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

TL;DR

This paper introduces a novel Mertens decomposition for $\\mathscr{Y}^{g,\xi}$-submartingale systems, providing new tools for control/stopping games and weakly constrained BSDEs, with applications to American option hedging.

Contribution

It presents a new Mertens decomposition for $\\mathscr{Y}^{g,\xi}$-submartingale systems without penalization, advancing the analysis of BSDEs with weak constraints at stopping times.

Findings

Established aggregation of $\\mathscr{Y}^{g,\xi}$-submartingale systems by right-lower semicontinuous processes.

Proved a Mertens decomposition for these systems using an original approach.

Applied the decomposition to characterize solutions of BSDEs with weak constraints at stopping times.

Abstract

We first introduce the concept of $\mathscr{Y}^{g,\xi}$-submartingale systems, where the nonlinear operator $\mathscr{Y}^{g,\xi}$ 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 $\mathscr{Y}^{g,\xi}$-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 $\mathscr{Y}^{g,\xi}$-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.