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dc.contributor.authorBouchard, Bruno
dc.contributor.authorElie, Romuald
dc.contributor.authorRéveillac, Anthony
dc.date.accessioned2012-10-24T13:55:55Z
dc.date.available2012-10-24T13:55:55Z
dc.date.issued2015
dc.identifier.issn0091-1798
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/10494
dc.language.isoenen
dc.subjectoptimal control
dc.subjectstochastic target
dc.subjectBackward stochastic differential equations
dc.subject.ddc519en
dc.titleBSDEs with weak terminal condition
dc.typeArticle accepté pour publication ou publié
dc.contributor.editoruniversityotherCentre 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.abstractenWe 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.cityParisen
dc.relation.isversionofjnlnameAnnals of Probability
dc.relation.isversionofjnlvol43
dc.relation.isversionofjnlissue2
dc.relation.isversionofjnldate2015
dc.relation.isversionofjnlpages572-604
dc.relation.isversionofdoi10.1214/14-AOP913
dc.relation.isversionofjnlpublisherInstitute of Mathematical Statistics
dc.subject.ddclabelProbabilités et mathématiques appliquéesen
dc.description.submittednonen
dc.description.submittedoui
dc.description.submittedpublicationAnnals of Probability
dc.description.ssrncandidatenon
dc.description.halcandidateoui
dc.description.readershiprecherche
dc.description.audienceInternational
dc.relation.Isversionofjnlpeerreviewedoui
dc.date.updated2020-07-05T21:54:27Z


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