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

TL;DR

This paper establishes a broad Doob-Meyer decomposition for $g$-supermartingale systems without requiring right-continuity, extending classical results and enabling new applications in BSDEs with constraints.

Contribution

It introduces a general Doob-Meyer decomposition for $g$-supermartingale systems without right-continuity, broadening the scope of previous results.

Findings

Generalized Doob-Meyer decomposition for $g$-supermartingales

Derived optional decomposition theorem for $g$-supermartingales

Provided a dual formulation for constrained BSDEs

Abstract

We provide a general Doob-Meyer decomposition for $g$-supermartingale systems, which does not require any right-continuity on the system. In particular, it generalizes the Doob-Meyer decomposition of Mertens (1972) for classical supermartingales, as well as Peng's (1999) 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.