BSDEs with mean reflection

TL;DR

This paper introduces a new class of backward stochastic differential equations (BSDEs) with mean reflection constraints on the distribution of the solution, extending classical reflected BSDEs to include distributional constraints and applications in risk management.

Contribution

The paper establishes well-posedness of BSDEs with mean reflection constraints and extends the framework to static risk measures, with applications to super hedging under risk constraints.

Findings

Proved existence and uniqueness of solutions with mean reflection constraints.

Extended the framework to static risk measures on the solution.

Applied the theory to super hedging with risk management constraints.

Abstract

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.