Many real-world systems exhibit noisy evolution; interpreting their
finite-time behavior as arising from continuous-time processes (in the It\^o or
Stratonovich sense) has led to significant success in modeling and analysis in
a variety of fields. Here we argue that a class of differential equations where
evolution depends nonlinearly on a random or effectively-random quantity may
exhibit finite-time stochastic behavior in line with an equivalent It\^o
process, which is of great utility for their numerical simulation and
theoretical analysis. We put forward a method for this conversion, develop an
equilibrium-moment relation for It\^o attractors, and show that this relation
holds for our example system. This work enables the theoretical and numerical
examination of a wide class of mathematical models which might otherwise be
oversimplified due to a lack of appropriate tools.Comment: 13 pages, 6 figure