Drift estimation for discretely sampled SPDEs
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Drift estimation for discretely sampled SPDEs Igor Cialenco1,2 · Francisco Delgado-Vences3,4 · Hyun-Jung Kim1,2 Received: 24 April 2019 / Revised: 2 October 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The aim of this paper is to study the asymptotic properties of the maximum likelihood estimator (MLE) of the drift coefficient for fractional stochastic heat equation driven by an additive space-time noise. We consider the traditional for stochastic partial differential equations statistical experiment when the measurements are performed in the spectral domain, and in contrast to the existing literature, we study the asymptotic properties of the maximum likelihood (type) estimators (MLE) when both, the number of Fourier modes and the time go to infinity. In the first part of the paper we consider the usual setup of continuous time observations of the Fourier coefficients of the solutions, and show that the MLE is consistent, asymptotically normal and optimal in the mean-square sense. In the second part of the paper we investigate the natural time discretization of the MLE, by assuming that the first N Fourier modes are measured at M time grid points, uniformly spaced over the time interval [0, T ]. We provide a rigorous asymptotic analysis of the proposed estimators when N → ∞ and/or T , M → ∞. We establish sufficient conditions on the growth rates of N , M and T , that guarantee consistency and asymptotic normality of these estimators. Keywords Fractional stochastic heat equation · Parabolic SPDE · Stochastic evolution equations · Statistical inference for SPDEs · Drift estimation · Discrete sampling · High-frequency sampling
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Hyun-Jung Kim [email protected] https://sites.google.com/view/hyun-jungkim Igor Cialenco [email protected] http://math.iit.edu/∼igor Francisco Delgado-Vences [email protected]
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Department of Applied Mathematics, Illinois Institute of Technology, Chicago, USA
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Present Address: 10 W 32nd Str, Building REC, Room 220, Chicago, IL 60616, USA
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Catedra Conacyt and Instituto de Matemáticas, UNAM, Mexico City, Mexico
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Present Address: Alameda de Leon 2, altos, Centro, 68000 Oaxaca de Juarez, Mexico
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Stoch PDE: Anal Comp
Mathematics Subject Classification 60H15 · 65L09 · 62M99
1 Introduction Undoubtedly, the stochastic partial differential equations (SPDEs) serve as a modern powerful modeling tool in describing the evolution of dynamical systems in the presence of spatial-temporal uncertainties with particular applications in fluid mechanics, oceanography, temperature anomalies, finance, economics, biological and ecological systems, and many other applied disciplines. Major breakthrough results have been established on the general analytical theory for SPDEs, such as existence, uniqueness and regularity properties of the solutions. For an in depth discussion of the theory of SPDEs and their various applications, we refer to recent monographs [14,15]. In contrast, the investigation of inverse problems for SPDEs, and in particular parameter estim
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