The Mathematics of Financial Modelingand Investment Management

6-ConceptsProbability Page 178 Wednesday, February 4, 2004 3:00 PM

178 The Mathematics of Financial Modeling and Investment Management

for all A ∈B , B ∈B. This definition generalizes in obvious ways to any

number of variables and therefore to the components of a random vec-

tor. It can be shown that if the components of a random vector are inde-

pendent, the joint probability distribution is the product of distributions.

Therefore, if the variables (X 1 ,...,Xn) are all mutually independent, we

can write the joint d.f. as a product of marginal distribution functions:

n

Fx( 1 , , ...xn)= ∏FXj ()xj

j = 1

It can also be demonstrated that if a d.f. admits a joint p.d.f., the

joint p.d.f. factorizes as follows:

n

fx( 1 , , ...xn)= ∏fXj ()xj

j = 1

Given the marginal p.d.f.s the joint d.f. can be recovered as follows:

x 1 xn

Fx( 1 , , ...xn) = ∫... ∫ fu( 1 , , ...un)du 1 ...dun

• ∞ –∞
  x 1 xn n

= ∫ ...∫ ∏fXj()uj du 1 ...dun

• ∞ –∞ j =^1
  n xj

= ∏∫fXj ()ujduj

j = (^1) – ∞
n

= ∏FX()xj

j

j = 1

STOCHASTIC PROCESSES

Given a probability space (Ω,ℑ,P) a stochastic process is a parameterized

collection of random variables {Xt}, t ∈[0,T] that are measurable with

respect to ℑ. The parameter t is often interpreted as time. The interval in

which a stochastic process is defined might extend to infinity in both

directions.