STOCHASTIC

INTRODUCTION
The first part of this book is devoted to technical prerequisites tor the study
of stochastic optimization models. We have selected articles and included
exercises that appear to provide the necessary background for the study of
the specific financial models discussed in the remainder of the book. The
treatment in this part is, however, necessarily brief because of space limitations
; hence the reader may wish to consult some of the noted additional
references on some points.
The prerequisites for an in-depth study of stochastic optimization models
are expected utility theory, convex functions and nonlinear optimization
methods, and the embedded concepts of dynamic programming.
I. Expected Utility Theory
Fishburn's article presents a concise proof of a general expected utility
theorem. He considers a set of outcomes and probability distributions over
outcomes. The latter are termed horse lotteries. Given a set of reasonable
assumptions concerning the decision-maker's preferences over alternative
horse lotteries, Fishburn demonstrates the existence of a utility function and a
subjective probability measure over the states of the world such that the
decision-maker acts as if he maximized expected utility. The utility function
is continuous and is uniquely defined up to a positive linear transformation
(see Exercise CR-2 for examples). In some cases the utility function is bounded.
However, most utility functions are unbounded in at least one direction. For
such utility functions horse lotteries can be constructed that have arbitrarily
large expected utility and an arbitrarily small probability of receiving a
positive return. Such examples are called St. Petersburg paradoxes. They are
illustrated in Exercises CR-4 and ME-2. Further restrictions on the utility
function result if one makes additional assumptions concerning the decisionmaker's
behavior. For example, an investor who never prefers a fair gamble
to the status quo must have a concave utility function (Exercise CR-1). Such
investors will not simultaneously gamble and purchase insurance. Exercise
CR-6 is concerned with this classic Friedman-Savage paradox; see also the
papers by Yaari (1965a) and Hakansson (1970). 1 If the investor's preferences
for horse lotteries are independent of his initial wealth level, then the utility
function must be linear or exponential (Exercise CR-5). Similarly, if the
investor's preferences for proportional gambles are independent of initial
1 Throughout this book references cited by date will be found in the Bibliography at the
end of the book.
INTRODUCTION 3