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 defineWe 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 ofAlso, 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 ,