The multilinear algebra of the moments of distributions in the analysis of nonlinear stochastic systems

The differential equations are obtained for multilinear moment forms of the phase vector of a stochastic system described by the nonlinear stochastic Ito equation. The equations are derived for mixed moment forms of the process whose phase vector takes values in the product space. Multilinear cumulant forms are defined, and the link is established between the moments and cumulants. The issues of approximate solution of an infinite set of equations for the moments are discussed. The exact solution is given to the equations for the moments of a specific two-dimensional bilinear system.