# Passage time and historiacl maximum of geometric Brownian Motion

I need to prove two ineqaulities about a geometric Brownian Motion:

for $$v>0$$, $$c>0$$, define

$$R_c:=\inf \{t>0:e^{vW(t)-\frac12v^2t}=c\}$$ as the first passage time of a geometric BM where $$W(t)$$ is a standard BM.

I want to show that $$\mathbb P[R_c<\infty]=\frac1c, \mathbb ER_c=\frac{2\log c}{v^2}$$ Thank you so much!