Backward stochastic differential equations with rough drivers

TL;DR

This paper extends backward stochastic differential equations (BSDEs) to drivers with very low regularity by employing rough path analysis, establishing existence, uniqueness, and stability results for these generalized BSDEs.

Contribution

It introduces a framework for BSDEs driven by rough signals, broadening the class of drivers and solutions to include highly irregular time-dependent functions.

Findings

Established continuity of BSDE solutions in rough path metrics.

Proved existence and uniqueness of BSDEs with rough drivers.

Extended Lyons' limit theorem to this new context.

Abstract

Backward stochastic differential equations (BSDEs) in the sense of Pardoux-Peng [Backward stochastic differential equations and quasilinear parabolic partial differential equations, Lecture Notes in Control and Inform. Sci., 176, 200--217, 1992] provide a non-Markovian extension to certain classes of non-linear partial differential equations; the non-linearity is expressed in the so-called driver of the BSDE. Our aim is to deal with drivers which have very little regularity in time. To this end we establish continuity of BSDE solutions with respect to rough path metrics in the sense of Lyons [Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14, no. 2, 215--310, 1998] and so obtain a notion of "BSDE with rough driver". Existence, uniqueness and a version of Lyons' limit theorem in this context are established. Our main tool, aside from rough path analysis, is the stability theory for quadratic BSDEs due to Kobylanski [Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab., 28(2):558--602, 2000].