I was just wondering why the function f used in Dynkin's formula must have compact support? What do you get by this?
Dynkin's formula:
Let f $\in$ $C_0^2$($R^n$). Suppose $\tau$ is a stopping time, $E^x$[$\tau$] < $\infty$. Then
$E^x$[f($X_\tau$)] = f(x) + $E^x$[$\int_0^\tau$ Af($X_s$)ds]
As mentioned in 4.2 proposition in 5.4 Shreve-Karatzas
if the $f$ is not compactly supported, then we still get local martingale i.e. we get Dynkin's formula but after taking minimum with some sequence of stopping times $\tau_{k}$. The compactly supported assumption is needed in order to get finiteness for generic stopping time. I suppose depending on the process you could do with less because you just need to able to apply the dominated convergence theorem.