Workshop

Junior Female Researchersin Probability 2024

Berlin, July 3–5, 2024

Contents
Introduction

1

Timetable

3

Keynote speakers

5

Amandine Véber: A spatial measure-valued model for chemical reaction networks inheterogeneous systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5Anita Winter: Convergence of the Aldous-Broder chain on the high dimensional torus .

5

Invited talks

7

Alexandra Carpentier: On optimal ranking in crowd-sourcing problems in several scenarios

7

Julia Eisenberg: Insurance - A Wide Field to Apply Probability . . . . . . . . . . . . .

8Sophie Langer: Is Statistical Theory Essential for Understanding Deep Learning? . . .

8Makiko Sasada: Probabilistic aspects of the box-ball system . . . . . . . . . . . . . .

9Contributed talks

11

Mathilde André: Genealogies in multitype frequency-dependent branching processes .

11

Burcu Aydogan: Optimal portfolio strategies in jump-diffusion markets with relative
performance concerns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12Zsuzsanna Bősze: Estimating the rate of horizontal gene transfer in a host-parasite model 12
Hannah Dopmeyer: The two-size Wright–Fisher model: an analysis via (uniform) renewal
theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

Evangelia Dragazi: Improved model-free bounds for multi-asset options using optionimplied information and deep learning . . . . . . . . . . . . . . . . . . . . . . .

13

Annabel Gros: From 3 to infinity: a microscopic analysis of variable-speed branching
Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

Laura Hucker: Early stopping for conjugate gradients in statistical inverse problems .

14

Céline Kerriou: Random Friend Trees . . . . . . . . . . . . . . . . . . . . . . . . . .

15

Marilyn Korfhage: Existence of a robust phase in the Poisson Boolean model with convex
grains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15Valeriya Kotelnikova: A law of the iterated logarithm for iterated random walks, withapplication to random recursive trees . . . . . . . . . . . . . . . . . . . . . . .

16

iii

Angeliki Koutsimpela: Tagged particles and size-biased dynamics in mean-field interacting particle systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Kata Kubatovics: Branching processes in nearly degenerate varying environment . . .
Sofiia Lukashevych: Likelihood function derivatives for a linear mixed model . . . . .
Sophia-Marie Mellis: Coalescents with migration in the moderate regime . . . . . . .
Rivka Mitchell: Globally centered discrete snakes . . . . . . . . . . . . . . . . . . . .
Yana Mokanu: On ergodicity of Lévy-type processes in Rd . . . . . . . . . . . . . . .
Martha Nansubuga: On stochastic differential equations with jumps of Marcus-type . .
Francesca Pistolato: Limit theorems for p-domain functionals of stationary Gaussian
random fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Beatriz Salvador: From duality to correlations . . . . . . . . . . . . . . . . . . . . . .
Viktorya Vardanyan: Mean Field Games with Terminal State Constraints . . . . . . . .
Yuqiong Wang: Optimal tournament design in continuous time . . . . . . . . . . . . .
Johanna Weinberger: Stochastic Differential Equations Involving the Local Time of the
Unknown Process Driven by Stable Processes . . . . . . . . . . . . . . . . . . .
Xiaohan Zhu: On the pitchfork bifurcation for the Φ42 equation . . . . . . . . . . . . .
Speaker Index

16
17
17
19
19
20
20
21
22
22
23
23
24
25

iv

Introduction
It is our great pleasure to welcome you to the workshop Junior female researchers in probability.
We hope you enjoy illustrative talks and an interactive and inspiring exchange and networking.

Venue
The workshop will take place in presence at TU Berlin (HBS) at Hardenbergstraße 16–18, 10623
Berlin.

Presentations
The talks of the keynote and invited speakers each last one hour, and the contributed talks each last
20 minutes including questions.

WebsiteYou can find further information on the web page of the workshop: https://wias-berlin.de/workshops/JFRP24/.

OrganisersPeter Bank, Laura Körber, Gemma Lucia Sedrakjan, Weile Weng (TU Berlin)
Marta Dai Pra, Dörte Kreher, Janine Steck, Maite Wilke Berenguer (HU Berlin)
Ana Djurdjevac (FU Berlin)
César Zarco-Romero, Alexander Zass (WIAS Berlin)

1

Timetable
Registration Wednesday 8:30–8:50
Welcome Wednesday 8:50–9:00

9:00–10:00

WEDNESDAY 3

THURSDAY 4

Véber

Winter

10:00–10:40

coffee break

FRIDAY 5

9:00–10:00

Langer

10:00–10:40

coffee break

10:40–11:00

Mellis

Kerriou

10:40–11:00

Bősze11:00–11:20

Dopmeyer

Korfhage

11:00–11:20

Hucker

11:20–11:40

AndréKubatovics

11:20–11:40

Lukashevych

11:40–12:00

SalvadorGros

11:40–12:00

Zhu

12:00–13:30

lunch break

12:00–13:30

lunch break

13:30–13:50

Wang

Koutsimpela

13:30–13:50

Weinberger

13:50–14:10

Nansubuga

Mitchell

13:50–14:50

Carpentier

14:10–14:30

Vardanyan

Kotelnikova

14:50–15:00

closing remarks

14:30–14:50

Aydogan

Pistolato

14:50–15:30

coffee break

15:30–15:50

Dragazi

Mokanu15:50–16:50

Eisenberg

Sasada17:00–18:30

Q&A session

19:00

conference dinner3Keynote speakers
A spatial measure-valued model for chemical reaction networks in heterogeneous
systems
Amandine Véber
University Paris-Cité

