Let $M$ be a continuous martingale. Denote by $E$ the event that its total quadratic variation is finite, i.e.


$$E := \{\langle M, M \rangle\_\infty < \infty\}.$$


**Question:** Is it true that as $t \to \infty$, $M\_t$ converges almost surely on $E$?



Does a continuous martingale converge almost surely on the event that its quadratic variation is finite?
It is true even for local Martingales, see Proposition 1.26 page 124 in [1].


Here is an intuitive way to understand it:
The proof of the Dambis-Dubins-Schwarz theorem [2, 3] (see also [1,4]) implies that (on an enlarged probability space) we can write $M\_t=B\_{\langle M, M \rangle\_t}$ for some Brownian motion $B$, and the existence of the limit follows from continuity of $B$.


[1] Revuz, Daniel, and Marc Yor. Continuous martingales and Brownian motion. Vol. 293. Springer, 1990.


[2] Dubins, Lester E., and Gideon Schwarz. "On continuous martingales." Proceedings of the National Academy of Sciences 53, no. 5 (1965): 913-916.


[3] Dambis, Karl E. "On the decomposition of continuous submartingales." Theory of Probability & Its Applications 10, no. 3 (1965): 401-410.


[4] <https://www.iam.uni-bonn.de/fileadmin/user_upload/gubinelli/stochastic-analysis-ss20/sa-ss20-script-6.pdf> See Theorem 1 page 2.


