BSDEs with weak terminal condition

TL;DR

This paper introduces a new class of BSDEs with a weak terminal condition, providing a non-Markovian PDE formulation, analyzing their properties, and relating them to dual control problems, extending previous results on hedging under constraints.

Contribution

It defines BSDEs with weak terminal conditions, derives their representation, and explores their properties, extending prior work on quantile hedging and loss-constrained hedging.

Findings

Established a representation for minimal solutions of BSDEs with weak terminal conditions.

Proved continuity and convexity of the minimal value with respect to the threshold parameter.

Connected the minimal value to a dual optimal control problem in Meyer form.

Abstract

We introduce a new class of Backward Stochastic Differential Equations in which the $T$-terminal value $Y_{T}$ of the solution $(Y,Z)$ is not fixed as a random variable, but only satisfies a weak constraint of the form $E[\Psi(Y_{T})]\ge m$, for some (possibly random) non-decreasing map $\Psi$ and some threshold $m$. We name them \textit{BSDEs with weak terminal condition} and obtain a representation of the minimal time $t$-values $Y_{t}$ such that $(Y,Z)$ is a supersolution of the BSDE with weak terminal condition. It provides a non-Markovian BSDE formulation of the PDE characterization obtained for Markovian stochastic target problems under controlled loss in Bouchard, Elie and Touzi \cite{BoElTo09}. We then study the main properties of this minimal value. In particular, we analyze its continuity and convexity with respect to the $m$-parameter appearing in the weak terminal condition, and show how it can be related to a dual optimal control problem in Meyer form. These last properties generalize to a non Markovian framework previous results on quantile hedging and hedging under loss constraints obtained in F\"{o}llmer and Leukert \cite{FoLe99,FoLe00}, and in Bouchard, Elie and Touzi \cite{BoElTo09}.