In this talk, we shall introduce a measure-valued Markov process modelling a finite system ofbiochemical reactions taking place in a continuous, compact, space. In its general form, molecules
need to be close to each other to react; some reactions or chemical species may be localised in space;some species are abundant (with a number O(N ) of molecules) while others may be rare (withO(1) molecules). We shall first show that, as we let N tend to infinity, a suitable normalisation of
the counting measure describing the state of the system converges to a measure-valued piecewisedeterministic Markov process. In a second part, we shall describe a Central Limit Theorem
associated to this convergence.
This is joint work with Lea Popovic (Concordia University).

Convergence of the Aldous-Broder chain on the high dimensional torus
Anita Winter
University of Duisburg-Essen

The CRT is the scaling limit of the UST on the complete graph. The Aldous Broder chain
on a connected regular graph G = (V, E) is a MC with values in the space of rooted trees withvertices in V that has the UST on G as its invariant distribution. In Evans, Pitman and Winter (2006)
the so-called root growth with regrafting process (RGRG) was constructed. It was further shownthat the suitable rescaled Aldous Broder chain on the complete graph converges to the RGRG
weakly with respect to the GH-topology. It was shown in Peres and Revelle (2005) that (up toa dimension depending constant factor) the CRT is also the Gromov-weak scaling limit of the
UST on the d-dimensional torus, d 5. This result was recently strengthens in Archer, Nachmias
and Shalev (2024) to convergence with respect to the GH-weak topology, and therefore also withrespect to the GH-topology. In this talk we show that the suitable rescaled Aldous Broder chain on
the high-dimensional torus also converges to the RGRG weakly with respect to the GH-topology
when initially started in the trivial rooted tree.
Joint work with Osvaldo Angtuncio Hernandez and Gabriel Berzunza Ojeda.

5

Invited talks
On optimal ranking in crowd-sourcing problems in several scenarios
Alexandra Carpentier
University of Potsdam

Consider a crowd sourcing problem where we have n experts and d tasks. The average abilityof each expert for each task is stored in an unknown matrix M, from which we have incomplete
and noise observations. We make no (semi) parametric assumptions, but assume that the experts
can be perfectly ordered: so that if an expert A is better than an expert B, the ability of A is higher
than that of B for all tasks. We either assume the same for the task, or not, depending on the
scenario. This implies that if the matrix M, up to permutations of its rows and columns, is either
isotonic, or bi-isotonic. We focus on the problem of recovering the optimal ranking of the experts
and/or of the tasks, in l2 norm. We will consider this problem with some side-information - i.e.
when the ordering of the tasks (if it exists) is known to the statistician - or not. In other words, we
aim at estimating the suitable permutation of the rows of M. We provide a minimax-optimal andcomputationally feasible method for this problem in three scenarios of increasing difficulty: known
order of the task, unknown order of the tasks, no order of the tasks. The algorithms we provide
are based on hierarchical clustering, PCA, change-point detection, and exchange of informations
among the clusters.
This talk is based on a joint ongoing work with Emmanuel Pilliat, Maximilian Graf and Nicolas
Verzelen.

7

Insurance - A Wide Field to Apply Probability
Julia Eisenberg
TU Vienna

When considering a life or a non-life insurance portfolio, it is a natural wish to reduce the riskor to maximise the profit. However, the perception of risk depends on the risk measure chosen. The
spectrum of possible risk measures is wide and includes, for instance, ruin probability, expecteddiscounted cash flow, expected utility of terminal wealth and many more.
In this talk, we consider life and non-life insurance frameworks where the company’s surplus
is modelled by a stochastic process. The set of admissible controls is designed in line with thelong-term aims of the insurer. These strategies along with the overall target of the insurer are
formulated in terms of a stochastic control problem. We discuss possibilities to find the optimal
strategy and the value function solving the respective control problems.

Is Statistical Theory Essential for Understanding Deep Learning?
Sophie Langer
University of Twente

The remarkable practical success of deep learning has revealed some major surprises from atheoretical perspective. Despite the non-convex loss landscape of neural networks, gradient methods
consistently find near-optimal solutions and overparameterized networksthose with significantly
more parameters than the sample sizeexhibit great generalization results. These observations
cannot be fully explained by existing statistical theory: According to the bias-variance trade-off,
overparameterized networks should perform worse, and statistical studies typically focus on an
empirical risk minimizer, excluding the training process, such as gradient descent, from theiranalysis.
Given these discrepancies, a natural question arises: Is statistical theory even necessary tounderstand deep learning? In this talk, we will explore the role of statistical theory in understanding
deep learning, as well as its limitations and the necessity to connect with other research areas.

