#### Prof. Dr. Jeannette H. C. Woerner

Stochastics and Analysis

TU Dortmund University

##### Contact

- jwoerner[at]mathematik[dot]uni-dortmund[dot]de
- +49 0231 755 3055
- personal website

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**Limit theorems for Bessel and Dunkl processes of large dimensions and free convolutions**

Voit, M. and Woerner, J.H.C.*Stochastic Processes and their Applications*143 (2022)We study Bessel and Dunkl processes (Xt,k)t≥0 on RN with possibly multivariate coupling constants k≥0. These processes describe interacting particle systems of Calogero–Moser–Sutherland type with N particles. For the root systems AN−1 and BN these Bessel processes are related with β-Hermite and β-Laguerre ensembles. Moreover, for the frozen case k=∞, these processes degenerate to deterministic or pure jump processes. We use the generators for Bessel and Dunkl processes of types A and B and derive analogues of Wigner's semicircle and Marchenko–Pastur limit laws for N→∞ for the empirical distributions of the particles with arbitrary initial empirical distributions by using free convolutions. In particular, for Dunkl processes of type B new non-symmetric semicircle-type limit distributions on R appear. Our results imply that the form of the limiting measures is already completely determined by the frozen processes. Moreover, in the frozen cases, our approach leads to a new simple proof of the semicircle and Marchenko–Pastur limit laws for the empirical measures of the zeros of Hermite and Laguerre polynomials respectively. © 2021 Elsevier B.V.view abstract 10.1016/j.spa.2021.10.005 **The differential equations associated with Calogero-Moser-Sutherland particle models in the freezing regime**

Voit, M. and Woerner, J.H.C.*Hokkaido Mathematical Journal*51 (2022)Multivariate Bessel processes describe Calogero-Moser-Sutherland particle models. They depend on a root system and a multiplicity k. Recently, several stochastic limit theorems for k → ∞ were derived where the limits depend on the solutions of associated ODEs in these freezing regimes. In this paper we study these ODEs which are singular on the boundaries of their domains. We prove that for arbitrary initial conditions on the boundary, the ODEs have unique solutions in their domains for t > 0. © 2022, Hokkaido Mathematical Journal. All Rights Reserved.view abstract 10.14492/HOKMJ/2020-307 **Functional central limit theorems for multivariate Bessel processes in the freezing regime**

Voit, M. and Woerner, J.H.C.*Stochastic Analysis and Applications*39 (2021)Multivariate Bessel processes (Formula presented.) describe interacting particle systems of Calogero-Moser-Sutherland type and are related with β-Hermite and β-Laguerre ensembles. They depend on a root system and a multiplicity k. Recently, several limit theorems were derived for (Formula presented.) with fixed starting point. Moreover, the SDEs of (Formula presented.) were used to derive strong laws of large numbers for (Formula presented.) with starting points of the form (Formula presented.) with x in the interior of the Weyl chambers. Here we provide associated almost sure functional central limit theorems which are locally uniform in t. The Gaussian limit processes admit explicit representations in terms of the solutions of associated deterministic ODEs. © 2020 Taylor & Francis Group, LLC.view abstract 10.1080/07362994.2020.1786402 **Inference for fractional Ornstein-Uhlenbeck type processes with periodic mean in the non-ergodic case**

Shevchenko, R. and Woerner, J.H.C.*Stochastic Analysis and Applications*(2021)In the paper we consider the problem of estimating parameters entering the drift of a fractional Ornstein-Uhlenbeck type process in the non-ergodic case, when the underlying stochastic integral is of Young type. We consider the sampling scheme that the process is observed continuously on (Formula presented.) and (Formula presented.) For known Hurst parameter (Formula presented.) i.e. the long range dependent case, we construct a least-squares type estimator and establish strong consistency. Furthermore, we prove a second order limit theorem which provides asymptotic normality for the parameters of the periodic function with a rate depending on H and a non-central Cauchy limit result for the mean reverting parameter with exponential rate. For the special case that the periodicity parameter is the weight of a periodic function, which integrates to zero over the period, we can even improve the rate to (Formula presented.). © 2021 Taylor & Francis Group, LLC.view abstract 10.1080/07362994.2021.1942916 **Martingale estimation functions for Bessel processes**

