14. Doktorand*innentreffenStochastik

01.-03. August 2018[v1.2.1 – 31/07/18]

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Contents

Timetable ii

Program 1Wednesday 01. August . . . . . . . . . . . . . . . . . . . . 2Thursday 02. August . . . . . . . . . . . . . . . . . . . . . 4Friday 03. August . . . . . . . . . . . . . . . . . . . . . . . 6

Abstracts 9

Exhibition: Women of Mathematics 37

Further information 41Weblinks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Participants 43

Map 46

iii

Program

Sessions

The main part of the program consists of 2× 6 sessions of talks of20-25 minutes and one plenary talk by Alessandra Caraceni.

Important: If you are giving a talk, please make sure to provideyour slides before the session begins. Each room is equipped witha beamer, laptop, presenter and blackboards.

Special program

Wednesday afternoon we have prepared a special program fo-cused on women in mathematics. Despite being focused on women,this part of the program is aimed at at all particitpants.

There will be two plenary talks by Dr. Tatjana Chudjakow fromd-fine and by Dr. Caraceni Alessandra from the University ofBath. Both speakers will share their experiences with a career inthe industry and in academia respectively.

The program is concluded in the evening with a reception at theopening of the photo exhibition Women of Mathematics throughoutEurope. A gallery of portraits.

Social events

On Thursday we have planned a tour to the UNESCO world her-itage site Zeche Zollverein. After the tours we will meet at therestaurant die kokerei close to Zeche Zollverein for a conferencedinner.

1

Wednesday 01. August

11:00-12:00 Registration and SnacksRoom WSC-S-U-3.02

12:00-12:30 OpeningRoom WSC-S-U-4.02

Optimal Control Statistical InferenceChair Marvin Kettner Chair Sebastian KerstingRoom WSC-S-U-3.03 Room WSC-S-U-3.01

12:30-13:00 Sascha Pascal Nolte 27 Nick Kloodt 19Robust optimal stopping with-out time-consistency

Testing in Transformation Mod-els

13:00-13:30 Tobias Huebner 15 Viktor Schulmann 33Solving stopping-problems withan empirical dual optimizationapproach

Inference of Stopping Timeswith Application to ParticleLifetime Estimation

13:30-14:00 Berenice Anne Neumann 26 Lubna Amro 10Solution techniques for meanfield games with finite state andaction spaces

Multiplication-CombinationTests for Incomplete PairedData

14:00-14:30 CoffeeRoom WSC-S-U-3.02

2

Problems in Probability StatisticsChair Roland Meizis Chair Viktor SchulmannRoom WSC-S-U-3.03 Room WSC-S-U-3.01

14:30-15:00 Tobias Sohr 34 Kira Alhorn 9How to Break Impulse ControlProblems Down into SomethingEasier

Optimal designs for frequentistmodel averaging

15:00-15:30 Leon Ramzews 29 Mirko Alexander Jakubzik17

Unlabelled Set Partitions Applications of a minimum dis-tance estimator for specific self-exciting point processes

15:30-16:00 Christina Zou 36 Matthias Reuber 30The construction of Root’s solu-tion to Skorokhod embedding

Modellierung der Ab-hängigkeitsstruktur stündlicherStrahlungsdaten mithilfe vonCopula basierten Zeitreihen-modellen

16:00-16:30 CoffeeRoom WSC-S-U-3.02

Special Program: Women in MathRoom WSC-S-U-4.02

16:30-17:30 Tatjana Chudjakow Plenary Talk17:30-18:00 Coffee WSC-S-U-3.0218:00-19:00 Alessandra Caraceni Plenary Talk19:00-19:30 Women in Math Exhibition Opening19:30 Women in Math Exhibition Snacks and Drinks

3

Thursday 02. August

Stochastic Analysis Calculus of Stocastic Pro-cesses

Chair Anselm Hudde Chair Nicole HufnagelRoom WSC-S-U-3.03 Room WSC-S-U-3.01

9:00-9:30 Robert Link 23 Moritz Otto 28Existence and uniqueness of so-lutions of infinite dimensionalKolmogorov equations

Functional Poisson approxima-tion of thinned Poisson pro-cesses

9:30-10:00 Yannic Bröker 11 Kevin Musielak 25Central limit theorem and lo-calization for the stochastic heatequation in d ≥ 3

Studying Markov Processesthrough its Symbol

10:00-10:30 Lars Schmitz 32 Zoubir Dahmani 13About the front of the inhomoge-neous Fisher-KPP equation andits linearisation, the parabolicAnderson model

Mixed operators of fractionalcalculus and applications

10:30-11:00 CoffeeRoom WSC-S-U-3.02

Plenary SessionRoom WSC-S-U-4.02

11:00-12:00 Alessandra Caraceni 13The geometry of large planar maps: about random

quadrangulations of the half-plane

12:00-14:00Lunch

4

Probability on Graphs Finance and InsuranceChair Leonid Kolesnikov Chair Berenice Anne Neu-

mannRoom WSC-S-U-3.03 Room WSC-S-U-3.01

14:00-14:30 Markus Schepers 31 Gregor Leimcke 22The local clustering coefficientin hyperbolic random graphs

Optimal control in insurancemathematics under partial in-formation

14:30-15:00 Johannes Ehlert 14 Johannes Wiesel 35Existence of infinite randomloops on trees

A unified framework to robustmodelling of financial marketsin discrete time

15:00-15:30 Kilian Matzke 24 Felix-Benedikt Liebrich 22The Triangle Condition for theRandom Connection Model inHigh Dimensions

Efficient allocations under prob-abilistic sophistication: a unify-ing approach

15:30-16:00 Carina Betken 10Sedentary Random Waypoint Cancelled

17:30-19:30Tour Zollverein

from 19:30 Dinner

5

Friday 03. August

Probability Theory in Ap-plications

Stochastic Processes

Chair Johannes Wiesel Chair Carina BetkenRoom WSC-S-U-3.03 Room WSC-S-U-3.01

9:00-9:30 Daniel Pieper 29 Nicole Hufnagel 16Altruistic defense traits instructured populations: Many-demes limit in the sparseregime

