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Efficient parameter estimation for parabolic SPDEs based on a log-linear model for realized volatilities

Markus Bibinger, Patrick Bossert

TL;DR

The paper tackles parameter estimation for a one-dimensional parabolic SPDE observed at high frequency in time and space. It introduces a log-linear model for realized volatilities across spatial points and develops least-squares estimators for the curvature parameter and an intercept tied to the volatility, achieving asymptotically efficient rates with feasible confidence intervals. The authors prove central limit theorems for the estimators, derive explicit asymptotic variance-covariance matrices, and demonstrate substantial finite-sample efficiency gains over existing M-estimators through numerical comparisons and Monte Carlo simulations. The approach combines spectral SPDE analysis with a regression-based framework, offering practical and theoretically grounded tools for calibrating SPDEs from high-frequency data.

Abstract

We construct estimators for the parameters of a parabolic SPDE with one spatial dimension based on discrete observations of a solution in time and space on a bounded domain. We establish central limit theorems for a high-frequency asymptotic regime. The asymptotic variances are shown to be substantially smaller compared to existing estimation methods. Moreover, asymptotic confidence intervals are directly feasible. Our approach builds upon realized volatilities and their asymptotic illustration as response of a log-linear model with spatial explanatory variable. This yields efficient estimators based on realized volatilities with optimal rates of convergence and minimal variances. We demonstrate efficiency gains compared to previous estimation methods numerically and in Monte Carlo simulations.

Efficient parameter estimation for parabolic SPDEs based on a log-linear model for realized volatilities

TL;DR

The paper tackles parameter estimation for a one-dimensional parabolic SPDE observed at high frequency in time and space. It introduces a log-linear model for realized volatilities across spatial points and develops least-squares estimators for the curvature parameter and an intercept tied to the volatility, achieving asymptotically efficient rates with feasible confidence intervals. The authors prove central limit theorems for the estimators, derive explicit asymptotic variance-covariance matrices, and demonstrate substantial finite-sample efficiency gains over existing M-estimators through numerical comparisons and Monte Carlo simulations. The approach combines spectral SPDE analysis with a regression-based framework, offering practical and theoretically grounded tools for calibrating SPDEs from high-frequency data.

Abstract

We construct estimators for the parameters of a parabolic SPDE with one spatial dimension based on discrete observations of a solution in time and space on a bounded domain. We establish central limit theorems for a high-frequency asymptotic regime. The asymptotic variances are shown to be substantially smaller compared to existing estimation methods. Moreover, asymptotic confidence intervals are directly feasible. Our approach builds upon realized volatilities and their asymptotic illustration as response of a log-linear model with spatial explanatory variable. This yields efficient estimators based on realized volatilities with optimal rates of convergence and minimal variances. We demonstrate efficiency gains compared to previous estimation methods numerically and in Monte Carlo simulations.
Paper Structure (11 sections, 4 theorems, 71 equations, 5 figures)

This paper contains 11 sections, 4 theorems, 71 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Curvature estimation
  3. Asymptotic log-linear model for realized volatilities and least squares estimation
  4. Numerical illustration and simulations
  5. Numerical comparison of asymptotic variances
  6. Monte Carlo simulation study
  7. Proofs of the theorems
  8. Preliminaries
  9. Proof of Theorem 1
  10. Proof of Theorem 3
  11. Proof of Theorem 2

Key Result

Theorem 1

Grant Assumptions assump-obs and cond with $y_1=\delta$, $y_m=1-\delta$, and $\delta\in(0,1/2)$. Then, the estimator kappahat satisfies, as $n\to\infty$, the central limit theorem (clt)

Figures (5)

  • Figure 1: Top panel: Comparison of asymptotic variances of $\hat{\kappa}_{n,m}$ from \ref{['kappahat']} (for known $\sigma_0^2$), $\hat{\kappa}_{n,m}^{LS}$ from \ref{['lskappa']} and the estimator from trabs, for $\delta=0{.}05$, and for different values of $\kappa$. Lower panel: Ratio of asymptotic variances of new method using \ref{['lskappa']} and \ref{['lsint']} vs. trabs, left for estimating $\kappa$, right for $\sigma_0^2$.
  • Figure 2: Comparison of empirical distributions of normalized estimation errors for $\kappa$ from simulation with $n=1000$, $m=11$, $\sigma_0^2=1$, $\kappa=1$ in the left two columns and $\kappa=6$ in the right two columns. Grey is for $\hat{\kappa}_{n,m}^{LS}$, brown for the estimator by trabs and yellow for $\hat{\kappa}_{n,m}$.
  • Figure 3: Comparison of empirical distributions of normalized estimation errors for $\sigma_0^2$ from simulation with $n=1000$, $m=11$, $\sigma_0^2=1$, $\kappa=1$ in the left panel and $\kappa=6$ in the right panel. Grey is for $(\hat{\sigma}_{0}^{2})^{LS}$, and brown for the estimator by trabs.
  • Figure 4: QQ-normal plots for normalized estimation errors for $\kappa$ from simulation with $n=1000$, $m=11$, $\sigma_0^2=1$, $\kappa=1$ in the left panel and $\kappa=6$ in the right panel. Brown (top) is the estimator from trabs, dark grey is for \ref{['lskappa']} and yellow (bottom) for \ref{['kappahat']}.
  • Figure 5: QQ-normal plots for normalized estimation errors for $\sigma_0^2$ from simulation with $n=1000$, $m=11$, $\sigma_0^2=1$, $\kappa=1$ in the left panel and $\kappa=6$ in the right panel. Brown (top) is the estimator from trabs and dark grey is for the estimator using \ref{['lsint']}.

Theorems & Definitions (6)

  • Example 1
  • Theorem 1
  • Theorem 2
  • Example 2
  • Theorem 3
  • Corollary 4