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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