The Mathematics of Financial Modelingand Investment Management

6-ConceptsProbability Page 174 Wednesday, February 4, 2004 3:00 PM

174 The Mathematics of Financial Modeling and Investment Management

The integral can be defined not only on Ω but on any measurable

set G. In order to define the integral over a measurable set G, consider

the indicator function IG, which assumes value 1 on each point of the

set G and 0 elsewhere. Consider now the function f · IG. The integral

over the set G is defined as

∫ fMd = ∫ fI⋅ G Md

G Ω

The integral ∫ fMd is called the indefinite integral of f.

G

Given a σ-algebra ℑ, suppose that G and M are two measures and

that a function f exists such that for A ∈ ℑ

GA() = ∫ fMd

A

In this case G is said to have density f with respect to M.

The integrals in the sense of Riemann and in the sense of Lebesgue-

Stieltjes (see Chapter 4 on calculus) are special instances of this more

general definition of the integral. Note that the Lebesgue-Stieltjes inte-

gral was defined in Chapter 4 in one dimension. Its definition can be

extended to n-dimensional spaces. In particular, it is always possible to

define the Lebesgue-Stieltjes integral with respect to a n-dimensional dis-

tribution function. We omit the definitions which are rather technical.^8

Given a probability space (Ω,ℑ,P) and a random variable X, the

expected value of X is its integral with respect to the probability measure P

E[X] = ∫ XPd

Ω

where integration is extended to the entire space.

DISTRIBUTIONS AND DISTRIBUTION FUNCTIONS

Given a probability space (Ω,ℑ,P) and a random variable X, consider a set

A ∈ B 1. Recall that a random variable is a real-valued measurable func-

(^8) For details, see Yuan Shih Chow and Henry Teicher, Probability Theory: Second
Edition (New York: Springer, 1988).