Statistical inference for stochastic parabolic equations : a spectral approach

A parameter estimation problem is considered for a stochastic par- abolic equation driven by additive Gaussian noise that is white in time and space. The estimator is of spectral type and utilizes a finite number of the spatial Fourier coefficients of the solution. The asymptotic properties of the estimator are studied as the number of the Fourier coefficients increases, while the observation time and the noise intensity are fixed. A necessary and sufficient condition for consistency and asymptotic normality of the estimator is derived in terms of the eigenvalues of the operators in the equation, and a detailed proof is provided. Other estimation problems are briefly surveyed.

[1]  D. Dawson.  Qualitative behavior of geostochastic systems , 1980 .

[2]  B. Rao.  Estimation for Some Stochastic Partial Differential Equations Based on Discrete Observations II , 2000 .

[3]  S. Aihara.  Regularized maximum likelihood estimate for infinite dimensional parameter in stochastic parabolic systems , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[4]  Sergey V. Lototsky,et al.  Spectral asymptotics of some functionals arising in statistical inference for SPDEs , 1999 .

[5]  Y. Kutoyants.  Statistical Inference for Ergodic Diffusion Processes , 2004 .

[6]  Identification of a Discontinuous Parameter in Stochastic Parabolic Systems , 1998 .

[7]  A. Bagchi,et al.  Parameter identification for hyperbolic stochastic systems , 1991 .

[8]  Igor Cialenco,et al.  Parameter estimation in diagonalizable bilinear stochastic parabolic equations , 2007 .

[9]  N. Krylov,et al.  Statistics and control of stochastic processes , 1985 .

[10]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Theory of martingales , 1989 .

[11]  Jaroslav Hájek.  A property of $J$-divergences of marginal probability distributions , 1958 .

[12]  A. Bagchi,et al.  Infinite dimensional parameter identification for stochastic parabolic systems , 1989 .

[13]  Asymptotic Properties of the Maximum Likelihood Estimator for Stochastic PDEs Disturbed by Small Noise , 1999 .

[14]  Jacob Feldman,et al.  Equivalence and perpendicularity of Gaussian processes , 1958 .

[15]  B. Rozovskii.  Stochastic Evolution Systems , 1990 .

[16]  Boris Rozovskii,et al.  Maximum likelihood estimators in the equations of physical oceanography , 1996 .

[17]  The Bernstein-von Mises theorem and spectral asymptotics of Bayes estimators for parabolic SPDEs , 2002, Journal of the Australian Mathematical Society.

[18]  Prakasa Rao.  Nonparametric inference for parabolic stochastic partial differential equations , 2001 .

[19]  A. Shiryayev,et al.  Statistics of Random Processes Ii: Applications , 2000 .

[20]  B. Rao.  Estimation for some Stochastic Partial Differential Equations Based on Discrete Observations , 2000 .

[21]  Yacine Aït-Sahalia.  Likelihood Inference for Diffusions: A Survey , 2006 .

[22]  G. Adomian,et al.  New contributions to the solution of transport equations in porous media , 1996 .

[23]  S. Aihara.  Maximum likelihood estimate for discontinuous parameter in stochastic hyperbolic systems , 1994 .

[24]  W. Loges.  Girsanov's theorem in Hilbert space and an application to the statistics of Hilbert space- valued stochastic differential equations , 1984 .

[25]  国田 寛.  Stochastic flows and stochastic differential equations , 1990 .

[26]  Yu Safarov,et al.  The Asymptotic Distribution of Eigenvalues of Partial Differential Operators , 1996 .

[27]  B.L.S.Prakasa Rao.  Bayes estimation for some stochastic partial differential equations , 2000 .

[28]  J. B. Walsh,et al.  An introduction to stochastic partial differential equations , 1986 .

[29]  Asymptotic statistical inference for a stochastic heat flow problem , 1985 .

[30]  A. Bagchi,et al.  STOCHASTIC HYPERBOLIC DYNAMICS FOR INFINITE‐DIMENSIONAL FORWARD RATES AND OPTION PRICING , 2003 .

