Nonparametric calibration for stochastic reaction-diffusion equations based on discrete observations

TL;DR

This paper develops a nonparametric method for estimating reaction functions and parameters in stochastic reaction-diffusion equations from discrete data, achieving optimal convergence rates and adaptivity.

Contribution

It introduces a novel nonparametric estimator for the reaction function and extends quadratic variation estimators to nonlinear SPDEs, enabling simultaneous calibration.

Findings

Estimator achieves nonparametric convergence rates.

Joint estimation of diffusivity and volatility is rate-optimal.

Method works without prior parameter knowledge.

Abstract

Nonparametric estimation for semilinear SPDEs, namely stochastic reaction-diffusion equations in one space dimension, is studied. We consider observations of the solution field on a discrete grid in time and space with infill asymptotics in both coordinates. Firstly, we derive a nonparametric estimator for the reaction function of the underlying equation. The estimate is chosen from a finite-dimensional function space based on a least squares criterion. Oracle inequalities provide conditions for the estimator to achieve the usual nonparametric rate of convergence. Adaptivity is provided via model selection. Secondly, we show that the asymptotic properties of realized quadratic variation based estimators for the diffusivity and volatility carry over from linear SPDEs. In particular, we obtain a rate-optimal joint estimator of the two parameters. The result relies on our precise analysis of the H\"older regularity of the solution process and its nonlinear component, which may be of its own interest. Both steps of the calibration can be carried out simultaneously without prior knowledge of the parameters.