Analytical modeling of distributions with invariant measure in stochastic systems with unsolved derivatives

Exact and approximate analytical modeling methods for stochastic processes with invariant measure in Gaussian and non-Gaussian stochastic systems with unsolved derivatives are considered. The methods are based on the linear regression approximation of nonlinear functions with unsolved derivatives and reduction to stochastic Ito differential equations. Two exact methods for analytical modeling of one- and multidimensional distributions with invariant measure are described. Special attention is paid to normal approximation and parametrization methods. A test example for Duffing equation nonlinear in second derivative is given. The stationary and nonstationary regimes and asymptotic stability are investigated. The method of normal approximation for one- and two-dimensional distributions is accurate enough for engineering applications. Some generalizations concerning numerical analytical modeling are considered.