The Mathematics of Financial Modelingand Investment Management

6-ConceptsProbability Page 184 Wednesday, February 4, 2004 3:00 PM

184 The Mathematics of Financial Modeling and Investment Management

from a logical point of view, the primitive concept is that of states and

events. The evolution of time has to be defined on the primitive struc-

ture—it cannot simply be imposed on random variables. In practice, fil-

trations become an important concept when dealing with conditional

probabilities in a continuous environment. As the probability that a

continuous random variable assumes a specific value is zero, the defini-

tion of conditional probabilities requires the machinery of filtration.

CONDITIONAL PROBABILITY AND CONDITIONAL EXPECTATION

Conditional probabilities and conditional averages are fundamental in

the stochastic description of financial markets. For instance, one is gen-

erally interested in the probability distribution of the price of an asset at

some date given its price at an earlier date. The widely used regression

models are an example of conditional expectation models.

The conditional probability of event Agiven event Bwas defined

earlier as

PA( B)= PA( ∩B)

------------------------

PB()

This simple definition cannot be used in the context of continuous ran-

dom variables because the conditioning event (i.e., one variable assum-

ing a given value) has probability zero. To avoid this problem, we

condition on σ-algebras and not on single zero-probability events. In

general, as each instant is characterized by a σ-algebra ℑt, the condition-

ing elements are the ℑt.

The general definition of conditional expectation is the following.

Consider a probability space (Ω,ℑ,P) and a σ-algebra G contained in ℑ

and suppose that Xis an integrable random variable on (Ω,ℑ,P). We

define the conditional expectation of Xwith respect to G, written as

E[X|G], as a random variable measurable with respect to G such that

∫ EX[ G]dP= ∫ XPd

G G

for every set G ∈G. In other words, the conditional expectation is a

random variable whose average on every event that belongs to G is

equal to the average of Xover those same events, but it is G-measurable