On quadratic variations of the fractional-white wave equation

TL;DR

This paper investigates the quadratic variations of a stochastic wave equation driven by space-white and fractional-in-time noise, deriving limit theorems for different Hurst parameters and constructing estimators.

Contribution

It extends previous work by analyzing the time component of the quadratic variations and establishing limit theorems for various Hurst parameters in the fractional noise setting.

Findings

Proved central and noncentral limit theorems for quadratic variations.

Constructed consistent parameter estimators based on the limit theorems.

Analyzed rectangular quadratic variations in the white noise case with a central limit theorem.

Abstract

This paper studies the behaviour of quadratic variations of a stochastic wave equation driven by a noise that is white in space and fractional in time. Complementing the analysis of quadratic variations in the space component carried out by M. Khalil and C. A. Tudor (2018) and by R. Shevchenko, M. Slaoui and C. A. Tudor (2020), it focuses on the time component of the solution process. For different values of the Hurst parameter a central and a noncentral limit theorems are proved, allowing to construct consistent parameter estimators and compare them to the finding in the space-dependent case. Finally, rectangular quadratic variations in the white noise case are studied and a central limit theorem is demonstrated.