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

Abstract

Let y^ɛ(·) denote a random process whose bandwidth, loosely speaking, goes to ∞ as ɛ → 0. Consider the family of differential equations x^ɛ=g(x^ɛ,y^ɛ)+f(x^ɛ,y^ɛ)/α(ɛ), where α(ɛ) → 0 as ɛ → 0. The question of interest is: does the sequence {x^ɛ(·)} 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 x^ɛ(·). In any case, in practice approximations to properties of the x^ɛ(·) 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.