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In this paper, we study a new type of BSDE, where the distribution of theY-component of the solution is required to satisfy an additional constraint,written in terms of the expectation of a loss function. This constraint isimposed at any deterministic time t and is typically weaker than the classicalpointwise one associated to reflected BSDEs. Focusing on solutions (Y, Z, K)with deterministic K, we obtain the well-posedness of such equation, in thepresence of a natural Skorokhod type condition. Such condition indeed ensuresthe minimality of the enhanced solution, under an additional structuralcondition on the driver. Our results extend to the more general framework wherethe constraint is written in terms of a static risk measure on Y. Inparticular, we provide an application to the super hedging of claims underrunning risk management constraint.

BSDEs with mean reflection