On Z-mean reflected BSDEs

TL;DR

This paper investigates the existence of supersolutions to backward stochastic differential equations with mean-reflections on the Z component, highlighting differences from Y component reflections and establishing conditions for solutions.

Contribution

It provides new conditions for supersolutions with stochastic increasing processes in Z-reflected BSDEs and clarifies the time-inconsistency issues involved.

Findings

Supersolutions with deterministic K are generally not possible.

Conditions for supersolutions with stochastic K are established.

Minimal supersolutions are shown to be actual solutions.

Abstract

In this paper we provide conditions for the existence of supersolutions to BSDEs with mean-reflections on the $Z$ component. We show that, contrary to BSDEs with mean-reflections on the $Y$ component, we cannot expect a supersolution with a deterministic increasing process $K$. Nonetheless, we give conditions for the existence of a supersolution for a stochastic component $K$ and under various constraints. We formalize some previous arguments on the time-inconsistency of such problems, proving that a minimal supersolution is necessarily a solution in our framework.