A new Mertens decomposition of ${\mathcal{Y}}^{g,\xi }$-submartingale systems. Application to BSDEs with weak constraints at stopping times

Abstract

We first introduce the concept of ${\mathcal{Y}}^{g,\xi }$-submartingale systems, where the nonlinear operator ${\mathcal{Y}}^{g,\xi }$ corresponds to the first component of the solution of a reflected BSDE with generator $g$ and lower obstacle $\xi$. We first show that, in the case of a left-limited right-continuous obstacle, any ${\mathcal{Y}}^{g,\xi }$-submartingale system can be aggregated by a process which is right-lower semicontinuous. We then prove a \textit{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 \textit{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 ${\mathcal{Y}}^{g,\xi }$-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.