$M(t)$ is a local continuous martingale and $\tau_{k}$ is a localizing sequence $\lim\limits_{n \rightarrow \infty}M(t\wedge \tau_{n})=M(t)$. I know that this is true for continuous martingales, but why do we need the continuity here. Does it not work without continuity ?
2026-02-23 18:45:19.1771872319
Why do we need continuity for the convergence of the localized martingale?
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