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Approximation of solutions to differential equations with random inputs by diffusion processes

  • Harold J. Kushner
Part I: Survey Lectures
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 16)

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.

Keywords

Stochastic Differential Equation Weak Convergence Automatic Gain Control Adaptive Array Finite Dimensional Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • Harold J. Kushner
    • 1
  1. 1.Division of Applied MathematicsBrown UniversityProvidence

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