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.
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