Estimation of the Brownian dimension of a continuous It\^{o} process

TL;DR

This paper develops asymptotic statistical methods to estimate the minimal dimension of the Wiener process driving a continuous Itô process observed at discrete times, helping identify the underlying stochastic complexity.

Contribution

It introduces new asymptotic testing procedures to determine the Brownian dimension of a multidimensional Itô process from discrete observations.

Findings

Procedures effectively estimate the Brownian dimension.

Methods applicable to high-dimensional processes.

Theoretical validation of asymptotic properties.

Abstract

In this paper, we consider a $d$-dimensional continuous It\^{o} process which is observed at $n$ regularly spaced times on a given time interval $[0,T]$. This process is driven by a multidimensional Wiener process and our aim is to provide asymptotic statistical procedures which give the minimal dimension of the driving Wiener process, which is between 0 (a pure drift) and $d$. We exhibit several different procedures, all similar to asymptotic testing hypotheses.