# Expectations of Brownian motion (simple, I hope)

1. Sep 24, 2011

### AxiomOfChoice

Let $B_t$ be Brownian motion in $\mathbb R$ beginning at zero. I am trying to find expressions for things like $E[(B^n_s - B^n_t)^m]$ for $m,n\in \mathbb N$. So, for example, I'd like to know $E[(B^2_s - B^2_t)^2]$ and $E[(B_s - B_t)^4]$. Here are the only things I know:
1. $$E[B_t^{2k}] = \frac{(2k)!}{2^k \cdot k!}$$
2. $$E[B_s B_t] = \min(s,t)$$
3. $$E[(B_s - B_t)^2] = |s-t|$$
4. Brownian motion has independent increments.
But I'm having a hard time getting expressions for the expectations I listed in the start of the question using these facts. Can someone help?

2. Sep 24, 2011

### AxiomOfChoice

For example, in trying to compute $E[(B_s - B_t)^4]$, one comes up against things like $4E[B_s^3B_t]$. How in the world am I supposed to deal with that?