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We provide an existence and uniqueness theory for an extension of backwardSDEs to the second order. While standard Backward SDEs are naturally connectedto semilinear PDEs, our second order extension is connected to fully nonlinearPDEs, as suggested by Cheridito et.al. In particular, we provide a fullynonlinear extension of the Feynman-Kac formula. Unlike the earlier papers, thealternative formulation of this paper insists that the equation must hold undera non-dominated family of mutually singular probability measures. The keyargument is a stochastic representation, suggested by the optimal controlinterpretation, and analyzed in our accompanying paper

Wellposedness of Second Order Backward SDEs