Suppose (W i t )t>0, i ∈ {1, 2} are two independent Wiener processes. Let $\tau = inf (t > 0 : |W_{t}2 | = a) $ for a > 0. Show that the random variable W 1 τ has a density f (x) = (2a cosh(πx/(2a)))−1.
I do not know how to solve this exercise
I think you can use that $f(t)=(\E \Phi(t/\sqrt{\tau} ))'=\E (1/\sqrt{\tau})(1/\sqrt{2\pi})\exp(-t^2/(2\tau))$ And computing this expectancy knowing the definition of $\tau$. But I dont know how this can follow...