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Occupation densities for stochastic integral processes in the second Wiener chaos
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  • Published: March 1992

Occupation densities for stochastic integral processes in the second Wiener chaos

  • Peter Imkeller1 nAff2 

Probability Theory and Related Fields volume 91, pages 1–24 (1992)Cite this article

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Summary

The two-point distributions of Skorohod integral processes in the second Wiener chaos are mainly described by a Hilbert-Schmidt operatorT giving the mutual interaction of infinitely many Gaussian components and by simple multiplication operators. So are the Fourier transforms of their occupation measures. This enables us to use the well known Fourier analytic criterion discovered and elaborated by Berman to derive integral conditions for the existence of their occupation densities in terms of associated Hilbert-Schmidt operators. IfT is a trace class operator, we get a necessary and sufficient criterion, if it is not, still a sufficient one. In a case in which the interaction is particularly simple, we verify the appropriate integral condition and show that the results are essentially beyond the reach of “enlargement of filtrations” techniques of semimartingale theory.

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Author notes
  1. Peter Imkeller

    Present address: Mathematisches Institut der LMU, Theresienstrasse 39, W-8000, München 2, FRG

Authors and Affiliations

  1. Department of Mathematics, University of B.C., 1984 Mathematics Road, V6T 1Y4, Vancouver, B.C., Canada

    Peter Imkeller

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  1. Peter Imkeller
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Imkeller, P. Occupation densities for stochastic integral processes in the second Wiener chaos. Probab. Th. Rel. Fields 91, 1–24 (1992). https://doi.org/10.1007/BF01194487

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  • Received: 13 April 1990

  • Revised: 13 May 1991

  • Issue Date: March 1992

  • DOI: https://doi.org/10.1007/BF01194487

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Keywords

  • Fourier Transform
  • Mathematical Biology
  • Multiplication Operator
  • Integral Process
  • Berman
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