No Access
ASYMPTOTIC PROPERTIES OF THE MAXIMUM LIKELIHOOD ESTIMATOR FOR STOCHASTIC PARABOLIC EQUATIONS WITH ADDITIVE FRACTIONAL BROWNIAN MOTION
https://doi.org/10.1142/S0219493709002610Cited by:53
Abstract
A parameter estimation problem is considered for a diagonalizable stochastic evolution equation using a finite number of the Fourier coefficients of the solution. The equation is driven by additive noise that is white in space and fractional in time with the Hurst parameter H ≥ 1/2. The objective is to study asymptotic properties of the maximum likelihood estimator as the number of the Fourier coefficients increases. A necessary and sufficient condition for consistency and asymptotic normality is presented in terms of the eigenvalues of the operators in the equation.
AMSC: Primary 60H15, Secondary 62F12
References
-
L. C. Andrews , Special Functions for Engineers and Applied Mathematicians ( Macmillan Co. , 1985 ) . Google Scholar - , Stochastic Processes: A Volume in Honor of G. Kallianpur, eds.
S. Cambanis (Springer, 1992) pp. 149–160. Google Scholar - Probab. Th. Relat. Fields 103, 143 (1995), DOI: 10.1007/BF01204212. Crossref, ISI, Google Scholar
-
I. A. Ibragimov and R. Z. Khas′minskiĭ , Statistical Estimation, Applications of Mathematics 16 ( Springer-Verlag , 1981 ) . Crossref, Google Scholar -
V. A. Il′in , Spectral Theory of Differential Operators ( Consultants Bureau , New York , 1995 ) . Crossref, Google Scholar -
J. Jacod and A. N. Shiryaev , Limit Theorems for Stochastic Processes , 2nd edn. ,Grundlehren der Mathematischen Wissenschaften 288 ( Springer , 2003 ) . Crossref, Google Scholar - Stat. Inference Stoch. Process. 5, 229 (2002), DOI: 10.1023/A:1021220818545. Crossref, Google Scholar
- Statist Inference Stoch. Process. 3, 173 (2000), DOI: 10.1023/A:1009923431187. Crossref, Google Scholar
-
S. G. Kreĭn , Yu. Ī. Petunīn and E. M. Semënov , Interpolation of Linear Operators ,Translations of Mathematical Monographs 54 ( Amer. Math. Soc. , 1982 ) . Google Scholar -
R. S. Liptser and A. N. Shiryayev , Statistics of Random Processes, II: Applications , 2nd edn. ( Springer-Verlag , 2001 ) . Google Scholar - Appl. Math. Optim. 67, 401 (2008). ISI, Google Scholar
- Bernoulli 5, 571 (1999), DOI: 10.2307/3318691. Crossref, ISI, Google Scholar
- Probab. Th. Relat. Fields 118, 251 (2000), DOI: 10.1007/s440-000-8016-7. Crossref, ISI, Google Scholar
- Th. Stoch. Process. 10, 116 (2004). Google Scholar
-
Yu. Safarov and D. Vassiliev , The Asymptotic Distribution of Eigenvalues of Partial Differential Operators ,Translations of Mathematical Monographs 155 ( Amer. Math. Soc. , 1997 ) . Google Scholar -
M. A. Shubin , Pseudodifferential Operators and Spectral Theory , 2nd edn. ( Springer-Verlag , 2001 ) . Crossref, Google Scholar - Probab. Th. Relat. Fields 127, 186 (2003), DOI: 10.1007/s00440-003-0282-2. Crossref, ISI, Google Scholar
- Ann. Statist. 35, 1183 (2007), DOI: 10.1214/009053606000001541. Crossref, ISI, Google Scholar