8Probabilistic aspects of the box-ball system
Makiko Sasada
University of Tokyo

The box-ball system (BBS), introduced by Takahashi and Satsuma in 1990, is a cellular
automaton that exhibits solitonic behaviour. The BBS has been studied from various viewpoints
such as tropical geometry, combinatorics, and discrete integrable systems (cf.[6]). As a newperspective, research on probabilistic approaches to this system has been rapidly expanding inrecent years, including the application of the Pitman transform, analysis of invariant measures andscaling limits ([1, 2, 4, 5]). At the same time, in the field of mathematical physics, the macroscopic
behavior of integrable systems has been found to have universality that differs from chaotic systems([3]). The BBS is one of the few models that is mathematically tractable to verify such universality.
In this talk, I will introduce these new research topics that have been spreading over the past several
years from the basics.

References[1] D. A. Croydon, T. Kato, M. Sasada, and S. Tsujimoto, Dynamics of the box-ball system withrandom initial conditions via Pitmans transformation, Mem. Amer. Math. Soc, 283, 1398
(2023).
[2] D. A. Croydon and M. Sasada, Invariant measures for the box-ball system based on stationary
Markov chains and periodic Gibbs measures, J. Math. Phys. 60 (2019), 083301.
[3] B. Doyon, Lecture notes on Generalised Hydrodynamics, SciPost Phys. Lect. Notes 18 (2020).
[4] P. A. Ferrari and D. Gabrielli, BBS invariant measures with independent soliton components,
Electron. J. Probab. 25 1 - 26, (2020).
[5] P. A. Ferrari, C. Nguyen, L. Rolla, and M. Wang, Soliton decomposition of the box-ball system,
Forum of Mathematics, Sigma (2021).
[6] R. Inoue, A. Kuniba, and T. Takagi, Integrable structure of box-ball systems: crystal, Bethe
ansatz, ultradiscretization and tropical geometry, J. Phys. A 45 (2012), no. 7, 073001, 64.

9Contributed talks
Genealogies in multitype frequency-dependent branching processes
Mathilde André
Collège de France - University of Vienna

Our work delves into the universality class of a very celebrated entity in population genetics:
Kingman’s coalescent. Introduced in [Kin82], it serves as baseline models for panmictic, neutral
populations. More generally, Λ-coalescents catalog the genealogies of constant-sized exchangeable
populations models known as Cannings models. We establish a broad class of regulated multitypeindividual-based models extending beyond exchangeable and fix-sized populations, yet for which
the scaling limit of the genealogies sampled at a given time is still Kingman’s coalescent. Thus, ourresult strides towards refining the intuition that neutral, Cannings-like genealogies can arise fromcomplex interactions.
We prove convergence in distribution of the genealogies for the Gromov-Weak topology, using
a change of measure and building on the multiple spine decomposition formalism developed by[FRS23]. The underlying purpose is to formulate a versatile methodology to derive scaling limitsof genealogies within structured populations subject to type-frequency interactions. This method
streamlines computations on forest-valued processes to a fine-grained analysis of the simpler
stochastic process driving the type frequencies in the population. Kingman’s coalescence rates arerecovered from the contribution of the spinal immigration to this frequency process.
This is joint work with Félix Foutel-Rodier (University of Oxford) and Emmanuel Schertzer(University of Vienna).

References[FRS23] Félix Foutel-Rodier and Emmanuel Schertzer. Convergence of genealogies through spinal
decomposition with an application to population genetics. Probability Theory and Related
Fields, 187(3):697–751, 2023.
[Kin82] John F. C. Kingman. The coalescent. Stochastic Processes and their Applications,13(3):235–248., 1982.

11

Optimal portfolio strategies in jump-diffusion markets with relative performance
concerns
Burcu Aydogan
RWTH Aachen University

We work on a portfolio management problem for one agent and a large group of agents underrelative performance concerns in jump-diffusion markets with the CRRA utility function. We havestochastic optimal control problems for both the representative agent and the group to determine
what the group does and the agent’s optimal proportion in the portfolio relative to the group’s
performance. Our framework assumes that the agent’s performance does not affect the group, while
the group affects the agent’s utility. Moreover, we investigate special cases where all agents in themarket are homogeneous in their risk aversion and relative performances. We explore the qualitative
behavior of the agent and show some numerical results depending on her relative performance
consideration and risk tolerance degree. Further, we compare the optimal strategies with small andmarket benchmark portfolios under under jump-diffusion markets assumption.
This is a joint work with Mogens Steffensen from University of Copenhagen.

References[BM24] B. Aydogan and M. Steffensen. Optimal investment strategies under the relative performance in jump-diffusion markets. Decisions in Economics and Finance, Under Review,2024.
[LZ19] D. Lacker and T. Zariphopoulou. Mean-field and n-agent games for optimal investment
under relative performance criteria. Mathematical Finance, Under Review, 29:1003–1038,
2019.