Girsanov’s Theorem for Besselprocesses

9:30-10:00 Alexander Wapenhans 34 Marvin Kettner 18The White Knight Model - Prop-agation of Malware on a D2DNetwork

Persistence Probabilities of Au-toregressive Processes

10:00-10:30 Sebastian Kersting 18 Jan-Erik Lübbers 24Estimation of an improved sur-rogate model in uncertaintyquantification by neural net-works

Displacement of biased randomwalk in a one-dimensional per-colation model

10:30-11:00 CoffeeRoom WSC-S-U-3.02

6

Limit Theorems Statistical MechanicsChair Yannic Bröker Chair Lars SchmitzRoom WSC-S-U-3.03 Room WSC-S-U-3.01

11:00-11:30 Geronimo Rojas 31 Florian Henning 14On the rate of convergence ofsymmetric Feller processes oncompact metric spaces via occu-pation time

Gibbs-non-Gibbs transition inthe fuzzy Potts model with aKac-type interaction: Closingthe Ising gap

11:30-12:00 Lukas Knichel 20 Leonid Kolesnikov 21The Rate of Convergence to aGamma Target on the WienerSpace

Critical 1-arm exponent for theIsing model on regular trees

12:00-12:30 Stephan Bussmann 12 Daniel Meißner 25Central limit theorems in aboolean model with dependen-cies

Spin-flip dynamics for theCurie-Weiss-Potts model

12:30-13:00 ClosingRoom WSC-S-U-4.02

7

Abstracts

Optimal designs for frequentist model averaging WSC-S-U-3.01

Wed 14:30Kira Alhorn

Technische Universität Dortmund

We consider the problem of designing experiments for the estima-tion of a target in regression analysis if there is uncertainty aboutthe parametric form of the regression function. A new optimal-ity criterion is proposed, which minimizes the asymptotic meansquared error of the frequentist model averaging estimate by thechoice of an experimental design. Necessary conditions for theoptimal solution of a locally and Bayesian optimal design prob-lem are established. The results are ilustrated in several examplesand it is demonstrated that Bayesian optimal designs can yield areduction of the mean squared error of the model averaging esti-mator up to 45%.

9

Multiplication-Combination Tests for IncompletePaired DataWSC-S-U-3.01

Wed 13:30 Lubna AmroUniversität Ulm

We consider statistical procedures for hypothesis testing of realvalued functionals of matched pairs with missing values. In or-der to improve the accuracy of existing methods, we propose anovel multiplication combination procedure. Dividing the ob-served data into dependent (completely observed) pairs and inde-pendent (incompletely observed) components, it is based on com-bining separate results of adequate tests for the two sub datasets.Our methods can be applied for parametric as well as semi- andnonparametric models and make efficient use of all available data.In particular, the approaches are flexible and can be used to testdifferent hypotheses in various models of interest. This is exem-plified by a detailed study of mean- as well as rank-based ap-proaches. Extensive simulations show that the proposed proce-dures are more accurate than existing competitors. A real data setillustrates the application of the methods.

Sedentary Random WaypointWSC-S-U-3.03

Thu 15:30 Carina BetkenUniversität Osnabrück

We study three fundamental problems in a probabilistic model forlarge random networks. More specifically we look at an extensionof the random waypoint mobility model, in which we assign a siteto each participant around which the participant’s movements arecentered and to which the walker returns now and then. We areinterested in the following three questions: detection (the time un-til some target point gets in contact with a node of the network),

10

coverage (the time until a certain region has been completely dis-covered by the nodes of the network) and percolation (the timeuntil a given node first belongs to the infinite component of thenetwork).

Central limit theorem and localization for thestochastic heat equation in d ≥ 3 WSC-S-U-3.03

Thu 9:30Yannic BrökerUniversität Münster

We consider the spatially smoothened stochastic heat equation(SHE)

duε,t =12

∆uε,tdt + βεd−2

2 uε,tdBε,t

in d ≥ 3. Here B is a space-time white noise and the parame-ter β, known as the inverse temperature, captures the strength ofthe noise. The Feynman-Kac formula relates the solution uε,t tothe partition function of the Brownian directed polymer in a ran-dom environment. In a recent work by Mukherjee, Shamov andZeitouni, using techniques from Gaussian multiplicative chaos, itwas shown that, for β small enough, the solution uε,t of the SHEconverges as ε → 0 in distribution to a strictly positive randomvariable, while for β large, these converge in probability to zero.We investigate the behavior of the actual polymer measure, showthat, for β small, the endpoint of the polymer satisfies an almostsure (quenched) central limit theorem, while for β large, the end-point is localized in random regions of the space. The methodsof our proof are based on the translation-invariant compactifica-tion theory developed by Mukherjee and Varadhan. This is a jointwork with my supervisor Chiranjib Mukherjee.

11

Central limit theorems in a boolean model withdependenciesWSC-S-U-3.03

Fri 12:00Stephan Bussmann

Universität Osnabrück

A boolean model usually consists of random compact sets as-signed to Poisson points in Rn. In existing literature the distribu-tion of the sets is independent from the distribution of the points,which is an unrealistic assumption in many applications.We try to remedy that by introducing a random map generator.The sets are then chosen according to the positions of the pointson the generated map.We build on the work of [1] and [2] and try to apply their resultsin our setting.

References

[1] Günter Last and Mathew Penrose. Lectures on the Pois-son Process. Cambridge University Press, 2017.

[2] Raphaël Lachièze-Rey and Giovanni Peccati. NewBerry-Esseen bounds for functionals of binomial point pro-cesses . The Annals of Applied Probability, 2017.

