A prediction theory for a coloured noise-driven stochastic differential system

The standard approaches to analyse the Itô stochastic differential system are the Fokker–Planck equation and multi-dimensional Itô differential rule. In contrast to the Itô stochastic differential system, this paper develops a mathematical theory of a coloured noise process-driven stochastic differential system. More precisely, the coloured noise process-driven stochastic differential equation x˙t=f(xt)+g(xt) ξt is the subject of investigations. The statistical properties of the input noise process ξt are stationary and finite, non-zero, relatively smaller correlation time. The notion of “stochastic equivalence” coupled with stochastic differential rule plays the key role to develop the theory of this paper. The theory of the paper will be of use to analysing and control of dynamical systems embedded in the coloured noise environment.