• Bernoulli
  • Volume 6, Number 4 (2000), 633-651.

Chaotic Kabanov formula for the Azéma martingales

Nicolas Privault, Josep Lluís Solé, and Josep Vives

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We derive the chaotic expansion of the product of nth- and first-order multiple stochastic integrals with respect to certain normal martingales. This is done by application of the classical and quantum product formulae for multiple stochastic integrals. Our approach extends existing results on chaotic calculus for normal martingales and exhibits properties, relative to multiple stochastic integrals, polynomials and Wick products, that characterize the Wiener and Poisson processes.

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Bernoulli, Volume 6, Number 4 (2000), 633-651.

First available in Project Euclid: 8 April 2004

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Azéma martingales multiple stochastic integrals product formulae


Privault, Nicolas; Lluís Solé, Josep; Vives, Josep. Chaotic Kabanov formula for the Azéma martingales. Bernoulli 6 (2000), no. 4, 633--651.

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  • [1] Attal, S. (1998) Classical and quantum stochastic calculus. In L. Hudson and J.M. Lindsay (eds), Quantum Probab. Commun. 10, pp. 1-52. River Edge: World Scientific.
  • [2] Attal S. and Lindsay J.M. (1997) Quantum stochastic calculus. A new formulation. Preprint.
  • [3] Biane, P. (1995) Calcul stochastique non-commutatif. In P. Bernard (ed.), Lectures on Probability Theory. École d'Été de Probabilités de Saint-Flour XXIII - 1993, Lecture Notes in Math. 1608. Berlin: Springer-Verlag.
  • [4] Dellacherie, C., Maisonneuve, B. and Meyer, P.A. (1992) Probabilités et Potentiel. Processus de Markov (fin). Complements de Calcul Stochastique. Paris: Hermann.
  • [5] Dermoune, A., Krée, P. and Wu, L. (1988) Calcul stochastique non adapté par rapport à la mesure de Poisson. In J. Azéma, P.A. Meyer and M. Yor (eds), Séminaire de Probabilités XXII, Lecture Notes in Math. 1321. Berlin: Springer-Verlag.
  • [6] Emery, M. (1989) On the Azéma martingales. In J. Azéma, P.A. Meyer and M. Yor (eds), Séminaire de Probabilités XXIII, Lecture Notes in Math. 1372, pp. 66-87. Berlin: Springer-Verlag.
  • [7] Kabanov, Y.M. (1975) On extended stochastic integrals. Theory Probab. Appl., 20, 710-722.
  • [8] Ma, J., Protter, P. and San Martin, J. (1998) Anticipating integrals for a class of martingales. Bernoulli, 4, 81-114.
  • [9] Meyer, P.A. (1976) Un cours sur les intégrales stochastiques. In P.A. Meyer (ed.), Séminaire de Probabilités X, Lecture Notes in Math. 511. Berlin: Springer-Verlag.
  • [10] Nualart, D. and Vives, J. (1990) Anticipative calculus for the Poisson process based on the Fock space. In J. Azéma, P.A. Meyer and M. Yor (eds), Séminaire de Probabilités XXIV 1988/89, Lecture Notes in Math. 1426, pp. 154-165. Berlin: Springer-Verlag.
  • [11] Parthasarathy, K.R. (1990) Azéma martingales and quantum stochastic calculus. In R.R. Bahadur (ed.) Proceedings of the R.C. Bose Memorial Symposium, pp. 551-569. Singapore: Wiley Eastern.
  • [12] Privault, N. (1996) On the independence of multiple stochastic integrals with respect to a class of martingales. C. R. Acad. Sci. Paris, Sér. I, 323, 515-520.
  • [13] Russo, F. and Vallois, P. (1998) Product of two multiple stochastic integrals with respect to a normal martingale. Stochastic Process. Appl., 73, no. 1.
  • [14] Surgailis, D. (1984) On multiple Poisson stochastic integrals and associated Markov semi-groups. Probab. Math. Statist., 3, 217-239.
  • [15] Urbanik, K. (1967) Some prediction problems for strictly stationary processes. In L. LeCam and J. Neyman (eds), Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 2, pp. 235-258. Berkeley: University of California Press.
  • [16] Üstünel, A.S. and Zakai, M. (1990) On the structure on independence on Wiener space. J. Funct. Anal., 90, 113-137.
  • [17] Utzet, F. (1992) Les processus à accroissements indépendants et les équations de structure. In J. Azéma, P.A. Meyer and M. Yor (eds), Séminaire de Probabilités XXVI, Lecture Notes in Math. 1526, pp. 405-409. Berlin: Springer-Verlag.
  • [18] Yor, M. (1997) Some Aspects of Brownian Motion (Part II). Birkhäuser.