16. DoktorandInnentreffen der Stochastik

Virtuelle Konferenz

Programm. Das 16. DoktorandInnentreffen der Stochastik findet am 29. und 30. Juli 2021 statt. Neben vielen spannenden Vorträgen der Doktorandinnen und Doktoranden hat das Programm auch außerhalb der Mathematik einiges zu bieten: Durch die Konferenzplattform gather.town entsteht eine soziale Atmosphäre die zu persönlichen Gesprächen zwischen den Vorträgen einlädt. Am Donnerstagabend werden wir ein virtuelles Get-Together organisieren. Dabei wird es online Brettspiele, wissenschaftliche speed-dates und viel Raum zum Austausch geben.

Die Vortragsdauer der Fachvorträge beträgt 20 Minuten. Anschließend ist Zeit für die Technik sowie Fragen eingeplant.

Zeitplan.

Donnerstag, 29.07.2021
13:30-14:00		Begrüßung
14:00-14:25	Heide Langhammer	A Large-Deviations Approach to the Phase Transition in Inhomogeneous Random Graphs
14:25-14:50	Daniel Willhalm	Upper Large Deviations for Power-Weighted Edge Lengths in Spatial Random Networks
14:50-15:15	Yanjia Bai	Refined Large Deviation Principle for Branching Brownian Motion Conditioned to Have a Low Maximum
15:15-15:30		Pause
15:30-15:55	Boris Aleksandrov	Goodness-of-Fit Tests for Poisson Count Time Series Based on the Stein-Chen Identity
15:55-16:20	Sebastian Neblung	Sliding and Disjoint Blocks Estimators for Extremes of Time Series
16:20-16:45	Maximilian Steffen	High-Dimensional Multi-Index Regression Models
16:45-		Get-Together

Freitag, 30.07.2021
10:00-10:25	Dennis Malcherczyk	Sign Depths: Asymptotics, Computation and Open Problems
10:25-10:50	Celeste Mayer	Whittle Estimation for Equidistantly Observed Lévy-Driven MCARMA Processes
10:50-11:00		Pause
11:00-11:25	Stefan Wagner	Stochastic Self-Duality for Consistent Particle Systems on General State Spaces
11:25-11:50	Glib Verovkin	Stationary Limits of Renewal Shot Noise Processes
11:50-14:00		Mittagspause
14:00-14:25	Emanuel Rapsch	A Game of Change under Uncertainty. Disruption or Regulation?
14:25-14:50	Julia Steinmetz	Asymptotic Theory for Mack's Model
14:50-15:15	Simon Breneis	Markovian Approximations of Stochastic Volatility Models
15:15-15:40		Verabschiedung

A Large-Deviations Approach to the Phase Transition in Inhomogeneous Random Graphs

Speaker: Heide Langhammer

Abstract: In 1960, P. Erdős and A. Rényi discovered that a certain family of random graphs exhibits a phase transition that marks the emergence of a giant (connected) component whose size is of the same order as the total number of vertices. We are studying the same phenomenon, but in a more general model. In an inhomogeneous random graph, \(N\) vertices are equipped with certain types and random edges are set independently of each other with a probability that depends only on the types of the incident vertices. The events of high probability in this model for large \(N\) were studied by Bollobás, Janson and Riordan in 2006 with the help of branching processes. We pursue a different approach and derive a large deviation principle (LDP) for the decomposition of this graph into its components. The phase transition can be recovered by a thorough analysis of the rate functional, but beyond that we can use the LDP to describe how certain unlikely events are typically realized in this model. Apart from that, we hope to apply a similar methodology to study spatial coagulation processes that describe successive merging events of a large number of particles in a compact space. (Based on joint work with Luisa Andreis, Wolfgang König and Robert Patterson.)

Upper Large Deviations for Power-Weighted Edge Lengths in Spatial Random Networks

Speaker: Daniel Willhalm

Abstract: We study large-volume asymptotics of the sum of power-weighted edge lengths \(\sum_{e \in E}|e|^\alpha\) in a spatial random network constructed on a Poisson point process in a bounded sampling window and for \(\alpha\) larger than the dimension \(d\). We develop a framework such that, for suitable graphs, we can study the upper large deviations for this functional, i.e., the probability that it exceeds its expectation by a factor of \((1+r)\). A graph fits in the framework under some monotonicity and continuity conditions of the score function, representing the power-weighted edge lengths of one node. The rate function is given as an optimization problem and conditioned on this unlikely event, statements are possible about how the graph is structured. In particular, there is a condensation phenomenon, meaning that the excess is caused by a negligible portion of Poisson points. The upper large deviations and the statements about condensation are applied to the \(k\)-nearest neighbor graph and the relative neighborhood graph.

