ABSTRACT
The question of how phenomena evolve over time is central to a broad range of fields
within the social, natural, and physical sciences. The behavior of such phenomena is
governed by established laws of the underlying field that typically describe the rates
at which it and related quantities evolve over time. The measurement of all param-
eters is subject to noise. As such, a precise mathematical description involves the
formulation of so-called stochastic evolution equations whose complexity depends
to a large extent on the realism of the model. We focus in this chapter on models in
which the evolution equation is generated by a system of ordinary differential equa-
tions with finitely many independent sources of multiplicative noise. We are in search
of an abstract paradigm into which all of these models are subsumed as special cases.
Once established, we will study the rudimentary properties of the abstract paradigm
and subsequently apply the results to each model. Some standard references used
throughout this chapter are [11, 62, 178, 193, 244, 251, 287, 302, 318, 333, 375,
397, 399].
