In this work, we consider a prototype stochastic dynamic model for dynamic processes in biological, chemical, economic, financial, medical, military, physical and technological sciences. The dynamic model is described by Lévy-type nonlinear stochastic differential equation. The model validation is established by the usage of Lyapunov-like function. The basic innovative idea is to transform a nonlinear Lévy-type nonlinear stochastic differential into a simpler stochastic differential equation that is easily tested for the existence and uniqueness theorem. Using the nature of Lyapunov-like function, the existence and uniqueness of solution of the original Lévy-type nonlinear stochastic differential equation is established. The main idea of the proof is based on the property of the one-to-one and onto transformation. As the byproduct of the analysis, it is shown that the closed-form implicit solution of transformed stochastic differential equation is a positive martingale. Furthermore, using the change of measure, a Girsanov-type theorem for Lévy-type nonlinear stochastic dynamic model is established.
On the approximate moment equations of a nonlinear stochastic differential equation
