Let $(\nu_t)_{t \in [0,1]}$ be Borel probability measures on a stochastic basis $(\Omega,\mathcal{F},(\mathcal{F}_{t \in [0,1]})_t,\mathbb{P})$.

Does there exist a *semi-martingale* $(X_t)_{t\in[0,1]}$ on this stochastic basis, such that $X_t\sim \nu_t$ for everty $t \in [0,1]$?  If not, what conditions are needed on the measures $\nu_{\cdot}$ for this to be possible?