The Annals of Probability

The determinant of the iterated Malliavin matrix and the density of a pair of multiple integrals

David Nualart and Ciprian A. Tudor

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Abstract

The aim of this paper is to show an estimate for the determinant of the covariance of a two-dimensional vector of multiple stochastic integrals of the same order in terms of a linear combination of the expectation of the determinant of its iterated Malliavin matrices. As an application, we show that the vector is not absolutely continuous if and only if its components are proportional.

Article information

Source
Ann. Probab., Volume 45, Number 1 (2017), 518-534.

Dates
Received: February 2014
Revised: November 2014
First available in Project Euclid: 26 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1485421338

Digital Object Identifier
doi:10.1214/15-AOP1015

Mathematical Reviews number (MathSciNet)
MR3601655

Zentralblatt MATH identifier
1364.60071

Subjects
Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus
Secondary: 60G15: Gaussian processes

Keywords
Multiple stochastic integrals Wiener chaos iterated Malliavin matrix covariance matrix absolute continuity

Citation

Nualart, David; Tudor, Ciprian A. The determinant of the iterated Malliavin matrix and the density of a pair of multiple integrals. Ann. Probab. 45 (2017), no. 1, 518--534. doi:10.1214/15-AOP1015. https://projecteuclid.org/euclid.aop/1485421338


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References

  • [1] Nourdin, I., Nualart, D. and Poly, G. (2013). Absolute continuity and convergence of densities for random vectors on Wiener chaos. Electron. J. Probab. 18 no. 22, 19.
  • [2] Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality. Cambridge Tracts in Mathematics 192. Cambridge Univ. Press, Cambridge.
  • [3] Nourdin, I. and Rosiński, J. (2014). Asymptotic independence of multiple Wiener–Itô integrals and the resulting limit laws. Ann. Probab. 42 497–526.
  • [4] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Springer, Berlin.
  • [5] Tudor, C. A. (2013). The determinant of the Malliavin matrix and the determinant of the covariance matrix for multiple integrals. ALEA Lat. Am. J. Probab. Math. Stat. 10 681–692.