Solving Random Differential Equations by RVT Technique and Lagrange-Bürmann theorem via densities: Implementation and simulation

The first probability density function associated with the solution of a stochastic process defined by differential equations can often be obtained by the Random Variable Transformation (RVT) method. The application of this method allows us to obtain, as a solution, a function which depends on random inputs (parametric-type stochastic process). However, the RVT method requires to calculate the inverse mapping of the transformation that involves the solution itself. This seriously limits the application of RVT technique in many problems. In this work, we take advantage of the Lagrange-Bürmann theorem in order to give an alternative method to compute the inverse of an analytical mapping in the setting of random differential equations. A detailed description of the implementation is given. Finally, a numerical experiment is shown in order to highlight the helpfulness of combining RVT and Lagrange-Bürmann theorem in practice.