The Mathematics of Financial Modelingand Investment Management

6-ConceptsProbability Page 199 Wednesday, February 4, 2004 3:00 PM

Concepts of Probability 199

In particular, a bivariate normal distribution factorizes in a linear

regression as follows:

σxz, (σxz, )^2

(XZ= z) ∼N μx – ----------(μz – z), σ^2 x – ------------------

σ^2 z σ^2 z

σxz, σxz,

gz()= EX[ Z= z]= μx – ----------μz + ----------z

σ^2 z σ^2 z

SUMMARY

■ Probability is a set function defined over a class of events where events

are sets of possible outcomes of an experiment. A probability space is a

triple formed by a set of outcomes, a σ-algebra of events, and a proba-

bility measure.

■ A random variable is a real-valued function defined over the set of out-

comes such that the inverse image of any interval is an event. n-dimen-

sional random vectors are functions from the set of outcomes into the

n-dimensional Euclidean space with the property that the inverse image

of n-dimensional generalized rectangles is an event.

■ Stochastic processes are time-dependent random variables.

■ An information structure is a collection of partitions of events associ-

ated to each instant of time that become progressively finer with the

evolution of time. A filtration is an increasing collection of σ-algebras

associated to each instant of time.

■ The states of the economy, intended as full histories of the economy,

are represented as a probability space. The revelation of information

with time is represented by information structures or filtrations. Prices

and other financial quantities are represented by adapted stochastic

processes.

■ By conditioning is meant the change in probabilities due to the acqui-

sition of some information. It is possible to condition with respect to

an event if the event has nonzero probability. In general terms, condi-

tioning means conditioning with respect to a filtration or an informa-

tion structure.

■ A martingale is a stochastic process such that the conditional expected

value is always equal to its present value. It embodies the idea of a fair

game where today’s wealth is the best forecast of future wealth.