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).
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H. Mete Soner research was partly supported by the European Research Council under the grant 228053-FiRM. Financial support from the ETH Foundation and Swiss Finance Institute are also gratefully acknowledged.
N. Touzi research was supported by the Chair Financial Risks of the Risk Foundation sponsored by Société Générale, the Chair Derivatives of the Future sponsored by the Fédération Bancaire Française, and the Chair Finance and Sustainable Development sponsored by EDF and Calyon.
J. Zhang research was supported in part by NSF grants DMS 06-31366 and DMS 10-08873.
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Soner, H.M., Touzi, N. & Zhang, J. Wellposedness of second order backward SDEs. Probab. Theory Relat. Fields 153, 149–190 (2012). https://doi.org/10.1007/s00440-011-0342-y
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DOI: https://doi.org/10.1007/s00440-011-0342-y
Keywords
- Backward SDEs
- Non-dominated family of mutually singular measures
- Viscosity solutions for second order PDEs