We consider a simple mean reverting diffusion process, with piecewise constant drift and diffusion coefficients, discontinuous at a fixed threshold. We discuss estimation of drift and diffusion parameters from discrete observations of the process, with a generalized moment estimator and a maximum likelihood estimator. We develop the asymptotic theory of the estimators when the time horizon of the observations goes to infinity, considering both cases of a fixed time lag (low frequency) and a vanishing time lag (high frequency) between consecutive observations. In the setting of low frequency observations and infinite time horizon we also study the convergence of three local time estimators, that are already known to converge to the local time in the setting of high frequency observations and fixed time horizon. We find that these estimators can behave differently, depending on the assumptions on the time lag between observations.

翻译：我们考虑一个具有分段常数漂移和扩散系数的简单均值回复扩散过程，该系数在固定阈值处不连续。基于过程的离散观测，我们分别采用广义矩估计法和最大似然估计法讨论漂移参数和扩散参数的估计问题。当观测时间跨度趋于无穷时，我们建立了估计量的渐近理论，同时考虑相邻观测之间时间间隔固定（低频情形）和趋于零（高频情形）两种情况。在低频观测且时间跨度无限的情形下，我们还研究了三种局部时估计量的收敛性——这些估计量在高频观测且时间跨度固定时已知收敛于局部时。研究发现，这些估计量的收敛行为可能因观测时间间隔的不同假设而呈现差异。