I wonder whether there exists a continuous-time martingale satisfying the following: let $\mu$ be a centered probability measure whose support is compact (one can assume, if necessary, that support of $\mu$ is a finite set contained in an arbitrarily small neighborhood of zero).
Then, Does there exist a (bounded) martingale $X_t$ such that $X_0=0$ and it has terminal law $\mu$ (either at some finite terminal time $T$ or at infinity), and that its quadratic variation $dXdX$ is DETERMINISTIC?
Thank you.