Abstract
Let yɛ(·) denote a random process whose bandwidth, loosely speaking, goes to ∞ as ɛ → 0. Consider the family of differential equations xɛ=g(xɛ,yɛ)+f(xɛ,yɛ)/α(ɛ), where α(ɛ) → 0 as ɛ → 0. The question of interest is: does the sequence {xɛ(·)} converge in some sense and if so which, if any, ordinary or Itô differential equation does it satisfy? Normally, the limit is taken in the sense of weak convergence. The problem is of great practical importance, for such questions arise in many practical situations arising in many fields. Often the limiting equation is nice and can be treated much more easily than can the xɛ(·). In any case, in practice approximations to properties of the xɛ(·) are usually sought in terms of ɛ and some limit. To illustrate these points, as well as a related stability problem, we give a practical example which arises in the theory of adaptive arrays of antennas.
The topic of convergence has seen much work, starting with the fundamental papers of Wong and Zakai, and followed by others, including Khazminskii, Papanicolaou and Kohler, etc. From a non-probabilistic point of view, it has been dealt with by McShane and Sussmann. In this paper, we discuss a rather general and efficient method of getting the correct limits. The idea exploits some general semigroup approximation results of Kurtz, and often not only gets better results than those obtained by preceding methods, but is also easier to use.
This research was supported in part by the Air Force Office of Scientific Research under AF-AFOSR 76-3063, in part by the National Science Foundation under NSF-Eng 77-12946, and in part by the Office of Naval Research under N0014-76-C-0279-P0002.
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References
E. Wong and M. Zakai, "On the relationship between ordinary and stochastic differential equations", Int. J. Engin. Science, (1965), 3, 213–229.
E. Wong and M. Zakai, "On the convergence of ordinary integrals to stochastic integrals", Ann. Math. Statist. 36, 1560–1564.
R. Z. Khasminskii, "A limit theorem for solutions of differential equations with random right hand sides", Theory of Prob. and Applic., (1966), 11, 390–406.
G. C. Papanicolaou and W. Kohler, "Asymptotic theory of mixing stochastic ordinary differential equations", Comm. Pure and Appl. Math., (1974), 27, 641–668.
G. C. Papanicolaou, "Some probabilistic problems and methods in singular perturbations", Rocky Mountain Journal of Math., (1976), 6, 653–674.
G. Blankenship and G. C. Papanicolaou, "Stability and control of stochastic systems with wide-band noise disturbances", SIAM J. on Appl. Math., (1978), 34, 437–476.
H. J. Kushner, "Jump diffusion approximations for ordinary differential equations with random right hand sides", submitted to SIAM J. on Control; see also LCDS Report 78-1, September 1978, Brown University.
T. G. Kurtz, "Semigroups of conditional shifts and approximation of Markov processes", Ann. Prob., (1975), 4, 618–642.
IEEE Tras. on Antennas and Propagation, (1976), AP-24. Special Issue on Adaptive Antenna Arrays.
L. E. Brennan, E. L. Pugh and I. S. Reed, "Control-loop noise in adaptive antenna arrays", IEEE Trans. on Aerospace and Electronic Systems, (1971), AES-7, 254–262.
I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes, (1965), Saunders, Philadelphia.
P. Billingsley, Convergence of Porbability Measures, (1968), John Wiley and Sons, New York.
G. L. Blankenship and G. C. Papanicolaou, "Stability and control of stochastic systems with wide-band noise disturbances", (1978), preprint.
H. J. Kushner, Probability Methods for Approximations for Elliptic Equations and Optimal Stochastic Control Problems, Academic Press, New York, 1977.
E. F. Infante, "On the stability of some linear autonomous random systems", (1968), ASME J. Appl. Mech., 35, 7–12.
F. Kozin and C. M. Wu, "On the stability of linear stochastic differential equations", (1973), ASME J. Appl. Mech., 40, 87–92.
G. Blankenship, "Stability of linear differential equations with random coefficients", (1977), IEEE Trans. on Automatic Control, AC-22, 834–838.
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© 1979 Springer-Verlag
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Kushner, H.J. (1979). Approximation of solutions to differential equations with random inputs by diffusion processes. In: Kohlmann, M., Vogel, W. (eds) Stochastic Control Theory and Stochastic Differential Systems. Lecture Notes in Control and Information Sciences, vol 16. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0009381
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DOI: https://doi.org/10.1007/BFb0009381
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