12

The geometry of large planar maps: about randomquadrangulations of the half-plane Plenary

WSC-S-U-4.02

11:00Alessandra CaraceniUniversity of Bath

The investigation of large random combinatorial objects has spawneda very active research field centred around local and scaling limitsfor different classes of random planar maps. After an introductionwhich will get us acquainted with some fundamental and fasci-nating objects in the field, such as the Continuum Random Treeand the Brownian Map, we shall focus on the Uniform InfiniteQuadrangulation of the Half-Plane (UIHPQ), which arises as alocal limit of random quadrangulations with a boundary whosesize and perimeter are sent to infinity. Using tools from both com-binatorics and stochastic analysis, a wide range of results can beobtained about different features of the UIHPQ’s geometry, someof which have interesting implications for an annealed model ofself-avoiding walks on large random quadrangulations.

Mixed operators of fractional calculus andapplications WSC-S-U-3.01

Thu 10:00Zoubir Dahmani

University of Mostaganem

In this talk, we establish new mixed integral operators that gener-alise some recent results on fractional calculus. We prove some oftheir properties, then we discuss some of their applications.

13

Existence of infinite random loops on treesWSC-S-U-3.03

Thu 14:30 Johannes EhlertTechnische Universität Darmstadt

We consider a model of random loops on graphs that generalizesthe Random Interchange/Random Stirring Model and has con-nections to the Quantum Heisenberg Ferro- and Antiferromagnetas well as to the Quantum XY-Model.

An interesting question is, whether or not there is an infiniteloop with positive probability. To resolve this problem we intendto find sufficient conditions depending only on the underlyingparameters. Since on general graphs – and even on Zd – this isquite challenging, we restrict ourselves to trees. The talk will giveinsights on how we can exploit the additional properties of loopsin this situation.

Gibbs-non-Gibbs transition in the fuzzy Pottsmodel with a Kac-type interaction: Closing the Ising

gapWSC-S-U-3.01

Fri 11:00 Florian HenningRuhr-Universität Bochum

We complete the investigation of the Gibbs properties (with re-spect to the notion of sequential Gibbsianness) of the fuzzy Pottsmodel on the d-dimensional torus with Kac-type interaction whichwas started by B. Jahnel and C. Külske. As our main result, weextend the previous sharpness result of mean-field bounds forthe case of all fuzzy classes being of size at least three to cover allpossible cases of fuzzy transformations, allowing also for the oc-currence of Ising classes. The closing of this previously left openIsing-gap involves an analytical argument showing uniqueness

14

of minimizing profiles for certain non-homogeneous conditionalvariational problems.

Based on joint work with Richard Kraaij and Christof Külske

Solving stopping-problems with an empirical dualoptimization approach WSC-S-U-3.03

Wed 13:00Tobias HuebnerUniversität Duisburg-Essen

In this talk we give a new method to solve the standard-stoppingproblem

supτ∈T

E [Zτ ]

numerically.The starting point is the well-known dual representation

supτ∈T

E [Zτ ] = infM∈A

E

[sup

t∈[0,T](Zt −Mt)

]= sup

t∈[0,T](Zt −M∗t ) a.s.

Here, A denotes the set of all martingales starting in zero. M∗

stands for the martingale part of the Doob-Meyer-decompositionof the Snell-Envelope of Zt.The task is to find a proper approximation of M∗.One idea is to select a subclass Aψ := Mψ|ψ ∈ Ψ ⊂ A ofparameterized martingales for a proper estimation, based on amonte-carlo-type-method.Here we present a refinement of the so called empirical dual opti-mization approach introduced in Belomestny [AAP 2013].We give convergence rates depending on the space of parameter-functions Ψ and its entropy-numbers. Results of empirical-process-theory are used to calculate these convergence-rates.Furthermore we give an outlook how to extend this method to

15

non-standard stopping-problems, where the ordinary conditionalexpectation is replaced by some non-linear functional. We are in-terested in problems where we cannot invoke the established dy-namic programming principle to solve standard stopping-problems.

Girsanov’s Theorem for Bessel processesWSC-S-U-3.01

Fri 9:00Nicole Hufnagel

Technische Universität Dortmund

Girsanov’s Theorem for the Brownian motion plays an importantrole within the financial mathematics and for the construction ofestimators for diffusions. This raises the question of transferingthis result to Dunkl processes, which are generalizations of theBrownian motion. We can tackle this question by considering theabsolute value of a Dunkl process, a continuous Bessel process.In particular, a Dunkl process is the unique martingale, whoseabsolute value is a Bessel process, which is helpful for statementsabout the original process.

16

Applications of a minimum distance estimator forspecific self-exciting point processes WSC-S-U-3.01

Wed 15:00Mirko Alexander Jakubzik

Technische Universität Dortmund

In this contribution based on a paper by Kopperschmidt and Stutepublished in 2013 we study minimum distance estimation forself-exciting point processes. In a semi-parametric modelling ap-proach we invoke the compensator of a counting process givenby the Doob-Meyer decomposition to predict its qualitative be-haviour. The introduced minimum distance estimator yields con-sistent and asymptotically gaussian distributed estimates for theparametric part of the predictor.

While the main results concerning these asymptotic propertiesare due to Kopperschmidt and Stute, we augment the range ofapplications by discussing models convenient for practical usage.This approach encompasses load-sharing systems commonly em-ployed in civil engineering that also allow for damage accumu-lation. We derive formulae to explicitly compute the covariancematrix of the minimum distance estimator and establish the corre-sponding confidence sets in the case of specific self-exciting pointprocesses, e.g. the class of shifted birth processes. Furthermorewe provide simulation based methods to obtain approximate con-fidence sets whenever the exact covariance matrix can not be de-termined. In conclusion, prediction intervalls for the underlyingcounting process are deduced from these confidence sets and de-bated in view of simulation studies as well as real data obtainedthrough a civil engineering experiment that took place at TU Dort-mund University.

17

Estimation of an improved surrogate model inuncertainty quantification by neural networksWSC-S-U-3.03

Fri 10:00 Sebastian KerstingTechnische Universität Darmstadt

Quantification of uncertainty of a technical system is often basedon a surrogate model of a corresponding simulation model. Inany application the simulation model will not describe the realityperfectly, and consequently the surrogate model will be imper-fect. We propose a method to combine observed data from thetechnical system with simulated data from the imperfect simu-lation model in order to estimate an improved surrogate modelconsisting of multi-layer feedforward neural networks, and showthat under suitable assumptions this estimate is able to circum-vent the curse of dimensionality. Based on this improved surro-gate model we show a rate of convergence result for density esti-mates. The practical usefulness of the newly proposed estimatesis demonstrated by using them to predict the uncertainty of a lat-eral vibration attenuation system with piezo–elastic supports.

