The Mathematics of Financial Modelingand Investment Management

8-Stochastic Integrals Page 229 Wednesday, February 4, 2004 12:50 PM

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Stochastic Integrals 229

In the initial definition of a Brownian motion, we assumed that a fil-

tration ℑ t was given and that the Brownian motion was adapted to the

filtration. In the present construction, however, we reverse this process.

Given that the process we construct has normally distributed, station-

ary, independent increments, we can define the filtration ℑ t as the filtra-

tion ℑ B

t generated by Bt(ω ). The independence of the increments of the

Brownian motion guarantee the absence of anticipation of information.

Note that if we were given a filtration ℑ t larger than the filtration ℑ Bt ,

Bt(ω ) would still be a Brownian motion with respect to ℑ t.

As we will see in Chapter 10 when we cover stochastic differential

equations, there are two types of solutions of stochastic differential equa-

tions—strong and weak—depending on whether the filtration is given or

generated by the Brownian motion. The implications of these differences

for economics and finance will be discussed in the same section.

The above construction does not specify uniquely the Brownian

motion. In fact, there are infinite stochastic processes that start from the

same point and have the same finite dimensional distributions but have

totally different paths. However it can be demonstrated that only one

Brownian motion has continuous paths a.s. Recall that a.s. means

almost surely, that is, for all paths except a set of measure zero. This

process is called the canonical Brownian motion. Its paths can be identi-

fied with the space of continuous functions.

The Brownian motion can also be constructed as the continuous limit

of a discrete random walk. Consider a simple random walk Wi where i are

discrete time points. The random walk is the motion of a point that moves

∆ x to the right or to the left with equal probability ¹
₂ at each time incre-

ment ∆ x. The total displacement Xi at time i is the sum of i independent

increments each distributed as a Bernoulli variable. Therefore the random

variable X has a binomial distribution with mean zero and variance:

∆^2 x

∆ t

Suppose that both the time increment and the space increment

approach zero: ∆ t → 0 and ∆ x → 0. Note that this is a very informal

statement. In fact what we mean is that we can construct a sequence of

n

random walk processes Wi , each characterized by a time step and by a

time displacement. It can be demonstrated that if

∆^2 x

---------- σ →

∆ t