A real-valued stochastic process $B=(B_t,t\ge 0)$ is called Brownian motion $:\Leftrightarrow$
- $B_0=0$
- $B$ has independent and stationary increments, i.e. $$\left(X_{t_i}-X_{t_{i-1}}\right)_{i=1,\ldots,n}\;\;\;\text{are independent}$$ for all $n\in\mathbb{N}$ and $0\le t_0<\ldots<t_n$ and $$\mathcal{L}\left[X_{s+t}-X_s\right]=\mathcal{L}\left[X_t-X_0\right]$$ for all $s,t\ge 0$ (where $\mathcal{L}[X]$ denotes the distribution of $X$)
- $B_t\text{ ~ }\mathcal{N}_{0,t}$, i.e. $B_t$ is normally-distributed with expectation $0$ and variance $t$, for all $t>0$
- The paths $t\mapsto B_t$ are almost surely continuous
Now, I've read that $B$ is a martingal. My first question was: "Martingal with respect to which filtration?". I assume we're considering the generated filtration $$\mathbb{F}=(\mathcal{F}_t,t\ge 0)\;\;\;\text{with }\mathcal{F}_t=\sigma(B_0,\ldots,B_t)$$ However, the proofs that I've seen so far generally start with: "Since the increments $X_t-X_s$ are independent of $\mathcal{F}_s$ for all $t>s$, $\ldots$"
But why is this true? I absolutely don't get it.