Cubification of σπ-SDE and exact moment equations

For an Itô-like Stochastic Differential Equation (SDE) system, with drift and diffusion that are formal polynomials of the independent variables, we show that all moments satisfy an infinite, countable, set of linear ordinary differential equations. This result is achieved by means of the exact cubification of the SDE, which consists in a set of deterministic transformations of the state variables, giving place to a new SDE with further finitely many state variables. Exact cubification can be considered as an extension to the ‘stochastic case’ of the exact quadratization of deterministic nonlinear systems, available in the literature. An example is finally shown, taken from systems biology, in which, for a basic chemical reaction network the exact moment equation is written down, and an approximate solution is calculated through a moment closure method.