Bernoulli

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

Statistical estimation of the Oscillating Brownian Motion

Antoine Lejay and Paolo Pigato

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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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.bj/1524038763

Digital Object Identifier
doi:10.3150/17-BEJ969

Mathematical Reviews number (MathSciNet)
MR3788182

Zentralblatt MATH identifier
06869885

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

Citation

Lejay, Antoine; Pigato, Paolo. Statistical estimation of the Oscillating Brownian Motion. Bernoulli 24 (2018), no. 4B, 3568--3602. doi:10.3150/17-BEJ969. https://projecteuclid.org/euclid.bj/1524038763


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