The Mathematics of Financial Modelingand Investment Management

8-Stochastic Integrals Page 228 Wednesday, February 4, 2004 12:50 PM

228 The Mathematics of Financial Modeling and Investment Management

( H 1 ...× × H ) = μ t 1 ... ...× × H

σ –^1 ()

μ tσ (), , ...tσ () m , , t )

1 m m 1

( H

σ –^1 () m

for all permutations σ on {1,2,...,k}, and

μ t 1 ... ( H 1 ...× × Hk) = μ t 1 , , , , , t ( H 1 ...× × Hk × R

n

...× × R

n

, , tk ... tk tk + 1 ... m )

for all m. The Kolmogorov extension theorem states that, if the above

conditions are satisfied, then there is (1) a probability space (Ω ,ℑ ,P) and

(2) a stochastic process that admits the probability measures

μ t

1 ...

( H 1 ...× × H ) = PXt

1

, , t m [( ∈ H 1 , , ... Xtm ∈ Hm), Hi ∈ Bn]

m

as finite dimensional distributions.

The construction is lengthy and technical and we omit it here, but it

should be clear how, with an appropriate selection of finite-dimensional

distributions, the Kolmogorov extension theorem can be used to prove

the existence of Brownian motions. The finite-dimensional distributions

of a one-dimensional Brownian motion are distributions of the type

μ t 1 , , ...tk ( H 1 ...× × Hk)

= ∫ p txx( ,, 1 ) pt( 2 – t 1 , x 1 , x 2 )... pt( k – tk – 1 ,xk – 1 , xk) dx 1 ... dxk

H 1 ...× × Hk

where

-^1 ---

(^2) xy–^2
p txy( ,,) = ( 2 π t) exp 
 – ----------------- 2 t -
and with the convention that the integrals are taken with respect to the
Lebesgue measure. The distribution p(t,x,x 1 ) in the integral is the initial
distribution. If the process starts at zero, p(t,x,x 1 ) is a Dirac delta, that
is, it is a distribution of mass 1 concentrated in one point.
It can be verified that these distributions satisfy the above consis-
tency conditions; the Kolmogorov extension theorem therefore ensures
that a stochastic process with the above finite dimensional distributions
exists. It can be demonstrated that this process has normally distributed
independent increments with variance that grows linearly with time. It
is therefore a one-dimensional Brownian motion. These definitions can
be easily extended to a n-dimensional Brownian motion.