Temporal quartic variation for non-linear stochastic heat equations with piecewise constant coefficients

TL;DR

This paper investigates the temporal quartic variation of solutions to non-linear stochastic heat equations with piecewise constant coefficients driven by white noise, establishing theoretical properties and practical estimators.

Contribution

It introduces a new analysis of the temporal quartic variation for such equations and proposes a consistent estimator based on these findings.

Findings

Existence and uniqueness of mild solutions are proven.

Asymptotic behavior of the temporal quartic variation is characterized.

A consistent estimator for the model parameters is derived.

Abstract

We consider a stochastic partial differential equation with piecewise constant coefficients driven by a multiplicative space-time white noise. The existence and uniqueness of the mild solution in Walsh sense is established. We mainly study the limit behavior of the temporal quartic variation of the mild solution. As an application, we deduce a consistent estimator based on corresponding results.