Electronic Communications in Probability

Moment identities for Skorohod integrals on the Wiener space and applications

Nicolas Privault

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Abstract

We prove a moment identity on the Wiener space that extends the Skorohod isometry to arbitrary powers of the Skorohod integral on the Wiener space. As simple consequences of this identity we obtain sufficient conditions for the Gaussianity of the law of the Skorohod integral and a recurrence relation for the moments of second order Wiener integrals. We also recover and extend the sufficient conditions for the invariance of the Wiener measure under random rotations given in A. S. Üstünel and M. Zakai Prob. Th. Rel. Fields 103 (1995), 409-429.

Article information

Source
Electron. Commun. Probab., Volume 14 (2009), paper no. 11, 116-121.

Dates
Accepted: 19 February 2009
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465234720

Digital Object Identifier
doi:10.1214/ECP.v14-1450

Mathematical Reviews number (MathSciNet)
MR2481671

Zentralblatt MATH identifier
1189.60113

Subjects
Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus
Secondary: 60G30: Continuity and singularity of induced measures

Keywords
Malliavin calculus Skorohod integral Skorohod isometry Wiener measure random isometries

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Privault, Nicolas. Moment identities for Skorohod integrals on the Wiener space and applications. Electron. Commun. Probab. 14 (2009), paper no. 11, 116--121. doi:10.1214/ECP.v14-1450. https://projecteuclid.org/euclid.ecp/1465234720


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References

  • D. Nualart. The Malliavin calculus and related topics. Probability and its Applications, Springer-Verlag, Berlin, second edition, 2006.
  • A.S. Üstünel and M. Zakai. Random rotations of the Wiener path. Prob. Th. Rel. Fields 103 (1995), 409-429.
  • A.V. Skorokhod. On a generalization of a stochastic integral. Theor. Probab. Appl. XX (1975), 219-223.
  • S. Watanabe. Lectures on Stochastic Differential Equations and Malliavin Calculus. Tata Institute of Fundamental Research, 1984.