In the discrete case, we have the following theorem: Let $M_n$ be a squared-integrable martingale with quadratic variation $\langle M\rangle$. Then, $\displaystyle\frac{M_n}{\langle M\rangle_n}\rightarrow 0, a.s. $ on the event $\{\langle M\rangle_\infty=\infty\}$. Do we have a continuous version for that? Say:
Let $M_t$ be a continuous squared-integrable martingale with quadratic variation $\langle M\rangle_t$. Then, as $t$ goes to infinity $\displaystyle\frac{M_t}{\langle M\rangle_t}\rightarrow 0,$ a.s. on the $\{\langle M\rangle_t=\infty\}$.
If we have this version, could you recommend any reference?
Please refer to the book: Liptser, R. Sh. and A. N. Shiryayev, Theory of Martingales, Kluwer Academic Publishers, 1989. you could find a similar result about $ \dfrac{M_t}{\langle M\rangle_t}\to 0 $ a.s. on the $\{\langle M\rangle_t=\infty\}$ at page 141 of this book.