We consider the optimal stopping problem with non-linear f-expectation (induced by a BSDE) without making any regularity assumptions on the pay-off process \xi. We show that the value family can be aggregated by an optional process Y. We characterize the process Y as the \mathcal{E}^f-Snell envelope of \xi. We also establish an infinitesimal characterization of the value process Y in terms of a Reflected BSDE with \xi as the obstacle. This characterization is established by first showing existence and uniqueness for the Reflected BSDE with irregular obstacle and also a comparison theorem.
This talk is based on joint work with Miryana Grigorova, Peter Imkeller and Marie-Claire Quenez.