Hufnagel, N. and Woerner, J.H.C.*Statistical Inference for Stochastic Processes*(2021)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. © 2021, The Author(s).view abstract 10.1007/s11203-021-09250-8 **Some Martingales Associated With Multivariate Bessel Processes**

Kornyik, M. and Voit, M. and Woerner, J.*Acta Mathematica Hungarica*163 (2021)We study Bessel processes on Weyl chambers of types A and Bon RN. Using elementary symmetric functions, we present several space-timeharmonicfunctions and thus martingales for these processes (Xt)t≥0which areindependent from one parameter of these processes. As a consequence, pt(y):=E(∏i=1N(y-Xti)) can be expressed via classical orthogonal polynomials. Suchformulas on characteristic polynomials admit interpretations in random matrixtheory where they are partially known by Diaconis, Forrester, and Gamburd. © 2020, Akadémiai Kiadó, Budapest, Hungary.view abstract 10.1007/s10474-020-01096-5 **Ordinal patterns in long-range dependent time series**

Betken, A. and Buchsteiner, J. and Dehling, H. and Münker, I. and Schnurr, A. and Woerner, J.H.C.*Scandinavian Journal of Statistics*(2020)We analyze the ordinal structure of long-range dependent time series. To this end, we use so called ordinal patterns which describe the relative position of consecutive data points. We provide two estimators for the probabilities of ordinal patterns and prove limit theorems in different settings, namely stationarity and (less restrictive) stationary increments. In the second setting, we encounter a Rosenblatt distribution in the limit. We prove more general limit theorems for functions with Hermite rank 1 and 2. We derive the limit distribution for an estimation of the Hurst parameter H if it is higher than 3/4. Thus, our theorems complement results for lower values of H which can be found in the literature. Finally, we provide some simulations that illustrate our theoretical results. © 2020 The Authors. Scandinavian Journal of Statistics published by John Wiley & Sons Ltd on behalf of The Board of the Foundation of the Scandinavian Journal of Statistics.view abstract 10.1111/sjos.12478 **Low-frequency estimation of continuous-time moving average Lévy processes**

Belomestny, D. and Panov, V. and Woerner, J.H.C.*Bernoulli*25 (2019)In this paper, we study the problem of statistical inference for a continuous-time moving average Lévy process of the form Z t = R K(t − s)dL s , t ∈ R, with a deterministic kernel K and a Lévy process L. Especially the estimation of the Lévy measure ν of L from low-frequency observations of the process Z is considered. We construct a consistent estimator, derive its convergence rates and illustrate its performance by a numerical example. On the mathematical level, we establish some new results on exponential mixing for continuous-time moving average Lévy processes. © 2019 ISI/BS.view abstract 10.3150/17-BEJ1008 **Estimating drift parameters in a fractional Ornstein Uhlenbeck process with periodic mean**

Dehling, H. and Franke, B. and Woerner, J.*Statistical Inference for Stochastic Processes*20 (2017)We construct a least squares estimator for the drift parameters of a fractional Ornstein Uhlenbeck process with periodic mean function and long range dependence. For this estimator we prove consistency and asymptotic normality. In contrast to the classical fractional Ornstein Uhlenbeck process without periodic mean function the rate of convergence is slower depending on the Hurst parameter H, namely (Formula presented.). © 2016 Springer Science+Business Media Dordrechtview abstract 10.1007/s11203-016-9136-2 **A mathematical analysis of the Gumbel test for jumps in stochastic volatility models**

