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Topological properties of convex order in Wasserstein metric spaces

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Abstract

Martingale optimal transportation has gained significant attention in mathematical finance due to its applications in pricing and hedging. A key distinguishing factor between martingale optimal transportation and traditional optimal transportation is the concept of a peacock, which refers to a sequence of measures satisfying the convex order property. In the realm of traditional optimal transportation, the Wasserstein geometry, induced by a transportation problem with the p-th power of distance as the cost, provides valuable geometric insights. This motivates us to investigate the differences between Wasserstein geometries with and without the martingale constraint. As a first step, this paper focuses on studying the topological properties of convex order, with the aim of establishing a foundational understanding for further exploration of the geometric properties of martingale Wasserstein geometry.

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References

  1. Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel (2008)

    Google Scholar 

  2. Ambrosio, L., Pratelli, A.: Existence and stability results in the \(L^1\) theory of optimal transportation, Optimal transportation and applications (Martina Franca, 2001), 123–160, Lecture Notes in Math., 1813, Springer, Berlin (2003)

  3. Arjovsky, M., Chintala, S., Bottou, L.: Wasserstein generative adversarial networks. In: Proceedings of the 34th International Conference on Machine Learning, PMLR, vol. 70, pp. 214–223 (2017)

  4. Athreya, K.B., Lahiri, S.N.: Measure Theory and Probability Theory. Springer Texts in Statistics, Springer, New York (2006)

    Google Scholar 

  5. Beiglböck, M., Henry-Labordère, P., Penkner, F.: Model-independent bounds for option prices-a mass transport approach. Finance Stoch. 17, 477–501 (2013)

    Article  MathSciNet  Google Scholar 

  6. Billingsley, P.: Convergence of Probability Measures. John Wiley & Sons, Hoboken (2013)

    Google Scholar 

  7. Bonnans, J.F., Tan, X.: A model-free no-arbitrage price bound for variance options. Appl. Math. Optim. 68, 43–73 (2013)

    Article  MathSciNet  Google Scholar 

  8. Carmona, R., Delarue, F.: Probabilistic analysis of mean-field games. SIAM J. Control Optim. 51, 2705–2734 (2013)

    Article  MathSciNet  Google Scholar 

  9. Carrillo, J.A., Toscani, G.: Asymptotic \(L^1\)-decay of solutions of the porous medium equation to self-similarity. Indiana Univ. Math. J. 49, 113–142 (2000)

    Article  MathSciNet  Google Scholar 

  10. Claisse, J., Guo, G., Henry-Labordère, P.: Some results on Skorokhod embedding and robust hedging with local time. J. Optim. Theory Appl. 179, 569–597 (2018)

    Article  MathSciNet  Google Scholar 

  11. Cullen, M., Maroofi, H.: The fully compressible semi-geostrophic system from meteorology. Arch. Ration. Mech. Anal. 167, 309–336 (2003)

    Article  MathSciNet  Google Scholar 

  12. Dolinsky, Y., Soner, H.M.: Martingale optimal transport and robust hedging in continuous time. Probab. Theory Relat. Fields 160, 391–427 (2014)

    Article  MathSciNet  Google Scholar 

  13. Fathi, A.: Weak KAM Theorem in Lagrangian Dynamics, to appear on Cambrige studies in Advanced Mathematics. Cambrige University Press, Cambrige (2008)

    Google Scholar 

  14. Galichon, A., Henry-Labordere, P., Touzi, N.: A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options. Ann. Appl. Probab. 24, 312–336 (2014)

    Article  MathSciNet  Google Scholar 

  15. Henry-Labordere, P.: Model-Free Hedging: A Martingale Optimal Transport Viewpoint. Chapman and Hall/CRC, Boca Raton (2017)

