I was hoping someone could answer a few basic Brownian Motion questions.
A Brownian Motion, $B_t$, (let us assume defined on the continuous path space with the Brownian Measure starting at $0$) has the following properties:
1) For $0 \leq t_0 < t_1, ..., t_n$ we have $B_{t_1} - B_{t_0}, ..., B_{t_n} - B_{t_{n-1}}$ are independent.
2) For $t \geq s, B_t - B_s$ is $N(0,t-s)$.
3) The map $t \rightarrow B_t$ is almost surely continuous.
4) $B_0 = 0$
Let $\mathcal{F}_t$ be the "infinitesimal peak into the future" filtration. I have seen the following claims.
a) $B_t - B_s$ is independent of $\mathcal{F}_s$
b) $\mathbb{E}(B_t - B_s| \: \mathcal{F}_s$) is $N(0,t-s)$. EDIT: This is incorrect
c) $B_t$ is a martingale wrt $\mathcal{F}_t$.
Could someone explain a) and b) for me? I believe a) follows from 1) but I am not 100% certain. b) is stating that the unconditional distribution (property 1)) and the conditional distribution are the same, which I find confusing.
My problem with c) is the following: If $B_t$ is a martingale wrt $\mathcal{F}_t$, why is it not the case that
$\mathbb{E}(B_t - B_s| \: \mathcal{F}_s) = \mathbb{E}(B_t|\: \mathcal{F}_s) -\mathbb{E}(B_s| \: \mathcal{F}_s) = B_s - B_s = 0$? (thus contradicting b))