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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The random variable \(\xi\) can take values in a more general space without any difficulty.
- 2.
The upper script “pr” stands for progressive.
- 3.
A set \(A \subset \varOmega \times [0,\infty [\) is thin if, for all ω ∈ Ω the set A(ω) is countable.
- 4.
The upper script “i ” stands for initial.
References
B. Acciaio, C. Fontana, C. Kardaras, Arbitrage of the first kind and filtration enlargements in semimartingale financial models. Preprint (2014) [arXiv:1401.7198]
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)
A. Aksamit, T. Choulli, J. Deng, M. Jeanblanc, Non-arbitrage under a class of honest times. Preprint (2014) [arXiv:1404.0410]
A. Aksamit, T. Choulli, J. Deng, M. Jeanblanc, Non-arbitrage up to random horizon for semimartingale models, long version. Preprint (2014) [arXiv:1310.1142v2]
A. Aksamit, T. Choulli, J. Deng, M. Jeanblanc, Non-arbitrage up to random horizon for semimartingale models, short version. Preprint (2014) [arXiv:1310.1142]
J. Amendinger, Initial enlargement of filtrations and additional information in financial markets. Ph.D. thesis, Technischen Universität Berlin, 1999
T. Choulli, J. Deng, J. Ma, How non-arbitrage, viability and numéraire portfolio are related. Finance Stochast (2014). arXiv:1211.4598v3
F. Delbaen, W. Schachermayer, A general version of the fundamental theorem of asset pricing. Math. Ann. 300(1), 463–520 (1994)
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)
C. Fontana, No-arbitrage conditions and absolutely continuous changes of measure. Arbitrage, Credit Inf. Risks, Peking Univ. Ser. Math. 6, 3–18 (2014)
C. Fontana, M. Jeanblanc, S. Song, On arbitrages arising from honest times. Finance Stochast. 18, 515–543 (2014)
A. Grorud, M. Pontier, Insider trading in a continuous time market model. Int. J. Theor. Appl. Finance 1(03), 331–347 (1998)
A. Grorud, M. Pontier, Asymmetrical information and incomplete markets. Int. J. Theor. Appl. Finance 4(02), 285–302 (2001)
H. Hulley, M. Schweizer, M6-on minimal market models and minimal martingale measures, in Contemporary Quantitative Finance (Springer, New York, 2010), pp. 35–51
P. Imkeller, Random times at which insiders can have free lunches. Stochastics 74(1–2), 465–487 (2002)
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
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
M. Jeanblanc, M. Yor, M. Chesney, Mathematical Methods for Financial Markets (Springer, New York, 2009)
T. Jeulin, Semi-martingales et grossissement d’une filtration (Springer, New York, 1980)
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
Y. Kabanov, On the FTAP of Kreps-Delbaen-Schachermayer, in Statistics and control of stochastic processes (Moscow, 1995/1996) (1997), pp. 191–203
Y. Kabanov, C. Kardaras, S. Song, On local martingale deflators and market portfolios (2014) [arXiv:1501.04363]
Karatzas, I., Kardaras, C. The numéraire portfolio in semimartingale financial models. Finance Stochast. 11(4), 447–493 (2007)
C. Kardaras, Market viability via absence of arbitrage of the first kind. Finance Stochast. 16(4), 651–667 (2012)
C. Kardaras, On the stochastic behaviour of optional processes up to random times. Ann. Appl. Probab. 25(2), 429–464 (2015)
K. Larsen, G. Žitković, On utility maximization under convex portfolio constraints. Ann. Appl. Probab. 23(2), 665–692 (2013)
E. Platen, A benchmark approach to finance. Math. Finance 16(1), 131–151 (2006)
P. Protter, Stochastic Integration and Differential Equations: Version 2.1, vol. 21 (Springer, New York, 2004)
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)
J. Ruf, Hedging under arbitrage. Math. Finance 23(2), 297–317 (2013)
M. Schweizer, K. Takaoka, A note on the condition of no unbounded profit with bounded risk. Finance Stochast. 28(2), 393–405 (2013)
S. Song, Grossissement de filtration et problèmes connexes. Ph.D. thesis, Université Paris VI, 1987
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]
C. Stricker, M. Yor, Calcul stochastique dépendant d’un paramètre. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 45(2), 109–133 (1978)
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
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-319-18585-9_9
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-18584-2
Online ISBN: 978-3-319-18585-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)