I'm tasked with solving an optimal stopping problem relating to stochastic process representing a firms profit namely $X_t = X_0 + \mu t + \sigma Wt$ where $X_0, \mu$ and $\sigma$ are constants.
specifically I need to find $V(x,\mu) = \sup_\tau E^x [\int_{0}^{\tau} e^-rs Xs \,ds]$ which after some calculations reduces to $F(x) - \inf_\tau [ E^x [ $$e^-r\tau$$ F(x_\tau) ]]$
F(x) can now be found using Monte Carlo methods, however working out $\inf_\tau [ E^x [ e^-r\tau F(x_\tau) ]$ is proving to be very tricky. So far I've been trying to approximate it using a binomial tree and the Nelson Ramaswamy scheme ( I've succesfully approximated $X_s$ using Nelson Ramaswamy and have been able to work out $\inf_\tau [ E^x [ e^-r\tau X_s ]$ using binomial trees much in the same way as one would for an American option). However I've been struggling to approximate the cdf of $X_s$ using binomial trees and so have struggled computing $\inf_\tau [ E^x [ e^-r\tau F(x_\tau) ]$
Any help would be greatly appreciated!
