Estimation of the lead-lag parameter between two stochastic processes driven by fractional Brownian motions

TL;DR

This paper introduces a new method to estimate the lead-lag parameter between two processes driven by fractional Brownian motions with Hurst parameter > 1/2, effective even with non-synchronous data.

Contribution

It proposes a consistent estimator for the lead-lag parameter that does not require knowledge of the Hurst parameters and handles irregular observations.

Findings

Estimator works without Hurst parameters.

Handles non-synchronous and irregular data.

Convergence rates are explicitly calculated.

Abstract

In this paper, we consider the problem of estimating the lead-lag parameter between two stochastic processes driven by fractional Brownian motions (fBMs) of the Hurst parameter greater than 1/2. First we propose a lead-lag model between two stochastic processes involving fBMs, and then construct a consistent estimator of the lead-lag parameter with possible convergence rate. Our estimator has the following two features. Firstly, we can construct the lead-lag estimator without using the Hurst parameters of the underlying fBMs. Secondly, our estimator can deal with some non-synchronous and irregular observations. We explicitly calculate possible convergence rate when the observation times are (1) synchronous and equidistant, and (2) given by the Poisson sampling scheme. We also present numerical simulations of our results using the R package YUIMA.