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Parameter estimation for the stochastic heat equation with multiplicative noise from local measurements

Josef Janák, Markus Reiß

Abstract

For the stochastic heat equation with multiplicative noise we consider the problem of estimating the diffusivity parameter in front of the Laplace operator. Based on local observations in space, we first study an estimator that was derived for additive noise. A stable central limit theorem shows that this estimator is consistent and asymptotically mixed normal. By taking into account the quadratic variation, we propose two new estimators. Their limiting distributions exhibit a smaller (conditional) variance and the last estimator also works for vanishing noise levels. The proofs are based on local approximation results to overcome the intricate nonlinearities and on a stable central limit theorem for stochastic integrals with respect to cylindrical Brownian motion. Simulation results illustrate the theoretical findings.

Parameter estimation for the stochastic heat equation with multiplicative noise from local measurements

Abstract

For the stochastic heat equation with multiplicative noise we consider the problem of estimating the diffusivity parameter in front of the Laplace operator. Based on local observations in space, we first study an estimator that was derived for additive noise. A stable central limit theorem shows that this estimator is consistent and asymptotically mixed normal. By taking into account the quadratic variation, we propose two new estimators. Their limiting distributions exhibit a smaller (conditional) variance and the last estimator also works for vanishing noise levels. The proofs are based on local approximation results to overcome the intricate nonlinearities and on a stable central limit theorem for stochastic integrals with respect to cylindrical Brownian motion. Simulation results illustrate the theoretical findings.
Paper Structure (17 sections, 17 theorems, 100 equations, 3 figures, 1 table)

This paper contains 17 sections, 17 theorems, 100 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The model
  3. Notations
  4. The stochastic heat equation
  5. The observation scheme
  6. Estimation methods and main results
  7. The additive noise estimator
  8. The multiplicative noise estimator
  9. The stabilised multiplicative noise estimator
  10. Confidence intervals
  11. Implementation and simulation results
  12. Proofs
  13. Fundamental asymptotics
  14. Approximation of quadratic variations and related terms
  15. Asymptotics of quadratic variations and related terms
  16. ...and 2 more sections

Key Result

Proposition 3.2

Let $(Y_\delta(t), 0\le t\le T)$ for $\delta\in(0,1)$ be $L^2(\Lambda)$-valued processes, progressively measurable with respect to the cylindrical Brownian filtration $({\mathscr F}_t)_{0\le t\le T}$ and satisfying $\int_0^T \| Y_\delta(t) \|^2 \, dt < \infty$. If then a stable limit theorem for the stochastic integrals holds as $\delta\to 0$: with an independent scalar Brownian motion $(B(t), 0

Figures (3)

  • Figure 1: Realisation of the stochastic heat equation with multiplicative noise $\sigma_2 (x) = 0.20 \times |x|^{0.8} + 0.01$ (left) and $\sigma_3 (x) = 10e^{-10 |x-2|} + 10 e^{-10 |x-4|}$ (right). The horizontal lines indicate the support of the kernel $K_{\delta,x_0}$.
  • Figure 2: Histograms of the estimators with red lines depicting the asymptotic densities. From left to right: ANE $\hat{\vartheta}_{\delta}$; MNE $\tilde{\vartheta}_{\delta}$; SMNE $\vartheta_{\delta}^{\star}$. From top to bottom: $\sigma_1$; $\sigma_2$; $\sigma_3$.
  • Figure 3: $\log_{10}$-$\log_{10}$ plot of RMSE for the estimators ANE $\hat{\vartheta}_{\delta}$ and SMNE $\vartheta_{\delta}^{\star}$ under multiplicative noise $\sigma_2(\cdot)$ and $\sigma_3(\cdot)$. The purple line with slope $1$ is added as reference.

Theorems & Definitions (40)

  • Definition 3.1
  • Proposition 3.2
  • Theorem 3.3
  • proof
  • Definition 3.4
  • Theorem 3.5
  • proof
  • Definition 3.6
  • Example 3.7
  • Theorem 3.8
  • ...and 30 more