Local martingales and filtration shrinkage
ESAIM: Probability and Statistics (2011)
- Volume: 15, page S25-S38
- ISSN: 1292-8100
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topFöllmer, Hans, and Protter, Philip. "Local martingales and filtration shrinkage." ESAIM: Probability and Statistics 15 (2011): S25-S38. <http://eudml.org/doc/197762>.
@article{Föllmer2011,
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.
},
author = {Föllmer, Hans, Protter, Philip},
journal = {ESAIM: Probability and Statistics},
keywords = {Filtration shrinkage; filtration expansion; martingales; semimartingales; local time; Brownian motion},
language = {eng},
month = {5},
pages = {S25-S38},
publisher = {EDP Sciences},
title = {Local martingales and filtration shrinkage},
url = {http://eudml.org/doc/197762},
volume = {15},
year = {2011},
}
TY - JOUR
AU - Föllmer, Hans
AU - Protter, Philip
TI - Local martingales and filtration shrinkage
JO - ESAIM: Probability and Statistics
DA - 2011/5//
PB - EDP Sciences
VL - 15
SP - S25
EP - S38
AB -
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.
LA - eng
KW - Filtration shrinkage; filtration expansion; martingales; semimartingales; local time; Brownian motion
UR - http://eudml.org/doc/197762
ER -
References
top- S. Blei and H.J. Engelbert, On Exponential Local Martingales Associated with Strong Markov Continuous Local Martingales. Stoch. Process. Appl.119 (2009) 2859–2880.
- P. Brémaud and M. Yor, Changes of Filtration and Probability Measures. Z. Wahrscheinlichkeitstheorie verw. Gebiete45 (1978) 269–295.
- U. Çetin, R. Jarrow, P. Protter and Y. Yildirim, Modeling credit risk with partial information. Ann. Appl. Probab.14 (2004) 1167–1178.
- A. Cox and D. Hobson, Local martingales, bubbles and option prices. Finance Stoch.9 (2005) 477–492.
- 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).
- H. Föllmer, The exit measure of a supermartingale. Z. Wahrscheinlichkeitstheorie verw. Gebiete21 (1972) 154–166.
- H. Föllmer, On the representation of semimartingales. Ann. Probab.1 (1973) 580–589.
- H. Föllmer, P. Protter and A.S. Shiryaev, Quadratic covariation and an extension of Itô's formula. Bernoulli1 (2005) 149–169.
- K. Itô and S. Watanabe, Transformation of Markov processes by multiplicative functionals. Ann. Inst. Fourier15 (1965) 15–30.
- R. Jarrow and P. Protter, Structural versus Reduced Form Models: A New Information Based Perspective. J. Invest. Manage.2 (2004) 34–43.
- R. Jarrow and P. Protter, Large Traders, Hidden Arbitrage, and Complete Markets. J. Bank. Financ.29 (2005) 2803–2810.
- R. Jarrow, P. Protter and D. Sezer, Information Reduction via Level Crossings in a Credit Risk Model. Finance Stoch.11 (2007) 195–212.
- 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.
- R. Jarrow, P. Protter and K. Shimbo, Asset price bubbles in an incomplete market. Math. Finance20 (2010) 145–185.
- 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.
- G. Johnson and L.L. Helms, Class (D) Supermartingales. Bull. Am. Math. Soc.69 (1963) 59–62.
- 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.
- M. Loewenstein and G.A. Willard, Rational equilibrium asset-pricing bubbles in continuous trading models. J. Econ. Theory91 (2000) 17–58.
- 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)
- S. Pal and P. Protter, Analysis of continuous strict local martingales via h-transforms. Stoch. Process. Appl.120 (2010) 1424–1443.
- P. Protter, A Partial Introduction to Financial Asset Pricing Theory. Stoch. Process. Appl.91 (2001) 169–203.
- 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).
- D. Sezer, Filtration shrinkage by level crossings of a diffusion. Ann. Probab.35 (2007) 739–757.
- C. Stricker, Quasimartingales, martingales locales, et filtrations naturelles. Z. Wahrscheinlichkeitstheorie verw. Gebiete39 (1977) 55–63.
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