The stochastic differential calculus for systems with non-Gaussian delta-correlated parametric loads

In the present work, an approach of differential calculus with parametric delta-correlated loads is presented again, after its first publication in 1993. Moreover, a brief review of the papers in literature, that had treated and used the Non-Gaussian Stochastic Differential Calculus (NGSDC), that can be considered as the generalization of what happens in the case of Gaussian white noise parametric actions. In the Gaussian case, two different stochastic differential approaches are in the literature, the Ito stochastic differential calculus (ISDC) and the Stratonovich one (SSDC). In the case of parametric external actions, they give different results and some correction terms, the so-called Wong-Zakai ones, must be considered for making uniform the results, representing the conversion adding terms in the transformation from ISDC to SSDC. When parametric actions are non-normal delta-correlated processes, the NGSDC has to be applied. It gives different results respect to both ISDC and SSDC. In any case, if the NGSDC is interpretated as made in the Di Paola-Falsone method (DFM), the expression of the correction terms to be used, in the conversion from ISDC to NGSDC, are expressed in terms of an infinite series that must be truncated after a certain order. It is shown that, if the closure order is two, the expression of these new correction terms coincides with the Wong-Zakai ones. • A brief review of the papers that, in the last thirty years, treated the response of systems excited by parametric delta-correlated actions, is applications given. • The efficiency of the stochastic differential calculus, as introduced by Di Paola-Falsone (1993), is confirmed by its numerous during these years. • The efficiency of the new correction terms, defined by the difference between the results of the Ito calculus and those of the new calculus, is always confirmed. • The Di Paola-Falsone correction terms have been always verified to be the generalization of the Wong-Zakai terms to the non-Gaussian inputs. • The applications of the non-Gaussian stochastic differential calculus are always more numerous, not only in Engineering and Mechanics science, but also in other fields, such as Financial Economy, Medical Sciences, Chemistry, Biology, and so on.