BSDEs with weak terminal condition
نویسندگان
چکیده
We introduce a new class of Backward Stochastic Differential Equations in which the T -terminal value YT of the solution (Y, Z) is not fixed as a random variable, but only satisfies a weak constraint of the form E[Ψ(YT )] ≥ m, for some (possibly random) non-decreasing map Ψ and some threshold m. We name them BSDEs with weak terminal condition and obtain a representation of the minimal time t-values Yt 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 [2]. 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 [6, 7], and in Bouchard, Elie and Touzi [2].
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