Abstract
In this note, we give a short overview of enlargement (or expansion) of filtration to help the reader who would like to have a survey on known results and some open questions. We try to select some papers (not all!) which contain important results. We present some applications in mathematical finance.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Aksamit, M. Jeanblanc and M. Rutkowski. Integral representations of martingales for progressive enlargements of filtrations. Stochastic Process. Appl.129(4), 1229–1258 (2019).
A. Aksamit, T. Choulli, J. Deng and M. Jeanblanc. Arbitrages in a progressive enlargement setting. In: Arbitrage, Credit and Informational Risks, C. Hillairet, M. Jeanblanc and Y. Jiao (eds.). pp. 53–86, World Scientific, Singapore (2014).
A. Aksamit and M. Jeanblanc. Enlargement of filtration with finance in view. Springer Brief (2017).
J. Amendinger. Initial enlargement of filtrations and additional information in financial markets. PhD thesis, Technischen Universität Berlin (1999).
J. Amendinger, D. Becherer and M. Schweizer. A monetary value for initial information in portfolio optimization. Finance and Stochastics7(1), 29–46 (2003).
S. Ankirchner. Information and Semimartingales. PhD thesis, Humboldt Universität Berlin (2005).
M.T. Barlow. Study of filtration expanded to include an honest time. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete44, 307–323 (1978).
T. Björk, T. Arbitrage Theory in Continuous Time. Second edition, Oxford University Press (2004).
Ch. Blanchet-Scalliet and M. Jeanblanc. Enlargement of filtration in discrete time. In: From Probability to Finance, Lecture Notes of BICMR Summer School on Financial Mathematics, pp. 71–144, Springer (2020).
P. Brémaud and M. Yor. Changes of filtration and of probability measures. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete17, 550–566 (2013).
G. Callegaro, M. Jeanblanc and B. Zargari. Carthagian enlargement of filtrations. ESAIM: Probability and Statistics17, 550–566 (2013).
T. Choulli and J. Deng. Non-arbitrage for Informational Discrete Time Market. Stochastics89(3–4), 1–26 (2017).
D. Coculescu, M. Jeanblanc and A. Nikeghbali. Default times, non arbitrage conditions and change of probability measures. Finance and Stochastics16(3), 513–535 (2012).
J.M. Corcuera and V. Valdivia. Enlargements of Filtrations and Applications. Preprint, arXiv:1201.5870 (2012).
F. Delbaen and W. Schachermayer. The Mathematics of Arbitrage. Springer, Berlin (2005).
C. Dellacherie, B. Maisonneuve and P.-A. Meyer. Probabilités et Potentiel, chapitres XVII–XXIV, Processus de Markov (fin). Compléments de calcul stochastique. Hermann, Paris (1992).
P. Di Tella and M. Jeanblanc. Martingale representation in the enlargement of the filtration generated by a point process. Stochastic Process. Appl.131, 103–121 (2021).
R.J. Elliott, M. Jeanblanc and M. Yor. On models of default risk. Math. Finance10, 179–196 (2000).
C. Fontana The strong predictable representation property in initially enlarged filtrations under the density hypothesis. Stochastic Process. Appl.128(3), 1007–1033 (2018).
C. Fontana, M. Jeanblanc and S. Song. On arbitrages arising with honest times. Finance Stoch.18, 515–543 (2014).
A. Grorud and M. Pontier. Insider trading in a continuous time market model. Int. J. Theor. Appl. Finance1, 331–347 (1998).
D. Gueye and M. Jeanblanc. Generalized Cox model for default times. Forthcoming in Mathematics Going Forward, Lecture Notes in Mathematics 2313, ch. 9, Springer.
C. Hillairet and Y. Jiao. Portfolio Optimization with Different Information Flow. ISTE Press, Elsevier (2017).
K. Itô. Extension of stochastic integrals. In: Proceedings of the International Sympsoium in Stochastic Differential Equations (RIMS, Kyoto Univ., Kyoto, 1976), pp. 95–109, Wiley (1976).
J. Jacod. Calcul stochastique et Problèmes de martingales. Lecture Notes in Mathematics 714, Springer-Verlag, Berlin (1979).
J. Jacod. Grossissement initial, hypothèse H′ et théorème de Girsanov. In: Grossissements de filtrations: exemples et applications, Lecture Notes in Mathematics, Séminaire de Calcul Stochastique 1982–83, vol. 1118, Springer-Verlag (1987).
M. Jeanblanc and L. Li. Characteristics and constructions of default times. SIAM J. Financial Math. 11(3), 720–749 (2020).
M. Jeanblanc, M. Yor and M. Chesney. Martingale Methods for Financial Markets. Springer-Verlag, Berlin (2007).
Th. Jeulin. Semi-martingales et grossissement de filtration. Lecture Notes in Mathematics 833, Springer-Verlag (1980).
Th. Jeulin and M. Yor. Grossissements de filtrations: exemples et applications. Lecture Notes in Mathematics 1118, Springer-Verlag (1985).
Y. Kabanov, C. Kardaras and S. Song. On Local Martingale Deflators and Market Portfolios. Finance and Stochastics20(4), 1097–1108 (2016).
Y. Kchia and P. Protter. Progressive Filtration Expansions via a Process, with Applications to Insider Trading. Int. J. Theor. Appl. Finan.18, 1550027 (2014).
D. Lando. On Cox processes and credit risky securities. Review of Derivatives Research2, 99–120 (1998).
B. Mallein and M. Yor. Exercices sur les temps locaux de semi-martingales continues et les excursions browniennes. Preprint, arXiv:1606.07118 (2016).
R. Mansuy and M. Yor. Random Times and Enlargements of Filtrations in a Brownian Setting. Lectures Notes in Mathematics 1873, Springer (2006).
L. Neufcourt and Ph. Protter. Expansion of a filtration with a stochastic process: the information drift. Preprint arXiv:1902.06780 (2019).
A. Nikeghbali. An essay on the general theory of stochastic processes. Probability Surveys3, 345–412 (2006).
P.E. Protter. Stochastic Integration and Differential Equations. Second edition, Springer, Berlin (2005).
S. Song. Local solution method for the problem of enlargement of filtration. Preprint, arXiv:1302.2862 (2013).
M. Yor. Some Aspects of Brownian Motion, Part II: Some Recent Martingale Problems. Lectures in Mathematics, ETH Zürich, Birkhäuser, Basel (1997).
Acknowledgements
The author thanks warmly the language editor for improving the English a lot and the two referees for providing some help to improve the paper.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Jeanblanc, M. (2023). Some Remarks on Enlargement of Filtration and Finance. In: Morel, JM., Teissier, B. (eds) Mathematics Going Forward . Lecture Notes in Mathematics, vol 2313. Springer, Cham. https://doi.org/10.1007/978-3-031-12244-6_9
Download citation
DOI: https://doi.org/10.1007/978-3-031-12244-6_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-12243-9
Online ISBN: 978-3-031-12244-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)