Random periodic solutions for a class of hybrid stochastic differential equations

We present the existence and uniqueness of random periodic path for stochastic dynamical systems generated by random switching stochastic differential equations (SDEs). These classes of SDEs arise as concrete models in molecular dynamics, biochemistry, climatology, wireless communications, financial mathematics, biological and artificial neural networks, etc. Random periodic processes are inevitable in these classes of stochastic dynamical systems due to the nonlinearity of their processing and the presence of time-dependent applied current. In our investigation, we employed Lyapunov second method and the theory of M-matrices, which are verifiable in terms of the coefficients of the SDE and the switching rates.