let's say I have a brownian motion $W_t$, and I know the value of $W_1$. Is there a way to simulate the hitting time of $W_t$ and a given function $f(t)$ ? For instance I know that if $f(t)$ is a constant, the first hitting time of a constant A has a density that is known explicitely, but I'm assuming that for more complex function f, there is a not always a closed formula ? In that case, is there a known algorithm to simulate the first time t, such that $f(t) = W_t$ ?
thanks!