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

Cialenco, Igor; Kim, Hyun-Jung; Lototsky, Sergey V.

doi:10.1007/s11203-019-09205-0

Cited by 6 publications

(9 citation statements)

References 18 publications

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“…However, it turns out that due to the fact that the finite difference approximation of the derivative is done at the critical spatial regularity, namely u(t, x) is only 3/2− Hölder continuous, a multiplicative adjustment has to be introduced toθ 2 N,M andσ 2 N,M to have consistency, and consequently asymptotic normality. This 'bias' was first noticed in the numerical experiments in [CKL20]. In the present paper we formally derive the value of this bias.…”

Section: Discrete Observations Of Umentioning

confidence: 60%

“…The existing literature on statistical inference for SPDEs usually deals with space-time noise. In [CKL20], the authors make the first attempt to study inverse problems for stochastic evolution equations driven by space-only noise, and the present paper contributes, in particular, to the efforts initiated therein. Specifically, we consider the one-dimensional stochastic heat equation driven by an additive space-only noise ∂u(t,x) ∂t = θ ∂ 2 u(t,x) ∂x 2 + σẆ (x), t > 0, x ∈ G ⊂ R, u(0, x) = 0, x ∈ G,…”

Section: Introductionmentioning

confidence: 98%

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A note on parameter estimation for discretely sampled SPDEs

Cialenco

Huang

2019

Stoch. Dyn.

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The main goal of this paper is to build consistent and asymptotically normal estimators for the drift and volatility parameter of the stochastic heat equation driven by an additive space-only white noise when the solution is sampled discretely in the physical domain. We consider both the full space R and the bounded domain (0, π). First, we establish the exact regularity of the solution and its spatial derivative, which in turn, using power-variation arguments, allows building the desired estimators. Second, we propose two sets of estimators, based on sampling either the spatial derivative or the solution itself on a discrete space-time grid. Using the so-called Malliavin-Stein's method, we prove that these estimators are consistent and asymptotically normal as the spatial mesh-size vanishes. More importantly, we show that naive approximations of the derivatives appearing in the power-variation based estimators may create nontrivial biases, which we compute explicitly. We conclude with some numerical experiments that illustrate the theoretical results.

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Section: Discrete Observations Of Umentioning

confidence: 60%

Section: Introductionmentioning

confidence: 98%

A note on parameter estimation for discretely sampled SPDEs

Cialenco

Huang

2019

Stoch. Dyn.

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“…The main difference in this case is that the optimal regularity is s * = 2γ + α − 1/2 (cf. [CKL20,CK20]).…”

Section: Semilinear Spdes On a Bounded Domainmentioning

confidence: 99%

“…One way to overcome this drawback, is to approximate the derivatives by using the discrete measurements of the solution itself, for example by finite differences. However, such approximations typically will yield a nontrivial and non-vanishing bias in the estimators -a phenomena noticed in [CKL20] through numerical experiments for SPDEs driven by space-only noise and with m = 1, and later in [CK20] the bias was explicitly given and the asymptotic properties of the estimator were formally proved. We built on these line of ideas, and we focus our study on discretely sampled (in space) of semilinear SPDEs.…”

Section: Introductionmentioning

confidence: 99%

Statistical analysis of discretely sampled semilinear SPDEs: a power variation approach

Cialenco¹,

Kim²,

Pasemann³

2021

Preprint

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Motivated by problems from statistical analysis for discretely sampled SPDEs, first we derive central limit theorems for higher order finite differences applied to stochastic process with arbitrary finitely regular paths. These results are proved by using the notion of ∆-power variations, introduced herein, along with the Hölder-Zygmund norms. Consequently, we prove a new central limit theorem for ∆-power variations of the iterated integrals of a fractional Brownian motion (fBm). These abstract results, besides being of independent interest, in the second part of the paper are applied to estimation of the drift and volatility coefficients of semilinear stochastic partial differential equations in dimension one, driven by an additive Gaussian noise white in time and possibly colored in space. In particular, we solve the earlier conjecture from [CKL20] about existence of a nontrivial bias in the estimators derived by naive approximations of derivatives by finite differences. We give an explicit formula for the bias and derive the convergence rates of the corresponding estimators. Theoretical results are illustrated by numerical examples.

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“…Unlike SPDEs with space-time noise, equations with purely spatial Gaussian noise are studied not so extensively in the statistical literature. We can mention only the papers [2,12,15], where the parameter estimation for equations with space-only white noise was investigated. At the same time, such type of noise is an important type of stationary perturbations, see discussion and examples in [27].…”

Section: Introductionmentioning

confidence: 99%