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

  • Kohei ChibaEmail author


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.


Fractional Brownian motion Lead–lag effect Non-synchronous observations Contrast estimation 

Mathematics Subject Classification

62M09 60G22 



The author would like to express deepest gratitude to Professor Nakahiro Yoshida for introducing him to this problem and for many valuable suggestions. He is also very grateful to two anonymous referees for their really helpful comments and suggestions. He could never have improved the results and the presentation of this paper without their helpful comments.This work was supported by JST CREST and the Program for Leading Graduate Schools, MEXT, Japan.

Compliance with ethical standards

Conflict of interest

The author has no conflicts of interest to declare.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoKomabaJapan

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