The Mathematics of Financial Modelingand Investment Management

6-ConceptsProbability Page 170 Wednesday, February 4, 2004 3:00 PM

170 The Mathematics of Financial Modeling and Investment Management

set is called the 1-dimensional Borel σ-algebra and is denoted by B. The

sets that belong to B are called Borel sets.

Now consider the n-dimensional Euclidean space Rn, formed by n-

tuples of real numbers. Consider the collection of all generalized rectan-

gles open to the left and closed to the right, for example, ((a 1 ,b 1 ] × ...

×(an,bn]). The σ-algebra generated by this collection is called the n-

dimensional Borel σ-algebra and is denoted by B n. The sets that belong

to B n are called n-dimensional Borel sets.

The above construction is not the only possible one. The B n, for any

value of n, can also be generated by open or closed sets. As we will see

later in this chapter, B n is fundamental to defining random variables. It

defines a class of subsets of the Euclidean space on which it is reasonable

to impose a probability structure: the class of every subset would be too

big while the class of, say, generalized rectangles would be too small. The

B n is a sufficiently rich class.

PROBABILITY

Intuitively speaking, probability is a set function that associates to every

event a number between 0 and 1. Probability is formally defined by a

triple (Ω,ℑ,P) called a probability space, where Ω is the set of all possi-

ble outcomes, ℑ the event σ-algebra, and P a probability measure.

A probability measure P is a set function from ℑ to R (the set of real

numbers) that satisfies three conditions:

■ Condition 1. 0 ≤ P(A), for all A ∈ ℑ

■ Condition 2. P(Ω) = 1

■ Condition 3. P(∪ Ai) = ∑P(Ai) for every finite or countable collection

of disjoint events {Ai} such that Ai ∈ ℑ

ℑ does not have to be a σ-algebra. The definition of a probability space

can be limited to algebras of events. However it is possible to demon-

strate that a probability defined over an algebra of events אcan be

extended in a unique way to the σ-algebra generated by .א

Two events are said to be independent if:

P(A ∩ B) = P(A)P(B)

The (conditional) probability of event A given event B, written as P(A|B),

is defined as follows: