Itô integral probability distribution

Question

I know in general this must not have an analytical expression in terms of common functions, but how do you (at least in theory) get the probability distribution of $X_t$ for a given $t$ in the integral

$$ X_t = \int_0^t f(t)\, \mathbb{d}W_t $$

Answer 1

As Did has commented, if $f(s)$ is deterministic, then $X_t$ will be a centered normal random variable with variance $$ \mathbb{E}(X_t^2) = \int^t_0 f(s)^2 \,\mathbb{d}s $$

For references, see this, pg 113.