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2019 Asymptotic expansion of Skorohod integrals
David Nualart, Nakahiro Yoshida
Electron. J. Probab. 24: 1-64 (2019). DOI: 10.1214/19-EJP310

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.

Citation

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David Nualart. Nakahiro Yoshida. "Asymptotic expansion of Skorohod integrals." Electron. J. Probab. 24 1 - 64, 2019. https://doi.org/10.1214/19-EJP310

Information

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

zbMATH: 07142913
MathSciNet: MR4029422
Digital Object Identifier: 10.1214/19-EJP310

Subjects:
Primary: 60F05, 60G22, 60H07, 62E20

JOURNAL ARTICLE
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