On the limitations of the Wiener path integral most probable path technique for solving nonlinear Itô stochastic differential equations

Abstract

The Wiener path integral (WPI) technique for determining the solution of a given nonlinear Itô stochastic differential equation (SDE) requires the solution of an Euler–Lagrange equation for the determination of the SDE’s associated most probable path. The derived most probable path family of functions is then used for approximating the SDE’s associated probability density function (PDF). In this paper, an example of modelling a wind turbine’s power is proposed where, although multiple most probable paths solutions exist for the associated SDE, only one can be used for the purpose of determining its solution PDF, however, the classical most probable path WPI methodology is not able of determining it. It is subsequently shown that using an enhanced semi-analytical version of the WPI based solution methodology results in the determination of the SDE’s exact solution PDF effortlessly, which can also be used for choosing the aforementioned most probable path solution which leads to the exact solution in a distributional sense.