Determining the change in $P(t)$ over the infinitesimal time $dt$

Question

A certain security has a price given by the following stochastic process: $$P(t) = S(t)e^{(r-q)r}, \hspace{20 pt} 0\le t \le T, \tau = T - t$$ where $S(t)$ is the price of a security following geometric Brownian motion. Can someone please help me determine the change in $P(t)$ over the infinitesimal time $dt.$ Also, is $P(t)$ a geometric Brownian motion?

Answer 1

Geometric Brownian motion is given as a differential equation. See here.

You can differentiate the function $P$ to get the desired $\frac{dP}{dt}$. You get $S'=\frac{dS}{dt}$ from the differential equation defining $S$.