Two consistent estimators for the Skew Brownian motion

Antoine Lejay (1, 2) , Ernesto Mordecki (3) , Soledad Torres (4)

1 TOSCA - TO Simulate and CAlibrate stochastic models
2 IECL - Institut Élie Cartan de Lorraine
3 CMAT - Centro de Matemática [Montevideo]
4 CIMFAV - Centro de Investigación y Modelamiento de Fenómenos Aleatorios – Valparaíso

The Skew Brownian motion is of primary importance in modeling diffusion in media with interfaces which arise in many domains ranging from population ecology to geophysics and finance. We show that the maximum likelihood estimator provides a consistent estimator of the parameter of a Skew Brownian motion observed at discrete times. The difficulties are that this process is only null recurrent and has a singular distribution with respect to the one of the Brownian motion. Finally, using the idea of the Expectation-Maximization algorithm, we show that the maximum likelihood estimator can be naturally interpreted as the expected number of positive excursions divided by the expected number of excursions.

Mots clés: Skew Brownian motion; maximum likelihood estimator (MLE) null recurrent process; Expectation-Maximization (EM) algorithm; excursion theory

Domaines: Probabilités [math.PR]

Antoine Lejay, Ernesto Mordecki, Soledad Torres. Two consistent estimators for the Skew Brownian motion. 2018. ⟨hal-01492853v3⟩