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.
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 .
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
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]).
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 .