  We establish some well-posedness and comparison results for BSDEs driven by
one- and multi-dimensional martingales. On the one hand, our approach is
largely motivated by results and methods developed in Carbone et al. (2008) and
El Karoui and Huang (1997). On the other hand, our results are also motivated
by the recent developments in arbitrage pricing theory under funding costs and
collateralization. A new version of the comparison theorem for BSDEs driven by
a multi-dimensional martingale is established and applied to the pricing and
hedging BSDEs studied in Bielecki and Rutkowski (2014) and Nie and Rutkowski
(2014). This allows us to obtain the existence and uniqueness results for
unilateral prices and to demonstrate the existence of no-arbitrage bounds for a
collateralized contract when both agents have non-negative initial endowments.
