The goal of this Lecture is to extend the domain of definition of the Itō integral with respect to Brownian motion. The idea is to use the fruitful concept of localization. We will then be interested in the wider class of processes for which it is possible to define a stochastic integral satisfying natural probabilistic properties. This will lead to the natural notion of semimartingales.

As before, we consider here a Brownian motion that is defined on a filtered probability space that satisfies the usual conditions.

**Definition.** *We define the space , as the set of the processes that are progressively measurable with respect to the filtration and such that for every ,
*

We first have the following fact:

**Lemma.** * Let . There exists an increasing family of stopping times for the filtration such that:*

- Almost surely,

Thanks to this Lemma, it is now easy to naturally define for . Indeed, let and let . According to the previous lemma, let us now consider an increasing sequence of stopping times such that:

- Almost surely,

Since the stochastic integral

exists. We may therefore define in a unique way a stochastic process such that:

- is a continuous stochastic process adapted to the filtration ;
- The stochastic process is a uniformly integrable martingale with respect to the filtration (because it is bounded in ).

This leads to the following definition:

**Definition.** * A stochastic process is called a local martingale (with respect to the filtration ) if there is a sequence of stopping times such that:*

- The sequence is increasing and almost surely satisfies ;
- For , the process is a uniformly integrable martingale with respect to the filtration .

Thus, as an example, if then the process is a local martingale. Of course, any martingale turns out to be a local martingale. But, as we will see it later, in general the converse is not true. The following Exercise gives a useful criterion to prove that a given local martingale is actually martingale.

**Exercise.**

Let be a continuous local martingale such that for ,

Show that is a martingale. As a consequence, bounded local martingales necessarily are martingales.

It is interesting to observe that if is a local martingale, then the sequence of stopping times may explicitly be chosen so that the resulting stopped martingales enjoy nice properties.

**Lemma.** * Let be a continuous local martingale on such that . Let
Then, for , the process is a bounded martingale.*

**Proof.** Let be a sequence of stopping times such that:

- The sequence is increasing and almost surely ;
- For every , the process is a uniformly integrable martingale with respect to the filtration .

For and , we have:

Letting leads then to the expected result

Since bounded martingales are of course square integrable, we easily deduce from the previous Lemma that the following result holds:

**Theorem.** *Let be a continuous local martingale on such that . Then, there is a unique continuous increasing process such that:*

- ;
- The process is a local martingale.

*Furthermore, for every and every sequence of subdivisions such that , the following limit holds in probability:
Moreover, if is a progressively measurable process such that for every , then we may define a stochastic integral such that the stochastic process is a continuous local martingale.*

At that point, we already almost found the widest class of stochastic processes with respect to which it was possible to naturally construct a stochastic integral. To go further in that direction, let us first observe that if we add a bounded variation process to a local martingale, then we obtain a process with respect to which a stochastic integral is naturally defined.

More precisely, if may be written under the form:

, where is a bounded variation process and where

is a continuous local martingale on such that , then if is a progressively measurable process such that for ,

we may define a stochastic integral as

where is simply understood as the Riemann-Stieltjes integral with respect to the process .

The class of stochastic processes that we obtained is called the class of semimartingales and, as we will see it later, is the most relevant one:

**Definition.**

Let be an adapted continuous stochastic process on the filtered probability space . We say that is a semimartingale with respect to the filtration if may be written as:

where is a bounded variation process and is a continuous local martingale such that . If it exists, the previous decomposition is unique.

**Exercise.*** Let be a continuous local martingale on the filtered probability space . Show that is a semimartingale.*

Since a bounded variation process has a zero quadratic variation, it is easy to prove the following theorem:

**Proposition.*** Let
be a continuous adapted semimartingale. For every and every sequence of subdivisions such that the following limit holds in probability:
We therefore call the quadratic variation of and denote .*

**Exercise.** * Let be a continuous semimartingale on the filtered probability space . If is a subdivision of the time interval $[0,T]$, we denote , where is such that . Let be a sequence of subdivisions of such that Show that the following limit holds in probability,
*

**Exercise.** * Let be a continuous semimartingale on . Let be a sequence of locally bounded and adapted processes almost surely converging toward 0 such that , where is a locally bounded process. Show that for , the following limit holds in probability
*

It already has been observed that in the Brownian case, though the stochastic integral is not an almost sure limit of Riemann sums, it is however a limit in probability of such sums. This may extended to semimartingales in the following way.

**Proposition.*** Let be a continuous and adapted process, let be a continuous and adapted semimartingale and let . For every sequence of subdivisions such that
the following limit holds in probability:
*

As we already suggested it, the class of semimartingales is actually the wider class of stochastic processes with respect to which we may define a stochastic integral that enjoys natural properties. Let us more precisely explain what the previous statement means.

Let us denote by the set of processes such that:

where are bounded stopping times and where the ‘s are random variable that are bounded and measurable with respect to . If is a continuous and adapted process and if , then we naturally define

We have the following theorem that we shall admit without proof:

**Proposition.*** Let be a continuous and adapted process. The process is a semimartingale if and only if for every sequence in that almost surely converges to 0, we have for every and ,
*

Awesome. This is by far the best note on local martingales I have seen.