Statistical analysis of non-linear diffusion process

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. We show that, under mild regularity conditions, the quasi-likelihood estimators of the parameters in the linear mean function of each regime are consistent and are asymptotically normal, whereas the threshold parameter is super consistent and weakly converges to some non-Gaussian continuous distribution. A notable feature is that the limiting distribution of the threshold parameter admits a closed-form probability density function, which enables the construction of its confidence interval; in contrast, for the discrete-time TAR models, the construction of the confidence interval for the threshold parameter has, so far, not been practically solved. A simulation study is provided to illustrate the asymptotic results. We also use the threshold model to estimate the term structure of a long time series of US interest rates. It is also of theoretical and practical interest that whether the observed process indeed satisfy the threshold model. In Chapter 3, we propose a likelihood ratio test scheme to test the existence of thresholds. It can test for non-linearity. Most importantly, we shall study how to price and predict value processes with nonlinear diffusion processes.be shown, under the null hypothesis of no threshold, the test statistics converges to a central Gaussian process asymptotically. Also the test is asymptotically powerful and the asymptotic distribution of the test statistic under the alternative hypothesis converge to a non-central Gaussian distribution. Further, the limiting distribution is the same as that of its discrete analogues for testing TAR(1) model against autoregressive model. Thus the upper percentage points of the asymptotic statistics for the discrete case are immediately applicable for our tests. Simulation studies are also conducted to show the empirical size and power of the tests. The application of our current method leads to more future work briefly discussed in Chapter 4. For example, we would like to extend our estimation methods to higher order and higher dimensional cases, use more general underlying mean processes, and most importantly, we shall study how to price and predict value processes with nonlinear diffusion processes.