BSDEs of counterparty risk

We study a BSDE with random terminal time that appears in the modeling of counterparty risk in finance. We model the terminal time as an invariant time, i.e. a time such that local martingales with respect to a reduced filtration and a possibly changed probability measure, once stopped right before that time, stay local martingales with respect to the original model filtration and probability measure. Using an Azéma supermartingale characterization of invariant times, we establish the equivalence between the original and a reduced BSDE.