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Enlargement of the Wiener filtration by an absolutely continuous random variable via Malliavin’s calculus
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  • Published: September 1996

Enlargement of the Wiener filtration by an absolutely continuous random variable via Malliavin’s calculus

  • Peter Imkeller1 

Probability Theory and Related Fields volume 106, pages 105–135 (1996)Cite this article

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Summary.

The analytic treatment of problems related to the asymptotic behaviour of random dynamical systems generated by stochastic differential equations suffers from the presence of non-adapted random invariant measures. Semimartingale theory becomes accessible if the underlying Wiener filtration is enlarged by the information carried by the orthogonal projectors on the Oseledets spaces of the (linearized) system.

We study the corresponding problem of preservation of the semimartingale property and the validity of a priori inequalities between the norms of stochastic integrals in the enlarged filtration and norms of their quadratic variations in case the random element F enlarging the filtration is real valued and possesses an absolutely continuous law. Applying the tools of Malliavin’s calculus, we give smoothness conditions on F under which the semimartingale property is preserved and a priori martingale inequalities are valid.

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Authors and Affiliations

  1. Laboratoire de Mathématiques, Université de Franche-Comté, 16, route de Gray, F-25030 Besançcon Cedex, France, , , , , , FR

    Peter Imkeller

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  1. Peter Imkeller
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Received: 12 April 1995 / In revised form: 7 March 1996

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Imkeller, P. Enlargement of the Wiener filtration by an absolutely continuous random variable via Malliavin’s calculus. Probab Theory Relat Fields 106, 105–135 (1996). https://doi.org/10.1007/s004400050059

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  • Issue Date: September 1996

  • DOI: https://doi.org/10.1007/s004400050059

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  • Mathematics Subject classification (1991): 60G48
  • 60H07
  • 60J65
  • 60H30
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