• Bernoulli
  • Volume 24, Number 4B (2018), 3568-3602.

Statistical estimation of the Oscillating Brownian Motion

Antoine Lejay and Paolo Pigato

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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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Bernoulli, Volume 24, Number 4B (2018), 3568-3602.

Received: January 2017
Revised: May 2017
First available in Project Euclid: 18 April 2018

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Zentralblatt MATH identifier

arcsine distribution Gaussian mixture local time occupation time Oscillating Brownian Motion Skew Brownian Motion


Lejay, Antoine; Pigato, Paolo. Statistical estimation of the Oscillating Brownian Motion. Bernoulli 24 (2018), no. 4B, 3568--3602. doi:10.3150/17-BEJ969.

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