Note: in 1905, Albert Einstein used the Brownian motion to prove the existence of atoms in one of his most famous contributions.

Wiener process

A Wiener process has the following characteristics:

(i) $W_0=0$;

(ii) $W_t$ is continuous;

(iii) the increments $\{W_{t_1}-W_{t_0}, W_{t_2}-W_{t_1},…,W_{t_k}-W_{t_{k-1}}\}$ are independent;

(iv) $W_{t}-W_{s} \sim \mathcal{N}(0,t-s)$ for any $0 \leq s \leq t \leq T$.

As a consequence of (i) and (iv) we can write a Wiener process as a normal variable with zero as expected value and time as variance:

\[W_t \sim \mathcal{N}(0,\sqrt{t}^2)\]

Brownian motion

A Brownian motion with drift $\mu$ and diffusion $\sigma^2$ is a random variable that depends on time and that solves the following SDE:

\[dX_t = \mu dt + \sigma dW_t\]

where $W_t$ is a standard Brownian motion (Wiener process).

The solution is:

\[X_t = \mu t + \sigma W_t\]

We write:

\[X_t \sim BM(\mu, \sigma^2)\]

Note: a Wiener process is often referred to a standard Brownian motion: $W_t \sim BM(0,1)$.