The Mathematics of Financial Modelingand Investment Management

6-ConceptsProbability Page 186 Wednesday, February 4, 2004 3:00 PM

186 The Mathematics of Financial Modeling and Investment Management

∞

fY ()y = ∫ fxy ( , )dx

• ∞

is assumed to be strictly positive.

In the discrete case, the conditional expectation is a random variable

that takes a constant value over the sets of the finite partition associated

to ℑt. Its value for each element of Ωis defined by the classical concept of

conditional probability. Conditional expectation is simply the average

over a partition assuming the classical conditional probabilities.

An important econometric concept related to conditional expecta-

tions is that of a martingale. Given a probability space (Ω,ℑ,P) and a fil-

tration {ℑt}, a sequence of ℑi-measurable random variables Xiis called a

martingale if the following condition holds:

EX[ i+ 1 ℑ ]i =Xi

A martingale translates the idea of a “fair game” as the expected value

of the variable at the next period is the present value of the same value.

MOMENTS AND CORRELATION

If Xis a random variable on a probability space (Ω,ℑ,P), the quantity

EX[ p], p> 0 is called the p-th absolute moment of X. If kis any posi-

tive integer, E[Xk], if it exists, is called the k-th moment. In the general

case of a probability measure Pwe can therefore write:

■ EX[ p] = Xpd P, p> 0, is the p-th absolute moment.

∫

Ω

■ EX[ k] = ∫ XkdP, if it exists for k positive integer, is the k-th moment.

Ω

In the case of discrete probabilities pi, Σpi= 1 the above expressions

become

EX[ p] = p

∑ xi pi

and