Author: Fei Su

Publisher:

Published: 2011

Total Pages: 106

ISBN-13:

In this paper, we study the problem of statistical inference of continuous-time diffusion processes and their higher-order analogues, and develop methods for modeling threshold diffusion processes in particular. The limiting properties of such estimators are also discussed. We also proposed the likelihood ratio test statistics for testing threshold diffusion process against its linear alternative. We begin in Chapter 1 with an introduction of continuous-time non-linear diffusion processes where I summarized the literature on model estimation. The most natural extension from affine to non-linear model would be piecewise linear diffusion process with piecewise constant variance functions. It can also be considered as a continuous-time threshold autoregressive model (CTAR), the continuous-time analogue of AR model for discrete-time time-series data. The order-one CTAR model is discussed in detail. The discussion is directed more toward the estimation techniques other than the mathematical details. Existing inferential methods (estimation and testing) generally assume known functional form of the (instantaneous) variance function. In practice, the functional form of the variance function is hardly known. So, it is important to develop new methods for estimating a diffusion model that does not rely on knowledge on the functional form of the variance function. In the second Chapter, we propose the quasi-likelihood method to estimate the parameters indexing the mean function of a threshold diffusion model without prior knowledge of its instantaneous variance structure. (and apply to other nonlinear diffusion models, which will be further investigated later.) We also explore the limiting properties of the quasi-likelihood estimators. We focus on estimating the mean function, after which the functional form of the instantaneous variance function can be explored and subsequently estimated from quadratic variation considerations.