Parameter Estimation of One-dimensional Ito Processes by LTDRM

Ito processes are processes commonly used as a mathematical model in many fields. In order to estimate the unknown parameters of an Ito process based on the maximum likelihood method, the transitional probability density function (PDF) of the Ito 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 Ito 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).