Backward stochastic differential equations with rough drivers

Abstract

Backward stochastic differential equations (BSDEs) in the sense of Pardoux–Peng [Lecture Notes in Control and Inform. Sci. 176 (1992) 200–217] provide a non-Markovian extension to certain classes of nonlinear partial differential equations; the nonlinearity 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 [Rev. Mat. Iberoam. 14 (1998) 215–310] 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 [Ann. Probab. 28 (2000) 558–602].