I am currently going through problems in Oksendal's intro SDE and stuck with this problem. I was wondering if I could get some help with it. I would sincerely appreciate if you would express out all the details, so that I can follow your thought process step by step.
So, the question goes like:
Getting an expression or value for the optimal stopping time for the following problem:
$$G^{*(s,p)} = sup_\tau E^{(s,p)}[\int^\infty_\tau e^{-\rho(s+t)}P_tdt-Ce^{-\rho(s+\tau)}]$$
where $$dPt = \mu dt + \sigma dBt$$ $$ P_0 = p $$ with $\mu, \sigma \neq 0$ constants.
Here's my idea about solving this question. I will have to come up with an expression for $Pt$ and then plug it into $\int^\infty_\tau e^{-\rho(s+t)}P_tdt$ to arrive at $\int^\tau_0 -e^{-\rho(s+t)}P_tdt$, which will give me:
$$G^{*(s,p)} = sup_\tau E^{(s,p)}[\int^\tau_0 -e^{-\rho(s+t)}P_tdt-Ce^{-\rho(s+\tau)}]$$
Then, I suspect that I will have to use the relationship $L\phi + f = 0$ with $f$ being $ -e^{-\rho(s+t)}$, $L$ being the generator, and $\phi$ being the entire expression in the brackets to get $T_D$ which will make $L\phi + f$ equal to zero.
Is this correct? or am I wildly off-course? Would you show me the entire process so that I can follow your thought process?
Oh, now, I think I understand the question. It seems like I was quite off the point. A combination of (10.3) and Theorem 10.4.1 seems to solve the question.
Namely, converting the given Brownian motion into a geometric one, then, into $dYt$ and $dZt$, correspondingly, and also, noticing that the given function following $f$ in the brackets satisfies the conditions in Theorem 10.4.1 seem to be the key points in this question. Not 100% sure, but these seem to make sense.