Parameter estimation for generalized mixed fractional stochastic heat equation

TL;DR

This work investigates parameter estimation for a stochastic heat equation driven by a generalized mixed fractional Brownian motion, using discrete-time observations of the solution. It develops moment-based estimators for the Hurst indices and noise intensities, proving strong consistency and asymptotic normality. Specifically, it provides a strongly consistent estimator for $H_1$ when $H_2$ is known, with $\sqrt{N}(\hat{H}_{1N}-H_1) \to N(0,\zeta^2)$, and, when $H_1$ and $H_2$ are known and distinct, consistent, asymptotically normal estimators for $\sigma_1^2$ and $\sigma_2^2$ via two-time moment equations; it also offers an alternative fourth-moment-based approach. These results advance parameter inference for SPDEs driven by generalized mixed fractional noise under discrete observations, with implications for modeling systems exhibiting mixed fractional dynamics and long-range dependence.

Abstract

We study the properties of a stochastic heat equation with a generalized mixed fractional Brownian noise. We obtain the covariance structure, stationarity and obtain bounds for the asymptotic behaviour of the solution. We suggest estimators for the unknown parameters based on discrete time observations and study their asymptotic properties.