Persistence Probabilities of AutoregressiveProcessesWSC-S-U-3.01

Fri 9:30 Marvin KettnerTechnische Universität Darmstadt

In this talk we consider a certain class of autoregressive pro-cesses and study the probability that such a process remains non-negative up to time N ∈ N, the so-called persistence probability.We are interested in the asymptotic behaviour of these probabili-ties. A recent result by Aurzada, Mukherjee and Zeitouni (2017)shows that under certain conditions this quantity decreases expo-nentially fast and that the rate of decay can be identified by the

18

largest eigenvalue of some integral operator. To date, this eigen-value and hence the desired rate of decay could be computed onlyin a few particular examples. We present a perturbation argumentto determine a series expansion of the eigenvalue (in the parame-ter of the autoregressive process) to deal with this problem.

Testing in Transformation Models WSC-S-U-3.01

Wed 12:30Nick Kloodt

Universität Hamburg

Almost every statistician is familiar with or at least has heard ofregression models. One possible extension of such models con-sists in transforming the dependent variable in order to satisfymodel assumptions or to improve the applicability for small sam-ple sizes. Doing so leads to several new statistical questions suchas existence, uniqueness or estimation of the so-called transfor-mation function.

After a brief introduction into transformation models in gen-eral, a nonparametric estimating technique for transformationfunctions is given. Subsequently, this estimator is used to con-struct a test for the null hypothesis of a parametric transformationfunction. Although all estimators, test statistics and correspond-ing asymptotic results are presented, instead of a detailed elab-oration it is rather given a rough idea of how to develop them.

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The Rate of Convergence to a Gamma Target on theWiener SpaceWSC-S-U-3.03

Fri 11:30Lukas Knichel

Ruhr-Universität Bochum

In 2005, Nualart and Peccati introduced the famous "fourth mo-ment theorem", which states that a sequence of random variablesinside a fixed Wiener chaos converges to a standard Gaussian ran-dom variable if and only if the fourth moments converge to 3 (thefourth moment of a standard Gaussian). The method of Gaus-sian analysis was later combined with Stein’s method (Nourdin &Peccati 2009) to prove quantitative versions of the fourth momenttheorem. In 2015, Nourdin and Peccati showed that the exact rateof convergence in total variation is determined by the third andfourth cumulants ("The optimal fourth moment theorem"). In par-ticular, they were able to remove the square root from the upperbound in the result from 2009. In this talk, we consider the casewhere the target is distributed according to a (centered) Gammadistribution. Under certain technical conditions, we are able toremove the square root in the upper bound from the previous re-sults, and even obtain a lower bound of the same order. This talkis based on the preprint arXiv:1806.03878.

20

Critical 1-arm exponent for the Ising model onregular trees WSC-S-U-3.01

Fri 11:30Leonid Kolesnikov

Ludwig-Maximilians-Universität München

We consider the ferromagnetic nearest-neighbor Ising model onregular trees (Bethe lattice), which is well-known to undergo aphase transition in the absence of an external magnetic field. Thebehavior of the model at critical temperature can be describedin terms of various critical exponents; one of them is the criti-cal 1-arm exponent ρ which characterizes the rate of decay of the(root) magnetization as a function of the distance to the bound-ary. The crucial quantity we analyze in this work is the thermalexpectation of the root spin on a finite subtree, where the expectedvalue is taken with respect to a probability measure related to thecorresponding finite-volume Hamiltonian with a fixed boundarycondition. The spontaneous magnetization, which is the limit ofthis thermal expectation in the distance between the root and theboundary (i.e., in the height of the subtree), is known to vanish atcriticality. We are interested in a quantitative analysis of the rateof this convergence in terms of the critical 1-arm exponent ρ.

21

Optimal control in insurance mathematics underpartial informationWSC-S-U-3.01

Thu 14:00 Gregor LeimckeKarlsruher Institut für Technologie

We consider the surplus process of an insurance company withseveral insurance classes and we suppose that the insurance com-pany is interested in an investment and reinsurance strategywhich maximizes the expected exponential utility of terminalwealth. The claim arrivals of the insurance classes are modelledby a multivariate point process with interdependencies betweenthe marginal point processes where the dependence modellingis reduced to the choice of thinning probabilities. We assumethat the thinning probabilities are unobservable. This leads to astochastic control problem under partial information. With thehelp of filter theory, it is possible to reduce this partially observ-able control problem to one with complete observation. Usingstochastic control theory, we identify an optimal investment andreinsurance strategy.

Efficient allocations under probabilisticsophistication: a unifying approachWSC-S-U-3.01

Thu 15:00 Felix-Benedikt LiebrichLudwig-Maximilians-Universität München

The problem of optimising individual utilities or risks aggregatedover a system of agents has been subject of extensive mathe-matical and economic research for decades. Particular attentionhas been paid to probabilistically sophisticated individual utilities,which means that they rank Savage acts — random variables —in a way which only depends on their distribution under a fixedreference probability measure. Under probabilistic sophistication

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and mild further conditions the optimisation problem turns outto have solutions which are comonotone allocations of an aggre-gated quantity. The talk presents a unifying framework in whichcomonotone solutions to maximising aggregated probabilisticallysophisticated individual utilities can be found. We strongly em-phasise clear-cut meta results in our approach. (Joint work withGregor Svindland)

Existence and uniqueness of solutions of infinitedimensional Kolmogorov equations WSC-S-U-3.03

Thu 9:00Robert LinkUniversität Duisburg-Essen

It is well known from the Feynman-Kac formula that a classicalsolution of the Kolmogorov backward equation can be written asthe expectation of the solution of the corresponding SDE. In 2015M. Hairer, M. Hutzenthaler, and A. Jentzen gave a finite dimen-sional example of a Kolmogorov backward equation with globallybounded and smooth coefficients and a smooth initial functionwith compact support such that the unique viscosity solution isnot locally Hölder continuous. Moreover, they proved in the finitedimensional case that under suitable assumption the Kolmogorovbackward equation has a unique viscosity solution which can berepresented as the expectation of the solution of the correspond-ing SDE.

