Electronic Journal of Probability

Quadratic variations for the fractional-colored stochastic heat equation

Soledad Torres, Ciprian Tudor, and Frederi Viens

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Using multiple stochastic integrals and Malliavin calculus, we analyze the quadratic variations of a class of Gaussian processes that contains the linear stochastic heat equation on $\mathbf{R}^{d}$ driven by a non-white noise which is fractional Gaussian with respect to the time variable (Hurst parameter $H$) and has colored spatial covariance of $\alpha $-Riesz-kernel type. The processes in this class are self-similar in time with a parameter $K$ distinct from $H$, and have path regularity properties which are very close to those of fractional Brownian motion (fBm) with Hurst parameter $K$ (in the heat equation case, $K=H-(d-\alpha )/4$ ). However the processes exhibit marked inhomogeneities which cause naive heuristic renormalization arguments based on $K$ to fail, and require delicate computations to establish the asymptotic behavior of the quadratic variation. A phase transition between normal and non-normal asymptotics appears, which does not correspond to the familiar threshold $K=3/4$ known in the case of fBm. We apply our results to construct an estimator for $H$ and to study its asymptotic behavior.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 76, 51 pp.

Accepted: 23 August 2014
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60H05: Stochastic integrals 60G18: Self-similar processes

multiple stochastic integral stochastic heat equation fractional Brownian motion Malliavin calculus non-central limit theorem quadratic variation Hurst parameter selfsimilarity statistical estimation

This work is licensed under a Creative Commons Attribution 3.0 License.


Torres, Soledad; Tudor, Ciprian; Viens, Frederi. Quadratic variations for the fractional-colored stochastic heat equation. Electron. J. Probab. 19 (2014), paper no. 76, 51 pp. doi:10.1214/EJP.v19-2698. https://projecteuclid.org/euclid.ejp/1465065718

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