Since a Brownian motion does not have absolutely continuous paths, we can not directly use the theory of Riemann-Stieltjes integrals to give a sense to integrals like for every continuous stochastic process . However, if is regular enough in the Hölder sense, then can still be constructed as a limit of Riemann sums by using a celebrated result of L.C. Young. In the sequel, we shall denote by the space of – Hölder continuous functions that are defined on an interval .
Theorem.(Young’s integral) Let and . If , then for every subdivision , whose mesh tends to 0, the Riemann sums
converge, when to a limit which is independent of the subdivision . This limit is denoted and called the Young’s integral of with respect to .
As a consequence of the previous result, we can therefore use Young’s integral to give a sense to the integral as soon as the stochastic process has -Hölder paths with . This is not satisfying enough, since for instance the integral is not even well defined. The alternative to using Riemann sums to and studying its almost sure convergence is to take advantage of the quadratic variation of the Brownian motion paths and use the full power of probabilistic methods.
In what follows, we consider a Brownian motion which is defined on a filtered probability space . is assumed to be adapted to the filtration . We also assume that satisfies the usual conditions as they were defined at the begining of Lecture 10. These assumptions imply in particular the following facts that we record here for later use :
- A limit (in or in probability) of adapted processes is still adapted;
- A modification of a progressively measurable process is still a progressively measurable process.
Proposition. Let be a standard Brownian motion. We denote by its natural filtration: and by the null sets of .
Show that the filtration satisfies the usual conditions.
We denote by the set of processes that are progressively measurable with respect to the filtration and such that
Exercise. Show that the space endowed with the norm
is a Hilbert space.
We now denote by 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 . The set is often called the set of simple previsible processes. We first observe that it is straightforward that
The following theorem provides the basic definition of the so-called Itō integral.
Theorem. (Itō integral) There is a unique linear map
- For ,
- For ,
The map is called the Itō integral and for , we will use the notation
Proof: Since endowed with the norm
is a Hilbert space, from the isometries extension theorem we just have to prove that
- For ,
- The set is dense in .
Let . Due to the independence of the Brownian motion increments, we have:
Let us now prove that is dense in . We proceed in several steps. As a first step, let us observe that the set of progressively measurable bounded processes is dense in . Indeed, for , the sequence converges to .
As a second step, we remark that if is a bounded process, then is a limit of bounded processes that are in and such that almost every paths are supported in a fixed compact set (consider the sequence ).
As a third step, if is a bounded process such that almost every paths are supported in a fixed compact set, then the sequence is seen to converge toward . Therefore, is a limit of left continuous and bounded processes that are in and such that almost every paths are supported in a fixed compact set.
Finally, it suffices to prove that if is a left continuous and bounded process such that almost every paths are supported in a fixed compact set, then is a limit of processes that belong to . This may be proved by considering the sequence: