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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 $E$?

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1 Answer 1

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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.

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  • $\begingroup$ Oh, I did not know that DDS could be used even when the quadratic variation was not infinite a.s. Guess I learnt something new today.. $\endgroup$
    – Nate River
    Nov 14, 2022 at 10:26
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    $\begingroup$ Note that in [1] Proposition 1.26 appears before DDS. In many expositions of DDS the quadratic variation is assumed infinite, since in that case there is no need to enlarge the probability space. $\endgroup$ Nov 14, 2022 at 12:19

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