High order asymptotic expansion for Wiener functionals

Ciprian A. Tudor and Nakahiro Yoshida

Stochastic Processes and their Applications, 2023, vol. 164, issue C, 443-492

Abstract: Combining the Malliavin calculus with Fourier techniques, we develop a high-order asymptotic expansion theory for general Wiener functionals. Our method gives an expansion of the characteristic functional and of the local density of a Wiener functional up to an arbitrary order. The asymptotic expansion is distributional. Except for the non-degeneracy of the limit covariance matrix, we do not assume any condition of non-degeneracy of the Malliavin covariance like a non-degeneracy condition for temporally local characteristic functions so far assumed in the theory for mixing processes, that corresponds to the Cramér condition in the classical setting. Moreover, our method does not require the Markovian property used in the mixing approach. An application to the stochastic wave equation with space–time white noise is discusses.

Keywords: Asymptotic expansion; Malliavin calculus; Central limit theorem; Cumulants; Wave equation (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:164:y:2023:i:c:p:443-492

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DOI: 10.1016/j.spa.2023.07.001

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