Electronic Journal of Probability

Quadratic variations for the fractional-colored stochastic heat equation

Soledad Torres, Ciprian Tudor, and Frederi Viens

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065718

Digital Object Identifier
doi:10.1214/EJP.v19-2698

Mathematical Reviews number (MathSciNet)
MR3256876

Zentralblatt MATH identifier
1314.60132

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

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

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

Citation

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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References

  • Balan, Raluca M.; Tudor, Ciprian A. The stochastic heat equation with fractional-colored noise: existence of the solution. ALEA Lat. Am. J. Probab. Math. Stat. 4 (2008), 57–87.
  • Berzin, Corinne; León, José R. Estimation in models driven by fractional Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008), no. 2, 191–213.
  • Chronopoulou, Alexandra; Tudor, Ciprian A.; Viens, Frederi G. Self-similarity parameter estimation and reproduction property for non-Gaussian Hermite processes. Commun. Stoch. Anal. 5 (2011), no. 1, 161–185.
  • Chronopoulou, Alexandra; Viens, Frederi G.; Tudor, Ciprian A. Variations and Hurst index estimation for a Rosenblatt process using longer filters. Electron. J. Stat. 3 (2009), 1393–1435.
  • Cialenco, Igor. Parameter estimation for SPDEs with multiplicative fractional noise. Stoch. Dyn. 10 (2010), no. 4, 561–576.
  • Cialenco, Igor; Lototsky, Sergey V.; Pospíšil, Jan. Asymptotic properties of the maximum likelihood estimator for stochastic parabolic equations with additive fractional Brownian motion. Stoch. Dyn. 9 (2009), no. 2, 169–185.
  • Coeurjolly, Jean-François. Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. Stat. Inference Stoch. Process. 4 (2001), no. 2, 199–227.
  • Dalang, Robert C. Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.'s. Electron. J. Probab. 4 (1999), no. 6, 29 pp. (electronic).
  • Eden, R.; Víquez, J.: Nourdin-Peccati analysis on Wiener and Wiener-Poisson space for general distributions. Preprint, (2012), 38 pp, ARXIV1202.6430
  • Fernique, Xavier. Fonctions aléatoires gaussiennes, vecteurs aléatoires gaussiens. (French) [Gaussian random functions, Gaussian random vectors] Université de Montréal, Centre de Recherches Mathématiques, Montreal, QC, 1997. iv+217 pp. ISBN: 2-921120-28-3
  • Fox, Robert; Taqqu, Murad S. Multiple stochastic integrals with dependent integrators. J. Multivariate Anal. 21 (1987), no. 1, 105–127.
  • Guyon, Xavier; León, José. Convergence en loi des $H$-variations d'un processus gaussien stationnaire sur ${\bf R}$. (French) [Convergence in law of the $H$-variations of a stationary Gaussian process in ${\bf R}$] Ann. Inst. H. Poincaré Probab. Statist. 25 (1989), no. 3, 265–282.
  • Hu, Yaozhong; Nualart, David. Parameter estimation for fractional Ornstein-Uhlenbeck processes. Statist. Probab. Lett. 80 (2010), no. 11-12, 1030–1038.
  • Istas, Jacques; Lang, Gabriel. Quadratic variations and estimation of the local Hölder index of a Gaussian process. Ann. Inst. H. Poincaré Probab. Statist. 33 (1997), no. 4, 407–436.
  • Kleptsyna, M. L.; Le Breton, A. Statistical analysis of the fractional Ornstein-Uhlenbeck type process. Stat. Inference Stoch. Process. 5 (2002), no. 3, 229–248.
  • Maejima, Makoto; Tudor, Ciprian A. Wiener integrals with respect to the Hermite process and a non-central limit theorem. Stoch. Anal. Appl. 25 (2007), no. 5, 1043–1056.
