In the same way that a stochastic integral with respect to Brownian motion was constructed, a stochastic integral with respect to square integrable martingales may be defined. We shall not repeat this construction, since it was done in the Brownian motion case, but we point out the main results without proofs.

Let be a continuous square integrable martingale on a filtered probability space that satisfies the usual conditions. We assume that and . Let us denote by the set of processes that are progressively measurable with respect to the filtration and such that

We still denote by the set of simple and predictable processes, that is the set of processes that may be written as:

where and where is a random variable that is measurable with respect to and such that . We define an equivalence relation on the set as follows:

and denote by

the set of equivalence classes. It is easy to check that endowed with the norm

is a Hilbert space.

**Theorem.*** There exists is a unique linear map
such that:*

- For ,
- For ,

*The map is called the Itō integral with respect to the continuous and square integrable martingale . We denote for ,
*

**Proposition.** *Let be a stochastic process which is progressively measurable with respect to the filtration and such that for every , . The process
is a square integrable martingale with respect to the filtration that admits a continuous modification.*

**Proposition.** *Let be a stochastic process which is progressively measurable with respect to the filtration and such that for every , . We have
*

**Proposition.*** Let be a stochastic process whose paths are left continuous. Let . For every sequence of subdivisions such that
the following convergence holds in probability:
*

**Proposition.*** Let us assume that where is a Brownian motion on and where . For ,
*

**Exercise.*** Let and be two square integrable martingales such that for every ,
For ,
*