Characteristic function equations for the state of dynamic systems with Gaussian, Poisson, and Lévy white noise

The Itô formula for semimartingales is applied to develop equations for the characteristic function of the state of linear and non-linear dynamic systems with Gaussian, Poisson, and Lévy white noise, viewed as the formal derivatives of Brownian, compound Poisson, and Lévy processes, respectively. These equations can be obtained if the drift and diffusion coefficient of a dynamic system are polynomials of the system state and the driving noise is Gaussian or Poisson. It was not possible to derive equations for the characteristic function for the state of systems driven by Lévy white noise. Numerical results are presented for dynamic systems with real-valued states driven by Gaussian, Poisson, and Lévy white noise processes.