  Parameter estimation for non-stationary stochastic differential equations
(SDE) with an arbitrary non-linear drift and non-linear diffusion is
accomplished in combination with a non-parametric clustering methodology. Such
a model-based clustering approach includes a quadratic programming (QP) problem
with equality and inequality constraints. We couple the QP problem to a
closed-form likelihood function approach based on suitable Hermite-expansions
to approximate the parameter values of the SDE model. The classification
problem provides a smooth indicator function, which enables us to recover the
underlying temporal parameter modulation of the one-dimensional SDE. As shown
by the numerical examples, the clustering approach recovers a hidden functional
relationship between the SDE model parameters and an additional auxiliary
process. The study builds upon this functional relationship to develop
closed-form, non-stationary, data-driven stochastic models for multiscale
dynamical systems in real-world applications.
