# No drift brownian motion problem

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"?)

• Can you clarify your last sentence? I don't understand the role of $\tau$. And when you say "hit the barrier faster" do you just mean "earlier"? – Nate Eldredge May 27 '15 at 15:53
• Yes, earlier is better. For $\tau$ that's just to identify the value of time at which G1 hit the barrier P1 – Clemente Cortile May 27 '15 at 16:19
• Ok, so if we let $\tau_1 = \inf\{t : G_1(t) \ge P_1\}$ and $\tau_2 = \inf\{t : G_2(t) \ge P_2\}$, your question is "What is $\mathbb{P}(\tau_1 < \tau_2)$?" Is that right? – Nate Eldredge May 27 '15 at 16:23
• I'm also confused by your definition of the processes $G_1, G_2$. Is $G_1$ supposed to satisfy the SDE $dG_1 = \sigma_1 G_1 dW$ for $W$ a Brownian motion? If so, then $G_1$ is not itself a Brownian motion. Why does $W$ have a subscript $g$? And are both processes $G_1, G_2$ supposed to be driven by the same Brownian motion $W$? – Nate Eldredge May 27 '15 at 16:26
• Yes they are. Thanks for the pointers I'll edit the question. – Clemente Cortile May 27 '15 at 16:51