In the previous Lecture, we proved that any martingale which is adapted to a Brownian filtration can be written as a stochastic integral. In this section, we prove that any martingale can also be represented as a time changed Brownian motion. To prove this fact, we give first first a characterization of the Brownian motion which is interesting in itself. In this section, we denote by a filtration that satisfies the usual conditions.
Proposition: (Levy’s characterization theorem) Let be a continuous local martingale such that and such that for every , . The process is a standard Brownian motion.
Proof. Let . By using Itō’s formula, we obtain that for ,
As a consequence, the process is a martingale and, from the above equality we get
The process is therefore a continuous process with stationary and independent increments such that is normally distributed with mean 0 and variance . It is thus a Brownian motion
The next proposition shows that continuous martingales behave in a nice way with respect to time changes.
Proposition Let be a continuous and increasing process such that for every , is a finite stopping time of the filtration . Let be a continuous martingale with respect to . The process is a local martingale with respect to the filtration . Moreover .
Proof. . By using localization, we may assume to be bounded. According to the Doob’s stopping theorem, we need to prove that for every bounded stopping time of the filtration , we have . But is obviously a bounded stopping time of the filtration and thus from Doob’s stopping theorem we have . The same argument shows that
Exercise. Let be an increasing and right continuous process such that for every , is a finite stopping time of the filtration . Let be a continuous martingale with respect to such that is constant on each interval . Show that the process is a continuous local martingale with respect to the filtration and that .
We can now prove the following nice representation result for martingales.
Theorem. ( Dambis, Dubins-Schwarz) Let be a continuous martingale such that and . There exists a Brownian motion , such that for every ,
Proof. Let . is a right continuous and increasing process such that for every , is a finite stopping time of the filtration and is obviously constant on each interval . From the previous exercise the process is a local martingale whose quadratic variation is equal to . From Levy’s characterization theorem, it is thus a Brownian motion
Exercise. Show that if is a continuous local martingale such that and , there exists a Brownian motion , such that for every ,
Let be a continuous adapted process and let be a Brownian motion. Show that for every , the process has Holder paths, where .
The study of the planar Brownian is deeply connected to the theory of analytic functions. The fundamental property of the Brownian curve is that it is a conformal invariant. The following proposition is easily proved as a consequence of Itō’s formula and of the Dambins-Dubins-Schwarz theorem. By definition, a complex Brownian motion is a process in the complex plane that can be decomposed as where and are independent Brownian motions.
Proposition.(Conformal invariance of the planar Brownian motion) Let be a complex Brownian motion and be an analytic function. Then
As a consequence, there exists a complex Brownian motion such that
To study the complex Brownian motion, it is useful to look at it in polar coordinates. It leads to the so-called skew-product decomposition of the complex Brownian motion.
Proposition. Let be a complex Brownian motion started at . There exists a complex Brownian motion such that
Proof. The proof is let as an exercise to the reader. The main idea is to prove, by using Itō’s formula, that and then to used the Dambins-Dubins-Schwarz theorem
Exercise. In the previous proposition, show that the process is independent from the process .
You will find below a video of a talk by Pr. Marc Yor concerning quadratic functionals of the planar Brownian motion. The talk was given at the University of Bristol in December 2008 for a special event.