BSDEs Driven by Multidimensional Martingales and Their Applications to Markets with Funding Costs

We establish some well-posedness and comparison results for BSDEs driven by one- and multidimensional martingales. On the one hand, our approach is largely motivated by results and methods developed by Carbone, Ferrario, and Santacroce in [Theory Probab. Appl., 52 (2008), pp. 304--314] and by El Karoui and Huang in [Backward Stochastic Differential Equations, Pitman Res. Notes in Math. Ser. 364, Longman, Harlow, 1997, pp. 27--36]. On the other hand, our results are also motivated by the recent developments in the theory of pricing of financial derivatives under funding costs and collateralization. A new version of the comparison theorem for BSDEs driven by a multidimensional martingale is established and applied to the pricing BSDEs studied by Bielecki and Rutkowski in [SIAM J. Financial Math., 6 (2015), pp. 594--655] and by Nie and Rutkowski in [Int. J. Theor. Appl. Finance, 18 (2015), 1550048], [Fair Bilateral Prices under Funding Costs and Exogenous Collateralization, Working paper, University of Sydney, Sydney, 2014], and [Finance Stoch., 20 (2016), pp. 855--900]. We establish the existence and uniqueness results for unilateral prices and demonstrate the existence of no-arbitrage bounds for a collateralized contract when both agents have nonnegative initial endowments.