STOCHASTIC

INTRODUCTION
This part of the book is concerned with stochastic dynamic models of
financial problems that are reducible to static models. That is, problems
where there exists a static program whose optimal solution provides the optimal
action to take in the first period and an optimal strategy for the remaining
periods conditional on preceding decision choices and realizations of random
variables. There are three natural classes of such models. First, the dynamic
nature of the model may be fictitious: Even though the model appears to
have a dynamic character there is only one decision period. The second category
concerns models in which there exists a deterministic equivalent (i.e., an
explicit deterministic program whose set of optimal solutions is the same as
that of the stochastic dynamic problem), but it is not generally possible to
find an explicit analytical characterization for the deterministic equivalent.
Models may be considered to be in this category whenever an optimal policy
exists and a backward induction type of dynamic programming argument is
valid. The major concern regarding such models is the determination of
qualitative properties of the optimal decision policy using the deterministic
equivalent. Third, the dynamic model may have the property that it is possible
to determine an explicit static deterministic equivalent whose solution provides
the requisite decision policy. Such a deterministic equivalent may depend on
decision variables from only one period or it may depend on decision variables
from several periods in a way that yields a static program. The original dynamic
program is said to have a myopic policy if the optimal decision in each period
can be determined without consideration of future decisions and random
events. The dynamic program is said to yield zero-order decision rules if the
optimal decision policy in each period is independent of the realizations of
random variable occurrences in all future periods.
I. Models that Have a Single Decision Point
The paper by Wilson illustrates a capital budgeting model in the first
category. He presents a method to add investment projects one at a time to
provide successively higher values of the expected utility of the returns obtained
over the n periods. It is assumed that the available project selection list is
known at the beginning of the time horizon. Generally, projects may be
commenced in any of several periods and they may have stochastic as well as
deterministic interconnections with projects beginning in any of the n periods.
Since projects are added one at a time, a static analysis is possible. Projects
are always accepted if they have the property that a financing plan paid for by
project revenues exists that will cover the project's costs for all states of the
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