Annals of Statistics

High-frequency analysis of parabolic stochastic PDEs

Carsten Chong

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We consider the problem of estimating stochastic volatility for a class of second-order parabolic stochastic PDEs. Assuming that the solution is observed at high temporal frequency, we use limit theorems for multipower variations and related functionals to construct consistent nonparametric estimators and asymptotic confidence bounds for the integrated volatility process. As a byproduct of our analysis, we also obtain feasible estimators for the regularity of the spatial covariance function of the noise.

Article information

Ann. Statist., Volume 48, Number 2 (2020), 1143-1167.

Received: June 2018
Revised: March 2019
First available in Project Euclid: 26 May 2020

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Mathematical Reviews number (MathSciNet)

Primary: 62M40: Random fields; image analysis 62G20: Asymptotic properties 60H15: Stochastic partial differential equations [See also 35R60]

High-frequency observations martingale limit theorems multipower variations stochastic heat equation SPDEs variation functionals volatility estimation


Chong, Carsten. High-frequency analysis of parabolic stochastic PDEs. Ann. Statist. 48 (2020), no. 2, 1143--1167. doi:10.1214/19-AOS1841.

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Supplemental materials

  • Supplement to “High-frequency analysis of parabolic stochastic PDEs”. This paper is accompanied by supplementary material in [14]. Section A in [14] gives some auxiliary results needed for the proofs in this paper. In Section B, some important estimates related to the heat kernel are derived. Sections C and D provide the details for the proof of Theorems 2.1 and 2.3, respectively. Finally, Section E contains the proofs for Sections 2.2 and 2.3.