So I need to find $P(max_{0\leq v \leq t} W(v) \geq 2$ and $min_{T_{2} \leq v \leq t} W(v) \leq -1)$.
$T_{q}$ is the first passage time to level q
My progress:
If the first condition does not hold, then the second one fails as well by definition, so lets assume the first one does not fail.
In this case using the formula for the maximum of browning motion and the definition of probability of multiple events, my original expression turns into, using the formula $P(A and B) = P(A)*P(B|A)$:
$P(T_{2} \leq t, min_{T_{2} \leq v \leq t} W(v) \leq -1) = \int_0^t P(min_{T_{2} \leq v \leq t} W(v) \leq -1 | T_{2} = s)*f(s)*ds $ where f(s) is a pdf of $T_{2}$
Now I'll use the strong Markov property: W(v) is a brownian motion and $T_{2}$ is a stopping time, then $B(v) = W(v+T_{2}) - W(T_{2})$ is also a brownian motion, formed from the old one, but starting anew from the stopping time moment and independent of old values, so, matching the things I got in my equation, I get, that $B(v-T_{2}) + 2$ is also a brownian motion. Putting the expression inside the integral and simultaneously subtracting the stopping time from the minimum constraints on v, I get this:
$\int_0^t P(min_{0 \leq v - s \leq t - s} B(v - s) \leq -3 | T_{2} = s)*f(s)*ds$
Now, the running minimum has the same distribution as the running maximum (I can just flip the signs everywhere I assume)
$\int_0^t P(max_{0 \leq v - s \leq t - s} B(v - s) \geq 3 | T_{2} = s)*f(s)*ds = \int_0^t P(T_{3} \leq t - s | T_{2} = s)*f(s)*ds $
Now i'm confused. How do I get rid of this condition $T_{2} = s$? Or is it already the case and these 2 events are independent? Going by the gut feeling I feel like I can just write the final result, which follows:
$\int_0^t \int_0^{t-s} f(m)*f(s)dmds $
Where f(s) is the pdf of stopping time at 2 and f(m) is a pdf of stopping time at 3.
Sadly I'm not sure if this solution is correct. I'm mainly concerned about the switching of minimum and maximum, getting rid of the conditional probability and the proper usage of strong markov property I hope someone can help me resolve this.