In this lecture note, we are going to discuss 7 topics, which is outlined as follows:
-
Conditioned Expectation
-
Discrete-Time Martingale
-
Continuous-Time Martingale
-
Stochastic Integral
-
Strong Solution of SDE
-
Weak Solution of SDE (optional)
-
Applications (optional)
References
-
Steven E. Shreve and Ioannis Karatres, Brownian Motion and Stochastic Calculus
-
Rick Durret, Probability: Theory and Examples
-
Patrick Billingslay: Probability and Measure
Topic 1: Conditioned Probability, Distribution and Expectations
Let
be a probability measure space, and
be a random variable. Actually,
is a functional from
to
satisfying that for all
,

This means every pre-image of a Borel set in the real number is in the
-algebra.
Mathematical expectation is defined by

where
represents the probability of the preimage of Borel set
, i.e.,

Ex: Prove that
is a measure on
.
Proof. Let us write the definition of a measure. The following is copied from Wikipedia measure (mathematics). If
is a measure, three conditions must hold, i.e.,
Let
be a
-algebra over
, a function
from
to the extended real number line is called a measure, if it satisfies the following properties:
-
Non-negativity
-
Null empty set
-
Countable additivity
First of all,
is a set function from
to
. Non-negativity follows from the non-negativity of
. To see the null empty set property, we check that if
implies that
. By the null empty set property of
, we arrive at the null empty set property of
. And finally, for countable additivity, suppose
are disjoint sets in
, then by the definition of
, we have

By the operation of set, and the disjointness of
we have

and finally by the countable additivity, we have

which arrive at the conclusion that satisfy countable additivity.
1.1 Sub-
-field and Information
In this subsection, an heuristic example is provided to explain the meaning of a sub-
-field. In general cases, a sub-
-field could be approximatedly understood as information.
Example: [Toss of a coin 3 times]
All the possible out come constructs the sample space
, which takes the form

After first toss, the sample space could be divided into two parts, as
, where
, and
.
We can consider the corresponding
-algebra:
, which stands for the ``information'' after the first toss. When a sample
is given, whose first experiment is a head, we can tell that
is not in
,
is in
,
is in
and
is not in
. And look it in another angle, we see that, this
-algebra contains all the possible situations for different ``first toss'' cases.
It is quite easy to generalize to the ``information''
-field after second toss
.
Generally speaking, if
is a sub-
-field of
, the information of
is understood as
for all
, one know whether
or not. In other word, the indicator function is well-defined
.
1.2 Conditional Probability
In this subsection, a theoretical treatment of conditional probability is concerned. As we know in the elementary probability theory, the nature definition for conditional probability is govened by the following equation

Therefore, it is natural to raise the question how to define
, where
is a sub-
-field?
Note: The lecture notes follows majorly from reference book 3---Patrick Billingsley's Probability and Measure, Section 33.
Sometimes,
is quite complicated. Thus, instead, we consider the simple case when
is generated by some disjoint
instead, where
. Therefore, we have
.
Then
can be defined pathwisely, by

for all sample
in sample space
.
Before we goto see the formal definition, we examine an example, which comes from the problem of predicting the telephone call probability.
Example: [Apply Simple Case to Computing Conditional Probability]
Consider a Poisson process
on measure space
. Let
and
. Compute
.
Solution. Recall some of the knowledge of Poisson process now. By Wikipedia Poisson Process, we have
-

-
Independent increments
-
Stationary increments
-
No counted occurrences are simultaneous
The result of this defintion is that
where
is the intensity.
Note: If
is a constant, this is the case of homogeneous Poisson process, which is also named as Lévy processes.
It follows that

for all
and
.
Another explaination is also need for
. This is a sigma field
Now, let
. Obviously, the union of all these set is the sample space
. Moreover, they are obviously disjoint. Then by the computation formula in the simple case, we have

To compute
and
, we have

and

which gives rise to the final result

Ex. Prove that
is
-measurable;
Proof. Recall the definition of
-measurable. If a random variable
is
-measurable, then for all
, we have its pre-image
. Or equivalently,
. In this case, we are going to prove that
. It reduced to the problem that
could be written as some union of
?
Since for all
,
is a constant.
Ex. Prove that
holds.
Proof. Note that the left hand side equals to
which is identical with the right hand side.
Now we go further to the general cases for
. Suppose,
is a probability measure space,
is a sub-
-field, event
. Then, we claim the conditional probability of
given
is a random variable satisfying (1)
is
-measurable; (2)
. The random variable exists and unique, by Radon-Nikodym Theorem.
Let
and 
Ex. Prove that
is a measure on
.
Proof.
is a function that
. Non-negative property and emptyset property are trivial. Countable additivity follows from the countable additivity of probability measure
.
Ex. Prove that
is absolute continuous with respect to
on
.
Proof. By Wikipedia, absolute continuity, we know that if
is absolute continuous with respect to
on
, this means that for all
and
implies that
. This is obvious, since
.
..
Ex. If
and
are
-measurable, and
, then
almost surely.
Proof.