A general Doob-Meyer-Mertens decomposition for g-supermartingale systems.

Summary We provide a general Doob-Meyer decomposition for g-supermartingale systems, which does not require any right-continuity on the system, nor that the filtration is quasi left-continuous. In particular, it generalizes the Doob-Meyer decomposition of Mertens [35] for classical supermartingales, as well as Peng's [40] version for right-continuous g-supermartingales. As examples of application, we prove an optional decomposition theorem for g-supermartingale systems, and also obtain a general version of the well-known dual formation for BSDEs with constraint on the gains-process, using very simple arguments.