  • Major, Péter. Tail behaviour of multiple random integrals and $U$-statistics. Probab. Surv. 2 (2005), 448–505.
  • Maslowski, Bohdan; Pospíšil, Jan. Ergodicity and parameter estimates for infinite-dimensional fractional Ornstein-Uhlenbeck process. Appl. Math. Optim. 57 (2008), no. 3, 401–429.
  • Maslowski, Bohdan; Pospíšil, Jan. Parameter estimates for linear partial differential equations with fractional boundary noise. Commun. Inf. Syst. 7 (2007), no. 1, 1–20.
  • Mueller, Carl; Wu, Zhixin. A connection between the stochastic heat equation and fractional Brownian motion, and a simple proof of a result of Talagrand. Electron. Commun. Probab. 14 (2009), 55–65.
  • Nualart, Eulalia; Viens, Frederi. The fractional stochastic heat equation on the circle: time regularity and potential theory. Stochastic Process. Appl. 119 (2009), no. 5, 1505–1540.
  • Ouahhabi, Hanae; Tudor, Ciprian A. Additive functionals of the solution to fractional stochastic heat equation. J. Fourier Anal. Appl. 19 (2013), no. 4, 777–791.
  • Nourdin, Ivan. Selected aspects of fractional Brownian motion. Bocconi & Springer Series, 4. Springer, Milan; Bocconi University Press, Milan, 2012. x+122 pp. ISBN: 978-88-470-2822-7; 978-88-470-2823-4
  • Nourdin, Ivan; Peccati, Giovanni. Normal approximations with Malliavin calculus. From Stein's method to universality. Cambridge Tracts in Mathematics, 192. Cambridge University Press, Cambridge, 2012. xiv+239 pp. ISBN: 978-1-107-01777-1
  • Nourdin, Ivan; Peccati, Giovanni. Noncentral convergence of multiple integrals. Ann. Probab. 37 (2009), no. 4, 1412–1426.
  • Nourdin, Ivan; Peccati, Giovanni. Stein's method on Wiener chaos. Probab. Theory Related Fields 145 (2009), no. 1-2, 75–118.
  • Nourdin, Ivan; Peccati, Giovanni. Cumulants on the Wiener space. J. Funct. Anal. 258 (2010), no. 11, 3775–3791.
  • Nourdin, Ivan; Rosinski, Jan. Asymptotic independence of multiple Wiener-Itô integrals and the resulting limit laws. Ann. Probab. 42 (2014), no. 2, 497–526.
  • Nualart, David. The Malliavin calculus and related topics. Second edition. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. xiv+382 pp. ISBN: 978-3-540-28328-7; 3-540-28328-5
  • Nualart, D.; Ortiz-Latorre, S. Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stochastic Process. Appl. 118 (2008), no. 4, 614–628.
  • Nualart, David; Peccati, Giovanni. Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33 (2005), no. 1, 177–193.
  • Pospíšil, Jan; Tribe, Roger. Parameter estimates and exact variations for stochastic heat equations driven by space-time white noise. Stoch. Anal. Appl. 25 (2007), no. 3, 593–611.
  • Prakasa Rao, B. L. S. Parametric estimation for linear stochastic differential equations driven by fractional Brownian motion. Random Oper. Stochastic Equations 11 (2003), no. 3, 229–242.
  • Swanson, Jason. Variations of the solution to a stochastic heat equation. Ann. Probab. 35 (2007), no. 6, 2122–2159.
  • Tudor, Ciprian A. Analysis of the Rosenblatt process. ESAIM Probab. Stat. 12 (2008), 230–257.
  • Tudor, C.A. and Viens, F.G.: Variations of the fractional Brownian motion via Malliavin calculus. Preprint, 2008, 13 pages.
  • Tudor, Ciprian A.; Viens, Frederi G. Statistical aspects of the fractional stochastic calculus. Ann. Statist. 35 (2007), no. 3, 1183–1212.
  • Tudor, Ciprian A.; Viens, Frederi G. Variations and estimators for self-similarity parameters via Malliavin calculus. Ann. Probab. 37 (2009), no. 6, 2093–2134.
  • Veillette, Mark S.; Taqqu, Murad S. Properties and numerical evaluation of the Rosenblatt distribution. Bernoulli 19 (2013), no. 3, 982–1005.