BSDEs with weak terminal condition
Bouchard, Bruno; Elie, Romuald; Réveillac, Anthony (2015), BSDEs with weak terminal condition, Annals of Probability, 43, 2, p. 572-604. 10.1214/14-AOP913
Type
Article accepté pour publication ou publiéDate
2015Journal name
Annals of ProbabilityVolume
43Number
2Publisher
Institute of Mathematical Statistics
Published in
Paris
Pages
572-604
Publication identifier
Metadata
Show full item recordAbstract (EN)
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.Subjects / Keywords
optimal control; stochastic target; Backward stochastic differential equationsRelated items
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