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 etal. (Commun. Pure Appl. Math. 60(7):1081-1110, 2007). In particular, we provide a fully nonlinear extension of the Feynman-Kac formula. Unlike (Cheridito etal. 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 etal. in Dual Formulation of Second Order Target Problems. arXiv:1003.6050, 2009
