# Potentials of class D

A potential $\pi_t$ is a positive supermartingale with the condition that $\mathbb{E}[\pi_t]\rightarrow 0$ as $t \rightarrow 0$. What are the necessary/sufficient conditions for a potential to be of Class D, as defined by Meyer (so that Doob-Meyer decomposition applies)?

We say a cadlag supermatingale Z is of Class D if the collection $( Z_T : \textrm{T is a finite stopping time})$ is uniformly integrable.

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It's not clear to me what this question is asking for. The last sentence states the necessary and sufficient conditions, so it answers itself in a sense. I assume that you are expecting or hoping for "simpler" conditions in the case of a nonnegative supermartigale. But, what kind of simpler conditions are desired? –  George Lowther Aug 8 '12 at 17:36
@GeorgeLowther, in particular I was thinking about some integrability type conditions that would be at least sufficient. Checking directly for Class D seems rather difficult... –  Grzenio Aug 9 '12 at 9:00