Threshold estimation for stochastic processes with small noise

Consider a process satisfying a stochastic differential equation with unknown drift parameter, and suppose that discrete observations are given. It is known that a simple least squares estimator (LSE) can be consistent, but unstable under finite samples when the noise process has jumps. We propose a filter to cut large shocks from data, and construct the same LSE from data selected by the filter. The proposed estimator can be asymptotically equivallent to the usual LSE, whose asymptotic distribution strongly depends on the noise process. However it is interesting to note that it could be asymptotically normal by choosing the filter suitably if the noise is a L\'evy process. We try to justify this phenomenon theoretically.