Ok, I'm new to stochastic calculus and I'm having some troubles with a simple exercise that I don't seem to get. Here it is:
Recalling that $\mathbb{E}[e^{W_t}]=e^{\frac{t}{2}}$ compute $\mathbb{E}[e^{W_t}|\mathcal{F}_s]$, with $s\leq t$.
Now the expectation that I'm supposed to remember is simply an expectation of a lognormal distribution with mean zero and variance $t$. I don't get the second point: how can the introduction of filtration affect the expected value? I would appreciate an explanation as general as possible, in order to avoid having problems with similar exercises even in the future... thanks!
$\mathbb{E}[e^{W_t}|\mathcal{F}_s]$
$=\mathbb{E}[e^{W_s}e^{W_t - W_s}|\mathcal{F}_s]$
$=e^{W_s} \mathbb{E}[e^{W_t - W_s}|\mathcal{F}_s]$ (why?)
$=e^{W_s} \mathbb{E}[e^{W_t - W_s}]$ (why?)
Now note that $W_t - W_s$ is a random variable distributed $N(0, t-s)$.
Do you remember moment generating functions?
As to how filtration can affect expectation, in general:
$E[X|Y] \neq E[X]$
where Y can be an event, random variable, $\sigma$-algebra or filtration.
Do you remember conditional expectation?