Refined Large Deviation Principle for Branching Brownian Motion Conditioned to Have a Low Maximum

Speaker: Yanjia Bai

Abstract: Conditioning a branching Brownian motion to have an atypically low maximum leads to a suppression of the branching mechanism. In this note, we consider a branching Brownian motion conditioned to have a maximum below \(\sqrt{2} \alpha t (\alpha < 1)\). We study the precise effects of an early/late first branching time and a low/high first branching location under this condition. We do so by imposing additional constraints on the first branching time and location. We obtain large deviation estimates, as well as the optimal first branching time and location given the additional constraints.

Goodness-of-Fit Tests for Poisson Count Time Series Based on the Stein-Chen Identity

Speaker: Boris Aleksandrov

Abstract: The Poisson distribution is characterized by the famous Stein-Chen identity. To test the null hypothesis of a Poisson marginal distribution, test statistics based on the Stein-Chen identity are proposed. For a wide class of Poisson count time series, the asymptotic distribution of different types of Stein-Chen statistics is derived.
The performance of the tests is analyzed with simulations.

Sliding and Disjoint Blocks Estimators for Extremes of Time Series

Speaker: Sebastian Neblung

Abstract: Extreme value estimators based on the observations of a time series are often constructed from blockwise defined statistics. These blocks can be specified as disjoint blocks \((X_t)_{(i-1)s+1\leq t\leq is}\), \(1\leq i\leq \lfloor n/s\rfloor\), of length \(s\), or alternatively as sliding blocks \((X_t)_{i\leq t\leq i+s-1}\), \(1\leq i\leq n-s+1\). Yet another approach are offered by runs estimators, which can be interpreted as a special type of sliding blocks estimators. While the asymptotic behavior of disjoint block statistics can be analyzed with known tools from the literature, for sliding block estimators corresponding tools are missing. We introduce a new unified framework for the asymptotic analysis of all three types of estimators. Moreover, we use this framework to show that the sliding blocks statistic leads to a smaller asymptotic variance than that from disjoint blocks.
We will discuss this unified, peak-over-threshold framework by the example of the extremal index, which is the reciprocal of the mean cluster length of extremes. The asymptotics of all three type of estimators for the extremal index will be established under the same conditions. In particular we will see that all three estimators have the same asymptotic variance.

High-Dimensional Multi-Index Regression Models

Speaker: Maximilian Steffen

Abstract: The multi-index model with sparse dimension reduction matrix is a popular approach to circumvent the curse of dimensionality in a high-dimensional regression setting. Building on the single-index analysis by Alquier and Biau (2013), we develop a PAC-Bayesian estimation method for a possibly miss-specified multi-index model with unknown active dimension and an orthogonal dimension reduction matrix. Our main result is a non-asymptotic oracle inequality, which shows that the estimation method adapts to the active dimension of the model, the sparsity of the dimension reduction matrix and the regularity of the link function. Under a Sobolev regularity assumption on the link function the estimator achieves the usual minimax rate of convergence and no additional price is paid for the unknown active dimension.
This talk is based on joint work with Mathias Trabs from Universität Hamburg.

Alquier, P. and Biau, G. (2013). Sparse single-index model. Journal of Machine Learning Research, 14.(1), 243–280.

Sign Depths: Asymptotics, Computation and Open Problems

Speaker: Dennis Malcherczyk

Abstract: The sign depths are a class of statistics to evaluate suggested parameters of regression models by considering the sign changes of the residuals. The depth of a model parameter denotes here how well this parameter fits for the data (similar to the concept of a likelihood function). We only assume that the errors of the model are independent with a continuous distribution, i.e., neither assumptions on the moments, homoscedasticity nor identical distributions are required.
Since these statistics are distribution free under the true parameter, statistical tests can be constructed for testing suggested model parameters. For larger sample sizes, the asymptotic distribution can be derived with Donsker's invariance principle (Functional Central Limit Theorem) by representing the sign depths as a functional of a symmetric random walk [2].
For convenient statistical applications, the computational costs of this method should be reduced as efficiently as possible. A naive computation by definition would have an undesirable time complexity of \(\Theta(n^K)\) where \(n\) denotes the sample size and \(K \geq 2\) is a freely choosable hyper-parameter in this method. By considering the block structures of the residuals' signs, a computation in linear time is possible for arbitrary \(K\). This computation based on the block structures yields some interesting theoretical statements. One intuitive assertion, which we have not proven completely yet for all \(K\), will be presented in this talk as a conjecture [1].

[1] Leckey, K., Malcherczyk, D., and Müller, C.H. (2020). Powerful generalized sign tests based on sign depth. Discussion papers SFB 823. URL: https://www.statistik.tu-dortmund.de/2630.html
[2] Malcherczyk, D., Leckey, K., and Müller, C.H. (2021). K-sign depth: From asymptotics to efficient implementation. Journal of Statistical Planning and Inference 215, 344-355.

