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On an Optional Semimartingale Decomposition and the Existence of a Deflator in an Enlarged Filtration

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Book cover In Memoriam Marc Yor - Séminaire de Probabilités XLVII

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2137))

Abstract

Given a reference filtration \(\mathbb{F}\), we consider the cases where an enlarged filtration \(\mathbb{G}\) is constructed from \(\mathbb{F}\) in two different ways: progressively with a random time or initially with a random variable. In both situations, under suitable conditions, we present a \(\mathbb{G}\)-optional semimartingale decomposition for \(\mathbb{F}\)-local martingales. Our study is then applied to the question of how an arbitrage-free semimartingale model is affected when stopped at the random time in the case of progressive enlargement or when the random variable used for initial enlargement satisfies Jacod’s hypothesis. More precisely, we focus on the No-Unbounded-Profit-with-Bounded-Risk (NUPBR) condition, also called non arbitrages of the first kind in the literature. We provide alternative proofs of some results from Aksamit et al. (Non-arbitrage up to random horizon for semimartingale models, short version, preprint, 2014 [arXiv:1310.1142]), incorporating a different methodology based on our optional semimartingale decomposition.

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Notes

  1. 1.

    The random variable \(\xi\) can take values in a more general space without any difficulty.

  2. 2.

    The upper script “pr” stands for progressive.

  3. 3.

    A set \(A \subset \varOmega \times [0,\infty [\) is thin if, for all ω ∈ Ω the set A(ω) is countable.

  4. 4.

    The upper script “i ” stands for initial.

References

  1. B. Acciaio, C. Fontana, C. Kardaras, Arbitrage of the first kind and filtration enlargements in semimartingale financial models. Preprint (2014) [arXiv:1401.7198]

    Google Scholar 

  2. A. Aksamit, T. Choulli, J. Deng, M. Jeanblanc, Arbitrages in a progressive enlargement setting. Arbitrage, Credit Inf. Risks, Peking Univ. Ser. Math. 6, 55–88 (2014)

    Google Scholar 

  3. A. Aksamit, T. Choulli, J. Deng, M. Jeanblanc, Non-arbitrage under a class of honest times. Preprint (2014) [arXiv:1404.0410]

    Google Scholar 

  4. A. Aksamit, T. Choulli, J. Deng, M. Jeanblanc, Non-arbitrage up to random horizon for semimartingale models, long version. Preprint (2014) [arXiv:1310.1142v2]

    Google Scholar 

  5. A. Aksamit, T. Choulli, J. Deng, M. Jeanblanc, Non-arbitrage up to random horizon for semimartingale models, short version. Preprint (2014) [arXiv:1310.1142]

    Google Scholar 

  6. J. Amendinger, Initial enlargement of filtrations and additional information in financial markets. Ph.D. thesis, Technischen Universität Berlin, 1999

    Google Scholar 

  7. T. Choulli, J. Deng, J. Ma, How non-arbitrage, viability and numéraire portfolio are related. Finance Stochast (2014). arXiv:1211.4598v3

    Google Scholar 

  8. F. Delbaen, W. Schachermayer, A general version of the fundamental theorem of asset pricing. Math. Ann. 300(1), 463–520 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  9. C. Dellacherie, P.A. Meyer, B. Maisonneuve, Probabilités et potentiel: Chapitres 17 à 24. Processus de Markov (fin), compléments de calcul stochastique (Hermann, Paris, 1992)

    Google Scholar 

  10. C. Fontana, No-arbitrage conditions and absolutely continuous changes of measure. Arbitrage, Credit Inf. Risks, Peking Univ. Ser. Math. 6, 3–18 (2014)

    Google Scholar 

  11. C. Fontana, M. Jeanblanc, S. Song, On arbitrages arising from honest times. Finance Stochast. 18, 515–543 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  12. A. Grorud, M. Pontier, Insider trading in a continuous time market model. Int. J. Theor. Appl. Finance 1(03), 331–347 (1998)

    Article  MATH  Google Scholar 

  13. A. Grorud, M. Pontier, Asymmetrical information and incomplete markets. Int. J. Theor. Appl. Finance 4(02), 285–302 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. H. Hulley, M. Schweizer, M6-on minimal market models and minimal martingale measures, in Contemporary Quantitative Finance (Springer, New York, 2010), pp. 35–51

