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Is it possible to decompose a discrete-time martingale $(M_n)$ uniquely into two processes $M_n=M_n^I+A_n$ where $(M^I_n)$ is a martingale with independent increments and $(A_n)$ is a martingale? If no, under which condition(s) on $(M_n)$, do we have this kind of decomposition?

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    I should be clear that in the examples of adding and subtracting another nontrivial martingale, it may require augmenting the filtration to which the process is adapted. :-)2012-10-10

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