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Mathematics and Financial Economics

, Volume 12, Issue 1, pp 111–134 | Cite as

Backward nonlinear expectation equations

  • Christoph Belak
  • Thomas Seiferling
  • Frank Thomas Seifried
Article
  • 226 Downloads

Abstract

Building on an abstract framework for dynamic nonlinear expectations that comprises g-, G- and random G-expectations, we develop a theory of backward nonlinear expectation equations of the form
$$\begin{aligned} X_t = {\mathcal {E}}_t \Bigl [{\textstyle \int _t^T} g(s,X) \mu ({\mathrm {d}}s) + \xi \Bigr ], \quad t \in [0,T]. \end{aligned}$$
We provide existence, uniqueness, and stability results and establish convergence of the associated discrete-time nonlinear aggregations. As an application, we construct continuous-time recursive utilities under ambiguity and identify the corresponding utility processes as limits of discrete-time recursive utilities.

Keywords

Backward stochastic differential equation Nonlinear expectation Random G-expectation Recursive utility Volatility uncertainty 

Mathematics Subject Classification

60G20 60H30 91B16 

JEL Classification

D81 D91 

Notes

Acknowledgements

We wish to thank Mete Soner, Holger Kraft, Rama Cont, Yuri Kabanov, and Keita Owari for comments and suggestions. Thomas Seiferling gratefully acknowledges financial support from Studienstiftung des Deutschen Volkes.

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Christoph Belak
    • 1
  • Thomas Seiferling
    • 2
  • Frank Thomas Seifried
    • 1
  1. 1.Department IV – MathematicsUniversity of TrierTrierGermany
  2. 2.Department of MathematicsUniversity of KaiserslauternKaiserslauternGermany

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