    Google Scholar 

  15. P. Imkeller, Random times at which insiders can have free lunches. Stochastics 74(1–2), 465–487 (2002)

    MATH  MathSciNet  Google Scholar 

  16. P. Imkeller, N. Perkowski, The existence of dominating local martingale measures. Finance Stoch. Published on line: 13 June 2015 doi:10.1007/s00780-015-0264-0

    Google Scholar 

  17. J. Jacod, Grossissement initial, hypothèse \((\mathcal{H}^{{\prime}})\) et théorème de Girsanov, in Grossissements de Filtrations: Exemples et Applications (Springer, New York, 1985), pp. 15–35

    Book  Google Scholar 

  18. M. Jeanblanc, M. Yor, M. Chesney, Mathematical Methods for Financial Markets (Springer, New York, 2009)

    Book  MATH  Google Scholar 

  19. T. Jeulin, Semi-martingales et grossissement d’une filtration (Springer, New York, 1980)

    MATH  Google Scholar 

  20. T. Jeulin, M. Yor, Grossissement d’une filtration et semi-martingales: formules explicites, in Séminaire de Probabilités XII (Springer, New York, 1978), pp. 78–97

    Google Scholar 

  21. Y. Kabanov, On the FTAP of Kreps-Delbaen-Schachermayer, in Statistics and control of stochastic processes (Moscow, 1995/1996) (1997), pp. 191–203

    Google Scholar 

  22. Y. Kabanov, C. Kardaras, S. Song, On local martingale deflators and market portfolios (2014) [arXiv:1501.04363]

    Google Scholar 

  23. Karatzas, I., Kardaras, C. The numéraire portfolio in semimartingale financial models. Finance Stochast. 11(4), 447–493 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  24. C. Kardaras, Market viability via absence of arbitrage of the first kind. Finance Stochast. 16(4), 651–667 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  25. C. Kardaras, On the stochastic behaviour of optional processes up to random times. Ann. Appl. Probab. 25(2), 429–464 (2015)

    Article  MathSciNet  Google Scholar 

  26. K. Larsen, G. Žitković, On utility maximization under convex portfolio constraints. Ann. Appl. Probab. 23(2), 665–692 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  27. E. Platen, A benchmark approach to finance. Math. Finance 16(1), 131–151 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  28. P. Protter, Stochastic Integration and Differential Equations: Version 2.1, vol. 21 (Springer, New York, 2004)

    Google Scholar 

  29. D.B. Rokhlin, On the existence of an equivalent supermartingale density for a fork-convex family of stochastic processes. Math. Notes 87(3–4), 556–563 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  30. J. Ruf, Hedging under arbitrage. Math. Finance 23(2), 297–317 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  31. M. Schweizer, K. Takaoka, A note on the condition of no unbounded profit with bounded risk. Finance Stochast. 28(2), 393–405 (2013)

    MathSciNet  Google Scholar 

  32. S. Song, Grossissement de filtration et problèmes connexes. Ph.D. thesis, Université Paris VI, 1987

    Google Scholar 

  33. S. Song, Local martingale deflators for asset processes stopped at a default time S τ or right before S τ−. Preprint (2014) [arXiv:1405.4474]

    Google Scholar 

  34. C. Stricker, M. Yor, Calcul stochastique dépendant d’un paramètre. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 45(2), 109–133 (1978)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

The authors are thankful to the Chaire Marchés en Mutation (Fédération Bancaire Française) for financial support and to Marek Rutkowski for valuable comments that helped to improve this paper.

We thank also the anonymous referee for his(her) helpful comments.

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques et Modélisation d’Évry (LaMME), Université d’Évry-Val-d’Essonne, UMR CNRS 8071, Évry, France

    Anna Aksamit & Monique Jeanblanc

  2. Mathematical and Statistical Sciences Department, University of Alberta, Edmonton, AB, Canada

    Tahir Choulli

Authors
  1. Anna Aksamit
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  2. Tahir Choulli
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  3. Monique Jeanblanc
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Corresponding author

Correspondence to Monique Jeanblanc .

Editor information

Editors and Affiliations

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

    Catherine Donati-Martin

  2. Campus scientifique, IECL, Vandœuvre-lès-Nancy, France

    Antoine Lejay

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

    Alain Rouault

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Aksamit, A., Choulli, T., Jeanblanc, M. (2015). On an Optional Semimartingale Decomposition and the Existence of a Deflator in an Enlarged Filtration. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) In Memoriam Marc Yor - Séminaire de Probabilités XLVII. Lecture Notes in Mathematics(), vol 2137. Springer, Cham. https://doi.org/10.1007/978-3-319-18585-9_9

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