Let $X_t,Y_t$ be $(\mathbb{P},\mathscr{F}_t)$-semi-maringales. Under what conditions does there exist a strict-martingale $Z_t>0$ such that $$ Z_t = X_t + Y_t? $$
Interest
In the case where $Z_t$ exist, I guess we can construct a measure change making $X_t$ into $X_t+Y_t$ using Girsanov....