In this Lecture we show that, remarkably, any square integrable integrable random variable which is measurable with respect to a Brownian motion, can be expressed as a stochastic integral with respect to this Brownian motion. A striking consequence of this result, which is known as Itō’s representation theorem, is that any square integrable martingale of a Brownian filtration has a continuous version.
Let be a Brownian motion. In the sequel, we consider the filtration which is the usual completion of the natural filtration of (such a filtration is called a Brownian filtration).
The following lemma is a straightforward consequence of Itō’s formula.
Lemma. Let be a locally square integrable function. The process is a square integrable martingale.
Proof. From Itō’s formula we have
The random variable is a Gaussian random variable with mean 0 and variance . As a consequence
and the process
is a martingale.
Lemma. Let be the set of compactly supported and piecewise constant functions , i.e. the set of functions that can be written as for some and . The family is total in .
Let such that for every ,
Let . We have for every ,
By analytic continuation, we see that
actually also holds for every . By using the Fourier transform, it implies that
Since were arbitrary, we conclude that . As a conclusion
We are now in position to state the representation theorem.
Theorem. For every , there is a unique progressively measurable process such that and
Proof. The uniqueness is immediate as a consequence of the Itō’s isometry for stochastic integrals. Let be the set of random variables such that there exists a progressively measurable process such that and From the above lemma, it is clear that contains the set of set of random variables
Since this set is total in , we just need to prove that is closed in . So, let be a sequence of random variables such that and in . There is a progressively measurable process such that and By using Itō’s isometry, it is seen that the sequence is a Cauchy sequence and therefore converges to a process which is seen to satisfy
As a consequence of the representation theorem, we obtain the following description of the square integrable martingales of the filtration .
Corollary. Let be a square integrable martingale of the filtration . There is a unique progressively measurable process such that for every , and In particular, admits a continuous version.
Exercise. Show that if is a local martingale of the filtration , then there is a unique progressively measurable process such that for every , and