| We study the problem of statistical inference of continuous-time diffusion processes, and develop methods for modeling threshold diffusion processes in particular. We derive the limiting properties of the quasi-likelihood estimator. We also propose a quasi-likelihood ratio test statistic for testing threshold diffusion process against linear alternative.;We begin in Chapter 1 with an introduction of continuous-time nonlinear diffusion processes where we summarized the literature on model estimation.;In Chapter 2, we propose the quasi-likelihood method to estimate the parameters indexing the mean function of a threshold diffusion model. 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.;In Chapter 3, we propose a quasi-likelihood ratio test scheme to test for the existence of thresholds. It can be shown that, under the null hypothesis of no threshold, the test statistic converges to a certain function of a central Gaussian process asymptotically. Under a sequence of local alternatives, the test statistic converges weakly to the same functional of, however, a non-central Gaussian process. The quasi-likelihood ratio test can be extended to decide the number of thresholds. Simulation studies are conducted to support the asymptotic results in both chapter 2 and Chapter 3. We also apply the estimation and test results to a set of US interest rate data.;In Chapter 4, we discuss risk-neutral pricing when the pricing process of the underlying asset follows a threshold diffusion process. We derive a risk-neutral measure which can be justified by the principle of minimum entropy.;In chapter 5, We briefly discuss some future work related to threshold diffusion processes. |
