In this lecture, we give study properties of the Itō integral that was defined in the previous lecture. The following proposition is easy to prove and its proof is left to the reader as an exercise.

**Proposition:** *Let ,
show that
and
*

Associated with Itō’s integral, we can construct an integral process, its fundamental property is that it is continuous martingale.

**Proposition:** * Let . The process
is a martingale with respect to the filtration that admits a continuous modification.
*

**Proof:**

We first prove the martingale property. If

is in , then for every ,

Thus if , the process

is a martingale with respect to the filtration . Since is dense in , and since it is easily checked that a limit in of martingales is still a martingale, we deduce the expected result.

We now prove the existence of a continuous version.

If , the continuity of the integral process easily stems from the continuity of the Brownian paths. Let and let be a sequence in that converges to . From Doob’s inequality, we have for and ,

.

There exists thus a sequence such that

From Borel-Cantelli lemma, the sequence of processes converges then almost surely uniformly to the process which is therefore continuous

As a straightforward consequence of the previous proposition and Doob’s inequalities, we obtain

**Proposition** *Let .*

- For every ,

For , the Riemann sums ,

need not to almost surely converge to . However the following proposition shows that under weak regularity assumptions we have a convergence in probability.

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

**Proof.**

Let us first assume that is bounded almost surely. We have

where . The Itō’s isometry and the Lebesgue dominated convergence theorem shows then that converges to in and therefore in probability. For general ‘s we can use a localization procedure. For , consider the random time

We have for every ,

This easily implies the convergence in probability.

**Exercise:**

- Show that for ,

What is surprising in this formula ? - Show that when , the sequence

converges to a random variable that shall be computed.