BSDEs with weak terminal condition

Bruno Bouchard (1, 2) , Romuald Elie (1, 2) , Anthony Réveillac (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 CREST - Centre de Recherche en Économie et Statistique

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}.

Mots clés: stochastic target; Backward stochastic differential equations; optimal control; stochastic target.

Domaines: Probabilités [math.PR]; Optimisation et contrôle [math.OC]

Bruno Bouchard, Romuald Elie, Anthony Réveillac. BSDEs with weak terminal condition. 2014. ⟨hal-00743348v2⟩