Wellposedness of second order backward SDEs

H. Mete Soner, Nizar Touzi, Jianfeng Zhang

Research output: Contribution to journalArticlepeer-review


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).

Original languageEnglish (US)
Pages (from-to)149-190
Number of pages42
JournalProbability Theory and Related Fields
Issue number1-2
StatePublished - Jun 2012


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

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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