In the talk I will generalize this result to infinite dimensionalHilbert spaces and SPDEs. Therefore I will use a more general no-tation of viscosity solution introduced by H. Ishii and show thatunder suitable assumptions the expectation of the solution of anSPDE is the unique viscosity solution of the corresponding Kol-mogorov backward equation.

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Displacement of biased random walk in aone-dimensional percolation modelWSC-S-U-3.01

Fri 10:00Jan-Erik Lübbers

Technische Universität Darmstadt

Suppose an ant is placed in a randomly generated, infinite maze.Having no orientation whatsoever, it starts to move along ac-cording to a nearest neighbour random walk. Now furthermore,suppose the maze is tilted, such that the ant makes a step alongthe slope with higher probability than in the opposite direction.Tracking the ant’s position, we are interested in the long-term be-haviour of the corresponding random walk.We study this model in the context that the maze is given by aone-dimensional percolation cluster. Depending on the bias pa-rameter λ of the walk, its linear speed converges almost surelytowards a deterministic value v. This limit exhibits a phase tran-sition from positive value to zero at a critical value of λ. Weinvestigate the typical order of fluctuations of the walk around vin the ballistic speed regime, and the order of displacement fromthe origin in the critical and subballistic speed regimes.

The Triangle Condition for the Random ConnectionModel in High DimensionsWSC-S-U-3.03

Thu 15:00Kilian Matzke

Ludwig-Maximilians-Universität München

We consider the random connection model, which is a continuumpercolation model. We discuss analogues to basic tools from dis-crete percolation theory to then adapt the lace expansion to fit theframework of the underlying continuum space Poisson point pro-cesses. This allows us to derive the triangle condition above the

24

upper critical dimension and furthermore to establish the infra-red bound. From this, mean-field behavior of the model can bededuced.

Spin-flip dynamics for the Curie-Weiss-Potts model WSC-S-U-3.01

Fri 12:00Daniel Meißner

Ruhr-Universität Bochum

We study Gibbs-non-Gibbs transitions of the Curie-Weiss-Pottsmodel under symmetric independent spin-flip. As usual this is re-lated to the study of minimizers of a rate function of a constrainedfirst layer model. Unlike in the respective Curie-Weiss settingnumerical studies indicate the existence of bad points above thecritical temperature of the initial model and possibly intermittentGibbsian behaviour.

Studying Markov Processes through its Symbol WSC-S-U-3.01

Thu 9:30Kevin Musielak

Universität Siegen

Under reasonable assumptions (e.g. Feller), we can associate anyMarkov process with the semigroup given by

Ttu(x) = Exu(Xt).

The generator A given by

limt0

Ttu− ut

25

(defined on the domain where above limit exists strongly) is apseudo-differential operator with a symbol q. It turns out thatthe symbol is given by

q(x, ξ) = − limt0

Ex (ei(Xt − x)′ξ)− 1

t.

For a Lévy process the (space homogenous) symbol is simply theLévy-Khintchine exponent. In this talk, we want to show possi-bilities to analyze some properties of the process (e.g. Hausdorffdimension, Hölder continuity, γ-variation, maximal inequalities)through its symbol. In the end, we present some generalizationsto the time-dependent case.

Solution techniques for mean field games with finitestate and action spacesWSC-S-U-3.03

Wed 13:30 Berenice Anne NeumannUniversität Hamburg

Mean field games have been introduced by Lasry and Lions aswell as Huang et al. as a model that describes dynamic gameswith a large number of players that incorporate explicit interac-tion, which has been extensively used since then for several so-cioeconomic applications. Up to now, the theoretical investiga-tions where done for continuous action spaces with mostly con-tinuous state spaces, but also finite action spaces have been con-sidered. The methodology for solving these problems cruciallyrelies on the standard assumption that for each population distri-bution the responding agent has a unique optimal control. If oneconsiders finite action spaces this assumption will only hold fortrivial models, that is the techniques used so far are no longer ap-plicable. As many applications nonetheless require the choice be-tween finitely many actions, we propose a model with finite state

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and action space, show existence of stationary solutions givenvery mild assumptions and propose solution techniques that al-low to compute all stationary mean field equilibria also those thatare randomized.

Robust optimal stopping without time-consistency WSC-S-U-3.03

Wed 12:30Sascha Pascal Nolte

Universität Duisburg-Essen

In the context of robust optimal stopping one aim is to prove min-imax identities of the form

infQ∈Q

supτ∈T

EQ[Yτ ] = supτ∈T

infQ∈Q

EQ[Yτ ]

where Y is an adapted process on some filtered probability space,T is a set of stopping times and Q is a set of probability measures.Such minimax results play an important role in financial mathe-matics, especially in the characterization of arbitrage-free pricesfor American options.Normally, the proof relies on the assumption that Q satisfies theproperty of “time-consistency” which can be regarded as an ex-tension of the “tower property” for conditional expectations to thewhole family of involved conditional expectations. The crucialpoint is that under this property it is possible to modify the dy-namic programming principle which is one of the standard tech-niques to solve ordinary stopping problems.Unfortunately, time-consistency is very restrictive. In this talk wepresent a different kind of conditions that ensure the desired min-imax result. The key is to impose a compactness assumption onthe set Q.Furthermore, we give a short outlook how to extend the method

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to robust Dynkin games. In this context the question is for whichsets Q and processes R the following identity holds

infτ∈T

supσ∈T

supQ∈Q

EQ[R(τ, σ)] = supQ∈Q

supσ∈T

infτ∈T

EQ[R(τ, σ)].

