Abstract
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.
Similar content being viewed by others
References
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).
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.
J. L. Doob,Stochastic Processes (John Wiley and Sons, New York, 1953).
A. V. Skorokhod,Studies in the Theory of Random Processes (Addison-Wesley, Reading, Mass., 1965).
R. L. Stratonovich, “A new representation for stochastic integrals and equations,”SIAM J. Control 4:362–371 (1966).
A. H. Gray and T. K. Caughey, “A controversy in problems involving random parametric excitation,”J. Math. Phys. 44(3):288–296 (1965).
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.
E. Wong and M. Zakai, “On the relation between ordinary and stochastic differential equations,”Intern. J. Eng. Sci. 3:213–229 (1965).
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.
Author information
Authors and Affiliations
Rights 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). https://doi.org/10.1007/BF01007481
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01007481