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Successive enlargement of filtrations and application to insider information

Part of: Mathematical finance Stochastic processes

Published online by Cambridge University Press:  08 September 2017

Christophette Blanchet-Scalliet ,
Caroline Hillairet  and
Ying Jiao
Christophette Blanchet-Scalliet*
Affiliation:
Université de Lyon
Caroline Hillairet*
Affiliation:
Ensae ParisTech
Ying Jiao*
Affiliation:
Université Claude Bernard - Lyon 1
*
* Postal address: Université de Lyon - CNRS, UMR 5208, Institut Camille Jordan-Ecole Centrale de Lyon, 36 avenue Guy de Collongue, 69134 Ecully Cedex, France. Email address: [email protected]
** Postal address: CREST, UMR 9194, Ensae ParisTech, Université Paris Saclay, France. Email address: [email protected]
*** Postal address: Université Claude Bernard - Lyon 1, Institut de Science Financière et d'Assurances, 50 Avenue Tony Garnier, 69007 Lyon, France. Email address: [email protected]
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Abstract

We model in a dynamic way an insider's private information flow which is successively augmented by a family of initial enlargement of filtrations. According to the a priori available information, we propose several density hypotheses which are presented in hierarchical order from the weakest to the strongest. We compare these hypotheses, in particular, with Jacod's one, and deduce conditional expectations under each of them by providing consistent expressions with respect to the common reference filtration. Finally, this framework is applied to a default model with insider information on the default threshold and some numerical illustrations are performed.

Keywords

Successive enlargementdensity hypothesisinsider informationcredit risk

MSC classification

Primary: 60G99: None of the above, but in this section
Secondary: 91G40: Credit risk
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 3 , September 2017 , pp. 653 - 685
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Amendinger, J. (2000). Martingale representation theorems for initially enlarged filtrations. Stoch. Process. Appl. 89, 101–116. CrossRefGoogle Scholar
[2] Amendinger, J., Imkeller, P. and Schweizer, M. (1998). Additional logarithmic utility of an insider. Stoch. Process. Appl. 75, 263–286. Google Scholar
[3] Biagini, F. and Øksendal, B. (2005). A general stochastic calculus approach to insider trading. Appl. Math. Optimization 52, 167–181. CrossRefGoogle Scholar
[4] Bielecki, T. R. and Rutkowski, M. (2002). Credit Risk: Modelling, Valuation and Hedging. Springer, Berlin. Google Scholar
[5] Bielecki, T. R., Jeanblanc, M. and Rutkowski, M. (2004). Modeling and valuation of credit risk. In Stochastic Methods in Finance (Lecture Notes Math. 1856), Springer, Berlin, pp. 27–126. Google Scholar
[6] Bilina Falafala, R. and Protter, P. (2015). Insider trading and risk. Submitted. Google Scholar
[7] Borodin, A. N. and Salminen, P. (1996). Handbook of Brownian Motion—Facts and Formulae. Birkhäuser, Basel. CrossRefGoogle Scholar
[8] Corcuera, J. M., Imkeller, P., Kohatsu-Higa, A. and Nualart, D. (2004). Additional utility of insiders with imperfect dynamical information. Finance Stoch. 8, 437–450. Google Scholar
[9] Elliott, R. J., Jeanblanc, M. and Yor, M. (2000). On models of default risk. Math. Finance 10, 179–195. CrossRefGoogle Scholar
[10] Föllmer, H. and Imkeller, P. (1993). Anticipation cancelled by a Girsanov transformation: a paradox on Wiener space. Ann. Inst. H. Poincaré Prob. Statist. 29, 569–586. Google Scholar
[11] Föllmer, H. and Schweizer, M. (1991). Hedging of contingent claims under incomplete information. In Applied Stochastic Analysis (Stoch. Monogr. 5), Gordon and Breach, New York, pp. 389–414. Google Scholar
[12] Grorud, A. and Pontier, M. (1998). Insider trading in a continuous time market model. Internat. J. Theoret. Appl. Finance 1, 331–347. Google Scholar
[13] Gumbel, E. J. (1961). Bivariate logistic distributions. J. Amer. Statist. Assoc. 56, 335–349. Google Scholar
[14] Hillairet, C. and Jiao, Y. (2012). Credit risk with asymmetric information on the default threshold. Stochastics 84, 183–198. Google Scholar
[15] Imkeller, P. (2002). Random times at which insiders can have free lunches. Stoch. Reports 74, 465–487. CrossRefGoogle Scholar
[16] Imkeller, P. (2003). Malliavin's calculus in insider models: additional utility and free lunches. Math. Finance 13, 153–169. Google Scholar
[17] Jacod, J. (1979). Calcul Stochastique et Problèmes de Martingales (Lecture Notes Math. 714). Springer, Berlin. CrossRefGoogle Scholar
[18] Jacod, J. (1985). Grossissement initial, hypothèse (H') et théorème de Girsanov. In Grossissements de Filtrations (Lecture Notes Math. 1118), Springer, Berlin, pp. 15–35. Google Scholar
[19] Jeulin, T. (1980). Semi-Martingales et Grossissement d'une Filtration (Lecture Notes Math. 833). Springer, Berlin. Google Scholar
[20] Jeulin, T. and Yor, M. (1978). Grossissement d'une filtration et semi-martingales: formules explicites. In Séminaire de Probabilités, XII (Univ. Strasbourg, 1976/1977; Lecture Notes Math. 649), Springer, Berlin, pp. 78–97. CrossRefGoogle Scholar
[21] Jeulin, T. and Yor, M. (eds) (1985). Grossissements de Filtrations: Exemples et Applications (Lecture Notes Math. 1118). Springer, Berlin. Google Scholar
[22] Kchia, Y. and Protter, P. (2015). Progressive filtration expansions via a process, with applications to insider trading. Internat. J. Theoret. Appl. Finance 18, 1550027. CrossRefGoogle Scholar
[23] Kchia, Y., Larsson, M. and Protter, P. (2013). Linking progressive and initial filtration expansions. In Malliavin Calculus and Stochastic Analysis (Springer Proc. Math. Statist. 34), Springer, New York, pp. 469–487. Google Scholar
[24] Mansuy, R. and Yor, M. (2006). Random Times and Enlargements of Filtrations in a Brownian Setting (Lecture Notes Math. 1873). Springer, Berlin. Google Scholar
[25] Meyer, P.-A. (1979). Une remarque sur le calcul stochastique dépendant d'un paramètre. In Séminaire de Probabilités, XIII (Lecture Notes Math. 721), Springer, Berlin, pp. 199–203. Google Scholar
[26] Protter, P. E. (2004). Stochastic Integration and Differential Equations. Springer, Berlin. Google Scholar
[27] Song, S. (1987). Grossissement de filtrations et problèmes connexes. Doctoral thesis. Université Paris VII. Google Scholar
[28] Song, S. (2013). Local solution method for the problem of enlargement of filtration. Preprint. Available at https://arxiv.org/abs/1302.2862. Google Scholar