The estimation of continuous time diffusion processes observed at discrete time intervals

Abstract

As the name implies, the aim of this dissertation, is to estimate the parameters of the diffusion process; a process which can be described by a stochastic differential equation. First, we shall define the term "stochastic differential equation and describe its components. During this process we will define Brownian motion, the stochastic integral and Ito's formula. Having defined the diffusion process, we shall next present two methods which will fit the diffusion equation to a given data set. That is, we will see how the drift and diffusion parameters can be estimated. These methods can be divided in two categories. The first method can be classified as a parametric method while the second method is classified as a nonparametric method. The parametric model is usually implemented when the functional form of the drift and diffusion equations are known in advance. However in general this important piece of information is totally missing because very often all we have in our possession is just the data set. As a result these methods cannot be used in general. In such cases the nonparametric models can be used instead of the parametric models. These methods do not require the user to specify the functional form. In this dissertation we will discuss a particular kind of nonparametric model, called the Kernel Regression model. In particular, we will study some of the statistical properties of the Nadaraya-Watson Estimator. Finally we will apply these models to three data sets.