Estimating the rate of horizontal gene transfer in a host-parasite modelZsuzsanna Bősze
Georg August University Göttingen, Institute for Mathematical Stochastics

Horizontal gene transfer (HGT) is the movement of genetic material between organisms otherthan by the transmission from parent to its offspring. The presence of HGT during the evolutionof symbiotic systems (hosts and parasites) can lead to significant incongruences between thephylogenetic trees of the hosts and the parasites separately. Here we introduce a theoreticalmodel for the evolution of a host-parasite system with HGTs as well as an estimator for the rateof HGTs using only the information coming from the separate phylogenetic trees. We present
some of its mathematical properties regarding existence, consistency and asymptotic normality.
Moreover, using this estimator, we introduce a new stochastic algorithm that takes the two separate
phylogenetic trees as an input and returns a possible joint evolutionary scenario of the two levels.
This is joint work with Anja Sturm.

12

The two-size Wright–Fisher model: an analysis via (uniform) renewal theory
Hannah Dopmeyer
Bielefeld University

Consider a population with two types of (one-dimensional) individuals, where type is interpreted
as size (length): large individuals are of size 1 and small individuals are of size ϑ, ϑ ∈ (0, 1).
Each generation has an available space of length G. To form a new generation, individuals fromthe current generation are sampled one by one, and if there is at least some available space,
they reproduce and their offspring are added to the new generation. The probability of sampling
an individual whose offspring is small is given by µG (x), where x is the proportion of small
individuals in the current generation. We call this stochastic model in discrete time the two-size
Wright–Fisher model. The function µG can be used to model mutation and/or various forms offrequency-dependent selection. Denoting by (Xt )t≥0 the frequency process of small individuals,
we show convergence on the evolutionary time scale Gt to the solution of the SDEp
dXt = − κXt (1 − Xt ) + µ(Xt ) dt + Xt (1 − Xt )(1 − κXt ) dBt ,where κ = 1 − ϑ, µ(x) = lima→∞ G(µG (x) − x), and B is a standard Brownian motion.
To prove this statement, the dynamics inside one generation of the model are seen as a renewal
process, with the population size as the first-passage time τ (G) above level G. Methods from(uniform) renewal theory are applied and in particular a uniform version of Blackwell’s renewal
theorem (for binary, non-arithmetic random variables) is established.
This project is joint work with Gerold Alsmeyer (University of Münster) and Fernando Cordero(Bielefeld University, BOKU University).

Improved model-free bounds for multi-asset options using option-implied
information and deep learning
Evangelia Dragazi
National Technical University of Athens

We consider the computation of model-free bounds for multi-asset options in a setting thatcombines dependence uncertainty with additional information on the dependence structure. Morespecifically, we consider the setting where the marginal distributions are known and partial information, in the form of known prices for multi-asset options, is also available in the market. We provide
a fundamental theorem of asset pricing in this setting, as well as a superhedging duality that allows
to transform the maximization problem over probability measures in a more tractable minimization
problem over trading strategies. The latter is solved using a penalization approach combined with adeep learning approximation using artificial neural networks. The numerical method is fast andthe computational time scales linearly with respect to the number of traded assets. We finally
examine the significance of various pieces of additional information. Empirical evidence suggeststhat “relevant” information, i.e. prices of derivatives with the same payoff structure as the target
payoff, are more useful that other information, and should be prioritized in view of the trade-off
between accuracy and computational efficiency.
13

References[1] Bartl Daniel, Kupper Michael, Lux Thibaut, Papapantoleon Antonis and Eckstein
Stephan. Marginal and dependence uncertainty: bounds, optimal transport, and sharpness.
arXiv.1709.00641, 2017.
[2] Lux Thibaut and Papapantoleon Antonis. Improved Fréchet−Hoeffding bounds on d-copulas
and applications in model-free finance. arXiv.1602.08894, 2016.
[3] Stephan Eckstein and Michael Kupper. Computation of optimal transport and related hedging
problems via penalization and neural networks. arXiv.1802.08539, 2019

From 3 to infinity: a microscopic analysis of variable-speed branching Brownian
motionAnnabel Gros
University of BonnThe extremes of variable speed branching Brownian motion (BBM) exhibit a phase transition
when the “speed function”, which describes the time-inhomogeneous variance, is the identity
function. In this talk we study this transition more closely by choosing piecewise linear, concave
speed functions converging to the identity function from above. We show that the logarithmiccorrection of the order of the maximum interpolates smoothly between the correction of standard
3
BBM, 2√log(t), and the correction for BBM with piecewise linear speed functions. A key part of2
the proof is the precise control of the path of an extremal particle.
Based on joint work with Alexander Alban (U Mainz), Anton Bovier (U Bonn) and Lisa
Hartung (U Mainz).

