The Annals of Probability

Backward stochastic differential equations with rough drivers

Joscha Diehl and Peter Friz

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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].

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Ann. Probab., Volume 40, Number 4 (2012), 1715-1758.

First available in Project Euclid: 4 July 2012

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Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H15: Stochastic partial differential equations [See also 35R60]

Backward stochastic differential equations backward doubly stochastic differential equations rough path analysis stochastic partial differential equations viscosity theory


Diehl, Joscha; Friz, Peter. Backward stochastic differential equations with rough drivers. Ann. Probab. 40 (2012), no. 4, 1715--1758. doi:10.1214/11-AOP660.

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