Some new results on time-varying Stratonovich and Itô bilinear stochastic systems

The bilinear stochasticity of dynamical systems is attributed to the input–output coupling term,where the input is a random input and the state is the output of dynamical systems.Stochastically influenced bilinear systems are described via bilinear stochastic differential equations.In this paper,first we construct a mathematical method for the closed-form solution to a scalar Stratonovich time-varying bilinear stochastic differential equation driven by a vector random input as well as the Itôcounterpart.Second,the analytic results of the paper are applied to an electrical circuit that assumes the structure of a bilinear stochastic dynamic circuit.The noise analysis of the bilinear dynamic circuit is achieved by deriving the mean and variance equations as well.The theory of this paper hinges on the‘Stratonovich calculus’,conversion of the Stratonovich integral into the Itôintegral and characteristic function of the vector Brownian motion.The results of this paper will be useful for research communities looking for estimation and control of bilinear stochastic differential systems.