Statistical estimation of the Oscillating Brownian Motion

Antoine Lejay 1, 2 Paolo Pigato 1, 2
2 TOSCA - TO Simulate and CAlibrate stochastic models
CRISAM - Inria Sophia Antipolis - Méditerranée , IECL - Institut Élie Cartan de Lorraine : UMR7502
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, Bernoulli Society for Mathematical Statistics and Probability, 2018, 24 (4B), pp.3568-3602. ⟨10.3150/17-BEJ969⟩. ⟨hal-01430794v3⟩



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