A stochastic process is said to be standard Brownian motion (or a Wiener process) if
- almost surely.
- has independent increments, i.e. for , the random variables
- For ,
- The map is almost surely continuous.
In this post I follow the method in Sidney Resnick’s Adventures in Stochastic Processes to construct Brownian motion on using i.i.d. standard normal random variables. The key to this construction is the following lemma:
Lemma 1 Suppose and are random variables such that
Then there exists a random variable such that
Proof: Define and suppose is independent of . Define by
Then as , we have
so that and are uncorrelated and hence independent (as they are normally distributed).
be independent random variables with
Let , , and define using so that and are i.i.d. random variables. Suppose
are defined such that
For each , define such that
and the sequence has independent increments. For each and define the processes
and is linear on each interval . Let be the maximum deviation of and on , partitioned by the intervals , that is,
For each we have
where , and so
For we compute
where and is the probability density of . Let , then we have
from the first Borel-Cantelli lemma we have
This implies that the sequence of functions on is Cauchy, as for
Since is complete, we conclude that exists, with probability , and by construction is Brownian motion on .