INTRODUCTION

In the study of the classification and recognition of mathematical processes,

dichotomies can be introduced in several different fashions. Two of the most

useful have been "deterministic vs. stochastic" and "descriptive vs. variational."

In recent years, we have seen greater and greater emphasis laid upon stochastic

and decision processes, and even a blending of the two in the form of modern

control theory.

The gradual shift of interest from the classical deterministic descriptive process

has been largely a consequence of the continuing challenges to the mathematician

to explain phenomena, produce numbers, and guide research in such fields as

physics, engineering, economics, biology, medicine and operations research.

Combinations of complexity and uncertainty have forced a more frequent use of

probabilistic concepts.

In this volume, the emphasis is upon stochastic processes, divided a bit unevenly

between descriptive and decision aspects. In dealing with random effects, we

can use deterministic or stochastic equations, just as in dealing with deterministic

effects we can employ stochastic or deterministic equations or in treating descriptive

processes, we can use either descriptive or variational equations. The choice of

a particular analytic formulation is a tribute to the existing analytic techniques and

the computational resources available, as well as a reflection of the underlying

physical process and the prevalent scientific philosophy.

The papers by Adomian, Bharucha-Reid, Hoffman, Keller, Richardson, and

Twersky deal with various and overlapping aspects of the general theory of

stochastic equations, principally ordinary and partial differential equations. Much

of this work is motivated by particular equations arising in wave propagation and

scattering theory, as the papers by Hoffman, Keller, and Twersky indicate. The

objective of obtaining specific analytic results leads to the development of pertur-

bation techniques and the consideration of questions of closure, as, for example,

in Richardson's paper.

The paper by Bellman deals with stochastic iteration, a topic which arises from

the consideration of stochastic differential equations in very much the same way

as classical iteration theory arises from deterministic differential equations. A

particularly important type of stochastic iteration arises in the theory of stochastic

approximation. This approach is used by Gray to study a design problem arising

from circuits with random components. In their papers, Blackwell and Derman

use dynamic programming, the first to obtain estimates in probability theory

by identification with a stochastic decision process, and the second to treat an

interesting class of sequential decision processes of Markovian type.

Classical estimation and identification problems arising from stochastic processes

are treated by Parzen and Root using new and powerful approaches.

vii