Looking at a problem set for an introduction to Brownian motion: I think it's the syntax that's getting me confused, but I'm a bit stumped on the following question (I suspect that knowing what's going on with this one will let the rest fall into place in my mind)...
Anyway, without further ado:
Let $W={W_t:t≥0}$ be a Brownian motion. Find $\mathbb{E}(W_t(W_{t+1} + W_{t+2}))$.
Thanks in advance, all!
Well, in these sorts of things there's not really much to do except try to exploit the independence of the increments. So here, $W_{t+2}=W_{t}+W_{t+2}-W_{t}$ and $W_{t+1}=W_t+W_{t+1}-W_t$. Thus, we get
$$ \mathbb{E}W_t(W_{t+1}+W_{t+2})=2\mathbb{E} W_t^2+\mathbb{E}W_t(W_{t+2}-W_{t})+\mathbb{E}W_t(W_{t+1}-W_t) $$ And by independence and Gaussianity, we thus get
\begin{align} \mathbb{E}W_t(W_{t+1}+W_{t+2})&=2\mathbb{E} W_t^2+\mathbb{E}W_t\left(\mathbb{E}(W_{t+2}-W_{t})+\mathbb{E}(W_{t+1}-W_t)\right)\\ &=2t, \end{align} since the other terms are $0$.