# Lehrstuhl IV Stochastik und Analysis

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Item Limit theorems and statistical inference for Bessel and Dunkl processes(2022) Hufnagel, Nicole; Woerner, Jeannette; Voit, MichaelShow more 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.Show more Item Martingale estimation functions for Bessel processes(2021-08-04) Woerner, Jeannette; Hufnagel, NicoleShow more In this paper we derive martingale estimating functions for the dimensionality parameter of a Bessel process based on the eigenfunctions of the diffusion operator. Since a Bessel process is non-ergodic and the theory of martingale estimating functions is developed for ergodic diffusions, we use the space-time transformation of the Bessel process and formulate our results for a modified Bessel process. We deduce consistency, asymptotic normality and discuss optimality. It turns out that the martingale estimating function based of the first eigenfunction of the modified Bessel process coincides with the linear martingale estimating function for the Cox Ingersoll Ross process. Furthermore, our results may also be applied to estimating the multiplicity parameter of a one-dimensional Dunkl process and some related polynomial processes.Show more Item Estimation of stopping times for stopped self-similar random processes(2021-03-01) Schulmann, ViktorShow more Let X=(Xt)t≥0 be a known process and T an unknown random time independent of X. Our goal is to derive the distribution of T based on an iid sample of XT. Belomestny and Schoenmakers (Stoch Process Appl 126(7):2092–2122, 2015) propose a solution based the Mellin transform in case where X is a Brownian motion. Applying their technique we construct a non-parametric estimator for the density of T for a self-similar one-dimensional process X. We calculate the minimax convergence rate of our estimator in some examples with a particular focus on Bessel processes where we also show asymptotic normality.Show more Item Limit theorems and statistical inference for solutions of some stochastic (partial) differential equations(2019) Shevchenko, Radomyra; Woerner, Jeannette; Reiß, MarkusShow more The starting point for the thesis is an Ornstein-Uhlenbeck type stochastic differential equation dXt=(L(t)-cXt)dt+dBt with a real number value at time zero. The driving process B is a fractional Brownian motion with Hurst parameter H between 0.5 and 1, L is a bounded periodic deterministic function and c is a real valued parameter. In the thesis the task of estimation of L and c is solved when the solution process X is observed continuously. In the first part L is considered to be a linear combination of known functions with the same period, and the vector of the coefficients together with the parameter c is estimated using the least squares method for c<0, i.e. in the non-ergodic setting. We show strong consistency of the estimator as well as its asymptotic normality in the first p components. For the last component a noncentral limit theorem is proved. In the second part we consider the ergodic setting, i.e. c>0, and construct a nonparametric estimator for the function L. The idea for construction is to decompose L with respect to a known orthonormal basis and to estimate coefficients of a finite number of summands, letting this number tend to infinity afterwards. For this estimator L2 consistency is shown and its rate of convergence is calculated. The next part is dealing with a similar equation, this time driven by a Rosenblatt process, a non-Gaussian process with the covariance structure of a fractional Brownian motion with Hurst parameter H between 0.5 and 1. We consider again the setting in which L is a linear combination of known functions and c is positive, i.e. the ergodic case. We construct three estimators for the vector of parameters and study their first and second order asymptotics. The last part of the thesis deals with a different problem. There the stochastic wave equation is considered driven by a random field that is white in space and fractional in time with Hurst parameter H between 0.5 and 1. The objects we investigate are so-called empirical power variations of the solution, i.e. averages over the weighted differences of the solution process raised to some integer powers, where the weights fulfil certain conditions. We show central and noncentral limit theorems for these averages, determine in some cases the third order asymptotics in terms of the Wasserstein distance and use these results to define strongly consistent and asymptotically normal estimators for H.Show more Item Limit theorems for random walks on non-compact Grassmann manifolds with growing dimensions(2019) Artykov, Merdan; Voit, Michael; Veselić, IvanShow more