# zbMATH — the first resource for mathematics

A general Doob-Meyer-Mertens decomposition for $$g$$-supermartingale systems. (English) Zbl 1343.60050
Summary: We provide a general Doob-Meyer decomposition for $$g$$-supermartingale systems, which does not require any right-continuity of the system, nor that the filtration is quasi left-continuous. In particular, it generalizes the Doob-Meyer decomposition of J.-F. Mertens [Z. Wahrscheinlichkeitstheor. Verw. Geb. 22, 45–68 (1972; Zbl 0236.60033)] for classical supermartingales, as well as S. Peng’s [Probab. Theory Relat. Fields 113, No. 4, 473–499 (1999; Zbl 0953.60059)] 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 formulation for BSDEs with constraints on the gains-process, using very simple arguments.

##### MSC:
 60G48 Generalizations of martingales 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
Full Text: