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