the Fokker-Planck-Kolmogorov equations, the Ito and Stratonovich stochastic calculi, and the problem of modeling physical systems.

This is a preview of subscription content, access via your institution.


  1. M. C. Wang and G. E. Uhlenbeck, “On the theory of the Brownian motion II”Rev. Mod. Phys. 17:323–342 (1945); reproduced in the book,Selected Papers on Noise and Stochastic Processes, N. Wax, ed. (Dover, New York, 1954).

    Google Scholar 

  2. J. F. Barrett, “Application of Kolmogorov's equations to randomly disturbed automatic control systems,” in:Automatic and Remote Control (Proceedings of the First International Congress of IFAC) (Butterworths, London, 1961), pp. 724–733.

    Google Scholar 

  3. J. L. Doob,Stochastic Processes (John Wiley and Sons, New York, 1953).

    Google Scholar 

  4. A. V. Skorokhod,Studies in the Theory of Random Processes (Addison-Wesley, Reading, Mass., 1965).

    Google Scholar 

  5. R. L. Stratonovich, “A new representation for stochastic integrals and equations,”SIAM J. Control 4:362–371 (1966).

    Google Scholar 

  6. A. H. Gray and T. K. Caughey, “A controversy in problems involving random parametric excitation,”J. Math. Phys. 44(3):288–296 (1965).

    Google Scholar 

  7. S. R. McReynolds, “A new approach to stochastic calculus,” paper presented at the Seminar on Guidance Theory and Trajectory Analysis, NASA Electronics Research Center, May 1967.

  8. E. Wong and M. Zakai, “On the relation between ordinary and stochastic differential equations,”Intern. J. Eng. Sci. 3:213–229 (1965).

    Google Scholar 

  9. T. Kailath, “The Ito stochastic integral,” talk presented to the Los Angeles chapter of the IEEE Professional Group on Information Theory, University of Southern California, February 1968.

Download references

Author information

Authors and Affiliations


Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mortensen, R.E. Mathematical problems of modeling stochastic nonlinear dynamic systems. J Stat Phys 1, 271–296 (1969).

Download citation

  • Received:

  • Issue Date:

  • DOI:

Key words

  • stochastic differential equations
  • continuous-time Markov processes
  • Fokker-Planck-Kolmogorov equations
  • Ito and Stratonovich stochastic calculi
  • modeling physical systems