Electronic Journal of Probability

Asymptotic expansion of Skorohod integrals

Abstract

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

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

Dates
Accepted: 29 April 2019
First available in Project Euclid: 31 October 2019

https://projecteuclid.org/euclid.ejp/1572508843

Digital Object Identifier
doi:10.1214/19-EJP310

Citation

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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