e are
considering development of some market model during a time interval
At the time
we know nothing about the future and we represent this fact with the trivial
algebra
The
is the event space. It is the full description of what may happen in the
model. By the time moment
random variables of the model have certain realizations. One may make
particular statements about such realizations that are perfectly verifiable
from the point of view of information available at time
.
Such statements are represented by subsets of
and constitute an algebra
,
(see section (
Operations on sets
)
for explanation of the "algebra" term in this context). Similarly, for a time
moment
we form an algebra
.
Since market participants do not forget information,
is a subset of
.
A family of such algebras
is called "filtration" or "flow of information".
Example
Consider a discrete random walk
described by the picture. The process
starts at the point A at time
.
By the time
the process may be observed at point
or
.
Such outcome is uncertain and these are all possibilities at the time
.
Similarly, if the process is at point
then it may jump up to
or down to
.
We introduce the
notation
The highlighted path then is described by the elementary random event
.
At the initial time moment
our knowledge is given by the trivial algebra
with
being enumeration of everything that may
happen:
and
being the event that never happens. At the time
we know where the process went at
.
Hence,
where the
means "take all intersections, unions and complements of the arguments and put
them together into the algebra
".
For example,
contains the set
The information at
is represented by the algebra
.
For example,
contains the set
.
The information at
is represented by the algebra
,
,
,
,
,
,
,
,
.
By definition, a process
is adapted to a filtration
if for any particular
the
is
measurable.
Equivalently,
is adapted to
if for any
the algebra
contains the full description of the path
for all times up to
.
For a particular process
one may form a family
such that for any
the algebra
is the minimal algebra that makes the random variable
measurable
(for any
the
is the minimal description of all possible realizations of
).
Such a family
is called "the filtration generated by the
".
The notation
is commonly used to describe the generated filtration.
Suppose
is adapted to
and
The
is
measurable
(because
is sufficient to describe
,
hence, it contains description of
as well). Regularly (unless
is deterministic during
),
is not
measurable
because
is lacking structure to describe
.
However, we may create a crude adjustment of
to
by taking the conditional expectation
for every set
from
This creates a mapping from
to the range of
.
A proper restriction of such mapping is an
measurable
random variable that we denote
The same (in "almost sure" sense) object
could be introduced by requiring that the variable
by definition, would be
measurable
and satisfy
for any set
from
.
The
condition
written in the
form


(Chain rule)

is called "the chain rule".
We will say that the random variable
is independent from
if for any smooth function
we have
.
A random variable
is called the
stopping
time if for any
the random event
belongs to the
.
In other words,
is a stopping time if at any moment
we are able to tell with certainty which of the statements
and
is true.
