Skip to main content
SpringerLink
Log in

Root to Kellerer

  • Chapter
  • First Online:
Séminaire de Probabilités XLVIII

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2168))

Abstract

We revisit Kellerer’s Theorem, that is, we show that for a family of real probability distributions (μ t ) t ∈ [0, 1] which increases in convex order there exists a Markov martingale (S t ) t ∈ [0, 1] s.t. S t  ∼ μ t .

To establish the result, we observe that the set of martingale measures with given marginals carries a natural compact Polish topology. Based on a particular property of the martingale coupling associated to Root’s embedding this allows for a relatively concise proof of Kellerer’s theorem.

We emphasize that many of our arguments are borrowed from Kellerer (Math Ann 198:99–122, 1972), Lowther (Limits of one dimensional diffusions. ArXiv e-prints, 2007), Hirsch-Roynette-Profeta-Yor (Peacocks and Associated Martingales, with Explicit Constructions. Bocconi & Springer Series, vol. 3, Springer, Milan; Bocconi University Press, Milan, 2011), and Hirsch et al. (Kellerer’s Theorem Revisited, vol. 361, Prépublication Université dÉvry, Columbus, OH, 2012).

Mathematics Subject Classification (2010): Primary 60G42, 60G44; Secondary 91G20

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    An early article to study this continuum time version of the martingale optimal transport problem is the recent article [10] of Kallblad et al.

  2. 2.

    Hobson’s solution [7] can be seen as an extension of the Azema-Yor embedding to the case of a general starting distribution.

References

  1. M. Beiglböck, N. Juillet, On a problem of optimal transport under marginal martingale constraints. Ann. Probab. 44 (1), 42–106 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. B. Dupire, Pricing with a smile. Risk 7 (1), 18–20 (1994)

    Google Scholar 

  3. I. Gyöngy, Mimicking the one-dimensional marginal distributions of processes having an Itô differential. Probab. Theory Relat. Fields 71 (4), 501–516 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  4. P. Henry-Labordere, N. Touzi, An explicit Martingale version of Brenier’s theorem. Finance Stochast. 20 (3), 635–668 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. F. Hirsch, C. Profeta, B. Roynette, M. Yor, Peacocks and Associated Martingales, with Explicit Constructions. Bocconi & Springer Series, vol. 3 (Springer, Milan; Bocconi University Press, Milan, 2011)

    Google Scholar 

  6. F. Hirsch, B. Roynette, M. Yor, Kellerer’s Theorem Revisited, vol. 361 (Prépublication Université dÉvry, Columbus, OH, 2012)

    MATH  Google Scholar 

  7. D. Hobson, The maximum maximum of a martingale, in Séminaire de Probabilités, XXXII. Lecture Notes in Mathematics, vol. 1686 (Springer, Berlin, 1998), pp. 250–263

    Google Scholar 

  8. D. Hobson, M. Klimmek, Model independent hedging strategies for variance swaps. Finance Stochast. 16 (4), 611–649 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. D. Hobson, A. Neuberger, Robust bounds for forward start options. Math. Financ. 22 (1), 31–56 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. S. Källblad, X. Tan, N. Touzi, Optimal Skorokhod embedding given full marginals and Azema-Yor peacocks. Ann. Appl. Probab. (2015, to appear)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. H.G. Kellerer, Integraldarstellung von Dilationen, in Transactions of the Sixth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes (Tech. Univ., Prague, 1971; Dedicated to the Memory of Antonín Špaček) (Academia, Prague, 1973), pp. 341–374

    Google Scholar 

  13. T. Liggett, Continuous Time Markov Processes. Graduate Studies in Mathematics, vol. 113 (American Mathematical Society, Providence, RI, 2010). An introduction.

    Google Scholar 

  14. G. Lowther, Limits of one dimensional diffusions. Ann. Probab. 37 (1), 78–106 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. G. Lowther, Fitting martingales to given marginals (2008). arXiv:0808.2319

    Google Scholar 

  16. D. Revuz, M. Yor, Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, 3rd edn. (Springer, Berlin, 1999)

    Google Scholar 

  17. D.H. Root, The existence of certain stopping times on Brownian motion. Ann. Math. Stat. 40, 715–718 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  18. V. Strassen, The existence of probability measures with given marginals. Ann. Math. Stat. 36, 423–439 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  19. C. Villani, Optimal Transport. Old and New. Grundlehren der mathematischen Wissenschaften, vol. 338 (Springer, Berlin, 2009)

    Google Scholar 

Download references

Acknowledgements

Mathias Beiglböck and Florian Stebegg acknowledge support through FWF-projects P26736 and Y782-N25. Martin Huesmann acknowledges support through CRC 1060. We also thank the Hausdorff Research Institute for Mathematics (HIM) for its hospitality in spring 2015 and Nicolas Juillet and Christophe Profeta for many insightful comments.

Author information

Authors and Affiliations

  1. Institut für Stochastik und Wirtschaftsmathematik, Technische Universität Wien, Wiedner Hauptstraße 8, 1040, Wien, Austria

    Mathias Beiglböck

  2. Institut für Angewandte Mathematik, Rheinische Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Martin Huesmann

  3. Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, 10025, NY, USA

    Florian Stebegg

Authors
  1. Mathias Beiglböck
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Huesmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Florian Stebegg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mathias Beiglböck .

Editor information

Editors and Affiliations

  1. Laboratoire de Mathématiques, Université de Versailles-St Quentin, Versailles , France

    Catherine Donati-Martin

  2. Campus scientifique, IECL, Vandoeuvre-les-Nancy CEDEX, France

    Antoine Lejay

  3. Laboratoire de Mathématiques, Université de Versailles-St Quentin, Versailles, France

    Alain Rouault

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Beiglböck, M., Huesmann, M., Stebegg, F. (2016). Root to Kellerer. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLVIII. Lecture Notes in Mathematics(), vol 2168. Springer, Cham. https://doi.org/10.1007/978-3-319-44465-9_1

Download citation

Publish with us

Policies and ethics

Search

Navigation