4-PrincipCalculus Page 93 Friday, March 12, 2004 12:39 PM
Principles of Calculus 93■ The evolution of probability distributions is represented through differ-
ential equations. This is the case within the framework of calculus.
■ The evolution of random phenomena is represented through direct
relationships between stochastic processes. This is the case within the
framework of stochastic calculus.Stochastic calculus has been adopted as the preferred framework in
finance and economics. We will start with a review of the key concepts
of calculus and then introduce the concepts of its stochastic evolution.SETS AND SET OPERATIONS
The basic concept in calculus (and in the theory of probability) is that of
a set. A set is a collection of objects called elements. The notions of both
element and set should be considered primitive. Following a common
convention, let’s denote sets with capital Latin or Greek letters:
A,B,C,Ω... and elements with small Latin or Greek letters: a,b,ω. Let’s
then consider collections of sets. In this context, a set is regarded as an
element at a higher level of aggregation. In some instances, it might be
useful to use different alphabets to distinguish between sets and collec-
tions of sets.
Piling up sets and sets of sets is not as innocuous as it might seem; it
is effectively the source of subtle and basic fundamental logical contra-
dictions called antinomies. Mathematics requires that a distinction be
made between naive set theory, which deals with basic set operations,
and axiomatic set theory, which deals with the logical structure of set
theory. In working with calculus, we can stay within the framework of
naive set theory and thus consider only basic set operations.Proper Subsets
An element a of a set A is said to belong to the set A written as a ∈ A. If
every element that belongs to a set A also belongs to a set B, we say that
A is contained in B and write: A ⊂ B. We will distinguish whether A is a
proper subset of B (i.e., whether there is at least one element that
belongs to B but not to A) or if the two sets might eventually coincide.
In the latter case we write A ⊆ B.
For example, as explained in Chapter 2, in the United States there
are indexes that are constructed based on the price of a subset of com-
mon stocks from the universe of all common stock in the country. There
are three types of common stock (equity) indexes: