Estimation of quadratic variation for two-parameter diffusions

TL;DR

This paper establishes a central limit theorem for weighted quadratic variations of two-parameter Brownian motion and demonstrates that discretized quadratic variations serve as asymptotically normal estimators of the quadratic variation of two-parameter diffusions.

Contribution

It provides the first central limit theorem for weighted quadratic variations of two-parameter Brownian motion and shows their use as consistent estimators for quadratic variation in two-parameter diffusions.

Findings

Weighted quadratic variations are asymptotically normal.

Discretized quadratic variations estimate quadratic variation consistently.

The results apply to observations on regular grids.

Abstract

In this paper we give a central limit theorem for the weighted quadratic variations process of a two-parameter Brownian motion. As an application, we show that the discretized quadratic variations $\sum_{i=1}^{[n s]} \sum_{j=1}^{[n t]} | \Delta_{i,j} Y |^2$ of a two-parameter diffusion $Y=(Y_{(s,t)})_{(s,t)\in[0,1]^2}$ observed on a regular grid $G_n$ is an asymptotically normal estimator of the quadratic variation of $Y$ as $n$ goes to infinity.