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Ornstein-Uhlenbeck Pinball and the Poincaré Inequality in a Punctured Domain Read chapter Read first chapter

2018 | OriginalPaper | Chapter

13. On Martingale Chaoses

Author : B. Rajeev

Published in: Séminaire de Probabilités XLIX

Publisher: Springer International Publishing

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Abstract

We extend Wiener’s notion of ‘homogeneous’ chaos expansion of Brownian functionals to functionals of a class of continuous martingales via a notion of iterated stochastic integral for such martingales. We impose a condition of ‘homogeneity’ on the previsible sigma field of such martingales and show that under this condition the notions of purity, chaos representation property and the predictable representation property all coincide.

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Literature
1.
R.H. Cameron, W.T. Martin, Transformation of Wiener integrals under translations. Ann. Math. 45, 386–396 (1944)
2.
C. Dellacherie, P.A. Meyer, Probabilities and Potential (North Holland, Amsterdam, 1978)
3.
G. Di Nunno, B. Øksendal, F. Proske, White noise analysis for Lévy processes. J. Funct. Anal. 206, 109–148 (2004)
4.
G. Di Nunno, B. Øksendal, F. Proske, Malliavin Calculus for Lévy Processes with Applications to Finance (Springer, Berlin, 2009)
5.
M. Émery, On the Azéma martingales, in Séminaire de Probabilités XXIV. Lecture Notes in Mathematics, vol. 1426 (Springer, Berlin, 1990), pp. 442–447
6.
M. Émery, Quelques cas de représentation chaotique, in Séminaire de Probabilités XXV. Lecture Notes in Mathematics, vol. 1485 (Springer, Berlin, 1991), pp. 10–23
7.
M. Émery, Chaotic representation property of certain Azema martingales. Ill. J. Math. 50(2), 395–411 (2006)
8.
V. Fock, Konfigurationsraum und zweite Quantelung. Z. Phys. 75, 622–647 (1932)
9.
I.V. Girsanov, On transforming a certain class of stochastic processes by absolutely continuous substitution of measures. Theor. Prob. Appl. 5, 285–301 (1960)
10.
A. Goswami, B.V. Rao, Conditional expectation of odd chaos given even. Stoch. Stoch. Rep. 35, 213–214 (1991)
11.
C. Houdré, V. Pérez-Abreu, Chaos Expansions, Multiple Wiener-Itô Integrals and Their Applications (CRC, Boca Raton, 1994)
12.
N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes (North Holland, Amsterdam, 1981)
13.
K. Itô, Multiple Wiener integral. J. Math. Soc. Jpn. 3, 157–169 (1951)
14.
K. Itô, Spectral type of the shift transformation of differential processes with stationary increments. Trans. Am. Math. Soc. 81, 252–263 (1956)
15.
O. Kallenberg, Foundations of Modern Probability (Springer, Berlin, 2010)
16.
V. Mandrekar, P.R. Masani, Proceedings of the Nobert Wiener Centenary Congress, 1994. Proceedings of Symposia in Applied Mathematics, vol. 52 (American Mathematical Society, Providence, 1997)
17.
P.A. Meyer, Un cours sur les intégrales stochastiques, in Séminaire de Probabilités X. Lecture Notes in Mathematics, vol. 511 (Springer, Berlin, 1976), pp. 245–400
18.
D. Nualart, The Malliavin Calculus and Related Topics (Springer, Berlin, 1995)
19.
D. Nualart, W. Schoutens, Chaotic and predictable representation for Lévy processes. Stoch. Process. Appl. 90, 109–122 (2000)
20.
K.R. Parthasarathy, An Introduction to Quantum Stochastic Calculus. Monographs in Mathematics (Birkhäuser, Boston, 1992)
21.
N. Privault, Stochastic Analysis in Discrete and Continuous Settings (Springer, Berlin, 2009)
22.
B. Rajeev, Martingale representations for functionals of Lévy processes. Sankhya Ser. A, 77(Pt 2), 277–299 (2015)
23.
B. Rajeev, Iterated stochastic integrals and purity. Pre-print (2016)
24.
M.C. Reed, B. Simon, Methods of Modern Mathematical Physics, vol. II (Academic, New York, 1975)MATH
25.
D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, 3rd edn. (Springer, Berlin, 1999)CrossRef
26.
D.W. Stroock, Homogenous chaos revisited, in Séminaire de Probabilités XXI. Lecture Notes in Mathematics, vol. 1247 (Springer, Berlin, 1987), pp. 1–7
27.
D.W. Stroock, M. Yor, On extremal solutions of Martingale problems. Ann. Sci. École Norm. Sup. 13, 95–164 (1980)MathSciNetCrossRef
28.
Metadata
Title
On Martingale Chaoses
Author
B. Rajeev
Copyright Year
2018
Book
Séminaire de Probabilités XLIX

Print ISBN: 978-3-319-92419-9

Electronic ISBN: 978-3-319-92420-5

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-92420-5_13