Multiplicative cases from additive cases: Extension of Kolmogorov-Feller equation to parametric Poisson white noise processes

In this paper the response of nonlinear systems driven by parametric Poissonian white noise is examined.As is well known, the response sample function or the response statistics of a system driven by external white noise processes is completelydefined. Starting from the system driven by external white noise processes, when an invertible nonlinear transformation is applied, the transformedsystem in the new state variable is driven by a parametric type excitation. So this latter artificial system may be used as a tool to find out the propersolution to solve systems driven by parametric white noises. In fact, solving this new system, being the nonlinear transformation invertible, wemust pass from the solution of the artificial system (driven by parametric noise) to that of the original one (driven by external noise, that isknown). Moreover, introducing this invertible nonlinear transformation into the Itˆo’s rule for the original system driven by external input, one canderive the Itˆo’s rule for systems driven by a parametric type excitation, directly. In this latter case one can see how natural is the presence of theWong–Zakai correction term or the presence of the hierarchy of correction terms in the case of normal and Poissonian white noise, respectively.Direct transformation on the Fokker–Planck and on the Kolmogorov–Feller equation for the case of parametric input are found.