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

  • Anna Aksamit
  • Tahir Choulli
  • Monique Jeanblanc
Part of the Lecture Notes in Mathematics book series (LNM, volume 2137)


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.


Random Time Local Martingale Arbitrage Opportunity Progressive Enlargement Positive Random Variable 
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.



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.


  1. 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. 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. 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. 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. 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. 6.
    J. Amendinger, Initial enlargement of filtrations and additional information in financial markets. Ph.D. thesis, Technischen Universität Berlin, 1999Google Scholar
  7. 7.
    T. Choulli, J. Deng, J. Ma, How non-arbitrage, viability and numéraire portfolio are related. Finance Stochast (2014). arXiv:1211.4598v3Google Scholar
  8. 8.
    F. Delbaen, W. Schachermayer, A general version of the fundamental theorem of asset pricing. Math. Ann. 300(1), 463–520 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 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. 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. 11.
    C. Fontana, M. Jeanblanc, S. Song, On arbitrages arising from honest times. Finance Stochast. 18, 515–543 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    A. Grorud, M. Pontier, Insider trading in a continuous time market model. Int. J. Theor. Appl. Finance 1(03), 331–347 (1998)zbMATHCrossRefGoogle Scholar
  13. 13.
    A. Grorud, M. Pontier, Asymmetrical information and incomplete markets. Int. J. Theor. Appl. Finance 4(02), 285–302 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    H. Hulley, M. Schweizer, M6-on minimal market models and minimal martingale measures, in Contemporary Quantitative Finance (Springer, New York, 2010), pp. 35–51Google Scholar
  15. 15.
    P. Imkeller, Random times at which insiders can have free lunches. Stochastics 74(1–2), 465–487 (2002)zbMATHMathSciNetGoogle Scholar
  16. 16.
    P. Imkeller, N. Perkowski, The existence of dominating local martingale measures. Finance Stoch. Published on line: 13 June 2015 doi:10.1007/s00780-015-0264-0Google Scholar
  17. 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. 15–35CrossRefGoogle Scholar
  18. 18.
    M. Jeanblanc, M. Yor, M. Chesney, Mathematical Methods for Financial Markets (Springer, New York, 2009)zbMATHCrossRefGoogle Scholar
  19. 19.
    T. Jeulin, Semi-martingales et grossissement d’une filtration (Springer, New York, 1980)zbMATHGoogle Scholar
  20. 20.
    T. Jeulin, M. Yor, Grossissement d’une filtration et semi-martingales: formules explicites, in Séminaire de Probabilités XII (Springer, New York, 1978), pp. 78–97Google Scholar
  21. 21.
    Y. Kabanov, On the FTAP of Kreps-Delbaen-Schachermayer, in Statistics and control of stochastic processes (Moscow, 1995/1996) (1997), pp. 191–203Google Scholar
  22. 22.
    Y. Kabanov, C. Kardaras, S. Song, On local martingale deflators and market portfolios (2014) [arXiv:1501.04363]Google Scholar
  23. 23.
    Karatzas, I., Kardaras, C. The numéraire portfolio in semimartingale financial models. Finance Stochast. 11(4), 447–493 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    C. Kardaras, Market viability via absence of arbitrage of the first kind. Finance Stochast. 16(4), 651–667 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    C. Kardaras, On the stochastic behaviour of optional processes up to random times. Ann. Appl. Probab. 25(2), 429–464 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    K. Larsen, G. Žitković, On utility maximization under convex portfolio constraints. Ann. Appl. Probab. 23(2), 665–692 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    E. Platen, A benchmark approach to finance. Math. Finance 16(1), 131–151 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    P. Protter, Stochastic Integration and Differential Equations: Version 2.1, vol. 21 (Springer, New York, 2004)Google Scholar
  29. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    J. Ruf, Hedging under arbitrage. Math. Finance 23(2), 297–317 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    M. Schweizer, K. Takaoka, A note on the condition of no unbounded profit with bounded risk. Finance Stochast. 28(2), 393–405 (2013)MathSciNetGoogle Scholar
  32. 32.
    S. Song, Grossissement de filtration et problèmes connexes. Ph.D. thesis, Université Paris VI, 1987Google Scholar
  33. 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. 34.
    C. Stricker, M. Yor, Calcul stochastique dépendant d’un paramètre. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 45(2), 109–133 (1978)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Anna Aksamit
    • 1
  • Tahir Choulli
    • 2
  • Monique Jeanblanc
    • 1
  1. 1.Laboratoire de Mathématiques et Modélisation d’Évry (LaMME)Université d’Évry-Val-d’Essonne, UMR CNRS 8071ÉvryFrance
  2. 2.Mathematical and Statistical Sciences DepartmentUniversity of AlbertaEdmontonCanada

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