Early stopping for conjugate gradients in statistical inverse problems
Laura Hucker
Humboldt-Universität zu Berlin

We consider estimators obtained by applying the conjugate gradient algorithm to the normal
equation of a prototypical statistical inverse problem. For such iterative procedures, it is necessary
to choose a suitable iteration index to avoid under- and overfitting. Unfortunately, classical model
selection criteria can be prohibitively expensive in high dimensions. In contrast, it has beenshown for several methods that sequential early stopping can achieve statistical and computational
efficiency by halting at a fully data-driven index depending on previous iterates only. Residualbased stopping rules, similar to the discrepancy principle for deterministic problems, are wellunderstood for linear regularisation methods. However, in the case of conjugate gradients, theestimator depends nonlinearly on the observations, allowing for greater flexibility. This significantly
14

complicates the error analysis. We establish adaptation results for both the prediction and thereconstruction error in this setting.
This is joint work with Markus Reiß.

Random Friend Trees
Céline Kerriou
University of Cologne

We study a random recursive tree model featuring complete redirection called the random friend
tree and introduced by Saramäki and Kaski. Vertices are attached in a sequential manner one by one
by selecting an existing target vertex and connecting to one of its neighbours (or friends), chosen
uniformly at random. This model has interesting emergent properties, such as a highly skewed
degree sequence. In contrast to the preferential attachment model, these emergent phenomena
stem from a local rather than a global attachment mechanism. The structure of the resulting tree
is also strikingly different from both the preferential attachment tree and the uniform random
recursive tree: every edge is incident to a macro-hub of asymptotically linear degree, and with highprobability all but at most n9/10 vertices in a tree of size n are leaves. We prove various results on
the neighbourhood of fixed vertices and edges, and we study macroscopic properties such as thediameter and the degree distribution, providing insights into the overall structure of the tree. We
also present a number of open questions on this model and related models.

Existence of a robust phase in the Poisson Boolean model with convex grains
Marilyn Korfhage
University of Cologne

Consider a homogenous Poisson Point Process on Rd , equipped with convex grains, i.e. i.i.d.
copies of a random convex body that is rotationally-invariant in distribution. We define for such a
convex body a non-increasing sequence of diameters. The first diameter is the classical diameter ofthe convex body. The i-th diameter is then defined as the diameter of the orthogonal projectionof the body from the previous step along the (i−1)-st diameter onto the (d−i + 1)-dimensional
hyperplane. This projected body is then projected further when determining the next diameter. We
state several criteria on the diameter distribution and moment conditions for the volume of theconvex body that result in either a dense process, i.e. the whole space is covered by the grains, orrobustness, i.e. the union of the grains has an unbounded connected component for any intensityof the underlying Poisson process. Importantly, we do not impose any conditions on the joint
distribution of the diameters. If the grains are chosen to be Euclidean balls, it is known that density
and robustness are equivalent. We show in our general model that in any dimension d ≥ 2 there
exists grain distributions where robustness does not imply density.
Joint work with Peter Gracar and Peter Mörters.

15

A law of the iterated logarithm for iterated random walks, with application to
random recursive trees
Valeriya Kotelnikova
Taras Shevchenko National University of Kyiv

Consider a general branching process (a.k.a Crump-Mode-Jagers process) generated by anincreasing random walk whose increments have finite second moment. Let Yk (t) be the number ofindividuals in generation k ∈ N born in the time interval [0, t]. I shall discuss a law of the iterated
logarithm for Yk (t) with fixed k, as t → +∞. As a corollary, I shall also present a law of theiterated logarithm for the number of vertices at a fixed level k in a random recursive tree, as thenumber of vertices goes to ∞.
The talk is based on the joint article [IKK22+] with Olexander Iksanov (Kyiv) and Zakhar
Kabluchko (Münster).

References[IKK22+] O. Iksanov, Z. Kabluchko and V. Kotelnikova, A law of the iterated logarithm for
iterated random walks, with application to random recursive trees. Preprint (2022) available athttps://arxiv.org/abs/2212.13441

Tagged particles and size-biased dynamics in mean-field interacting particle systems
Angeliki Koutsimpela
University of Augsburg

We establish a connection between tagged particles and size-biased empirical processes ininteracting particle systems, in analogy to classical results on the propagation of chaos. In a meanfield scaling limit, the evolution of the occupation number on the tagged particle site converges to atime-inhomogeneous Markov process with non-linear master equation given by the law of large
numbers of size-biased empirical measures.
Joint work with S. Grosskinsky.

16Branching processes in nearly degenerate varying environment
Kata Kubatovics
University of Szeged

P
We investigate branching processes in varying environment, for which f n → 1 and ∞n=1 (1 −P∞f n )+ = ∞, n=1 (f n − 1)+ < ∞, where f n stands for the offspring mean in generation n. Since
subcritical regimes dominate, such processes die out almost surely, therefore to obtain a nontrivial
limit we consider two scenarios: conditioning on non-extinction, and adding immigration. In bothcases we show that the process converges in distribution without normalization to a nondegeneratecompound-Poisson limit law. We also prove functional limit theorems in the above cases. In
the conditional setup, the limiting process is a simple birth and death process conditioned on
non-extinction at time 1, while the process with immigration tends to a continuous time branchingprocess with immigration with a simple birth and death branching counterpart. The proofs rely on
the shape function technique, worked out by Kersting [K20].
This is a joint work with Péter Kevei.

