dc.contributor.author Bouchard, Bruno dc.contributor.author Elie, Romuald dc.contributor.author Réveillac, Anthony HAL ID: 745074 dc.date.accessioned 2012-10-24T13:55:55Z dc.date.available 2012-10-24T13:55:55Z dc.date.issued 2015 dc.identifier.issn 0091-1798 dc.identifier.uri https://basepub.dauphine.fr/handle/123456789/10494 dc.language.iso en en dc.subject optimal control dc.subject stochastic target dc.subject Backward stochastic differential equations dc.subject.ddc 519 en dc.title BSDEs with weak terminal condition dc.type Article accepté pour publication ou publié dc.contributor.editoruniversityother Centre de Recherche en Économie et Statistique (CREST) http://www.crest.fr/ INSEE – École Nationale de la Statistique et de l'Administration Économique;France dc.description.abstracten 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. 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 Föllmer and Leukert, and in Bouchard, Elie and Touzi. Finally, we observe a surprisingly strong connection between BSDEs with weak terminal condition and 2nd order BSDEs in the quasi linear case. dc.publisher.city Paris en dc.relation.isversionofjnlname Annals of Probability dc.relation.isversionofjnlvol 43 dc.relation.isversionofjnlissue 2 dc.relation.isversionofjnldate 2015 dc.relation.isversionofjnlpages 572-604 dc.relation.isversionofdoi 10.1214/14-AOP913 dc.relation.isversionofjnlpublisher Institute of Mathematical Statistics dc.subject.ddclabel Probabilités et mathématiques appliquées en dc.description.submitted non en dc.description.submitted oui dc.description.submittedpublication Annals of Probability dc.description.ssrncandidate non dc.description.halcandidate oui dc.description.readership recherche dc.description.audience International dc.relation.Isversionofjnlpeerreviewed oui dc.date.updated 2020-07-05T21:54:27Z
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