et
be a random event space (see the section
(
Filtration
)) and
be a standard normal variable (see the section
(
Normal variable
)) . The standard Brownian
motion
is a mapping
with the following properties:

The path
is continuous for every
.

The random variable
is independent from
(see the section (
Filtration
))
for every
,
.

For any
,
the random variable
is distributed as
:


(Brownian motion)


The argument
of the Brownian motion
is conventionally called "time".
