Brownian Motion

Definition

A Wiener process $W_t$ is characterized be the following properties

• $W_0=0$ Almost Surely
• $\forall t>0,W_{t+u}-W_t$ are independent of past values $W_s$ where $u\ge 0$ and $s<t$.
• $\forall 0\le s<t,W_t-W_s\sim N(0,t-s)$
• $W_t$ is Almost Surely continuous on $t\in [0,\infty )$