Backward and reflected backward stochastic differential equations under initial enlargement of Brownian filtrations

Source 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.