On Martingale Chaoses

  • B. Rajeev
Part of the Lecture Notes in Mathematics book series (LNM, volume 2215)


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.


Martingale representation Stochastic integral representation Chaos expansion CRP PRP Pure martingales 

Mathematics Subject Classification (2000)

Primary 60H10 60H15; Secondary 60J60 35K15 



The author would like to thank Michel Émery for extensive discussions over e-mail.


  1. 1.
    R.H. Cameron, W.T. Martin, Transformation of Wiener integrals under translations. Ann. Math. 45, 386–396 (1944)Google Scholar
  2. 2.
    C. Dellacherie, P.A. Meyer, Probabilities and Potential (North Holland, Amsterdam, 1978)Google Scholar
  3. 3.
    G. Di Nunno, B. Øksendal, F. Proske, White noise analysis for Lévy processes. J. Funct. Anal. 206, 109–148 (2004)Google Scholar
  4. 4.
    G. Di Nunno, B. Øksendal, F. Proske, Malliavin Calculus for Lévy Processes with Applications to Finance (Springer, Berlin, 2009)Google Scholar
  5. 5.
    M. Émery, On the Azéma martingales, in Séminaire de Probabilités XXIV. Lecture Notes in Mathematics, vol. 1426 (Springer, Berlin, 1990), pp. 442–447Google Scholar
  6. 6.
    M. Émery, Quelques cas de représentation chaotique, in Séminaire de Probabilités XXV. Lecture Notes in Mathematics, vol. 1485 (Springer, Berlin, 1991), pp. 10–23Google Scholar
  7. 7.
    M. Émery, Chaotic representation property of certain Azema martingales. Ill. J. Math. 50(2), 395–411 (2006)Google Scholar
  8. 8.
    V. Fock, Konfigurationsraum und zweite Quantelung. Z. Phys. 75, 622–647 (1932)Google Scholar
  9. 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)Google Scholar
  10. 10.
    A. Goswami, B.V. Rao, Conditional expectation of odd chaos given even. Stoch. Stoch. Rep. 35, 213–214 (1991)Google Scholar
  11. 11.
    C. Houdré, V. Pérez-Abreu, Chaos Expansions, Multiple Wiener-Itô Integrals and Their Applications (CRC, Boca Raton, 1994)Google Scholar
  12. 12.
    N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes (North Holland, Amsterdam, 1981)Google Scholar
  13. 13.
    K. Itô, Multiple Wiener integral. J. Math. Soc. Jpn. 3, 157–169 (1951)Google Scholar
  14. 14.
    K. Itô, Spectral type of the shift transformation of differential processes with stationary increments. Trans. Am. Math. Soc. 81, 252–263 (1956)Google Scholar
  15. 15.
    O. Kallenberg, Foundations of Modern Probability (Springer, Berlin, 2010)Google Scholar
  16. 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)Google Scholar
  17. 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. 245–400Google Scholar
  18. 18.
    D. Nualart, The Malliavin Calculus and Related Topics (Springer, Berlin, 1995)Google Scholar
  19. 19.
    D. Nualart, W. Schoutens, Chaotic and predictable representation for Lévy processes. Stoch. Process. Appl. 90, 109–122 (2000)Google Scholar
  20. 20.
    K.R. Parthasarathy, An Introduction to Quantum Stochastic Calculus. Monographs in Mathematics (Birkhäuser, Boston, 1992)Google Scholar
  21. 21.
    N. Privault, Stochastic Analysis in Discrete and Continuous Settings (Springer, Berlin, 2009)Google Scholar
  22. 22.
    B. Rajeev, Martingale representations for functionals of Lévy processes. Sankhya Ser. A, 77(Pt 2), 277–299 (2015)Google Scholar
  23. 23.
    B. Rajeev, Iterated stochastic integrals and purity. Pre-print (2016)Google Scholar
  24. 24.
    M.C. Reed, B. Simon, Methods of Modern Mathematical Physics, vol. II (Academic, New York, 1975)zbMATHGoogle Scholar
  25. 25.
    D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, 3rd edn. (Springer, Berlin, 1999)CrossRefGoogle Scholar
  26. 26.
    D.W. Stroock, Homogenous chaos revisited, in Séminaire de Probabilités XXI. Lecture Notes in Mathematics, vol. 1247 (Springer, Berlin, 1987), pp. 1–7Google Scholar
  27. 27.
    D.W. Stroock, M. Yor, On extremal solutions of Martingale problems. Ann. Sci. École Norm. Sup. 13, 95–164 (1980)MathSciNetCrossRefGoogle Scholar
  28. 28.
    N. Wiener, The homogeneous chaos. Am. J. Math. 60, 897–936 (1938)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Statistics and Mathematics UnitIndian Statistical Institute, Bangalore CentreBangaloreIndia

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