Backward and reflected backward stochastic differential equations under initial enlargement of Brownian filtrations
Abstract
We investigate the effect of initially enlarging a Brownian filtration by its terminal value on backward stochastic differential equations (BSDEs) and reflected BSDEs (RBSDEs). We establish the Brownian innovation representation and the predictable representation property in the enlarged filtration, highlighting the singular information drift arising from the semimartingale decomposition of the original Brownian motion. We derive transfer identities under suitable integrability conditions and provide explicit counterexamples showing that the resulting singular equations may fail to admit unique square-integrable solutions, including in the reflected case. We further develop a conditional parametrization of regular BSDEs and establish a conditional comparison principle for Snell envelopes, illustrating both the strict value of additional information and the failure of general pathwise domination. An exact projection identity is also obtained for zero-driver terminal payoffs. Finally, we discuss possible extensions to irregular obstacles and more general information structures.
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BibTeX
@misc{ouknine2026backward,
title = {{Backward and reflected backward stochastic differential equations under initial enlargement of Brownian filtrations}},
author = {Youssef Ouknine and Badr Elmansouri},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18180},
eprint = {2610.18180},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18180},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Backward and reflected backward stochastic differential equations under initial enlargement of Brownian filtrations
AU - Youssef Ouknine
AU - Badr Elmansouri
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18180
UR - https://arxiv.org/abs/2610.18180
AB - We investigate the effect of initially enlarging a Brownian filtration by its terminal value on backward stochastic differential equations (BSDEs) and reflected BSDEs (RBSDEs). We establish the Brownian innovation representation and the predictable representation property in the enlarged filtration, highlighting the singular information drift arising from the semimartingale decomposition of the original Brownian motion. We derive transfer identities under suitable integrability conditions and provide explicit counterexamples showing that the resulting singular equations may fail to admit unique square-integrable solutions, including in the reflected case. We further develop a conditional parametrization of regular BSDEs and establish a conditional comparison principle for Snell envelopes, illustrating both the strict value of additional information and the failure of general pathwise domination. An exact projection identity is also obtained for zero-driver terminal payoffs. Finally, we discuss possible extensions to irregular obstacles and more general information structures.
ER -