The Annals of Applied Probability

Abstract nonlinear filtering theory in the presence of fractional Brownian motion

L. Coutin and L. Decreusefond

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We develop the filtering theory in the case where both the signal and the observation are solutions of some stochastic differential equation driven by a multidimensional fractional Brownian motion. We show that the classical approach fails to give a closed equation for the filter and we develop another approach using an auxiliary process-valued semimartingale which solves this problem theoretically.

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Ann. Appl. Probab., Volume 9, Number 4 (1999), 1058-1090.

First available in Project Euclid: 21 August 2002

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Zentralblatt MATH identifier

Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H20: Stochastic integral equations

Filtering theory fractional Brownian motion Malliavin calculus stochastic differential equation


Coutin, L.; Decreusefond, L. Abstract nonlinear filtering theory in the presence of fractional Brownian motion. Ann. Appl. Probab. 9 (1999), no. 4, 1058--1090. doi:10.1214/aoap/1029962865.

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  • Bertoin, J. (1989). Sur une int´egrale pour les processus a variation born´ee. Ann. Probab. 17 1521-1535.
  • Ciesielski, Z., Kerky acharian, G. and Roy nette, B. (1993). Quelques espaces fonctionnels associ´es a des processus gaussiens. Studia Math. 107 171-204.
  • Coutin, L. and Decreusefond, L. (1997). Stochastic differential equations driven by a fractional Brownian motion. Unpublished manuscript.
  • Dai, W. and Hey de, C. C. (1996). It o's Formula with respect to fractional Brownian motion and its application. J. Appl. Stochastic Anal. 9 439-458.
  • Decreusefond, L. and ¨Ust ¨unel, A. S. (1999). Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 177-214.
  • Fey el, D. and de La Pradelle, A. (1999). On fractional Brownian processes. Potential Anal. 10 273-288.
  • F ¨ollmer, H. (1980). Calcul d'It o sans probabibilit´e. S´eminaire de probabilit´es. Lecture Notes in Math. 143-150. Springer, Berlin.
  • Kallianpur, G. (1980). Stochastic Filtering Theory. Springer, Berlin. Kleptsy na, M. L., Kloeden, P. E. and Anh, V. V. (1996a). Linear filtering with fractional Brownian motion. Preprint. Kleptsy na, M. L., Kloeden, P. E. and Anh, V. V. (1996b). Nonlinear filtering with fractional Brownian motion. Preprint.
  • Kuo, H. H. (1975). Gaussian Measures in Banach Spaces. Lecture Notes in Math. 463. Springer, Berlin.
  • Lepingle, D. and M´emin, J. (1978). Sur l'int´egrabilit´e uniforme des martingales exponentielles. Z. Warhsch. Verw. Gebiete 42 175-203.
  • Lin, S. J. (1996). Stochastic analysis of fractional Brownian motions. Stochastics. To appear.
  • Ly ons, T. (1994). Differential equations driven by rough signals I. An extension of an inequality of L. C. Young. Math. Res. Lett. 4 451-464.
  • Nikiforov, A. F. and Uvarov, V. B. (1988). Special Functions of Mathematical physics. Birkh¨auser, Boston.
  • Nualart, D. (1995). The Malliavin Calculus and Related Topics. Springer, Berlin.
  • Pardoux, E. (1989). Filtrage non lin´eaire et ´equations aux d´eriv´ees partielles. Ecole d'´et´e de Saint-Flour. Lecture Notes in Math. Springer, Berlin.
  • Revuz, D. and Yor, M. (1994). Continuous Martingales and Brownian Motion, 2nd ed. Springer, New York.
  • Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional Integrals and Derivatives. Gordon and Breach.
  • ¨Ust¨unel, A. S. (1986). Some comments on the filtering of diffusions and the Malliavin calculus. Stochastic Analy sis and Related Topics. Silivri.
  • ¨Ust¨unel, A. S. (1995). An Introduction to Analy sis on Wiener Space. Lecture Notes in Math. 1610. Springer, Berlin.
  • Young, L. C. (1936). An inequality of H¨older ty pe, connected with Stieltjes integration. Acta Math. 67 251-282.
  • Zakai, M. (1969). On the optimal filtering of diffusion processes. Z. Wahrsch. Verw. Gebiete 11 230-245.