The book deals with several closely related topics concerning
approxima tions and perturbations of random processes and their
applications to some important and fascinating classes of problems
in the analysis and design of stochastic control systems and
nonlinear filters. The basic mathematical methods which are used
and developed are those of the theory of weak con vergence. The
techniques are quite powerful for getting weak convergence or
functional limit theorems for broad classes of problems and many of
the techniques are new. The original need for some of the
techniques which are developed here arose in connection with our
study of the particular applica tions in this book, and related
problems of approximation in control theory, but it will be clear
that they have numerous applications elsewhere in weak convergence
and process approximation theory. The book is a continuation of the
author's long term interest in problems of the approximation of
stochastic processes and its applications to problems arising in
control and communication theory and related areas. In fact, the
techniques used here can be fruitfully applied to many other areas.
The basic random processes of interest can be described by
solutions to either (multiple time scale) Ito differential
equations driven by wide band or state dependent wide band noise or
which are singularly perturbed. They might be controlled or not,
and their state values might be fully observable or not (e. g., as
in the nonlinear filtering problem)."
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