Modeling and analysis of systems with nonlinear functional dependence on random quantities

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.