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what are conditions for satisfying $\sup_{t\in[0,T]} M^2_t<\infty$ a.s. where $M_t$ is locally square integrable martingale?

I have no idea, besides using Doob's maximal inequalities, but they give bounds for expectation. may be some conditions should be imposed on $<M>_\infty$?

c-walk
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    Well unless mistaken those conditions must be quite stricts since it is not true even for a simple Brownian motion as $sup_t B_t^2=\infty$ a.s. – TheBridge Oct 05 '20 at 06:48
  • @TheBridge what about $\sup_{t\in[0,T]} M^2_t<\infty$ a.s.? – c-walk Oct 06 '20 at 03:44
  • If the expectation is finite dimensional thanks to Doob's inequality (you need at least separabiliry for that), then the supremum is finite a.s. – zhoraster Oct 06 '20 at 04:27
  • @zhoraster $M$ is optional. so $E(M^2_t)=E(_t)<\infty$ is enough? – c-walk Oct 06 '20 at 06:02
  • Well, I guess Doob inequality holds for optional martingales, but I'm not totally sure. You should try to find this in literature. – zhoraster Oct 07 '20 at 09:44

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