Maximum entropy principle and non-stationary distributions of stochastic systems

The principle of maximum entropy (in its classical form), successfully applied in many fields (e.g. statistics, reliability, estimation), has recently been extended to analyze the systems governed by stochastic differential equations and especially to determining the stationary probability distribution of the solution process. In this paper we develop the maximum entropy approach to characterize non-stationary probability distributions of the solutions of stochastic systems. The variational problem for the entropy functional includes time-dependent constraints in the form of differential equations for moments. The general scheme of the method is given along with the effective treatment of a number of first and second order stochastic systems. The maximum entropy probability distributions are compared with the exact solutions and with the simulation results.