BSDEs with weak terminal condition

• Bouchard, Bruno ^[2] ; Elie, Romuald ^[1] ; Réveillac, Antony ^[3]
  1. [1] Paris Dauphine University

     Paris Dauphine University

     París, Francia

     
  2. [2] ENSAE- Paris Tech
  3. [3] University Paris-Est

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 43, Nº. 2, 2015, págs. 572-604
• Idioma: inglés
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• Resumen

  • 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) nondecreasing 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 [SIAM J. Control Optim. 48 (2009/10) 3123–3150]. 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 [Finance Stoch. 3 (1999) 251–273; Finance Stoch. 4 (2000) 117–146], and in Bouchard, Elie and Touzi (2009/10).