The level of self-organized criticality in oscillating Brownian motion: n-consistency and stable Poisson-type convergence of the MLE

Abstract

For some discretely observed path of oscillating Brownian motion with level of self-organized criticality ρ0, we prove in the infill asymptotics that the MLE is n-consistent, where n denotes the sample size, and derive its limit distribution with respect to stable convergence. As the transition density of this homogeneous Markov process is not even continuous in ρ0, the analysis is highly nonstandard. Therefore, interesting and somewhat unexpected phenomena occur: The likelihood function splits into several components, each of them contributing very differently depending on how close the argument ρ is to ρ0. Correspondingly, the MLE is successively excluded to lay outside a compact set, a 1/n-neighborhood and finally a 1/n-neighborhood of ρ 0 asymptotically. The crucial argument to derive the stable convergence is to exploit the semimartingale structure of the sequential suitably rescaled local log-likelihood function (as a process in time). Both sequentially and as a process in ρ, it exhibits a bivariate Poissonian behavior in the stable limit with its intensity being a multiple of the local time at ρ0.