Stochastic Differential Equations: Simulation, Parameter Estimation and Applications

Abstract

Stochastic differential equations (SDEs), including time-homogeneous Itoˆ diffusion processes, play an essential role in modelling phenomena in various fields, including physics, biology and finance. The parameters of the stochastic model are usually unknown in reality. Statistical inference on the unknown parameters of an Itoˆ diffusion process has continued to attract in- creasing attention in the last decades. Because in general, the maximum likelihood estimation is not directly applicable to the Itoˆ diffusion process, sue to the transition density usually not being available in closed form, an approximation to the transition density is developed. We aim to formulate a skew-normal approximation method motivated by the fact that the well- known Gaussian approximation method [Kessler, 1997] is inadequate in a skewed situation. The solution of an SDE, also known as the numerical method for solving the SDE, is crucial to model various phenomena. We built a simulation scheme of the two commonly used numerical methods for a general Itoˆ diffusion process across various grid widths in R. In addition to the numerical method simulation scheme, we extended the existing parameter estimation scheme [Lu et al., 2021] to the skew-normal method, and can be applied to a general Itoˆ diffusion process. In the practical implementation of our parameter estimation scheme, we applied the Gaussian approximation method and the skew-normal approximation method to estimate the parameters of two commonly used interest rate models, the Cox–Ingersoll–Ross model and the Vasicek model, for a 3-year Australian government bond yield data set. The accuracy is verified by simulating the sample paths of the estimated models using the numerical method simulation scheme for the general Itoˆ diffusion processes. The Vasicek model is demonstrated to exhibit a better performance as a model for the bond yield data under parametric bootstrap hypothesis testing.