Distribution of difference of hitting times of Brownian motion

Question

Let $T_a$ and $T_{-b}$ be two hitting times for a standard Brownian motion where $a, b > 0$. How can I compute the probability $P(T_a > T_{-b}) $ using the reflection principle of the BM?

Answer 1

Hint

Set $T=T_a\wedge T_{-b}$. Using reflection principle allows you to prove that $\mathbb E[T]<\infty $. Using Wald's identity gives $$0=\mathbb E[B_{T}]=a(1-\mathbb P\{T_a>T_{-b}\})-b\mathbb P\{T_a>T_{-b}\}.$$

I let you conclude.