Functional Poisson approximation of thinnedPoisson processesWSC-S-U-3.01

Thu 9:00Moritz Otto

Karlsruher Institut für Technologie

We consider stationary point processes that arise as a dependentthinning from a stationary Poisson process. Based on the assump-tion that the thinning depends only locally on the underlyingPoisson process, we derive a result for Poisson process approx-imation of an appropriate scaling of the thinned process. In itsproof we construct an adequate coupling between the thinnedprocess and a Palm version of itself. We discuss implications ofour result for the theory of extremes of random spatial structuresand present an application in the context of random geometricgraphs.

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Altruistic defense traits in structured populations:Many-demes limit in the sparse regime WSC-S-U-3.03

Fri 9:00Daniel PieperUniversität Duisburg-Essen

We discuss spatially structured Wright-Fisher type diffusions mod-elling the frequency of an altruistic defense trait. These diffusionsarise as the limit of spatial Lotka-Volterra type models with a hostpopulation and a parasite population, where one type of hostindividuals (the altruistic type) is more effective in defendingagainst the parasite but has a weak reproductive disadvantage.For the many-demes limit (mean-field approximation) hereof, weobtain a propagation of chaos result in the case where only afew diffusions start outside of an accessible trap. In this "sparseregime", the system converges in distribution to a forest of treesof excursions from the trap.

Unlabelled Set Partitions WSC-S-U-3.03

Wed 15:00Leon RamzewsLudwig-Maximilians-Universität München

We study combinatorial classes G which satisfy the multiset-con-struction in the unlabelled setting, i.e., any structure in G is com-posed of a multiset of unlabelled structures from an underlyingclass C. For example, graphs can be viewed as a multiset of con-nected graphs. Let Gn be drawn uniformly at random from theset of all structures in G having size n. A central parameter in thissetting is the distribution of the number of components κ(Gn) –the number of elements in the multiset –, which is known to con-verge in distribution. Here we determine the tails of κ(Gn), underthe rather general assumption that the ordinary generating func-tion of C is sub-exponential. Further, we prove a phenomenon,

29

which we call extreme condensation: when sampling an object ofsize n and with N components uniformly at random, it “typically”consists of one large component containing almost all atoms andN − 1 components of size o(1).

Modellierung der Abhängigkeitsstrukturstündlicher Strahlungsdaten mithilfe von Copula

basierten ZeitreihenmodellenWSC-S-U-3.01

Wed 15:30Matthias ReuberUniversität Siegen

Die weltweite Bedeutung von Strom aus erneuerbaren Energie-quellen, wie z.B. Photovoltaik, ist in den vergangenen Jahrenimmer weiter angestiegen. In vielen Anwendungsbereichen istinsbesondere die Modellierung des stündlichen Ertrages von In-teresse. Dabei besteht eine hohe Abhängigkeit zwischen Wertenaufeinanderfolgender Stunden, besonders in den oberen Quan-tilen (Upper Tail Dependence). Wir betrachten stündliche Strah-lungsdaten und stellen einen Ansatz vor, bei dem wir die mul-tivariaten Abhängigkeiten mit geeigneten Copulas beschreiben.Die Randverteilungen der einzelnen Stunden modellieren wirmit Beta-Verteilungen. Dazu benötigen wir geeignete obere unduntere Grenzen der stündlichen Strahlung, die wir durch eineKombination von einer Quantil-Regression und Methoden derExtremwerttheorie erhalten. Mit Hilfe eines Event-based Scoresevaluieren wir dann das Verhalten in den oberen Tails und stelleneinen Vergleich zu einfachen Benchmark-Modellen an.

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On the rate of convergence of symmetric Fellerprocesses on compact metric spaces via occupation

times WSC-S-U-3.03

Fri 11:00Geronimo Rojas

Universität Duisburg-Essen

We introduce a new distance in the space of càdlàg paths whichinduces the Skorohod topology. In contrast to the Skorohod dis-tance which takes off from the supremum norm by simultane-ously trying to match jumps points and their heights, our dis-tance evaluates the occupation times until hitting balls. We willapply this new notion of distance to symmetric Feller processeswith compact state space which can be associated with a resis-tance metric. This allows to obtain a bound on the speed of weakconvergence in path space in terms of the Gromov-Hausdorff-Prohorov distance of the corresponding resistance metric measurespaces.

The local clustering coefficient in hyperbolicrandom graphs WSC-S-U-3.03

Thu 14:00Markus Schepers

University of Groningen

The local clustering coefficient of a vertex of a graph measuresthe ratio of adjacent neighbours among all pairs of neighbours.Hyperbolic random graphs are given by a collection of pointsdistributed uniformly in a hyperbolic disk with edges betweennearby vertices. This model was invented by Krioukov et al. andhas been suggested as a suitable model for real-world networkssuch as the Internet. In this project we study the local clusteringcoefficient averaged over all vertices of degree k in the hyperbolic

31

random graph in the probabilistic limit (convergence in probabil-ity) as the number of vertices n tends to infinity. We consider boththe case of a fixed degree k, as well as a sequence of degrees (kn)tending to infinity. In the first case, we derive an exact analyticexpression, in the second case, we determine the leading term (in-cluding the multiplicative constant). (joint work with: NikolaosFountoulakis, Pim van der Hoorn, Tobias Müller)

About the front of the inhomogeneous Fisher-KPPequation and its linearisation, the parabolic

Anderson modelWSC-S-U-3.03

Thu 10:00 Lars SchmitzUniversität zu Köln

We consider the discrete-space, one-dimensional version of theinhomogeneous Fisher-KPP equation

∂

∂tv(t, x) =

12(4dv)(t, x) + ξ(x) · v(t, x)(1− v(t, x)),

(t, x) ∈ (0, ∞)×Z,

v(0, x) = 1(−∞,0](x),

(where (4d f )(t, x) := f (t, x − 1) − 2 f (t, x) + f (t, x + 1) is thediscrete Laplacian) and its linearisation (i.e. ξ · v instead of ξ ·v(1− v)), the parabolic Anderson model (PAM). We randomizethe model by assuming (ξ(x))x∈Z to be a field of i.i.d. positiverandom variables on some probability space (Ω,F, P), uniformlybounded away from 0 and +∞. We will look at the front of the so-lution m(t) := sup

x ∈ Z : v(t, x) ≥ 1

2

. Motivated by results in

the non-random, homogeneous setting (i.e. ξ ≡ 1), we conjecturethat there exists a deterministic constant C > 0 (probably depend-ing on the distribution of ξ), such that for P-almost all realisations

32

of ξ, the distance of both fronts is at most C · log t for all t ≥ t0(ξ).Both equations have a solution, which can be expressed in termsof branching random walks.

