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On martingale chaoses. (English) Zbl 1452.60027

Donati-Martin, Catherine (ed.) et al., Séminaire de probabilités XLIX. Cham: Springer. Lect. Notes Math. 2215, 475-494 (2018).
Summary: 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.
For the entire collection see [Zbl 1402.60004].

MSC:

60G44 Martingales with continuous parameter
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H05 Stochastic integrals
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[1] R.H. Cameron, W.T. Martin, Transformation of Wiener integrals under translations. Ann. Math. 45, 386-396 (1944) · Zbl 0063.00696
[2] C. Dellacherie, P.A. Meyer, Probabilities and Potential (North Holland, Amsterdam, 1978) · Zbl 0494.60001
[3] G. Di Nunno, B. Øksendal, F. Proske, White noise analysis for Lévy processes. J. Funct. Anal. 206, 109-148 (2004) · Zbl 1078.60054
[4] G. Di Nunno, B. Øksendal, F. Proske, Malliavin Calculus for Lévy Processes with Applications to Finance (Springer, Berlin, 2009) · Zbl 1080.60068
[5] M. Émery, On the Azéma martingales, in Séminaire de Probabilités XXIV. Lecture Notes in Mathematics, vol. 1426 (Springer, Berlin, 1990), pp. 442-447 · Zbl 0712.60048
[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 · Zbl 0754.60043
[7] M. Émery, Chaotic representation property of certain Azema martingales. Ill. J. Math. 50(2), 395-411 (2006) · Zbl 1110.60043
[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) · Zbl 0100.34004
[10] A. Goswami, B.V. Rao, Conditional expectation of odd chaos given even. Stoch. Stoch. Rep. 35, 213-214 (1991) · Zbl 0728.60059
[11] C. Houdré, V. Pérez-Abreu, Chaos Expansions, Multiple Wiener-Itô Integrals and Their Applications (CRC, Boca Raton, 1994) · Zbl 0839.00028
[12] N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes (North Holland, Amsterdam, 1981) · Zbl 0495.60005
[13] K. Itô, Multiple Wiener integral. J. Math. Soc. Jpn. 3, 157-169 (1951) · Zbl 0044.12202
[14] K. Itô, Spectral type of the shift transformation of differential processes with stationary increments. Trans. Am. Math. Soc. 81, 252-263 (1956) · Zbl 0073.35303
[15] O. Kallenberg, Foundations of Modern Probability (Springer, Berlin, 2010) · Zbl 0892.60001
[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 · Zbl 0374.60070
[18] D. Nualart, The Malliavin Calculus and Related Topics (Springer, Berlin, 1995) · Zbl 0837.60050
[19] D. Nualart, W. Schoutens, Chaotic and predictable representation for Lévy processes. Stoch. Process. Appl. 90, 109-122 (2000) · Zbl 1047.60088
[20] K.R. Parthasarathy, An Introduction to Quantum Stochastic Calculus. Monographs in Mathematics (Birkhäuser, Boston, 1992) · Zbl 0751.60046
[21] N. Privault, Stochastic Analysis in Discrete and Continuous Settings (Springer, Berlin, 2009) · Zbl 1185.60005
[22] B. Rajeev, Martingale representations for functionals of Lévy processes. Sankhya Ser. A, 77(Pt 2), 277-299 (2015) · Zbl 1325.60081
[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) · Zbl 0308.47002
[25] D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, 3rd edn. (Springer, Berlin, 1999) · Zbl 0917.60006
[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) · Zbl 0447.60034
[28] N. · JFM 64.0887.02
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