Let $\left\{ B\left( t \right) :t\geqslant 0 \right\} $ be the standard Brownian motion. For general curved boundary $b\left( t \right) $, the first hitting time is, $$\tau =\inf\left\{ t>0:W\left( t \right) \geqslant b\left( t \right) \right\} $$
This boundary $b\left( t \right) $ is continuously differentiable and satisfies $\underset{t\rightarrow 0^+}{\lim}b\left( t \right) >0$.
Now I want to ask, what is $$\mathbb{E}\left( \tau \right) $$ I want to know whether this problem has an analytical solution? Because I understand that the density of $\tau $ is not analytically tractable but requires numerical approximation. Ref
For some interesting exponential boundaries, it seems analytical results can be obtained here.