Inference of Stopping Times with Application toParticle Lifetime Estimation WSC-S-U-3.01

Wed 13:00Viktor Schulmann

Technische Universität Dortmund

Let X = (Xt)t≥0 be a known Markov process and T an unknownrandom time with a smooth Lebesgue density and independentof X. We present an estimator for the density of T based on i.i.d.samples of XT . For a Brownian motion X or, more generally, aLévy process on R such an estimator was given in Belomestnyand Schoenmakers (2015) and Belomestny and Schoenmakers(2016) using the Mellin and Laplace transforms. Applying theirtechniques we study this problem for Bessel processes or, moregenerally, Lévy processes on certain noncompact commutativehypergroups. We calculate the convergence rates of our estima-tors and show their asymptotic normality in some cases. An ap-plication to the estimation of a lifetime for certain particle modelsis discussed. [1] D. Belomestny, J. Schoenmakers (2016). Statisticalinference for time-changed Lévy processes via Mellin transformapproach. Stochastic Processes and their Applications 126, 2092–1222. [2] D. Belomestny, J. Schoenmakers (2015). Statistical Skoro-hod embedding problem: Optimality and asymptotic normality.Statistics & Probability Letters 104, 169–180.

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How to Break Impulse Control Problems Down intoSomething EasierWSC-S-U-3.03

Wed 14:30 Tobias SohrUniversität Hamburg

The purpose of impulse control problems usually is to shift downa stochastic process, say a Lévy process or a diffusion, countablemany times in order to maximize a payoff depending on the val-ues of the process before and after each shift. These problems areused for example to find the optimal way to manage a portfolioor to economize natural resources in a sustainable way. Some-thing easier (in this context) is an optimal stopping problem. Themost famous of these probably is the sectretary problem, but theyappear in noumerous fields of stochastics, ranging from optionpricing over game theory to sequential analysis. This talk willgive an short heuristic introduction to both stopping and impulsecontrol problems, thereafter emphasize how to use suitable stop-ping problems to characterize the value and the optimal strategyof some impulse control problems and close with some prospectsin the very new field of statistics for impulse control problems.

The White Knight Model - Propagation of Malwareon a D2D NetworkWSC-S-U-3.03

Fri 9:30 Alexander WapenhansWeierstraß-Institut für Angewandte Analysis und Stochastik

The use of D2D technology in future telecommunication systemspresents a great opportunity to increase connectivity as well ascapacity in networks under the pressure of accelerating demandfor faster and larger volume services. On the other hand, D2Dsystems present their own set of vulnerabilities, essentially com-ing from the lack of operated infrastructure. In this talk I want

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to present a proximity based infection process on a network,buildby users of an urban infrastructure driven D2D network.As the propagation of malware is not desirable, it is necessary todevise appropriate countermeasures.I will present one possible countermeasure – the White Knightmodel – and analyse it’s efficiency: How many White Knights arenecessary and how fast do they have to counter the malware tokeep the city safe?

Let X ⊂ R be the set of users and denote by ξ(t, x) ∈ S, I, G.the state of user x ∈ X at time t ∈ [0, ∞). Users are eitherSusceptible to the malware, Infected or have "Goodware" in-stalled.If we denote by I(t) the set of infected user, how fast does theinteraction between Goodware and malware has to be, such that

P

(inf

t∈R+t|I(t) = ∅ < ∞

)= 0,

e.g. the malware dies out almost surely?

A unified framework to robust modelling offinancial markets in discrete time WSC-S-U-3.01

Thu 14:30Johannes WieselUniversity of Oxford

We prove a Fundamental Theorem of Asset Pricing as well asa Superhedging Theorem in discrete time, which comprises thepathwise and quasisure formulation of [BN15] and [BFH+16].Furthermore we explain how to extend an M-quasisure super-hedging duality result on a set Ω to a pathwise duality withoutchanging the superhedging price.

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The construction of Root’s solution to SkorokhodembeddingWSC-S-U-3.03

Wed 15:30 Christina ZouUniversity of Oxford

A classical problem in stochastic analysis is the Skorokhod em-bedding problem: Given a Brownian motion and a probabilitymeasure, the task is to stop the trajectories of the process suchthat the terminal points are distributed according to the givenmeasure. One approach in order to determine a solution for theproblem is to construct it as a first hitting time of some set, e.g.the Root barrier. Rost proved in 1976 that Root’s solution has theminimal variance among the solutions to Skorokhod embeddingproblem using methods from potential theory. We are going to in-vestigate sufficient conditions such that such an embedding canbe made and provide a more explicit characterization of Root’ssolution for a more general class of Markov processes using thesolution of an obstacle problem.

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Exhibition: Women of Mathematics

We are happy that we could bring the photo exhibition Women ofMathematics throughout Europe. A gallery of portraits to Essen.

The following description of the project is from the homepageof the project Women of Mathematics womeninmath.net.

About the project

Entering the field of mathematics can be tough, and women oftenencounter specific obstacles. The exhibition offers a glimpse intothe world of mathematics through photographs (by Noel ToviaMatoff) and excerpts of interviews (by Sylvie Paycha and Sara Az-zali) of thirteen women mathematicians throughout Europe. Thiswebsite provides a platform for contact, exchange and mutual as-sistance.

This touring exhibition, whose starting point is the 7th ECMheld in July 2016 in Berlin, stems from the observation that nowa-days, women still find it difficult to embrace a career in the mathe-matical academic world and the disparity between the proportionof men and that of women among professional mathematicians isstill shamefully large.

