Cubification of nonlinear stochastic differential equations and approximate moments calculation of the Langevin Equation

For the class of Ito-type nonlinear Stochastic Differential Equations (SDE), where the drift and the diffusion are ??-functions (??-SDE), we prove that the (infinite) set of all moments of the solution satisfies a system of infinite ordinary differential equations (ODEs), which is always linear. The result is proven by showing first that a ??-SDE can be cubified, i.e. reduced to a system of SDE of larger (but still finite) dimension in general, where drifts and diffusions are at most third-degree polynomial functions. Our motivation for deriving a moment equation in closed form comes from systems biology, where second-order moments are exploited to quantify the stochastic variability around the steady-state average amount of the molecular players involved in a bio-chemical reaction framework. Indeed, the proposed methodology allows to write the moment equations in the presence of non-polynomial nonlinarities, when exploiting the Chemical Langevin Equations (which are SDE) as a model abstraction. An example is given, associated to a protein-gene production model, where non-polynomial nonlinearities are known to occur.