Advertisement

Linking Progressive and Initial Filtration Expansions

  • Younes Kchia
  • Martin Larsson
  • Philip ProtterEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 34)

Abstract

In this article, we study progressive filtration expansions with random times. We show how semimartingale decompositions in the expanded filtration can be obtained using a natural link between progressive and initial expansions. The link is, on an intuitive level, that the two coincide after the random time. We make this idea precise and use it to establish known and new results in the case of expansion with a single random time. The methods are then extended to the multiple time case, without any restrictions on the ordering of the individual times. Finally we study the link between the expanded filtrations from the point of view of filtration shrinkage. As the main analysis progresses, we indicate how the techniques can be generalized to other types of expansions.

Keywords

Random Time Conditional Density Local Martingale Progressive Expansion Initial Expansion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

Philip Protter was supported in part by NSF grant DMS-0906995.

References

  1. 1.
    Callegaro, G., Jeanblanc, M., Zargari, B.: Carthaginian enlargement of filtrations. ESAIM: Probability and Statistics, doi:10.1051/ps/2011162Google Scholar
  2. 2.
    Dellacherie, C., Meyer, P.A.: A propos du travail de Yor sur le grossissement des tribus. Séminaire de Probabilités XII, 70–77 (1976)Google Scholar
  3. 3.
    Dellacherie, C., Meyer, P.A.: Probabilities and Potential B, Theory of Martingales, North-Holland Publishing Company, Amsterdam (1982)zbMATHGoogle Scholar
  4. 4.
    Föllmer, H., Protter, P.: Local martingales and filtration shrinkage. ESAIM: Probability and Statistics, Available on CJO 2006 doi:10.1051/ps/2010023 (2011)Google Scholar
  5. 5.
    Guo, X., Zeng, Y.: Intensity process and compensator: a new filtration expansion approach and the Jeulin-Yor Theorem. The Ann. Appl. Probab. 18(1), 120–142 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Imkeller, P.: Malliavin’s calculus in insider models: additional utility and free lunches. Math. Finance 13, 153–169 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Imkeller, P., Pontier, M., and Weisz, F.: Free lunch and arbitrage possibilities in a financial market model with an insider. Stoch. Proc. Appl. 92, 103–130 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Jacod, J.: Grossissement initial, hypothèse (H’) et théorème de Girsanov. In: Lecture Notes in Mathematics, vol. 1118, pp. 15–35. Springer, Berlin (1987)Google Scholar
  9. 9.
    Jeanblanc, M., Le Cam, Y.: Progressive enlargement of filtrations with initial times. Stoch. Proc. Appl. 119, 2523–2543 (2009)zbMATHCrossRefGoogle Scholar
  10. 10.
    Jeulin, T., Yor, M. (eds.): Grossissement de filtrations: examples et applications. In: Lecture Notes in Mathematics, vol. 1118. Springer, Berlin (1985)Google Scholar
  11. 11.
    Kohatsu-Higa, A.: Models for insider trading with finite utility. In: Paris-Princeton Lectures on Mathematical Finance, Lecture Notes in Mathematics. Springer, Berlin (2004)Google Scholar
  12. 12.
    Protter, P.: Stochastic Integration and Differential Equations, 2nd edn. Springer, Heidelberg (2005)Google Scholar
  13. 13.
    Stricker, C.: Quasimartingales, martingales locales, semimartingales et filtration naturelle. Z. Wahrscheinlishkeitstheorie verw. Gebiete 39, 55–63 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Sulem, A., Kohatsu-Higa, A., Øksendal, B., Proske, F., Di Nunno, G.: Anticipative stochastic control for Lévy processes with application to insider trading. In: Mathematical Modeling and Numerical Methods in Finance, Elsevier (2008)Google Scholar
  15. 15.
    Yor, M.: Grossissement de filtrations et absolue continuité de noyaux. In: Jeulin, T., Yor, M. (eds.) Grossissements de filtrations: exemples et applications, Springer (1985)Google Scholar
  16. 16.
    Yor, M.: Inégalités entre processus minces et applications. C.R. Acad. Sci. Paris, Sér. A 286, 799–801 (1978)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.ANZ BankSingaporeSingapore
  2. 2.Swiss Finance Institute and Ecole Polytechnique Fédérale de LausanneLausanneSwitzerland
  3. 3.Statistics DepartmentColumbia UniversityNew YorkUSA

Personalised recommendations