Assume that $M = (M_t)_{t\geq 0}$ is a uniformly integrable continuous local martingale with $M_0 \in L^1$ and $\lim_{t\to\infty} M_t = M_\infty$ almost surely. By Fatou's Lemma we know that $E[M_\infty] \leq E[M_0]$.
Question: What are equivalent conditions for the equality $E[M_\infty] = E[M_0]$ to hold?
More information: A necessary condition for the above condition to hold is that $M$ is a (uniformly integrable) martingale. Moreover, Novikov's and Kazamaki's criterion provide sufficient conditions. Also, I am mostly interested in the case where $M$ is non-negative (and therefore automatically a supermartingale; then the equality actually holds iff $M$ is a martingale).