The thirteen women mathematicians portrayed here share withus their experience, thus serving as role models to stimulateyoung women scientists to trust their own strength. In present-ing mathematics through women mathematicians’ perspectivesand samples of their life stories, we hope to highlight the humanaspects of producing mathematics, making this discipline more

37

tangible and therefore more accessible to outsiders or newcom-ers.

Following the opening in Berlin, the exhibit has been travelingto more than sixty cities in and out of Europe, including SouthAmerica, Australia and Africa. This touring format originally isenvisaged as a networking opportunity and for which the projectwas awarded with the Humboldt Alumni Award 2015, has indeedproved to reinforce collaborations and exchanges between mathe-maticians in different European countries, and stimulate dialoguearound the themes of the exhibition between the general publicand mathematicians. The present exhibition has further triggeredother similar projects leading to extended versions of the exhibi-tion in various cities (Cambridge, Aachen, Kaiserslautern, Hei-delberg and even other continents, in Chile) where Portraits ofMathematicians (and computer scientists in the case of Heidel-berg) where will be added to the existing 14 panels.

About the creators

The exhibition and the catalogue (publishing house: Verlag amFluss) are the result of the joint efforts of the photographer NoelTovia Matoff and four mathematicians by Sylvie Paycha, SaraAzzali, Alexandra Antoniouk, Magdalena Georgescu, with theprecious help of Maria Hoffmann-Dartevelle, who translated intoGerman and Sara Munday, who proofread the interviews and,last but not least, our two inspired graphic designers Wenke Ne-unast/eckedesign (exhibition) and Gesine Krüger (catalogue).

Bringing this exhibition project to life turned out to be muchmore difficult than expected, for a project centered around womenissues does not find much support in a mathematical world stillvery much dominated by men. We have learned a lot from over-coming the many obstacles on the path to its realization.

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Sponsors

The following organizations sponsored the exhibition

• Alexander von Humboldt Foundation

• Bosch Foundation

• Maecenia Frankfurt Foundation

• University of Potsdam

• Berlin Mathematic School

• European Women in Mathematics

• TU Berlin

• TU Mathematics library

• French Embassy in Germany

• London Mathematical Society

Organizations which supported the project

• 7th European Congress of Mathematics

• European Mathematical Society

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Further information

Adresses

VenueUniversität Duisburg-EssenFakultät für MathematikThea-Leymann-Str. 945127 Essen

HotelHoliday Inn Express EssenThea-Leymann-Str. 1145127 Essen

ExcursionZeche ZollvereinGelsenkirchener Str. 18145309 Essen

Dinnerdie kokerei cafe&restaurantUNESCO-Welterbe ZollvereinKokereiallee 7145141 Essen

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Organizers

Fabian GerleAnselm HuddeRobert LinkSara Mazzonetto

Roland MeizisLuis OsorioClemens PrintzGeronimo Rojas

Weblinks

DTS2018 www.uni-due.de/dts2018

Fachgruppe Stochastik www.fg-stochastik.de

RTG2131 sites.google.com/site/rtg2131

d-fine www.d-fine.com

Fakultät für Mathematik www.uni-due.de/mathematik

die kokerei www.die-kokerei.de

Zeche Zollverein www.zollverein.de

Women of Mathematics womeninmath.net

Map of Essenwith annotationstiny.cc/dts_map

Latest versionof this program

tiny.cc/dts_program

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Participants

Alhorn Kira TU DortmundAmro Lubna Universität UlmBata Katharina Universität zu KölnBetken Carina Universität OsnabrückBrinker Leonie Universität zu KölnBröker Yannic Universität MünsterBu Yunqi Ruhr-Universität BochumBussmann Stephan Universität OsnabrückCaraceni Alessandra University of BathChudjakow Tatjana d-fineDahmani Zoubir Univ. of MostaganemDüren Yannick Ruhr-Universität BochumEbel Daniel Universität HamburgEhlert Johannes TU DarmstadtGerle Fabian Universität Duisburg-EssenGrauer Arne Universität zu KölnGrygierek Jens Universität OsnabrückHallmann Sebastian Christian-Albrechts-Universität zu KielHeindl Thomas Katholische Universität Eichstätt-IngolstadtHenning Florian Ruhr-Universität BochumHudde Anselm Universität Duisburg-EssenHuebner Tobias Universität Duisburg-EssenHufnagel Nicole TU DortmundJakubzik Mirko Alexander TU DortmundKersting Sebastian TU DarmstadtKettner Marvin TU DarmstadtKissel Sascha Ruhr-Universität BochumKloodt Nick Universität HamburgKnichel Lukas Ruhr-Universität BochumKolesnikov Leonid LMU MünchenKramer Anke Universität Duisburg-EssenLeimcke Gregor Karlsruher Institut für Technologie (KIT)Liebrich Felix-Benedikt LMU MünchenLink Robert Universität Duisburg-Essen

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Lübbers Jan-Erik TU DarmstadtLüchtrath Lukas Universität zu KölnMatzke Kilian LMU MünchenMazzonetto Sara Universität Duisburg-EssenMeißner Daniel Ruhr-Universität BochumMeizis Roland Universität Duisburg-EssenMusielak Kevin Universität SiegenNeumann Berenice Anne Universität HamburgNolte Sascha Pascal Universität Duisburg-EssenOtto Moritz Karlsruher Institut für Technologie (KIT)Pieper Daniel Universität Duisburg-EssenPrevost Alexis Universität zu KölnPrintz Clemens Universität Duisburg-EssenRamosaj Burim Universität UlmRamzews Leon LMU MünchenReisser Simon LMU MünchenReuber Matthias Universität SiegenRojas Geronimo Universität Duisburg-EssenSattler Paavo Universität UlmSchepers Markus University of GroningenSchmitz Lars Universität zu KölnSchulmann Viktor TU DortmundSohr Tobias Universität HamburgWapenhans Alexander WIAS BerlinWeinig Michael Universität Duisburg-EssenWiesel Johannes University of OxfordYaseen Saad TU DortmundZou Christina University of Oxford

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