A process $(B_{t})_{t\ge{0}}$ on $(\Omega,\mathcal{F},P)$ is called a standard Brownian motion if 1. $B_{0}=0$ 2. Increments are normally distributed: $$\forall 0\le s<t:B_{t}-B_{s}\sim \mathcal{N}(0,t-s)$$ 3. Independent Increments: $$\forall 0\le s<t:B_{t}-B_{s}\perp\!\!\!\perp\mathcal{F}_{s}^{B}$$where $$\mathcal{F}_{s}^{B}=\sigma(B_{u}:0\le u<s)$$and $\mathcal{F}_{t}^{B}$ is the natural filtration of $(B_{t})_{t\ge 0}$ 4. Continuity: $\forall\omega \in\Omega:t\mapsto B_{t}(\omega)$ is continuous in $\mathbb{R}^{+}$