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 , *

**Exercise.**

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
where .
*

**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.**

In the proof of Dambis, Dubins-Schwarz, C_t is NOT necessarily continuous (e.g., let M_t be a BM, staying put for exponential times at the rings of an exponential clock). Hence the proof here is very much incomplete, isn’t it?

Thanks, this is now corrected