Limit theorems and statistical inference for Bessel and Dunkl processes

Abstract

The starting point of this thesis is the estimation of the multiplicities of a Dunkl process. We note that a Dunkl process can be transformed into a Bessel process or Cox-Ingersoll-Ross process, respectively. We study estimators for these two processes, which we then apply to the Dunkl process. We present well-established methods for inference in cases where the likelihood function is unknown or too complicated. First, we concentrate on martingale estimators at low frequency data for the index parameter of a Bessel process and, as an extension, polynomial processes. Since these processes are non-ergodic and most results for inference are developed for stationary and ergodic diffusions, we transform them into processes with such properties by adding a mean reverting term. Taking a closer look at these estimators, we recognize that they depend only on a particular transformation of the data which is a realization of a Cox-Ingersoll-Ross process. Therefore, in the remainder of the thesis we focus on estimators for the Cox-Ingersoll-Ross process. For this, we apply the Gaussian quasi-likelihood method. In doing so, the density is approximated by the Gaussian density. Even if the Cox-Ingersoll-Ross process has a non-central chi-squared density which is far from being Gaussian, this local approximation works well. We introduce a known preliminary estimator, then prove asymptotic normality for one-step improvements towards the Gaussian quasi-maximum likelihood estimator. We show that all these estimators for the Cox-Ingersoll-Ross process are asymptotically equivalent to the Gaussian quasi-maximum likelihood estimator and compare them in a simulation study. A key difference of the martingale estimators and the estimators for the Cox-Ingersoll-Ross process lies in their asymptotic behaviour. The Gaussian quasi-maximum likelihood estimators converge only when the underlying Bessel process never hits the origin whereas the martingale estimators converge even in these cases. Of particular interest are multivariate generalizations of these critical cases, where the boundary of the Weyl chamber is then considered instead of the origin. For our final result, we present the Hausdorff dimension of the times when a multivariate Bessel process hits the Weyl chamber's boundary.