    Book  Google Scholar 

  16. Hirsch, F., Roynette, B.: A new proof of Kellerer’s theorem. ESAIM Probab. Stat. 16, 48–60 (2012)

    Article  MathSciNet  Google Scholar 

  17. Hirsch, F., Roynette, B.: On \({\mathbb{R} }^{d}\)-valued peacocks. ESAIM Probab. Stat. 17, 444–454 (2013)

    Article  MathSciNet  Google Scholar 

  18. Hou, Z., Obłój, J.: Robust pricing-hedging dualities in continuous time. Finance Stoch. 22, 511–567 (2018)

    Article  MathSciNet  Google Scholar 

  19. Hu, Z.: Foundations of Advanced Probability Theory and Limit Theory. Tsinghua University Press, Beijing (2014). (in Chinese)

    Google Scholar 

  20. Kantorovich, L.V.: On the transfer of masses. Dokl. Akad. Nauk. SSSR (in Russian) 37, 227–229 (1942)

    Google Scholar 

  21. Kantorovich, L.V.: On a problem of Monge. J. Math. Sci. 133, 1383–1383 (2006)

    Article  MathSciNet  Google Scholar 

  22. Kertz, R.P., Rösler, U.: Complete Lattices of Probability Measures with Applications to Martingale Theory. Game theory, optimal stopping, probability and statistics, 153–177, IMS Lecture Notes Monogr. Ser., 35, Inst. Math. Statist., Beachwood, OH (2000)

  23. Kellerer, H.G.: Markov-Komposition und eine Anwendung auf Martingale. Math. Ann. 198, 99–122 (1972)

    Article  MathSciNet  Google Scholar 

  24. Lowther, G.: Fitting Martingales to Given Marginals. arXiv:0808.2319v1 (2008)

  25. Monge, G.: Mémoire sur la thérie des déblais et des remblais. In: Historie del’Académie Royale des Sciences de Paris, pp. 666–704 (1781)

  26. Parthasarathy, Y.R.: Probability Measures on Metric Spaces, Probability and Mathematical Statistics No. 3. Academic Press Inc, New York-London (1967)

    Google Scholar 

  27. Rubner, Y., Tomasi, C., Guibas, L.J.: The earth mover’s distance as a metric for image retrieval. Int. J. Comput. Vis. 40, 99–121 (2000)

    Article  Google Scholar 

  28. Santambrogio, F.: Optimal transport for applied mathematicians. Birkäuser, NY 55, 58–63 (2015)

    Google Scholar 

  29. Villani, C.: Optimal Transport: Old and New, vol. 338. Springer-Verlag, Berlin (2009)

    Google Scholar 

  30. Villani, C.: Topics in Optimal Transportation, vol. 58. American Mathematical Society, Providence (2003)

    Google Scholar 

  31. Willard, S.: General Topology. Addison-Wesley Publishing Company, Boston (1970)

    Google Scholar 

  32. Wu, H., Cui, X.: Peacock geodesics in Wasserstein space. Differ. Geom. Appl. 77, 101764 (2021)

    Article  MathSciNet  Google Scholar 

  33. Yan, J.: Lecture Notes on Measure Theory. Science Press, Beijing (2004). (in Chinese)

    Google Scholar 

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Acknowledgements

The authors express their gratitude to Professor Xiaojun Cui from Nanjing University for his valuable discussions and suggestions.

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Authors and Affiliations

  1. School of Big Data, Changzhou University, Changzhou, 213164, China

    Hongbing Ju

  2. Department of Mathematics, Changzhou University, Changzhou, 213164, China

    Feng Wang & Hongguang Wu

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  1. Hongbing Ju
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  2. Feng Wang
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Correspondence to Feng Wang.

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Feng Wang was supported by the Natural Science Foundation of Jiangsu Province (Grant No. BK20201447), and Science and Technology Innovation Talent Support Project of Jiangsu Advanced Catalysis and Green Manufacturing Collaborative Innovation Center (Grant No. ACGM2022-10-02). Hongguang Wu was supported by the National Natural Science Foundation of China (Grant No. 12201073).

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Ju, H., Wang, F. & Wu, H. Topological properties of convex order in Wasserstein metric spaces. Ricerche mat (2024). https://doi.org/10.1007/s11587-024-00867-4

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