Brownian Motion

The Wiener process $W_t$ is characterized by four facts:

• $W_0=0$
• $W_t$ is almost surely continuous
• $W_t$ has independent increments
• $W_t-W_s\sim \mathcal{N}(0,t-s)$ (for $0\le s\le t$)

$\mathcal{N}(\mu ,{\sigma }^2)$ denotes the normal distribution with expected value $\mu$ and variance ${\sigma }^2$. The condition that it has independent increments means that if $0\le s_1<t_1\le s_2<t_2$ then $W_{t_1}-W_{s_1}$ and $W_{t_2}-W_{s_2}$ are independent random variables. In addition, for some filtration ${\mathcal{F}}_t$, $W_t$ is ${\mathcal{F}}_t$-measurable for all $t\ge 0$.