Two consistent estimators for the skew Brownian motion

¹ Université de Lorraine, IECL, UMR 7502, Vandœuvre-lès-Nancy, 54600, France.
² CNRS, IECL, UMR 7502, Vandœuvre-lés-Nancy, 54600, France.
³ Inria, Villers-lés-Nancy, 54600, France.
⁴ Facultad de Ciencias, Centro de Matemática. Iguá 4225, CP 11400, Montevideo, Uruguay.
⁵ CIMFAV, Facultad de Ingeniería, Universidad de Valparaíso, Valparaíso, Chile.

^* Corresponding author: antoine.lejay@inria.fr

Received: 19 May 2017
Accepted: 18 July 2018

Abstract

The skew Brownian motion (SBm) 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 procedure estimates consistently the parameter of an SBm observed at discrete times. The difficulties arise because the observed 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 total number of positive excursions divided by the expected number of excursions given the observations. The theoretical results are illustrated by numerical simulations.