8-Stochastic Integrals Page 225 Wednesday, February 4, 2004 12:50 PM
Stochastic Integrals 225BROWNIAN MOTION DEFINED
The previous section introduced Brownian motion informally as the
limit of a simple random walk when the step size goes to zero. This sec-
tion defines Brownian motion formally. The term “Brownian motion” is
due to the Scottish botanist Robert Brown who in 1828 observed that
pollen grains suspended in a liquid move irregularly. This irregular
motion was later explained by the random collision of the molecules of
the liquid with the pollen grains. It is therefore natural to represent
Brownian motion as a continuous-time stochastic process that is the
limit of a discrete random walk.
Let’s now formally define Brownian motion and demonstrate its
existence. Let’s first go back to the probabilistic representation of the
economy. Recall from Chapter 6 that the economy is represented as a
probability space (Ω,ℑ,P), where Ω is the set of all possible economic
states, ℑ is the event σ-algebra, and P is a probability measure. Recall
that the economic states ω ∈ Ω are not instantaneous states but repre-
sent full histories of the economy for the time horizon considered,
which can be a finite or infinite interval of time. In other words, the eco-
nomic states are the possible realization outcomes of the economy.
Recall also that, in this probabilistic representation of the economy,
time-variable economic quantities—such as interest rates, security prices
or cash flows as well as aggregate quantities such as economic output—
are represented as stochastic processes Xt(ω). In particular, the price and
dividend of each stock are represented as two stochastic processes St(ω)
and dt(ω).
Stochastic processes are time-dependent random variables defined
over the set Ω. It is critical to define stochastic processes so that there is no
anticipation of information, i.e., at time t no process depends on variables
that will be realized later. Anticipation of information is possible only
within a deterministic framework. However the space Ω in itself does not
contain any coherent specification of time. If we associate random vari-
ables Xt(ω) to a time index without any additional restriction, we might
incur in the problem of anticipation of information. Consider, for instance,
an arbitrary family of time-indexed random variables Xt(ω) and suppose
that, for some instant t, the relationship Xt(ω) = Xt+1(ω) holds. In this case
there is clearly anticipation of information as the value of the variable
Xt+1(ω) at time t+1 is known at an earlier time t. All relationships that lead
to anticipation of information must be treated as deterministic.
The formal way to specify in full generality the evolution of time and
the propagation of information without anticipation is through the con-
cept of filtration. Recall from Chapter 6 that the concept of filtration is
based on identifying all events that are known at any given instant. It is