Palmes, C. and Woerner, J.*Stochastic Analysis and Applications*34 (2016)This article gives an exhaustive mathematical analysis of the Gumbel test for additive jump components based on extreme value theory. The Gumbel test was first introduced by Lee and Mykland in 2008 from an economical point of view. They consider a continuous-time stochastic volatility model with a general continuous volatility process and observe it under a high-frequency sampling scheme. The test statistics based on the maximum of increments converges to the Gumbel distribution under the null hypothesis of no additive jump component and to infinity otherwise. Our article presents a moment method based technique that provides some deeper mathematical insights into the convergence and divergence case of the test statistics. In the non-jump case we are able to prove the convergence to the Gumbel distribution under greatly weak assumptions: The volatility process has to be merely pathwise Hölder continuous with an arbitrary random Hölder exponent and we have no restrictions concerning an additional drift term. Therefore, for example, we are allowing for long and short-range dependence. In the case of existing additive jumps, we give divergence results in a general semimartingale setting and investigate the speed of divergence depending on the jump activity. As a by-product of our analysis we also deduce an optimal pathwise estimator for the spot volatility process. Moreover, we provide a detailed simulation study that compares the power of the Gumbel test with the power of the jump test proposed by Barndorff–Nielsen and Shephard in 2006 for Hölder exponents close to zero. Finally, both tests are applied to a real dataset. © 2016 Taylor & Francis Group, LLC.view abstract 10.1080/07362994.2016.1182870 **The Gumbel test and jumps in the volatility process**

Palmes, C. and Woerner, J.*Statistical Inference for Stochastic Processes*19 (2016)In the framework of jump detection in stochastic volatility models the Gumbel test based on extreme value theory has recently been introduced. Compared to other jump tests it possesses the advantages that the direction and location of jumps may also be detected. Furthermore, compared to the Barndorff–Nielsen and Shephard test based on bipower variation the Gumbel test possesses a larger power. However, so far one assumption was that the volatility process is Hölder continuous, though there is empirical evidence for jumps in the volatility as well. In this paper we derive that the Gumbel test still works under the setting of finitely many jumps not exceeding a certain size. This maximal jump size depends on the relative sampling frequencies involved in the definition of the test statistics. Furthermore, we show that the given bound on the jump size is sharp and investigate the details of the phase transition at this critical bound. © 2015, Springer Science+Business Media Dordrecht.view abstract 10.1007/s11203-015-9127-8 **Statistical convergence of Markov experiments to diffusion limits**

Konakov, V. and Mammen, E. and Woerner, J.*Bernoulli*20 (2014)Assume that one observes the kth, 2kth,..., nkth value of a Markov chain X1,h,..., Xnk,h. That means we assume that a high frequency Markov chain runs in the background on a very fine time grid but that it is only observed on a coarser grid. This asymptotics reflects a set up occurring in the high frequency statistical analysis for financial data where diffusion approximations are used only for coarser time scales. In this paper, we show that under appropriate conditions the L1-distance between the joint distribution of the Markov chain and the distribution of the discretized diffusion limit converges to zero. The result implies that the LeCam deficiency distance between the statistical Markov experiment and its diffusion limit converges to zero. This result can be applied to Euler approximations for the joint distribution of diffusions observed at points Δ, 2Δ,..., nΔ. The joint distribution can be approximated by generating Euler approximations at the points Δk-1, 2Δk-1,..., nΔ. Our result implies that under our regularity conditions the Euler approximation is consistent for n → ∞ if nk-2 → 0. © 2014 ISI/BS.view abstract 10.3150/12-BEJ500 **A unifying approach to fractional lévy processes**

Engelke, S. and Woerner, J.*Stochastics and Dynamics*13 (2013)Starting from the moving average representation of fractional Brownian motion, there are two different approaches to constructing fractional Lévy processes in the literature. Applying L2-integration theory, one can keep the same moving average kernel and replace the driving Brownian motion by a pure jump Lévy process with finite second moments. Alternatively, in the framework of alpha-stable random measures, the Brownian motion is replaced by an alpha-stable Lévy process and the exponent in the kernel is reparametrized by H - 1/α. We now provide a unified approach taking kernels of the form $a((t - s)+ γ-(-s) + γ)+b((t - s)--γ-(-s) --γ)$, where γ can be chosen according to the existing moments and the Blumenthal-Getoor index of the underlying Lévy process. These processes may exhibit both long and short range dependence. In addition we will examine further properties of the processes, e.g., regularity of the sample paths and the semimartingale property. © 2013 World Scientific Publishing Company.view abstract 10.1142/S0219493712500177

#### continuous processes

#### fractional processes

#### high-frequency data

#### stochastics