In this below, some results on stochastic integration are given here based on [Karatzas and Shreve 1998]([KS98]).
Fix a probability space which satisfies the usual conditions.
Definition 1 [KS98, 1.5.1] Let
be a right-continuous martingale. We say that
is square-integrable if
for every
. If, in addition,
a.s., we write
(or
, if
is also continuous).
For any ,
is a nonnegative submartingale, hence of class DL, and so
has a unique Doob-Mayer decomposition
Now, is referred as quadratic variation of
. ([KS98, 1.5.3]) The cross-variation process
, for any
, is defined by
Following statements explains name of quadratic variation. Let be a partition of
, and
is
th variation over
. Then, we have, by [KS98, 1.5.8]
A metric structure is imposed on the space and its subspace
as follows, under which
is a complete metric space, and
is a closed subspace of
.
Definition 2 [KS98, 1.5.22] For any
and
, we define
We also set
Next, local martingale is defined.
Definition 3 [KS98,1.5.15] Let
be a (continuous) process. If there exists a nondecreasing sequence
of stopping times of
, such that
is a martingale for each
and
, then we say that
is a (continuous) local martingale; if, in addition,
a.s., we write
(respectively,
if
is continuous).
For measurable, -adapted process
, we define, for
Definition 4 [KS98, 3.2.1] Let
. Let
denote the set of equivalence classes of all measurable,
-adapted processes
, for which
for all
. We define a metric on
by
, where
Let
denote the set of equivalence classes of progressively measurable processes satisfying
for all
, and define a metric on
in the same way.
consists of those processes in
which are progressively measurable. When we wish to indicate the dependence on
, we write
and
.
For ,
is well defined for every
, and
. In addition, if the sample paths
of the quadratic variation process
are absolutely continuous functions of
for
-a.e.
, then
is well defined for all
, and all the properties of Proposition 6 hold in this case. (Remark 3.2.11 of [KS98]).
Definition 5 [KS98, 3.2.22] Let
. We denote by
the collection of equivalence classes of all measurable, adapted processes
satisfying
We denote by
the collection of equivalence classes of all progressively measurable processes satisfying this condition.
If , then both
and
are well defined, and we have
and
.
For and
, the stochastic integral of
with respect to
, denoted by
, is well defined, and
.
If a.e. path of the quadratic variation process
is an absolutely continuous function, we can choose integrands from wider class
, instead of
.
When and
, the integral
will not in general satisfy conditions involved with expectations, but only satisfy (1), (6), (7), and (11).
Proposition 6 [KS98, 3.2.10] For
and
, the stochastic integral
of
with respect to
satisfies following elementary properties: For any numbers of
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Also, its quadratic variation is given by
Further more, for any two stopping times
of the filtration
and any number
, we have
and in particular, for any number
in
,
where
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