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This question pertains to the topic titled: Is this local martingale a true martingale?

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"Using the Ito's formula I have shown that $X_t$ is a local martingale, because $\mathrm{d}X_t = \underline{\phantom{b_t}} \mathrm{d} B_t$",

Can someone please explain this opening phrase to the original question to me further? I can not get my head around why $X_t$ would not indeed be a true martingale if it is written in this form (all Ito integrals are martingales! (?)).

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    In order to make any statements about $\mathsf{E}\left(X_t| \mathcal{F}_s\right)$ one needs to make sure that the expectation is defined.2012-08-31

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