References[K20] Kersting, Götz. A unifying approach to branching processes in a varying environment.
Journal of Applied Probability, 57(1):196–220, 2020.

Likelihood function derivatives for a linear mixed model
Sofiia Lukashevych
Taras Shevchenko National University of Kyiv

Restricted Maximum Likelihood (REML) estimation is a statistical method used to estimateparameters in a mixed-effects model, especially in the range of linear mixed models. It serves
as an extension of the maximum likelihood estimation method, which aims to provide unbiased
and efficient parameter estimates, especially in scenarios involving correlated data. In the REML
approach, the likelihood function is adjusted to remove confounding parameters associated with
fixed effects. This modification improves the efficiency of parameter estimation, especially insituations where the emphasis is on the estimation of variance components or when fixed andrandom effects are considered in the model.
My research explores the properties of linear mixed models with simple random effects of theform:
yi = Xi β + Zi γi + i , i = 1, . . . , M(0.0.1)γi ∼ N (0, ψ), i ∼ N (0, σ 2 I),
where M is the number of distinct groups, each consisting of ni observations. Random effects γiand within-group errors i are independent across different groups and within the same group. β
17

is a p-dimensional vector of fixed effects, γi is a q-dimensional vector of random effects, and Xi
and Zi are known design matrices of dimensions ni × p and ni × q, of fixed and random effects
respectively. Vectors i represent within-group errors with a spherically Gaussian distribution.
Assuming a compound symmetry, autoregressive[1], Toeplitz and variance components inthe correlation structure of the matrix ψ governing the dependence among within-group errors,
analytical formulas for the first derivative and the second partial derivative of the restricted maximum
likelihood function with respect to the correlation parameters of the model are derived.
My research findings of analytical representation of derivatives facilitates the effective utilizationof numerical algorithms like Newton-Raphson or Levenberg-Marquardt.

References[1] Lukashevych, S., Yamnenko, R. Likelihood function derivatives for a linear mixed model withcompound symmetry assumption. Mohyla Mathematical Journal, 2024, pp.24–27.
[2] R. Maiboroda, Regresia: liniini modeli (VPC Kyivskyi universytet, 2007).
[3] Bates, D., Machler, M., Bolker, B. and Walker, S. “Fitting Linear Mixed-Effects Models Using
lme4. Journal of Statistical Software”, 67, 1-48 (2015).
[4] Lindstrom, M. J., Bates, D. M. “Newton-Raphson and EM Algorithms for Linear Mixed-Effects
Models for Repeated-Measures Data”. Journal of the American Statistical Association, 83,
10141022 (1988).
[5] SAS Institute Inc. SAS/STAT® 13.1. User’s Guide (Cary, NC: SAS Institute Inc, 2013).
[6] J. C. Pinheiro, D. M. Bates, Mixed-effect models in S and S-PLUS (Sringer: Statistics andComputing, 2000).
[7] S.R. Searle, G. Casella and C.E. McCulloch. Variance Components (John Wiley & Sons, 2009).

18

Coalescents with migration in the moderate regime
Sophia-Marie Mellis
Bielefeld University

Multi-type models have recently experienced renewed interest in the stochastic modeling ofevolution. This is partially due to their mathematical analysis often being more challenging thantheir single-type counterparts; an example of this is the site-frequency spectrum of a colony-based
population with moderate migration.
In this talk, we model the genealogy of such a population via a multi-type coalescent startingwith N (K) colored singletons with d ≥ 2 possible colors (colonies). The process is described bya continuous-time Markov chain with values on the colored partitions of the colored integers in{1, . . . , N (K)}; blocks of the same color coalesce at rate 1, while they are also allowed to change
color at a rate proportional to K (migration).
Given this setting, we study the asymptotic behavior, as K → ∞ at small times, of the vectorof empirical measures, whose i-th component keeps track of the blocks of color i at time t and of
the initial coloring of the integers composing each of these blocks. We show that, in the proper
time-space scaling, it converges to a multi-type branching process (case N (K) ∼ K) or a multitype Feller diffusion (case N (K)  K). Using this result, we derive an applicable representationof the site-frequency spectrum.
This is joint work with Fernando Cordero and Emmanuel Schertzer.

Globally centered discrete snakes
Rivka Mitchell
University of Oxford

We prove that under a global finite variance and a tail behaviour assumption on the displacements, any globally centered discrete snake on a Bienaymé tree whose offspring distribution is
critical and admits a finite third moment has the Brownian snake driven by a normalised Brownian
excursion as its scaling limit. Our proof relies on two perspectives of Bienaymé trees. To prove
convergence of finite dimensional distributions we rely on a line-breaking construction from a
recent paper of Addario-Berry, Blanc-Renaudie, Donderwinkel, Maazoun and Martin. To provetightness, we adapt a method used by Haas and Miermont with regard to Markov branching trees.
This is based on joint work in progress with Louigi Addario-Berry, Serte Donderwinkel, andChristina Goldschmidt.

