  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.
