Here is my question. I've done the first part, but I'm stuck on the second. If I can work out (/be advised) how to do the second, then I hope to be able to do the third similarly.
Please note: While I'd really appreciate help on this question, please don't just tell me the answer! I want to learn from the question, so I'd love it if someone would advise me of the next step to take.
I'm really unsure how to proceed at all. Let me write $$S_a^{-b} = \sup_{0 \le s \le t} (W_s - bs) \;\ \text{and} \;\ \{ T_{a,b} > t \} = \{ S_a^{-b} \le a \}.$$ So we have a nice (well, explicit) expression for $1(T_{a,b} < t)$. However, I can't work out how to combine this with the rest of the expectation. We can say $$ E(e^{\theta W_t} 1(T_{a,b} < t)) = E(e^{\theta W_t} \mid S_a^{-b} \le a)P(S_a^{-b} \le a),$$ and we have an explicit expression for the latter, but then this first expectation is equally difficult.
We know that $Z_t = \mathcal{E}(\theta W)_t = e^{\theta W_t - \theta^2 t/2}$ defines a UI martingale, and so I hoped to maybe use OST to convert the $e^{\theta W_t}$ into $e^{\theta t/2}$, but I can't see how to do that (or even if that would be a thing). Certainly deriving the third part from the second it looks like it would (/could) use this.
Any advise would be most appreciated. :)