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Is every submartingale a convex function of a martingale?

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  • $\begingroup$ Real values? Same filtration? $\endgroup$
    – Gerald Edgar
    Commented May 2, 2013 at 19:30
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    $\begingroup$ No, but there are results in that direction: projecteuclid.org/… $\endgroup$
    – Bill Johnson
    Commented May 2, 2013 at 19:31

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Not every. Every convex function is either bounded below or monotonic. The first kind cannot produce any submartingale that is not bounded below. The second kind cannot produce any submartingale that, at some point in time, might be certain to increase. It is easy to produce a submartingale satisfying both conditions.

If it is bounded below, $|x|-B$ will do. In other words, given a positive submartingale, we can add signs to make it a martingale: at each step, choose the probability of switching signs to cancel out the growth in expectation.

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