Given two same brownian motion with no drift and different variances:
$$dG_1= \sigma_1 G_1 dW $$ $$dG_2= \sigma_2 G_2 dW $$
and two barriers $P_1 > P_2$
assuming that $ \sigma_1 > \sigma_2 $ and given $τ_1=\inf [t:G_1(t)≥P_1] $; $τ_2= \inf [t:G_2(t)≥P_2]$
is it possible to find $P(τ_1<τ_2)$?
(the probability that a certain time $\tau\ $ "G1 will hit the barrier P1 earlier than G2 hits P2"?)