Level set and drift estimation from a reflected brownian motion with drift

Abstract

We estimate the drift and the level sets of the stationary distribution of a Brownian motion with drift, reflected in the boundary of a compact set S ⊂ Rd, 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, for example, when estimating the home range and the core area of an animal based on tracking data. Recent attempts to estimate the domain of a reflected Brownian motion have considered a uniform stationary distribution; however in this case the estimation of the core area, defined as a level set of the stationary distribution, is meaningless. We also give an estimator of the drift function, based on the increments of the process. In order to prove our results, we obtained several new theoretical properties of the reflected Brownian motion with drift, under fairly general assumptions. These properties allow us to perform the estimation for flexible regions close to reality. Lastly, the theoretical findings are illustrated using simulated and real-data examples.