Say, we have an SDE
$$ \mathrm d X_t = f(X_t) \mathrm d t + \sigma(X_t) \mathrm d W_t $$
where $W_t$ is a Wiener process.
Assuming a strong solution exists globally (so the 1st and 2nd moments should be bounded), what is exactly
$$\mathbb E [X_t]$$
from the computation standpoint?
In discrete-time processes, if we have transition pdfs, it's quite clear, but in time-continuous case it seems difficult.
I tried to look up a pdf of $X_t$ knowing that of the driving noise, but couldn't find anything.