Carthaginian enlargement of filtrations
ESAIM: Probability and Statistics, Tome 17 (2013), pp. 550-566.

This work is concerned with the theory of initial and progressive enlargements of a reference filtration 𝔽 F with a random time τ. We provide, under an equivalence assumption, slightly stronger than the absolute continuity assumption of Jacod, alternative proofs to results concerning canonical decomposition of an 𝔽 F -martingale in the enlarged filtrations. Also, we address martingales’ characterization in the enlarged filtrations in terms of martingales in the reference filtration, as well as predictable representation theorems in the enlarged filtrations.

DOI : 10.1051/ps/2011162
Classification : 60G46, 60-02
Mots clés : initial and progressive enlargements of filtrations, predictable projection, canonical decomposition of semimartingales, predictable representation theorem
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Callegaro, Giorgia; Jeanblanc, Monique; Zargari, Behnaz. Carthaginian enlargement of filtrations. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 550-566. doi : 10.1051/ps/2011162. http://www.numdam.org/articles/10.1051/ps/2011162/

[1] J. Amendinger, Initial Enlargement of Filtrations and Additional Information in Financial Markets. Ph.D. thesis, Technischen Universität Berlin (1999). | Zbl

[2] S. Ankirchner, S. Dereich and P. Imkeller, Elargement of filtrations, continuous Girsanov-type embeddings, Séminaire de probabilités XL (2007) 389-410. | MR | Zbl

[3] J. Azéma, Quelques applications de la théorie générale des processus, Invent. Math. 18 (1972). 293-336. | MR | Zbl

[4] M.T. Barlow, Study of filtration expanded to include an honest time. Z. Wahr. Verw. Gebiete 44 (1978) 307-323. | MR | Zbl

[5] T.R. Bielecki, M. Jeanblanc and M. Rutkowski, Credit Risk Modeling. CSFI Lect. Note Series. Osaka University Press (2009). | Zbl

[6] P. Brémaud, Point Processes and Queues: Martingale Dynamics. Springer-Verlag (1981). | MR | Zbl

[7] 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. | Numdam | MR | Zbl

[8] C. Dellacherie and P.-A. Meyer, Probabilités et Potentiel - Chapitres XXVII à XXIV, Processus de Markov. Hermann, Paris (1992). | Zbl

[9] N. El Karoui, M. Jeanblanc and Y. Jiao, What happens after a default: the conditional density approach. Stoch. Proc. Appl. 120 (2010) 1011-1032. | MR | Zbl

[10] H. Föllmer and P. Imkeller, Anticipation cancelled by a Girsanov transformation: a paradox on Wiener space. Ann. Inst. Henri Poincaré 29 (1993) 569-586. | Numdam | MR | Zbl

[11] 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. | MR | Zbl

[12] A. Grorud and M. Pontier, Insider trading in a continuous time market model. Int. J. Theor. Appl. Finance 1 (1998) 331-347. | Zbl

[13] A. Grorud and M. Pontier, Asymmetrical information and incomplete markets. Int. J. Theor. Appl. Finance 4 (2001) 285-302. | MR | Zbl

[14] Sh. He, J. Wang and J. Yan, Semimartingale theory and stochastic calculus. CRC Press (1992). | MR | Zbl

[15] J. Jacod, Grossissement initial, hypothèse (H′) et théorème de Girsanov, Lect. Notes Math., vol. 1118. Springer-Verlag (1985) 15-35. | Zbl

[16] M. Jeanblanc and Y. Le Cam, Progressive enlargement of filtrations with initial times. Stoch. Proc. Appl. 119 (2009) 2523-2543. | MR | Zbl

[17] 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. | MR | Zbl

[18] M. Jeanblanc, M. Yor and M. Chesney, Mathematical Methods in Financial Markets. Springer (2009). | Zbl

[19] T. Jeulin, Semimartingales et grossissement d'une filtration, Lect. Notes Math., vol. 833. Springer-Verlag (1980). | MR | Zbl

[20] Y. Kchia, M. Larsson and P. Protter, Linking progressive and initial filtration expansions, Working paper.

[21] S. Kusuoka, A remark on default risk models, Adv. Math. Econ. 1 (1999) 69-82. | MR | Zbl

[22] Sh. Song, Grossissement de filtration et problèmes connexes. Ph.D. thesis, Université Paris VI (1987).

[23] C. Stricker, Quasi-martingales, martingales locales et filtrations naturelles. Zeitschrift fur Wahr 39 (1977) 55-63. | MR | Zbl

[24] C. Stricker and M. Yor, Calcul stochastique dépendant d'un paramètre. Zeitschrift fur Wahr 45 (1978) 109-133. | MR | Zbl

[25] M. Yor, Grossissement de filtrations et absolue continuité de noyaux, Lect. Notes Math., vol. 1118. Springer-Verlag (1985) 7-14. | MR | Zbl

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