The Annals of Probability

The chaotic representation property of compensated-covariation stable families of martingales

Paolo Di Tella and Hans-Jürgen Engelbert

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In the present paper, we study the chaotic representation property for certain families ${\mathscr{X}}$ of square integrable martingales on a finite time interval $[0,T]$. For this purpose, we introduce the notion of compensated-covariation stability of such families. The chaotic representation property will be defined using iterated integrals with respect to a given family ${\mathscr{X}}$ of square integrable martingales having deterministic mutual predictable covariation $\langle X,Y\rangle$ for all $X,Y\in{\mathscr{X}}$. The main result of the present paper is stated in Theorem 5.8 below: If ${\mathscr{X}}$ is a compensated-covariation stable family of square integrable martingales such that $\langle X,Y\rangle$ is deterministic for all $X,Y\in{\mathscr{X}}$ and, furthermore, the system of monomials generated by ${\mathscr{X}}$ is total in $L^{2}(\Omega,\mathscr{F}^{\mathscr{X}}_{T},\mathbb{P})$, then ${\mathscr{X}}$ possesses the chaotic representation property with respect to the $\sigma$-field $\mathscr{F}^{\mathscr{X}}_{T}$. We shall apply this result to the case of Lévy processes. Relative to the filtration $\mathbb{F}^{L}$ generated by a Lévy process $L$, we construct families of martingales which possess the chaotic representation property. As an illustration of the general results, we will also discuss applications to continuous Gaussian families of martingales and independent families of compensated Poisson processes. We conclude the paper by giving, for the case of Lévy processes, several examples of concrete families ${\mathscr{X}}$ of martingales including Teugels martingales.

Article information

Ann. Probab., Volume 44, Number 6 (2016), 3965-4005.

Received: June 2014
Revised: September 2015
First available in Project Euclid: 14 November 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H05: Stochastic integrals 60G44: Martingales with continuous parameter 60G51: Processes with independent increments; Lévy processes
Secondary: 60G46: Martingales and classical analysis 60G57: Random measures

Square integrable martingales chaotic representation property Lévy processes Teugels martingales Hermitian polynomials Haar functions


Di Tella, Paolo; Engelbert, Hans-Jürgen. The chaotic representation property of compensated-covariation stable families of martingales. Ann. Probab. 44 (2016), no. 6, 3965--4005. doi:10.1214/15-AOP1066.

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  • [1] Azéma, J. (1985). Sur les fermés aléatoires. In Séminaire de Probabilités, XIX, 1983/84. Lecture Notes in Math. 1123 397–495. Springer, Berlin.
  • [2] Azéma, J. and Yor, M. (1989). Étude d’une martingale remarquable. In Séminaire de Probabilités XXIII. Lecture Notes in Math. 1372 88–130. Springer, Berlin.
  • [3] Cameron, R. H. and Martin, W. T. (1947). The orthogonal development of non-linear functionals in series of Fourier–Hermite functionals. Ann. of Math. (2) 48 385–392.
  • [4] Corcuera, J. M., Nualart, D. and Schoutens, W. (2005). Completion of a Lévy market by power-jump assets. Finance Stoch. 9 109–127.
  • [5] Di Tella, P. (2013). On the predictable representation property of martingales associated with Lévy processes. PhD thesis, Univ. Jena. Available at htpp://
  • [6] Di Tella, P. and Engelbert, H.-J. (2015). The predictable representation property of compensated-covariation stable families of martingales. Theory Probab. Appl. 60 99–130.
  • [7] Émery, M. (1989). On the Azéma martingales. In Séminaire de Probabilités, XXIII. Lecture Notes in Math. 1372 66–87. Springer, Berlin.
  • [8] Gihman, I. I. and Skorohod, A. V. (1974). The Theory of Stochastic Processes: Vol. 1. Die Grundlehren der Mathematischen Wissenschaften 210. Springer, Berlin.
  • [9] Itô, K. (1951). Multiple Wiener integral. J. Math. Soc. Japan 3 157–169.
  • [10] Itô, K. (1956). Spectral type of the shift transformation of differential processes with stationary increments. Trans. Amer. Math. Soc. 81 253–263.
  • [11] Jacod, J. (1979). Calcul Stochastique et Problèmes de Martingales. Lecture Notes in Math. 714. Springer, Berlin.
  • [12] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
  • [13] Kakutani, S. (1950). Determination of the spectrum of the flow of Brownian motion. Proc. Natl. Acad. Sci. USA 36 319–323.
  • [14] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • [15] Meyer, P. A. (1976). Un cours sur les intégrales stochastiques. In Séminaire de Probabilités, X (Seconde Partie: Théorie des Intégrales Stochastiques, Univ. Strasbourg, Strasbourg, Année Universitaire 1974/1975) Lecture Notes in Math. 511 245–400. Springer, Berlin.
  • [16] Nualart, D. and Schoutens, W. (2000). Chaotic and predictable representations for Lévy processes. Stochastic Process. Appl. 90 109–122.
  • [17] Peccati, G. and Taqqu, M. S. (2011). Wiener Chaos: Moments, Cumulants and Diagrams. Bocconi & Springer Series 1. Bocconi Univ. Press, Springer, Milan.
  • [18] Protter, P. E. (2005). Stochastic Integration and Differential Equations, 2nd ed. Stochastic Modelling and Applied Probability 21. Springer, Berlin.
  • [19] Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge.
  • [20] Segall, A. and Kailath, T. (1976). Orthogonal functionals of independent-increment processes. IEEE Trans. Inform. Theory IT-22 287–298.
  • [21] Solé, J. L., Utzet, F. and Vives, J. (2007). Chaos expansions and Malliavin calculus for Lévy processes. In Stochastic Analysis and Applications. Abel Symp. 2 595–612. Springer, Berlin.
  • [22] Wiener, N. (1938). The homogeneous chaos. Amer. J. Math. 60 897–936.
  • [23] Wojtaszczyk, P. (1997). A Mathematical Introduction to Wavelets. London Mathematical Society Student Texts 37. Cambridge Univ. Press, Cambridge.