  This paper addresses the problem of computing controllers that are correct by
design for safety-critical systems and can provably satisfy (complex)
functional requirements. We develop new methods for models of systems subject
to both stochastic and parametric uncertainties. We provide for the first time
novel simulation relations for enabling correct-by-design control refinement,
that are founded on coupling uncertainties of stochastic systems via
sub-probability measures. Such new relations are essential for constructing
abstract models that are related to not only one model but to a set of
parameterized models. We provide theoretical results for establishing this new
class of relations and the associated closeness guarantees for both linear and
nonlinear parametric systems with additive Gaussian uncertainty. The results
are demonstrated on a linear model and the nonlinear model of the Van der Pol
Oscillator.
