Let's suppose we owe an asset and its path is given by the following SDE: $$ S(t) = A(S, t) dt + B(S, t) d\widetilde{W}$$ where $\widetilde{W}$ is a Brownian motion (under risk neutral measure). What is the optimal strategy of selling this asset? What is the value of a portfolio consisting of this single asset?
My understanding is that there is a certain threshold the asset reaches that should trigger the sell. If the value of the portfolio is $V(S(t), t)$ then $$V(x, t) = \widetilde{\mathbb{E}}[e^{-\int_t^{\tau}R(s)ds}S(\tau)]$$ ($x = S(t)$, and $\tau$ is a stopping time for the threshold.)
My question: what is the PDE governing $V(x, t)$? How do I get (derive) it? What is the general logic of getting PDEs for american style derivatives? I understand the presentation Shreve gives about linear complementarity conditions in his book "Stochastic calculus for finance, vol II".
Another question (somewhat of a digression): if the SDE is in real world measure, how do we get PDEs? Only by transforming the measure through Girsanov's theorem?