The Mathematics of Financial Modelingand Investment Management

6-ConceptsProbability Page 193 Wednesday, February 4, 2004 3:00 PM

Concepts of Probability 193

This relationship can be extended to any number of variables.

In probability theory, given a random variable X, the following

expectation is called the characteristic function (c.f.) of the variable X

φX()t = Ee[ itX]= E[cos tX ]+ iE[sin tX]

If the variable X admits a d.f. FX(y), it can be demonstrated that the

following relationship holds:

∞ ∞ ∞

∫

φ itX

X()t = Ee[ x x

itX]= e dF

X()x = ∫ cos tx dFX()+ ∫sin tx dFX()

• ∞ –∞ –∞

In this case, the characteristic function therefore coincides with the Fou-

rier-Stieltjes transform. It can be demonstrated that there is a one-to-one

correspondence between c.d.s and d.f.s. In fact, it is well known that the

Fourier-Stieltjes transform can be uniquely inverted.

In probability theory convolution is defined, in a more general way,

as follows. Given two d.f.s FX(y) and FY(y), their convolution is defined

as:

∞

F ()u = (FXFY)()u = ∫ FX(uy– )dFY ()y

• ∞

It can be demonstrated that the d.f. of the sum of two variables X,Y

with d.f.s FX(y) and FY(y) is the convolution of their respective d.f.s:

∞

PX ( + Y ≤u) = FXY+ ()u = F ()u = (FXFY)()u = ∫ FX(uy– )dFY ()y

• ∞

If the d.f.s admits p.d.f.s, then the inversion formulas are those estab-

lished earlier. Inversion formulas also exist in the case that the d.f.s do

not admit densities but these are more complex and will not be given

here.^13

We can therefore establish the following property: the characteristic

function of the sum of n independent random variables is the product of

the characteristic functions of each of the summands.

(^13) See Chow and Teicher, Probability Theory.