19

On ergodicity of Lévy-type processes in Rd
Yana Mokanu
Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine,
Kyiv, Ukraine

In this talk we derive sufficient conditions for the ergodicity of the specific class of Lévy-type
processes. It is assumed that on test functions the generator of the respective semigroup admits therepresentationZ
Lf (x) = l(x)∇f (x) +f (x + u) − f (x) − ∇f (x)u1|u|≤1 ν(x, du),Rd \{0}where ν(x, du) is a Lévy-type measure and l : Rd → Rd . The conditions are given in terms of
the Lévy-type measure ν(x, ·) and applicable to the case, when the drift term is absent. The
Foster-Lyapunov criterion is used to prove the results.

References[BSW13] B. Böttcher, R. Schilling and J. Wang. Lévy-Type Processes: Construction, Approximation and Sample Path Properties. Springer, Lecture Notes in Mathematics vol. 2099 (Lévy
Matters III), Cham 2013.
[KM24] V. Knopova and Y. Mokanu. Ergodicity of Lévy-type processes in Rd . (In preparation).
[Kul17] A. Kulik. Ergodic Behaviour of Markov Processes. de Gruyter, Berlin, 2017.
[San16a] N. Sandrić. Ergodicity of Lévy-type processes. ESAIM: PS. 20 (2016), 154-177.

On stochastic differential equations with jumps of Marcus-type
Martha Nansubuga
HU Berlin

We study existence and uniqueness of stochastic differential equations (SDEs) driven bysemimartingales with jumps, where the jumps of the solution are obtained as small relaxation time
limits of fast curvilinear motions along the solution of a non-linear ordinary differential equation,and between the jumps, the solution to the SDE moves along the (coefficient) vector field in thesense of Marcus [1]. For stochastic integrals driven by a continuous process, stochastic integrals in
the Marcus sense coincide with stochastic integrals in the Stratonovich sense and satisfy a chain
rule for drivers with jumps. We study SDEs with time-dependent coefficients, which are comparedand checked in [2]. I will give an outlook on my ongoing research where such stochastic differential
equations are to be applied.
20

References[1] Steven I. Marcus. Modeling and approximation of stochastic differential equations driven bysemimartingales. Stochastics: An International Journal of Probability and Stochastic Processes
4, no.3 (1981): 223–245 Submitted. arXiv:2402.16701.
[2] Thomas G. Kurtz, Étienne Pardoux, Philip Protter. Stratonovich stochastic differential equations
driven by general semimartingales. Annales de lIHP Probabilités et statistiques vol. 31, no. 2,
pp. 351-377, 1995.

Limit theorems for p-domain functionals of stationary Gaussian random fields
Francesca Pistolato
University of Luxembourg

The topic of the talk are the p-domain functionals of stationary Gaussian random fields. Moreprecisely, given a Gaussian random field B = (Bx )x∈Rd , d ≥ 2, we study the asymptotic behaviourofZ
φ(Bx ) dx,
as ti → ∞, ∀i = 1, . . . , p,t1 D1 ×···×tp Dp

where Di ⊂ Rdi is bounded for any i = 1, . . . , p, and d1 + . . . + dp = d.
An important result to understand the behavior of such functionals is Breuer-Major theorem,that links the short-range dependence of the field, that is the integrability of (a power of) thecovariance function of B, together with the rescaled functional satisfying a Central Limit Theorem(CLT).
In this talk, under further assumptions on the covariance function, we will show how theasymptotic behaviour of p-domain functionals can be simply obtained from that of 1-domainfunctionals, explaining in a new light and in a more systematic way some results from the recent
literature. For instance, the result provides a new proof of the convergence of the qth variation of
the fractional Wiener sheet to a Gaussian distribution, proved in [2], and improves their estimate on
the rate of convergence. Moreover, it allows us to produce new examples of long-range dependent
fields satisfying CLTs. The talk is based on [1].

References[1] N. Leonenko, L. Maini, I. Nourdin and F. Pistolato. Limit theorems for p-domain functionalsof stationary Gaussian fields. Submitted. arXiv:2402.16701.
[2] A. Réveillac, M. Stauch and C. Tudor. Hermite variations of the fractional Brownian sheet.
Stoch. Dyn. 12(2012), no.3, 1150021, 21 pp.

21

From duality to correlations
Beatriz Salvador
Instituto Superior Técnico (IST), Lisbon, Portugal

In this talk, we introduce the concept of Stochastic Duality. From it, we will see an approach,based on duality, on how to bound the k-points centered correlation functions of a given interacting
particle system (IPS) when it has the duality property and also a specific type of duality function.
The problem of finding such bounds is an essential tool to derive the non-equilibrium fluctuationsof a boundary-driven IPS. We will see an application of those results for the symmetric simple
partial exclusion process, SEP(α), the symmetric simple inclusion process, SIP(α), and independent
particles, IRW, all considered with open boundary. The case k = 2 for SEP(α) is joint work with
Chiara Franceschini, Patrícia Gonçalves and Milton Jara [1] and the general case is a work inprogress with Patrícia Gonçalves.

