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.
Mots-clés : filtration shrinkage, filtration expansion, martingales, semimartingales, local time, brownian motion
@article{PS_2011__15__S25_0,
author = {F\"ollmer, Hans and Protter, Philip},
title = {Local martingales and filtration shrinkage},
journal = {ESAIM: Probability and Statistics},
pages = {S25--S38},
publisher = {EDP-Sciences},
volume = {15},
year = {2011},
doi = {10.1051/ps/2010023},
mrnumber = {2817343},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps/2010023/}
}
TY - JOUR AU - Föllmer, Hans AU - Protter, Philip TI - Local martingales and filtration shrinkage JO - ESAIM: Probability and Statistics PY - 2011 SP - S25 EP - S38 VL - 15 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2010023/ DO - 10.1051/ps/2010023 LA - en ID - PS_2011__15__S25_0 ER -
Föllmer, Hans; Protter, Philip. Local martingales and filtration shrinkage. ESAIM: Probability and Statistics, Tome 15 (2011), pp. S25-S38. doi : 10.1051/ps/2010023. http://archive.numdam.org/articles/10.1051/ps/2010023/
[1] and , On Exponential Local Martingales Associated with Strong Markov Continuous Local Martingales. Stoch. Process. Appl. 119 (2009) 2859-2880. | MR | Zbl
[2] and , Changes of Filtration and Probability Measures. Z. Wahrscheinlichkeitstheorie verw. Gebiete 45 (1978) 269-295. | MR | Zbl
[3] , , and , Modeling credit risk with partial information. Ann. Appl. Probab. 14 (2004) 1167-1178. | MR | Zbl
[4] and , Local martingales, bubbles and option prices. Finance Stoch. 9 (2005) 477-492. | MR | Zbl
[5] and , The Mathematics of Arbitrage. Springer-Verlag, Heidelberg (2006). | MR | Zbl
[6] and , Probabilités et Potentiel: Chapitres V à VIII, Théorie des Martingales. Hermann, Paris (1980). | MR | Zbl
[7] , The exit measure of a supermartingale. Z. Wahrscheinlichkeitstheorie verw. Gebiete 21 (1972) 154-166. | MR | Zbl
[8] , On the representation of semimartingales. Ann. Probab. 1 (1973) 580-589. | MR | Zbl
[9] , and , Quadratic covariation and an extension of Itô's formula. Bernoulli 1 (2005) 149-169. | Zbl
[10] and , Transformation of Markov processes by multiplicative functionals. Ann. Inst. Fourier 15 (1965) 15-30. | Numdam | MR | Zbl
[11] and , Structural versus Reduced Form Models: A New Information Based Perspective. J. Invest. Manage. 2 (2004) 34-43.
[12] and , Large Traders, Hidden Arbitrage, and Complete Markets. J. Bank. Financ. 29 (2005) 2803-2810.
[13] , and , Information Reduction via Level Crossings in a Credit Risk Model. Finance Stoch. 11 (2007) 195-212. | MR | Zbl
[14] , and , Asset price bubbles in a complete market. Adv. Math. Finance, in Honor of Dilip B. Madan (2006) 105-130. | Zbl
[15] , and , Asset price bubbles in an incomplete market. Math. Finance 20 (2010) 145-185. | MR | Zbl
[16] and , Inégalité de Hardy, semimartingales, et faux-amis, in Séminaire de Probabilités XIII. Lect. Notes Math. 721, Springer, Heidelberg, Berlin, New York (1979) 332-359. | Numdam | MR | Zbl
[17] and , Class (D) Supermartingales. Bull. Am. Math. Soc. 69 (1963) 59-62. | MR | Zbl
[18] , Krickeberg's decomposition for local martingales, in Séminaire de Probabilités VI. Lecture Notes in Math. 258, Springer, Heidelberg, Berlin, New York (1972) 101-103. | Numdam | MR | Zbl
[19] and , Rational equilibrium asset-pricing bubbles in continuous trading models. J. Econ. Theory 91 (2000) 17-58. | MR | Zbl
[20] , Probability and Potentials. Blaisdell, Waltham (1966). | MR | Zbl
[21] , 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. | Numdam | MR | Zbl
[22] , Martingales and stochastic integrals. Lect. Notes Math. 284, Springer, Heidelberg, Berlin, New York (1972). | MR | Zbl
[23] , Sur un théorème de C. Stricker, in Séminaire de Probabilités XI. Lecture Notes in Math. 581, Springer, Heidelberg, Berlin, New York (1977) 482-489. | Numdam | MR | Zbl
[24] and , The Martingale Property of Certain Local Martingales. arXiv:0905.3701 (2009) | MR | Zbl
[25] and , Analysis of continuous strict local martingales via h-transforms. Stoch. Process. Appl. 120 (2010) 1424-1443. | MR | Zbl
[26] , A Partial Introduction to Financial Asset Pricing Theory. Stoch. Process. Appl. 91 (2001) 169-203. | MR | Zbl
[27] , Stochastic Integration and Differential Equations, 2nd edition, Version 2.1. Springer-Verlag, Heidelberg (2005). | MR | Zbl
[28] and , Continuous martingales and Brownian motion, 3rd edition. Springer, Berlin-Heidelberg (1999). | MR | Zbl
[29] , Filtration shrinkage by level crossings of a diffusion. Ann. Probab. 35 (2007) 739-757. | MR | Zbl
[30] , Quasimartingales, martingales locales, et filtrations naturelles. Z. Wahrscheinlichkeitstheorie verw. Gebiete 39 (1977) 55-63. | MR | Zbl
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