Sub-sampling in parametric estimation of
stochastic differential equations from discrete data
In most of the work on estimation of stochastic differential
equations from observational and/or numerical data it is typically
assumed that the data can be accurately approximated by a stochastic
differential equation on any time-step. Necessity to sub-sample the
data arises when this is not the case, e.g. when it is desirable to
approximate statistical properties of a smooth trajectory by a
stochastic differential equation. In this case parametric estimation of
an SDE would yield incorrect results if the discrete data is too dense
in time. Therefore, the dataset has to be sub-sampled (i.e. rarefied).
We present two simple examples which demonstrate the issue if
sub-sampling. While the sub-sampling criteria can be rigorously
established for the first example, we also demonstrate that our
analysis can be potentially extended to other systems.