The Mathematics of Financial Modelingand Investment Management

6-ConceptsProbability Page 185 Wednesday, February 4, 2004 3:00 PM

Concepts of Probability 185

while Xis not. It is possible to demonstrate that such variables exist and

are unique up to a set of measure zero.

Econometric models usually condition a random variable given

another variable. In the previous framework, conditioning one random

variable Xwith respect to another random variable Ymeans condition-

ing Xgiven σ{Y} (i.e., given the σ-algebra generated by Y). Thus E[X|Y]

means E[X|σ{Y}].

This notion might seem to be abstract and to miss a key aspect of

conditioning: intuitively, conditional expectation is a function of the

conditioning variable. For example, given a stochastic price process, Xt,

one would like to visualize conditional expectation E[XtXs], s< tas a

function of Xsthat yields the expected price at a future date given the

present price. This intuition is not wrong insofar as the conditional

expectation E[XY] of Xgiven Yis a random variable function of Y.

For example, the regression function that will be explained later in this

chapter is indeed a function that yields the conditional expectation.

However, we need to specify how conditional expectations are

formed, given that the usual conditional probabilities cannot be applied

as the conditioning event has probability zero. Here is where the above

definition comes into play. The conditional expectation of a variable X

given a variable Y is defined in full generality as a variable that is measur-

able with respect to the σ-algebra σ(Y) generated by the conditioning

variable Yand has the same expected value of Yon each set of σ(Y). Later

in this section we will see how conditional expectations can be expressed

in terms of the joint p.d.f. of the conditioning and conditioned variables.

One can define conditional probabilities starting from the concept

of conditional expectations. Consider a probability space (Ω,ℑ,P), a sub-

σ-algebra G of ℑ, and two events A∈ ℑ, B∈ ℑ. If IA,IBare the indicator

functions of the sets A,B(the indicator function of a set assumes value 1

on the set, 0 elsewhere), we can define conditional probabilities of the

event A, respectively, given G or given the event Bas

P(AG) = E[IAG] P(AB) = E[IAIB]

Using these definitions, it is possible to demonstrate that given two ran-

dom variables Xand Ywith joint density f(x,y),the conditional density

of Xgiven Yis

fx( y) = fx y ---------------( , )-

fY () y

where the marginal density, defined as