A new Mertens decomposition of Y^g,ξ-submartingale systems. Application to BSDEs with weak constraints at stopping times

We first introduce the concept of Y^g,ξ-submartingale systems, where the nonlinear operator Y^g,ξ corresponds to the first component of the solution of a reflected BSDE with generator g and lower obstacle ξ. We first show that, in the case of a left-limited right-continuous obstacle, any Y^g,ξ-submartingale system can be aggregated by a process which is right-lower semicontinuous. We then prove a Mertens decomposition, by using an original approach which does not make use of the standard penalization technique. These results are in particular useful for the treatment of control/stopping game problems and, to the best of our knowledge, they are completely new in the literature. As an application, we introduce a new class of Backward Stochastic Differential Equations (in short BSDEs) with weak constraints at stopping times, which are related to the partial hedging of American options. We study the wellposedness of such equations and, using the Y^g,ξ-Mertens decomposition, we show that the family of minimal time-t-values Y_t, with (Y,Z) a supersolution of the BSDE with weak constraints, admits a representation in terms of a reflected backward stochastic differential equation.