I am reading Karatzas and Shreve's Brownian Motion and Stochastic Calculus.
Let $M_t$ be a continuous local martingale.
On page 157, it wrote that "because $\langle M\rangle_t = t$, we have $M \in \mathcal{M}_2^c$",
where $\mathcal{M}_2^c$ means the collection of continuous square integrable martingale.
Can you tell me why it is true?
It is true that a local martingale of class DL is a martingale.
However, I do not think that the condition there is concerned with class DL or uniformly integrability, because as you know, even a continuous, local martingales with uniformly integrability fail to be martingale.
Sincerely.
Best Answer
This is much easier than your comments suggest:
$$\mathbb{E}(M_t^2)=\mathbb{E}(\langle M \rangle_t)=t<\infty$$