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Probability Theory and Related Fields

, Volume 153, Issue 1–2, pp 149–190 | Cite as

Wellposedness of second order backward SDEs

  • H. Mete Soner
  • Nizar Touzi
  • Jianfeng Zhang
Article

Abstract

We provide an existence and uniqueness theory for an extension of backward SDEs to the second order. While standard Backward SDEs are naturally connected to semilinear PDEs, our second order extension is connected to fully nonlinear PDEs, as suggested in Cheridito et al. (Commun. Pure Appl. Math. 60(7):1081–1110, 2007). In particular, we provide a fully nonlinear extension of the Feynman–Kac formula. Unlike (Cheridito et al. in Commun. Pure Appl. Math. 60(7):1081–1110, 2007), the alternative formulation of this paper insists that the equation must hold under a non-dominated family of mutually singular probability measures. The key argument is a stochastic representation, suggested by the optimal control interpretation, and analyzed in the accompanying paper (Soner et al. in Dual Formulation of Second Order Target Problems. arXiv:1003.6050, 2009).

Keywords

Backward SDEs Non-dominated family of mutually singular measures Viscosity solutions for second order PDEs 

Mathematics Subject Classification (2000)

60H10 60H30 

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Departement MathematikETH (Swiss Federal Institute of Technology), Zürich and Swiss Finance InstituteZurichSwitzerland
  2. 2.CMAP, Ecole Polytechnique ParisPalaiseauFrance
  3. 3.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA

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