Stochastic response to additive Gaussian and Poisson white noises

The response process of a memoryless dynamical system subjected to a combination of Gaussian and Poisson white noises is completely characterized by its transition probability density function, which satisfies an integro-differential relation, the forward Kolmogorov-Feller equation. A boundary value problem for the characteristic function, Phi(x)(u, t), is obtained by taking the Fourier transform of the forward Kolmogorov-Feller equation, or directly from the state equations of the system using a generalized version of Ito's rule. The response processes for several two-state dynamical systems are found by numerically solving the transformed equations for the characteristic functions, followed by a numerical Fourier inversion to recover the probability density functions. Results are presented and compared with previously obtained solutions from the literature and from simulation, and the limitations of the algorithm are discussed.