The Mathematics of Financial Modelingand Investment Management

8-Stochastic Integrals Page 227 Wednesday, February 4, 2004 12:50 PM

Stochastic Integrals 227

The above conditions^2 state that the standard Brownian motion is a

stochastic process that starts at zero, has continuous paths and normally

distributed increments whose variance grows linearly with time. Note

that in the last condition the increments are independent of the σ-alge-

bra ℑs and not of the previous values of the process. As noted above,

this is because any single realization of the process has probability zero

and it is therefore impossible to use the standard concept of conditional

probability: conditioning must be with respect to a σ-algebra ℑs. Once

this concept has been firmly established, one might speak loosely of

independence of the present values of a process from its previous values.

It should be clear, however, that what is meant is independence with

respect to a σ-algebra ℑs.

Note also that the filtration ℑt is an integral part of the above defini-

tion of the Brownian motion. This does not mean that, given any proba-

bility space and any filtration, a standard Brownian motion with these

characteristics exists. For instance, the filtration generated by a discrete-

time continuous-state random walk is insufficient to support a Brown-

ian motion. The definition states only that we call a one-dimensional

standard Brownian motion a mathematical object (if it exists) made up

of a probability space, a filtration and a time dependent random vari-

able with the properties specified in the definition

However it can be demonstrated that Brownian motions exist by

constructing them. Several construction methodologies have been pro-

posed, including methodologies based on the Kolmogorov extension

theorem or on constructing the Brownian motion as the limit of a

sequence of discrete random walks. To prove the existence of the stan-

dard Brownian motion, we will use the Kolmogorov extension theorem.

The Kolmogorov theorem can be summarized as follows. Consider

the following family of probability measures

μt [(

1 , ..., t

(H 1 ... × × H ) = PXt

1

m ∈ H 1 , ..., Xt ∈ Hm), Hi ∈ Bn ]

m m

for all t 1 ,...,tk ∈ [0,∞), k ∈ N and where the Hs are n-dimensional Borel

sets. Suppose that the following two consistency conditions are satisfied

(^2) The set of conditions defining a Brownian motion can be more parsimonious. If a
process has stationary, independent increments and continuous paths a.s. it must
have normally distributed increments. A process with stationary independent incre-
ments and with paths that are continuous to the right and limited to the left (the cad-
lag functions), is called a Levy process. In Chapter 13 we will generalize Brownian
motion to α-stable Levy processes that admit distributions with infinite variance and/
or infinite mean.