Let $Y$ be an exponential random variable with rate parameter $\lambda$. Let $T_{a}$ be the first hitting time of a Brownian Motion. I want to find $ P(\min(T_{a}, T_{-a}) < Y) $
In order to compute this, I want to find a martingale, then apply the stopping theorem. Can you suggest me a martingale to begin with?
edit: I used the exponential martingale and I got stuck in this equation: $ E[e^{cB_{min(T_{a}, T_{-a})} - \frac{c^{2}}{2}min(T_{a}, T_{-a})}] = P(T_{a}
Then $ P(T_{a}>T_{-a}) = P(T_{a}
Then what to do?