[31]  Arunabha Bagchi,et al.  Parameter Estimation of Parabolic Type Factor Model and Empirical Study of US Treasury Bonds , 2005, System Modelling and Optimization.

[32]  Asymptotic Analysis of the Sieve Estimator for a Class of Parabolic SPDEs , 2000 .

[33]  Peter C. Kiessler,et al.  Statistical Inference for Ergodic Diffusion Processes , 2006 .

[34]  Modeling Term Structure Dynamics: An Infinite Dimensional Approach , 2004 .

[35]  Prakasa Rao.  Nonparametric inference for a class of stochastic partial differential equations based on discrete observations , 2002 .

[36]  Boris Rozovskii,et al.  On asymptotic properties of maximum likelihood estimators for parabolic stochastic PDE's , 1995 .

[37]  Yury A. Kutoyants,et al.  Identification of Dynamical Systems with Small Noise , 1994 .

[38]  Optimal filtering of stochastic parabolic equations , 2004 .

[39]  B. Rao.  Statistical Inference for Stochastic Partial Differential Equations , 2001 .

[40]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Statistics of random processes , 1977 .

[41]  D. Nualart.  The Malliavin Calculus and Related Topics , 1995 .

[42]  A. Shiryayev,et al.  Statistics of Random Processes I: General Theory , 1984 .

[43]  S. Lototsky,et al.  Asymptotic analysis of a kernel estimator for parabolic SPDE's with time-dependent coefficients , 2001 .

[44]  H. Heyer.  Statistics of random processes I: General theory , 1983 .

[45]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[46]  Aarnout Brombacher,et al.  Probability... , 2009, Qual. Reliab. Eng. Int..

[47]  S. S. De.  Stochastic model of population growth and spread. , 1987, Bulletin of mathematical biology.

[48]  Igor Cialenco,et al.  Asymptotic Properties of the Maximum Likelihood Estimator for Stochastic Parabolic Equations with Additive Fractional Brownian Motion , 2008, 0804.0407.

[49]  Yoshifumi Sunahara,et al.  Identification of an infinite dimensional parameter for stochastic diffusion equation , 1987, 26th IEEE Conference on Decision and Control.

[50]  R. Khasminskii,et al.  On the estimation of parameters for linear stochastic differential equations , 1999 .

[51]  T. E. Unny,et al.  Random evolution equations in hydrology , 1990 .

[52]  B. Rao.  Nonparametric Inference for a Class of Stochastic Partial Differential Equations II , 2001 .

[53]  René Carmona,et al.  Stochastic Partial Differential Equations: Six Perspectives , 1998 .

[54]  B. Rao.  ON SOME PROBLEMS OF ESTIMATION FOR SOME STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS , 2002 .

[55]  M. Shubin.  Pseudodifferential Operators and Spectral Theory , 1987 .

[56]  Sergey V. Lototsky,et al.  Parameter Estimation for Stochastic Evolution Equations with Non-commuting Operators , 1997 .

[57]  M. Landsberg,et al.  K. Yosida, Functional Analysis. (Die Grundlehren der mathematischen Wissenschaften, Band 123) XII + 458 S. Berlin/Heidelberg/New York 1965. Springer‐Verlag. Preis geb. DM 66, — , 1966 .

[58]  Nicole El Karoui,et al.  Identification of an infinite-dimensional parameter for stochastic diffusion equations , 1988 .

[59]  Mtw,et al.  Stochastic flows and stochastic differential equations , 1990 .

[60]  Asymptotic Properties of an Approximate Maximum Likelihood Estimator for Stochastic PDE's , 2001 .

[61]  Sergey V. Lototsky,et al.  Parameter Estimation for Stochastic Parabolic Equations: Asymptotic Properties of a Two-Dimensional Projection-Based Estimator , 2003 .

[62]  Consistency property of extended least-squares parameter estimation for stochastic diffusion equation , 1998 .