Facts about Processes of the Form $f(B_t,t)$
State some important facts about processes $X_t=f(B_t,t)$.
Answer
Martingality: Apply Itô’s lemma to the process and check that the drift term vanishes; when it does (and standard integrability holds), the process is a martingale. More generally, for any $f$, the process $f(B_t,t)$ minus its Itô drift is always a martingale.