Let $\{W_t:t \ge 0\}$ be a Brownian motion. Find for all $0 \le s < t$:
- $\mathbb{E}\left[W_s W_t \right]$
- $\mathbb{E}\left[W_s W_t^2 \right]$
- $\mathbb{E}\left[W_s^2 W_t^2 \right]$
- $\mathbb{E}\left[\left. e^{2W_t} \right| \mathcal{F}_s\right]$
Hint: Recall that the kurtosis for $Z \sim \mathcal{N}(0,1)$ is $\mathbb{E}\left[Z^4\right] = 3$.
I am confused. I think based on the theory, $\mathbb{E}\left[W_s W_t \right]$ should be $s$, and then I do not understand what I should do for $\mathbb{E}\left[W_s W_t^2 \right]$.
HINT
Recall that you want to use independent increments, so $$ \mathbb{E}[W_s W_t] = \mathbb{E}[W_s (W_s + W_{s,t})] = \mathbb{E}[W_s^2] + \mathbb{E}[W_s W_{s,t}] = s+\mathbb{E}[W_s] \mathbb{E}[ W_{s,t}] = s, $$ where $W_{s,t}$ is the process starting at $s$ and evolving until $t$.
Can you use a similar argument to solve others?