I am trying to solve the perpetual American option problem. Currently I'm following this (slide 9). The stock price is modelled as Ito's process.

$dS_t = (\mu-D_0)S_tdt\ +\ \sigma S_tdW_t $

where $D_0$ is the dividend. Basically the problem tries to find optimal stopping time to exercise the option i.e. we try to optimize the value function $V(S)$ w.r.t stopping time $\tau$.

$V(S) = sup_{\tau \in T}\ \mathbb{E}[e^{-r\tau}max(S_\tau -K,\ 0)] $

I'm stuck in deriving the partial derivative of $V(S)$ w.r.t. time.

In my opinion $V(S)$ can be written as follows (where we integrate over all the values of time considering the probability density function of time $u$ being the stopping time $\tau$)

$V(S) =sup_{\tau \in T}\ \int_0^\infty\ e^{-ru}max(S_u-K,\ 0)\ \mathbb{P}(\tau = u)\ du$

According to the reference (slide 9), the partial derivative of $V(S)$ w.r.t. time comes out to be $-rV(S)$ which is not very clear to me. So please help me out with in-depth proof, if possible.

Thanks :)


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