Volatility estimation for stochastic PDEs using high-frequency observations

TL;DR

This paper introduces a new method for estimating volatility in stochastic PDEs using high-frequency data, providing consistent estimators with a central limit theorem, and demonstrating effectiveness through simulations.

Contribution

It develops an explicit method of moments estimator for volatility in SPDEs, applicable in high-frequency regimes, with theoretical guarantees and simulation validation.

Findings

Estimator is consistent and asymptotically normal.

Method effectively estimates both volatility and differential operator parameters.

Simulation results confirm the estimator's practical performance.

Abstract

We study the parameter estimation for parabolic, linear, second-order, stochastic partial differential equations (SPDEs) observing a mild solution on a discrete grid in time and space. A high-frequency regime is considered where the mesh of the grid in the time variable goes to zero. Focusing on volatility estimation, we provide an explicit and easy to implement method of moments estimator based on squared increments. The estimator is consistent and admits a central limit theorem. This is established moreover for the joint estimation of the integrated volatility and parameters in the differential operator in a semi-parametric framework. Starting from a representation of the solution of the SPDE with Dirichlet boundary conditions as an infinite factor model and exploiting mixing-type properties of time series, the theory considerably differs from the statistics for semi-martingales literature. The performance of the method is illustrated in a simulation study.