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Carthaginian enlargement of filtrations∗

Published online by Cambridge University Press:  01 August 2013

Giorgia Callegaro ,
Monique Jeanblanc  and
Behnaz Zargari
Giorgia Callegaro
Affiliation:
Scuola Normale Superiore di Pisa, Piazza dei Cavalieri, 7 56126 Pisa, Italy. giogiocallegaro@gmail.com CREST, 15 Boulevard Gabriel Péri, 92254 Malakoff Cedex, France
Monique Jeanblanc
Affiliation:
Institut Europlace de Finance (EIF), Palais Brongniart, 28 Place de la Bourse, 75002 Paris, France; monique.jeanblanc@univ-evry.fr
Behnaz Zargari
Affiliation:
Sharif University of Technology, P.O. Box 11365-8639, Tehran, Iran; behnazzargari@yahoo.com
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Abstract

This work is concerned with the theory of initial and progressive enlargements of areference filtration \hbox{$\mathbb{F}$}F with a random time τ. We provide, under anequivalence assumption, slightly stronger than the absolute continuity assumption ofJacod, alternative proofs to results concerning canonical decomposition of an \hbox{$\mathbb{F}$}F-martingalein the enlarged filtrations. Also, we address martingales’ characterization in theenlarged filtrations in terms of martingales in the reference filtration, as well aspredictable representation theorems in the enlarged filtrations.

Keywords

Initial and progressive enlargements of filtrationspredictable projectioncanonical decomposition of semimartingalespredictable representation theorem
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 550 - 566
Copyright
© EDP Sciences, SMAI, 2013

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References

J. Amendinger, Initial Enlargement of Filtrations and Additional Information in Financial Markets. Ph.D. thesis, Technischen Universität Berlin (1999).
S. Ankirchner, S. Dereich and P. Imkeller, Elargement of filtrations, continuous Girsanov-type embeddings, Séminaire de probabilités XL (2007) 389–410.
Azéma, J., Quelques applications de la théorie générale des processus, Invent. Math. 18 (1972). 293–336. Google Scholar
Barlow, M.T., Study of filtration expanded to include an honest time. Z. Wahr. Verw. Gebiete 44 (1978) 307–323. Google Scholar
T.R. Bielecki, M. Jeanblanc and M. Rutkowski, Credit Risk Modeling. CSFI Lect. Note Series. Osaka University Press (2009).
P. Brémaud, Point Processes and Queues: Martingale Dynamics. Springer-Verlag (1981).
C.S. Chou and P.-A. Meyer, Sur la représentation des martingales comme intégrales stochastiques dans les processus ponctuels. Séminaire de probabilités IX (1975) 226–236.
C. Dellacherie and P.-A. Meyer, Probabilités et Potentiel – Chapitres XXVII à XXIV, Processus de Markov. Hermann, Paris (1992).
El Karoui, N., Jeanblanc, M. and Jiao, Y., What happens after a default: the conditional density approach. Stoch. Proc. Appl. 120 (2010) 1011–1032. Google Scholar
Föllmer, H. and Imkeller, P., Anticipation cancelled by a Girsanov transformation: a paradox on Wiener space. Ann. Inst. Henri Poincaré 29 (1993) 569–586. Google Scholar
D. Gasbarra, E. Valkeila and L. Vostrikova, Enlargement of filtration and additional information in pricing models: Bayesian approach, in From Stochastic Calculus to Mathematical Finance, edited by Y. Kabanov, R. Liptser and J. Stoyanov. Springer-Verlag (2006) 257–285.
Grorud, A. and Pontier, M., Insider trading in a continuous time market model. Int. J. Theor. Appl. Finance 1 (1998) 331–347. Google Scholar
Grorud, A. and Pontier, M., Asymmetrical information and incomplete markets. Int. J. Theor. Appl. Finance 4 (2001) 285–302. Google Scholar
Sh. He, J. Wang and J. Yan, Semimartingale theory and stochastic calculus. CRC Press (1992).
J. Jacod, Grossissement initial, hypothèse (H′) et théorème de Girsanov, Lect. Notes Math., vol. 1118. Springer-Verlag (1985) 15–35.
Jeanblanc, M. and Le Cam, Y., Progressive enlargement of filtrations with initial times. Stoch. Proc. Appl. 119 (2009) 2523–2543. Google Scholar
M. Jeanblanc and Y. Le Cam, Immersion Property and Credit Risk Modelling, in Optimality and Risk – Modern Trends in Mathematical Finance, edited by F. Delbaen, M. Rásonyi and C. Stricker. Springer (2010) 99–132.
M. Jeanblanc, M. Yor and M. Chesney, Mathematical Methods in Financial Markets. Springer (2009).
T. Jeulin, Semimartingales et grossissement d’une filtration, Lect. Notes Math., vol. 833. Springer-Verlag (1980).
Y. Kchia, M. Larsson and P. Protter, Linking progressive and initial filtration expansions, Working paper.
Kusuoka, S., A remark on default risk models, Adv. Math. Econ. 1 (1999) 69–82. Google Scholar
Sh. Song, Grossissement de filtration et problèmes connexes. Ph.D. thesis, Université Paris VI (1987).
Stricker, C., Quasi-martingales, martingales locales et filtrations naturelles. Zeitschrift fur Wahr 39 (1977) 55–63. Google Scholar
Stricker, C. and Yor, M., Calcul stochastique dépendant d’un paramètre. Zeitschrift fur Wahr 45 (1978) 109–133. Google Scholar
M. Yor, Grossissement de filtrations et absolue continuité de noyaux, Lect. Notes Math., vol. 1118. Springer-Verlag (1985) 7–14.