SOME EVOLUTION EQUATIONS FOR AN ORNSTEIN–UHLENBECK PROCESS-DRIVEN DYNAMICAL SYSTEM

Abstract

The statistical properties of the Ornstein–Uhlenbeck (OU) process, a colored noise process, confirm the real noise statistics, since the real noise process has finite, nonzero correlation time. For this reason, it seems worthwhile to develop the estimation-theoretic scenarios of dynamical systems embedded in the colored noise environment as well. Importantly, the application of the Itô theory is not straightforward to the dynamical system in which the OU variable is a driving input. The augmented solution vector approach coupled with the Itô stochastic differential rule plays the pivotal role to develop the theory of the OU process-driven Duffing–van der Pol (DvdP) system of this paper. Notably, the noise analysis of the Duffing–van der Pol system, especially from the estimation-theoretic viewpoint, under the colored noise influence is not available yet in literature. Numerical experimentations with three different sets of data are demonstrated to examine the efficacy of analytical findings of this paper. The results of this paper will be of interest to noise scientists, especially research communities in systems and control, looking for the estimation-theoretic scenarios of the colored noise-driven "vector" stochastic differential system.