Electronic Journal of Probability

Asymptotic expansion of Skorohod integrals

David Nualart and Nakahiro Yoshida

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Asymptotic expansion of the distribution of a perturbation $Z_n$ of a Skorohod integral jointly with a reference variable $X_n$ is derived. We introduce a second-order interpolation formula in frequency domain to expand a characteristic functional and combine it with the scheme developed in the martingale expansion. The second-order interpolation and Fourier inversion give asymptotic expansion of the expectation $E[f(Z_n,X_n)]$ for differentiable functions $f$ and also measurable functions $f$. In the latter case, the interpolation method connects the two non-degeneracies of variables for finite $n$ and $\infty $. Random symbols are used for expressing the asymptotic expansion formula. Quasi tangent, quasi torsion and modified quasi torsion are introduced in this paper. We identify these random symbols for a certain quadratic form of a fractional Brownian motion and for a quadratic from of a fractional Brownian motion with random weights. For a quadratic form of a Brownian motion with random weights, we observe that our formula reproduces the formula originally obtained by the martingale expansion.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 119, 64 pp.

Received: 22 April 2018
Accepted: 29 April 2019
First available in Project Euclid: 31 October 2019

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Primary: 60F05: Central limit and other weak theorems 60H07: Stochastic calculus of variations and the Malliavin calculus 60G22: Fractional processes, including fractional Brownian motion 62E20: Asymptotic distribution theory

asymptotic expansion Skorohod integral interpolation random symbol quasi tangent quasi torsion modified quasi torsion Malliavain covariance quadratic form fractional Brownian motion

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Nualart, David; Yoshida, Nakahiro. Asymptotic expansion of Skorohod integrals. Electron. J. Probab. 24 (2019), paper no. 119, 64 pp. doi:10.1214/19-EJP310. https://projecteuclid.org/euclid.ejp/1572508843

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