A note on parameter estimation for discretely sampled SPDEs

Delgado-Vences²,

Kim

2020

Stoch PDE: Anal Comp

Self Cite

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.

Section: Introductionsupporting

confidence: 81%

Drift estimation for discretely sampled SPDEs

Delgado-Vences²,

Kim

2020

Stoch PDE: Anal Comp

Self Cite

“…In [CH17] the authors further explore the quadratic variation method, starting with a simple and intuitively clear observation: the p-variation of a stochastic process is invariant with respect to smooth perturbations. Hence, if the p-variation of a process X can be computed by an explicit formula, and the parameter of interest enters non-trivially into this formula, one can derive consistent estimators of this parameter.…”

Section: Discrete Samplingmentioning

confidence: 99%

“…and also established an asymptotic normality result for this estimator. In contrast to [CH17] where the authors use elements of Malliavin calculus, the methods used in [BT17] are rooted in the mixing theory for Gaussian time series.…”

Section: Discrete Samplingmentioning

confidence: 99%

Statistical inference for SPDEs: an overview

Stat Inference Stoch Process

2018

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The aim of this work is to give an overview of the recent developments in the area of statistical inference for parabolic stochastic partial differential equations. Significant part of the paper is devoted to the spectral approach, which is the most studied sampling scheme under which the observations are done in the Fourier space over some finite time interval. We also discuss into details the practically important case of discrete sampling of the solution. Other relevant methodologies and some open problems are briefly discussed over the course of the manuscript.

“…In particular, there are only few works related to Bayesian statistics for infinite dimensional evolution equations [Bis02,Bis99,PR00]. As usual, studying SPDEs driven by multiplicative noise is more involved, and the parameter estimation problems for such equations are not an exception; the literature on this topic is also limited [CL09,Cia10,PT07,CH17,BT17].…”

Section: Introductionmentioning

confidence: 99%

Bayesian estimations for diagonalizable bilinear SPDEs

Ziteng

Stochastic Processes and their Applications

Gong

2020

Self Cite

The main goal of this paper is to study the parameter estimation problem, using the Bayesian methodology, for the drift coefficient of some linear (parabolic) SPDEs driven by a multiplicative noise of special structure. We take the spectral approach by assuming that one path of the first N Fourier modes of the solution is continuously observed over a finite time interval. First, we show that the model is regular and fits into classical local asymptotic normality framework, and thus the MLE and the Bayesian estimators are weakly consistent, asymptotically normal, efficient, and asymptotically equivalent in the class of loss functions with polynomial growth. Secondly, and mainly, we prove a Bernstein-Von Mises type result, that strengthens the existing results in the literature, and that also allows to investigate the Bayesian type estimators with respect to a larger class of priors and loss functions than that covered by classical asymptotic theory. In particular, we prove strong consistency and asymptotic normality of Bayesian estimators in the class of loss functions of at most exponential growth. Finally, we present some numerical examples that illustrate the obtained theoretical results.