I know that if $X_t$ is a semi-martingale then $X_t$ can be written as the sum of an martingale $M_t$ and a nice predictable process $A_t$.
I was wondering if the converse is false. That is possible to have two processes $Y_t$ and $Z_t$, each not being semi-martingales but their sum $$ Y_t+Z_t $$ is a non-deterministic semi-martingale? (My intuition says no, but maybe I'm overlooking something).
If not is it possible for the process $$ Y_tZ_t $$ to be a non-deterministic semi-martignale?