A half-wave rectifier circuit as a bilinear stochastic differential system

Bilinear stochastic differential systems have found striking applications in diverse fields, e.g. autonomous systems under noise influence, population models, theoretical biology and switched electrical networks. Because of their surprising applications, this paper develops the closed-form solution for a Stratonovich time-varying bilinear Stochastic Differential Equation (SDE). Subsequently, the analytic result of this paper is applied to a simple, but an appealing half-wave rectifier electronic circuit. Furthermore, we derive the mean and variance expressions of the state satisfying the rectifier SDE. The stochastic differential rule in combination with the concept of characteristic function of the Gaussian stochastic process plays the key role to develop the theory of this paper. The results of this paper will be useful to accomplish the noise analysis of bilinear systems arising from systems and control literature.