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November 2018 Statistical estimation of the Oscillating Brownian Motion
Antoine Lejay, Paolo Pigato
Bernoulli 24(4B): 3568-3602 (November 2018). DOI: 10.3150/17-BEJ969

Abstract

We study the asymptotic behavior of estimators of a two-valued, discontinuous diffusion coefficient in a Stochastic Differential Equation, called an Oscillating Brownian Motion. Using the relation of the latter process with the Skew Brownian Motion, we propose two natural consistent estimators, which are variants of the integrated volatility estimator and take the occupation times into account. We show the stable convergence of the renormalized errors’ estimations toward some Gaussian mixture, possibly corrected by a term that depends on the local time. These limits stem from the lack of ergodicity as well as the behavior of the local time at zero of the process. We test both estimators on simulated processes, finding a complete agreement with the theoretical predictions.

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Antoine Lejay. Paolo Pigato. "Statistical estimation of the Oscillating Brownian Motion." Bernoulli 24 (4B) 3568 - 3602, November 2018. https://doi.org/10.3150/17-BEJ969

Information

Received: 1 January 2017; Revised: 1 May 2017; Published: November 2018
First available in Project Euclid: 18 April 2018

zbMATH: 06869885
MathSciNet: MR3788182
Digital Object Identifier: 10.3150/17-BEJ969

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

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Vol.24 • No. 4B • November 2018
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