# PDE for holding a stock

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?

• Your definition of the porfolio value is a bit strange, you are just discounting the asset at the stopping time. Dec 23, 2019 at 20:13
• @Canardini Why is it strange? Isn't it a standard risk neutral valuation? Dec 24, 2019 at 16:59
• What is $V$ then ? What are you trying to value ? I read your formula as the value of receiving the stock at time $\tau$ Dec 24, 2019 at 17:01
• @Canardini I agree that with a stock it doesn't make a lot of sense as the discounting should produce the current stock price because of no arbitrage condition. But we can consider S as some general portfolio. Is it reasonable? Dec 30, 2019 at 2:56
• The portfolio will have the same property, you are discounting a tradable asset, therefore is a martingale. Dec 30, 2019 at 15:12