Nonlinear predictable representation and $L^1$-solutions of backward SDEs and second-order backward SDEs

TL;DR

This paper extends the theory of backward SDEs and their second-order versions under $L^1$-integrability conditions, establishing existence and uniqueness results without restrictive assumptions on the generator or data.

Contribution

It introduces $L^1$-solutions for backward and second-order backward SDEs, broadening the scope beyond previous $L ext{ln}L$ or sublinear conditions, and provides new existence and uniqueness results.

Findings

Established existence and uniqueness of $L^1$-solutions for backward SDEs.

Extended the predictable representation property to a nonlinear, path-dependent setting.

Bypassed previous restrictions on generator separability and integrability conditions.

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 $L^1$-type of 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 [Pen97], or strictly sublinear in the gradient variable $z$, see [BDHPS03], or that the final data satisfies an $L\ln L$-integrability condition, see [HT18]. We by-pass these conditions by defining $L^1$-integrability under the nonlinear expectation operator induced by the previously mentioned class of measures.