Parameter estimation in diagonalizable bilinear stochastic parabolic equations

Stat Inference Stoch Process

Kim

Lototsky

2019

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We study the statistical properties of stochastic evolution equations driven by space-only noise, either additive or multiplicative. While forward problems, such as existence, uniqueness, and regularity of the solution, for such equations have been studied, little is known about inverse problems for these equations. We exploit the somewhat unusual structure of the observations coming from these equations that leads to an interesting interplay between classical and non-traditional statistical models. We derive several types of estimators for the drift and/or diffusion coefficients of these equations, and prove their relevant properties.

Section: Additive Noisementioning

confidence: 96%

Statistical analysis of some evolution equations driven by space-only noise

Stat Inference Stoch Process

Kim

Lototsky

2019

Self Cite

“…Hence, once any two Fourier modes are observed at any time point T , the parameter can be found exactly, without any statistical procedure. In [CL09], we call such 'estimators' closed-from exact estimators. The fractional noise counterpart of these class of SPDEs is studied in [Cia10].…”

Section: Simple Multiplicative Noisementioning

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.

“…The authors decided to take powers of negative Laplace operator instead of a general positive defined linear operator that generates a scale of Hilbert spaces (cf. the setup from Cialenco and Lototsky [4]) with eigenfunctions forming a complete orthonormal system. The only essential difference in our analysis would be that the results and their derivation would be written in the terms of the asymptotics of the eigenvalues of that positive defined operator.…”

Section: Mathematical Settingmentioning

confidence: 99%

Hypothesis testing for stochastic PDEs driven by additive noise

Stochastic Processes and their Applications

2015

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We study the simple hypothesis testing problem for the drift coefficient for stochastic fractional heat equation driven by additive noise. We introduce the notion of asymptotically the most powerful test, and find explicit forms of such tests in two asymptotic regimes: large time asymptotics, and increasing number of Fourier modes. The proposed statistics are derived based on Maximum Likelihood Ratio. Additionally, we obtain a series of important technical results of independent interest: we find the cumulant generating function of the log-likelihood ratio; obtain sharp large deviation type results for $T\to\infty$ and $N\to\infty$