In this Lecture we present some basic properties of the Brownian motion paths.

**Proposition.** Let be a standard Brownian motion.

**Proof**

Since the process is also a Brownian motion, in order to prove that

we just have to check that

Let . From the scaling property of Brownian motion, we have

Therefore we have

Now, we may observe that

Since the process is a Brownian motion independent of , we have for ,

Therefore we get

Thus,

and we can deduce that

and

Since this holds for every , it implies that

By using this proposition we deduce the following proposition whose proof is let as an exercise to the reader.

**Proposition** (Recurrence property of Brownian motion)

Let be a Brownian motion. For every and ,

Martingale theory provides powerful tools to study Brownian motion. We list in the Proposition below some martingales that are naturally associated with the Brownian motion and that will play an important role in the sequel.

**Proposition.** Let be a standard Brownian motion. The following processes are martingales (with respect to their natural filtration):

- ;
- ;
- , .

**Proof**

- First, we note that for , because is a Gaussian random variable. Now for , , therefore
- For , and for , , therefore
- For , , because is a Gaussian random variable. Then we have for , , and therefore

The previous martingales may be used to explicitly compute the distribution of some functionals associated to Brownian motion.

**Proposition.** Let be a standard Brownian motion. We denote for ,

For every , we have

Therefore, the distribution of is given by the density function

**Proof**

Let . For , we denote by the almost surely bounded stopping time:

Applying Doob’s stopping theorem to the martingale yields:

But for ,

Therefore from Lebesgue dominated convergence theorem, by letting , we obtain

Since by continuity of the Brownian paths we have,

we conclude,

The formula for the density function of is obtained by inverting the previous Laplace transform