Nonlinear predictable representation and L¹-solutions of backward SDEs and second-order backward SDEs

Abstract

The theory of backward SDEs extends the predictable representation property of Brownian motion to the nonlinear framework, thus providing a path-dependent analog of fully nonlinear parabolic PDEs. In this paper, we consider backward SDEs, their reflected version, and their second-order extension, in the context where the final data and the generator satisfy L¹-integrability condition. Our main objective is to provide the corresponding existence and uniqueness results for general Lipschitz generators. The uniqueness holds in the so-called Doob class of processes, simultaneously under an appropriate class of measures. We emphasize that the previous literature only deals with backward SDEs, and requires either that the generator is separable in (y, z), see Peng (In Backward Stochastic Differential Equations (1997) 141-159 Longman), or strictly sublinear in the gradient variable z, see Briand, Delyon, Hu, Pardoux and Stoica (Stochastic Process. Appl. 108 (1) (2003) 109-129), or that the final data satisfies an LlnL-integrability condition, see Hu and Tang (Electron. Commun. Probab. 23 (2018) 27).We bypass these conditions by defining L¹-integrability under the nonlinear expectation operator induced by the previously mentioned class of measures.