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!


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