Characterization of optimal transport plans for the Monge-Kantorovich problem
HTML articles powered by AMS MathViewer
- by Walter Schachermayer and Josef Teichmann
- Proc. Amer. Math. Soc. 137 (2009), 519-529
- DOI: https://doi.org/10.1090/S0002-9939-08-09419-7
- Published electronically: September 9, 2008
- PDF | Request permission
Abstract:
We prove that $c$-cyclically monotone transport plans $\pi$ optimize the Monge-Kantorovich transportation problem under an additional measurability condition. This measurability condition is always satisfied for finitely valued, lower semi-continuous cost functions. In particular, this yields a positive answer to Problem 2.25 in C. Villani’s book. We emphasize that we do not need any regularity conditions as were imposed in the previous literature.References
- Luigi Ambrosio and Aldo Pratelli, Existence and stability results in the $L^1$ theory of optimal transportation, Optimal transportation and applications (Martina Franca, 2001) Lecture Notes in Math., vol. 1813, Springer, Berlin, 2003, pp. 123–160. MR 2006307, DOI 10.1007/978-3-540-44857-0_{5}
- W. Brannath and W. Schachermayer, A bipolar theorem for $L^0_+(\Omega ,\scr F,\mathbf P)$, Séminaire de Probabilités, XXXIII, Lecture Notes in Math., vol. 1709, Springer, Berlin, 1999, pp. 349–354. MR 1768009, DOI 10.1007/BFb0096525
- W. Schachermayer, Martingale measures for discrete-time processes with infinite horizon, Math. Finance 4 (1994), no. 1, 25–55. MR 1286705, DOI 10.1111/j.1467-9965.1994.tb00048.x
- Olav Kallenberg, Foundations of Modern Probability, Probability and its Applications, 2nd edition, Springer, New York, 2001.
- D. Ramachandran and L. Rüschendorf, A general duality theorem for marginal problems, Probab. Theory Related Fields 101 (1995), no. 3, 311–319. MR 1324088, DOI 10.1007/BF01200499
- A. Pratelli, About the sufficiency of the $c$-cyclical monotonicity for the optimal transport plans, personal communication, 2006.
- Lars Svensson, Sums of complemented subspaces in locally convex spaces, Ark. Mat. 25 (1987), no. 1, 147–153. MR 918383, DOI 10.1007/BF02384440
- Cédric Villani, Topics in optimal transportation, Graduate Studies in Mathematics, vol. 58, American Mathematical Society, Providence, RI, 2003. MR 1964483, DOI 10.1090/gsm/058
Bibliographic Information
- Walter Schachermayer
- Affiliation: Technical University Vienna, Wiedner Hauptstrasse 8–10, A-1040 Vienna, Austria
- Josef Teichmann
- Affiliation: Technical University Vienna, Wiedner Hauptstrasse 8–10, A-1040 Vienna, Austria
- MR Author ID: 654648
- Received by editor(s): February 15, 2006
- Received by editor(s) in revised form: August 24, 2007
- Published electronically: September 9, 2008
- Additional Notes: Financial support from the Austrian Science Fund (FWF) under grant P 15889, from the Vienna Science Foundation (WWTF) under grant MA13, and from the European Union under grant HPRN-CT-2002-00281 is gratefully acknowledged. Furthermore this work was financially supported by the Christian Doppler Research Association (CDG). The authors gratefully acknowledge a fruitful collaboration with and continued support by Bank Austria through CDG
- Communicated by: David Preiss
- © Copyright 2008 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 137 (2009), 519-529
- MSC (2000): Primary 49J45, 28A35
- DOI: https://doi.org/10.1090/S0002-9939-08-09419-7
- MathSciNet review: 2448572