Abstract

Itô processes are processes commonly used as a mathematical model in many _elds. In order to estimate the unknown parameters of an Itô process based on the maximum likelihood method, the transitional probability density function (PDF) of the Itô process is needed. In fact, the transitional PDF is the solution of a Fokker-Planck equation subject to an initial condition in which the transitional PDF is set to be coincided with the Dirac delta function at the initial time. In this research, we applied the numerical method called the Laplace transform dual reciprocity method (LTDRM) to approximate the solution of the Fokker-Planck equations, corresponding to a one-dimensional Itô process. The key idea of the LTDRM for solving this type of problems is to transform the Dirac delta function into the Laplace space and then use the dual reciprocity method (DRM) to solve the transformed equation without approximating the Dirac delta function. The Stehfests algorithm is used to convert the solutions back into the transitional PDF. We tested and ran experiments on the OU and CIR models by comparing with exact transitional PDF. The tests show that our results using LTDRM give a very accurate approximation and can be used in the maximum likelihood estimation (MLE).