Whittle Estimation for Equidistantly Observed Lévy-Driven MCARMA Processes

Speaker: Celeste Mayer

Abstract: In many financial, technical and physical applications, we are confronted with discretely observed data which results from a continuous process. In this talk we estimate the parameters of an equidistantly sampled multivariate continuous-time ARMA (MCARMA) process. The driving process of the MCARMA process is a Lévy process which allows exible margins in contrast to Brownian motion driven models which only have Gaussian margins. In this talk the Lévy process is supposed to have finite second moments. While Schlemm and Stelzer [1] studied the quasi-maximum likelihood estimator, we investigate the Whittle estimator. In particular, we present its asymptotic properties. A simulation study illustrates the theoretical results and enables a comparison of the Whittle estimator and the quasi maximum likelihood estimator.
joint work with: Vicky Fasen-Hartmann

[1] E. Schlemm, R. Stelzer: Quasi maximum likelihood estimation for strongly mixing state space models and multivariate Lévy-driven CARMA processes, Electronic Journal of Statistics (2012).

Stochastic Self-Duality for Consistent Particle Systems on General State Spaces

Speaker: Stefan Wagner

Abstract: In this talk we consider dualities of Markov processes. It is well-known that self-dualities in terms of orthogonal polynomials exist for exclusion processes, inclusion processes and independent random walkers. These processes represent particles evolving jumps on a discrete set. We try to generalize these self-dualities to consistent processes representing particles on a very general space, e.g. a Borel space. This leads to so-called intertwining relations. We make use of the theory of point processes, the theory of Lie algebras and methods of functional analysis.

Stationary Limits of Renewal Shot Noise Processes

Speaker: Glib Verovkin

Abstract: We investigate an asymptotic behaviour of centered renewal shot noise processes in the case, when second moment of an increment of an underlying random walk exists and the response function is square-integrable. We prove the convergence of finite-dimensional distributions of announced processes to a stationary limit process. We investigate the properties of the limit process and study its covariance structure. The talk is based on the paper [1].

[1] Marynych, A. and Verovkin, G. (2020). Stationary limits of renewal shot noise processes. Theory of Probability and Mathematical Statistics 101, pp. 67-83.

A Game of Change under Uncertainty. Disruption or Regulation?

Speaker: Emanuel Rapsch

Abstract: In the economics of the energy transition, an important question is which policy a regulator should adopt in order to stimulate irreversible investment of firms in a market economy. Typically, the firms face a trend (e.g. technological, consumer preference-related) whose future realisation is exposed to substantial uncertainty. This uncertainty, however, is not sufficiently observable from the regulator's perspective.

Following real options theory, we propose a stylised model for quantifying that uncertainty and, hence, its impact on both the firms' timing decisions and the regulator's policy. Mathematically, we interpret this objective in terms of a stopping game problem of Langrangian type for a Wright-Fisher diffusion as common noise. Feedback equilibria can be described in terms of a convenient coupled system of differential equations. Equilibria may be manifold, and hence the ability of the regulator to influence the game's output depends on the situation.

This can be illustrated via concrete examples arising in markets facing the energy transition, for instance the automotive industry. The model captures key features related to the strategic problem under uncertainty. However, some effects can of course not be addressed by such a simple model, and thus a short discussion of its possible extensions seems appropriate. At the expense of decreasing transparency, impulse controls and intrinsic noises could be included, calling for mean-field game approximation.

This project is based on a collaboration with my doctoral supervisor Christoph Belak.

Asymptotic Theory for Mack's Model

Speaker: Julia Steinmetz

Abstract: The distribution-free chain ladder reserving model by Mack (1993) belongs to the most popular approaches in non-life insurance mathematics. It was originally proposed to determine the first two moments of the reserve distribution. Together with a normal approximations, it is often applied to conduct statistical inference including the estimation of large quantiles of the reserve and determination of the reserve risk. However, for Mack's model, the literature lacks a rigorous justification of such a normal approximation for the reserve.

We propose a suitable stochastic framework which allows to derive meaningful asymptotic theory for Mack's model. For increasing number of accident years, we establish central limit theorems for the commonly used estimators in Mack's model. In particular, these results enable us to derive the limiting distribution of the reserve risk. First, it is split in two random parts that carry the process uncertainty and the estimation uncertainty, respectively. Surprisingly, by deriving their joint limiting distributions, we prove that the limiting distribution of the reserve risk will be usually non-Gaussian. This main result casts the common practice to use a normal approximation for the reserve in Mack's model into doubt. We illustrate our findings by simulations and illustrate that the limiting distributions of the reserve risk might deviate substantially from a Gaussian distribution.

Markovian Approximations of Stochastic Volatility Models

Speaker: Simon Breneis

Abstract: We consider rough stochastic volatility models where the variance process satisfies a stochastic Volterra equation with the fractional kernel, as in the rough Bergomi and the rough Heston model. In particular, the variance process is therefore not a Markov process or semimartingale, and has quite low Hölder-regularity. In practice, simulating such rough processes thus often results in high computational cost. To remedy this, we study approximations of stochastic Volterra equations using an \(N\)-dimensional ordinary stochastic differential equation. If the coefficients of the stochastic Volterra equation are Lipschitz, we show that these approximations converge strongly with superpolynomial rate in \(N\).

  

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