Level sets and drift estimation for reflected Brownian motion with drift

Abstract

We consider the estimation of the drift and the level sets of the stationary distri- bution of a Brownian motion with drift, reflected in the boundary of a compact set $S\subset R^d$ , departing from the observation of a trajectory of this process. We obtain the uniform consistency and rates of convergence for the proposed kernel based estimators. This problem has relevant applications in ecology, in estimating the home-range and the core-area of an animal based on tracking data. Recently, the problem of estimat- ing the domain of a reflected Brownian motion was considered in Cholaquidis, et al. (2016), when the stationary distribution is uniform and the estimation of the core-area, defined as a level set of the stationary distribution, is meaningless. As a by-product of our results, we obtain an estimation of the drift function. In order to prove our re- sults, some new theoretical properties of the reflected Brownian motion with drift are obtained, under fairly general assumptions. These properties allow us to perform the estimation for flexible regions close to reality. The theoretical findings are illustrated on simulated and real data examples.