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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)
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