​Backward stochastic​ ​differential​ ​equations extend the martingale representation theorem​ ​to the nonlinear setting. This can be seen as path-dependent counterpart of the extension​ ​from the heat equation to fully nonlinear parabolic equations in the Markov setting. This​ ​paper extends such a nonlinear representation to the context where the random variable​ ​of interest is measurable with respect to the information at a finite stopping time. We​ ​provide a complete wellposedness theory which covers the semilinear case (backward SDE),​ ​the semilinear case with obstacle (reflected backward SDE), and the fully nonlinear case​ ​(second order backward SDE).

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