## Lecture 12. The Brownian motion: Definition and basic properties

Definition: Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. A continuous real-valued process $(B_t)_{t \ge 0}$ is called a standard Brownian motion if it is a Gaussian process with mean function

$m(t)=\mathbb{E}(B_t)=0$

and covariance function

$R(s,t)=\mathbb{E}(B_sB_t)=\min (s,t).$

It is seen that $R(s,t)=\min (s,t)$ is a covariance function, because it is symmetric and for $a_1,...,a_n \in \mathbb{R}$ and $t_1,...,t_n \in \mathbb{R}_{\ge 0}$,

$\sum_{1 \le i,j \le n} a_i a_j \min (t_i,t_j) =\sum_{1 \le i,j \le n} a_i a_j \int_0^{+\infty} \mathbf{1}_{[0,t_i]} (s) \mathbf{1}_{[0,t_j]} (s) ds$

$= \int_0^{+\infty} \left( \sum_{i=1}^n a_i \mathbf{1}_{[0,t_i]} (s) \right)^2 ds \ge 0.$

The distribution probability of a standard Brownian motion is called the Wiener measure. It is probability measure on the space of continuous functions $[0, +\infty) \to \mathbb{R}$ (See Lecture 3).

Similarly, a $n$-dimensional stochastic process $(B_t)_{t \ge 0}$ is called a standard Brownian motion if

$(B_t)_{t \ge 0}=(B^1_t,\cdots,B^n_t)_{t \ge 0}$

where the processes $(B^i_t)_{t \ge 0}$ are independent standard Brownian motions.

Of course, the definition of Brownian motion is worth only because such an object exists.

Theorem.
There exist a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a stochastic process on it which is a standard Brownian motion.

Proof

From the Daniell-Kolmogorov existence theorem, there exists a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a Gaussian process $(X_t)_{t \ge 0}$ on it, whose mean function is $0$ and covariance function is

$\mathbb{E} (X_s X_t)=\min (s,t).$

We have for $n \ge 0$ and $0 \le s \le t$:

$\mathbb{E} \left( (X_t - X_s)^{2n} \right)=\frac{(2n)!}{2^n n!} (t-s)^n.$

Therefore, by using the Kolmogorov continuity theorem, there exists a modification $(B_t)_{t \ge 0}$ of $(X_t)_{t \ge 0}$ whose paths are locally $\gamma$-Hölder if $\gamma \in [0,\frac{n-1}{2n} )$ $\square$

From the previous proof, we also easily deduce that the paths of a standard Brownian motion are locally $\gamma$-Hölder for every $\gamma <\frac{1}{2}$. It will later be shown that they are not $\frac{1}{2}$-Hölder (It is a consequence of the law of iterated logarithm).

The following exercises give some first basic properties of Brownian motion. In these exercises, $(B_t)_{t \ge 0}$ is a standard one-dimensional Brownian motion.

Exercise.
Show the following properties:

• $B_0=0$ a.s.;
• For any $h \geq 0$, the process $(B_{t+h} - B_h)_{t \ge 0}$ is a standard Brownian motion;
• For any $t>s\geq 0$, the random variable $B_t -B_s$ is independent of the $\sigma$-algebra $\sigma(B_u, u \le s )$.

Exercise.(Symmetry property of the Brownian motion)

• Show that the process $(-B_t)_{t \ge 0}$ is a standard Brownian motion.
• More generally, show that if

$(B_t)_{t \ge 0}=(B^1_t,\cdots,B^d_t)_{t \ge 0}$

is a $d$-dimensional Brownian motion and if $M$ is an orthogonal $d \times d$ matrix, then $(MB_t)_{t \ge 0}$ is a standard Brownian motion.

Exercise.(Scaling property of the Brownian motion)

Show that for every $c >0$, the process $(B_{ct})_{t \geq 0}$ has the same distribution as the process $(\sqrt{c} B_t)_{t \geq 0}$.

Exercise.(Time inversion property of Brownian motion)

• Show that almost surely, $\lim_{t \to +\infty} \frac{B_t}{t} =0$.
• Deduce that the process $(t B_{\frac{1}{t}})_{t \geq 0}$ has the same law as the process $( B_t)_{t \geq 0}$.

Exercise.(Non-canonical representation of Brownian motion)

• Show that for $t \ge 0$, the Riemann integral $\int_0^t \frac{B_s}{s} ds$ almost surely exists.
• Show that the process $\left( B_t-\int_0^t \frac{B_s}{s}ds\right)_{t \ge 0}$ is a standard Brownian motion.

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