We begin with a simple symmetric random walk
starting in 0 (
). The increments
are i.i.d. with :
By shortening the period of time of two successive observations we
accelerate the process. Simultaneously, the increments of the
process become smaller during the shorter period of time. More
precisely, we consider a stochastic process
in continuous time which increases or decreases in a
time step
with probability
by
Between these jumps the process is constant (alternatively we
could interpolate linearly). At time
the
process is:
where the increments
are
mutually independent and take the values
or
with probability
respectively. From Section
4.1 we know:
Now, we let
and
become smaller. For the
process in the limit to exist in a reasonable sense,
must be finite. On the other hand,
should not converge to 0, since the process would
then not be random any more. Hence, we must choose:

such that
If
is small, then
is large. Thus, the
random variable
of the ordinary symmetric random walk is
approximately
N
distributed, and therefore for all
(not only for
such that
):
Thus the limiting process
which we
obtain from
for
has the
following properties:
- (i)
is
N
distributed for all
- (ii)
-
has independent increments,
i.e. for
is independent of
(since
the random walk
defining
has independent increments).
- (iii)
- For
the increment
is
N
distributed, i.e. its distribution depends
exclusively on the length
of the time interval in which the
increment is observed (this follows from (i) and (ii) and the properties
of the normal distribution).
A stochastic process
in continuous time
satisfying (i)-(iii) is called Wiener process or Brownian motion starting
in 0 (
). The standard Wiener process resulting from
will be denoted by
For this process it
holds for all
As for every stochastic process in continuous time, we can
consider a path or realization of the Wiener process as a randomly chosen function of time. With some major mathematical
instruments it is possible to show that the paths of a Wiener
process are continuous with probability 1:

is continuous as a function of
That is to say, the Wiener process has no jumps. But
fluctuates heavily: the paths are continuous but highly erratic.
In fact, it is possible to show that the paths are not
differentiable with probability 1.
Being a process with independent increments the Wiener process is
markovian. For
it holds
i.e.
depends only on
and on the increment
from time
to time
:
Using properties (i)-(iii), the distribution of
given the
outcome
can be formulated explicitly. Since the increment
is
N
distributed,
is
N
distributed given
:
Proceeding from the assumption of a random walk
with drift
instead of a symmetric
random walk results in a process
which is no more
zero on average, but
For
we obtain for all
:
The limiting process is a Wiener process
with drift or trend
It results from the
standard Wiener process:
Hence, it behaves in the same way as the standard Wiener process
but it fluctuates on average around
instead of 0. If
the process is increasing linearly on average, and if
it is decreasing linearly on average.