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Local martingales and filtration shrinkage

Published online by Cambridge University Press:  15 October 2010

Hans Föllmer  and
Philip Protter
Hans Föllmer
Affiliation:
Humboldt Universität, Berlin, Germany
Philip Protter
Affiliation:
Supported in part by NSF Grant DMS-0906995; Statistics, Columbia University, New York, NY, 10027, USA. protter@stat.columbia.edu
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Abstract

A general theory is developed for the projection of martingale related processes onto smaller filtrations, to which they are not even adapted. Martingales, supermartingales, and semimartingales retain their nature, but the case of local martingales is more delicate, as illustrated by an explicit case study for the inverse Bessel process. This has implications for the concept of No Free Lunch with Vanishing Risk, in Finance.

Keywords

Filtration shrinkagefiltration expansionmartingalessemimartingaleslocal timeBrownian motion
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 15: Supplement: In honor of Marc Yor , 2011 , pp. S25 - S38
Copyright
© EDP Sciences, SMAI, 2011

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References

Blei, S. and Engelbert, H.J., Exponential Local Martingales Associated, On with Strong Markov Continuous Local Martingales. Stoch. Process. Appl. 119 (2009) 2859–2880. CrossRef
Brémaud, P. and Yor, M., Changes of Filtration and Probability Measures. Z. Wahrscheinlichkeitstheorie verw. Gebiete 45 (1978) 269–295. CrossRef
Çetin, U., Jarrow, R., Protter, P. and Yildirim, Y., Modeling credit risk with partial information. Ann. Appl. Probab. 14 (2004) 1167–1178.
Cox, A. and Hobson, D., Local martingales, bubbles and option prices. Finance Stoch. 9 (2005) 477–492. CrossRef
F. Delbaen and W. Schachermayer, The Mathematics of Arbitrage. Springer-Verlag, Heidelberg (2006).
C. Dellacherie and P.A. Meyer, Probabilités et Potentiel: Chapitres V à VIII, Théorie des Martingales. Hermann, Paris (1980).
Föllmer, H., The exit measure of a supermartingale. Z. Wahrscheinlichkeitstheorie verw. Gebiete 21 (1972) 154–166. CrossRef
Föllmer, H., On the representation of semimartingales. Ann. Probab. 1 (1973) 580–589. CrossRef
Föllmer, H., Protter, P. and Shiryaev, A.S., Quadratic covariation and an extension of Itô's formula. Bernoulli 1 (2005) 149–169. CrossRef
Itô, K. and Watanabe, S., Transformation of Markov processes by multiplicative functionals. Ann. Inst. Fourier 15 (1965) 15–30. CrossRef
Jarrow, R. and Protter, P., Structural versus Reduced Form Models: A New Information Based Perspective. J. Invest. Manage. 2 (2004) 34–43.
Jarrow, R. and Protter, P., Large Traders, Hidden Arbitrage, and Complete Markets. J. Bank. Financ. 29 (2005) 2803–2810. CrossRef
Jarrow, R., Protter, P. and Sezer, D., Information Reduction via Level Crossings in a Credit Risk Model. Finance Stoch. 11 (2007) 195–212. CrossRef
R. Jarrow, P. Protter and K. Shimbo, Asset price bubbles in a complete market. Adv. Math. Finance, in Honor of Dilip B. Madan (2006) 105–130.
Jarrow, R., Protter, P. and Shimbo, K., Asset price bubbles in an incomplete market. Math. Finance 20 (2010) 145–185. CrossRef
T. Jeulin and M. Yor, Inégalité de Hardy, semimartingales, et faux-amis, in Séminaire de Probabilités XIII. Lect. Notes Math. 721, Springer, Heidelberg, Berlin, New York (1979) 332–359.
Johnson, G. and Helms, L.L., Class (D) Supermartingales. Bull. Am. Math. Soc. 69 (1963) 59–62. CrossRef
N. Kazamaki, Krickeberg's decomposition for local martingales, in Séminaire de Probabilités VI. Lecture Notes in Math. 258, Springer, Heidelberg, Berlin, New York (1972) 101–103.
Loewenstein, M. and Willard, G.A., Rational equilibrium asset-pricing bubbles in continuous trading models. J. Econ. Theory 91 (2000) 17–58. CrossRef
P.A. Meyer, Probability and Potentials. Blaisdell, Waltham (1966).
P.A. Meyer, La mesure de H. Föllmer en théorie des surmartingales, in Séminaire de Probabilités VI. Lecture Notes in Math. 258, Springer, Heidelberg, Berlin, New York (1971) 118–129.
P.A. Meyer, Martingales and stochastic integrals. Lect. Notes Math. 284, Springer, Heidelberg, Berlin, New York (1972).
P.A. Meyer, Sur un théorème de C. Stricker, in Séminaire de Probabilités XI. Lecture Notes in Math. 581, Springer, Heidelberg, Berlin, New York (1977) 482–489.
A. Mijatović and M. Urusov, The Martingale Property of Certain Local Martingales. arXiv:0905.3701 (2009)
Pal, S. and Protter, P., Analysis of continuous strict local martingales via h-transforms. Stoch. Process. Appl. 120 (2010) 1424–1443. CrossRef
Protter, P., Partial Introduction, A to Financial Asset Pricing Theory. Stoch. Process. Appl. 91 (2001) 169–203. CrossRef
P. Protter, Stochastic Integration and Differential Equations, 2nd edition, Version 2.1. Springer-Verlag, Heidelberg (2005).
D. Revuz and M. Yor, Continuous martingales and Brownian motion, 3rd edition. Springer, Berlin-Heidelberg (1999).
Sezer, D., Filtration shrinkage by level crossings of a diffusion. Ann. Probab. 35 (2007) 739–757. CrossRef
Stricker, C., Quasimartingales, martingales locales, et filtrations naturelles. Z. Wahrscheinlichkeitstheorie verw. Gebiete 39 (1977) 55–63. CrossRef