References[1] C. Franceschini, P. Gonçalves, M. Jara, B. Salvador (2023): Non-equilibrium fluctuations for
SEP(α) with open boundary, submitted and online at https://arxiv.org/abs/2308.09144.

Mean Field Games with Terminal State Constraints
Viktorya Vardanyan
University of Trento

We study a mean field game (MFG) with state dynamics described by stochastic differential
equations affected by both idiosyncratic and common noise, and subject to the constraint thatthe terminal state variable belongs to a nonempty convex closed set. Moreover, the mean fieldinteraction enters through both state and control in the dynamics and the costs. Inspired by the workof Ji and Zhou [JZ], we establish an auxiliary MFG problem and derive the stochastic maximum
principle (SMP) for an auxiliary optimization problem under fixed flows. By introducing a suitableforward-backward stochastic differential equation (FBSDE) of conditional McKean-Vlasov type,
we prove the existence of its solution and verify that it constitutes a MFG equilibrium. Additionally,
we apply our findings to MFGs of optimal liquidation and of optimal investment with a quadratic
relative performance criterion.
This is based on a joint work with Luciano Campi (University of Milan) and Luca Di Persio(University of Verona).

References[JZ] S. Ji and X.Y. Zhou. A maximum principle for stochastic optimal control with terminal state
constraints, and its applications. Communications in Information and Systems , 6(4):321–338,
2006.
22

Optimal tournament design in continuous time
Yuqiong Wang
Duisburg-Essen

We consider the problem of finding the optimal scheme for a knock-out tournament with 2n players,aiming at determining the top player. In each game in the tournament, one observes a Brownian
motion with drift where the drift is a Bernoulli random variable whose value reflects the players’
abilities. However, these observations come at a cost: a fixed penalty c > 0 per unit time ofobservation. We formulate and solve an optimal control problem to minimise the probability ofmaking a mistake while keeping the cost of observation low. Additionally, we discuss a few variantsof the problem and some possible generalisations. This is joint work with Erik Ekström.

Stochastic Differential Equations Involving the Local Time of the Unknown Process
Driven by Stable Processes
Johanna Weinberger
Department of Data and Decision Sciences, Technion - Israel Institute of Technology

Stochastic differential equations involving the local time of the unknown process with Brownian
forcing were thoroughly discussed by Jean-François Le Gall in his seminal paper [LG84] from1984. A special case of these equations are of the formZ
Xt = x0 +`xt µ(dx) + Bt , t ≥ 0,
R

where B is a standard Brownian motion, x0 ∈ R, µ is a Radon measure and ` is the local time of
X given by the Tanaka’s formula. In this talk, we formulate an analogous equation in dimensiond = 1 , where B is replaced by an α-stable process L with α ∈ (1, 2) and discuss weak andstrong existence and uniqueness of solutions, as well as the equivalence to singular SDEs with driftµ. To this end, we define a local time ` by deriving a Tanaka-type formula for semimartingales
whose martingale part is an α-stable process with α ∈ (1, 2). We show that ` coincides with theoccupation density of X for solutions to singular SDEs under mild assumptions on the drift µ. The
talk is based on joint work with Leonid Mytnik.

References[LG84] J. F. Le Gall. One-dimensional stochastic differential equations involving the local timesof the unknown process. Stochastic analysis and applications, Proc. int. Conf., Swansea 1983,
Lect. Notes Math. 1095, 51-82 (1984).

23

On the pitchfork bifurcation for the Φ42 equation
Xiaohan Zhu
Universität Zürich

In this talk, we present a bifurcation analysis for the Φ42 equation∂t ϕ = ∆ϕ + αϕ − ϕ3 + 3ϕ · ∞ + ξ,
where ϕ : R+ × T2 → R, ξ is space-time white noise on R+ × T2 , and α ∈ R. While the noise-freesystem exhibits a pitchfork bifurcation, the introduction of noise disrupts the bifurcation, leading
to a phenomenon known as “synchronisation by noise”. However, with finite-time Lyapunov
exponents, we can still detect the underlying pitchfork bifurcation. Specifically, a change in thesign of the finite-time Lyapunov exponents implies a shift in local stability on a finite time scale,offering a new perspective on the dynamics of the Φ42 equation.

24

Speaker Index
André, Mathilde, 11

Lukashevych, Sofiia, 17

Aydogan, Burcu, 12Mellis, Sophia-Marie, 19Bősze, Zsuzsanna, 12

Mitchell, Rivka, 19

Carpentier, Alexandra, 7

Mokanu, Yana, 20

Dopmeyer, Hannah, 13

Nansubuga, Martha, 20

Dragazi, Evangelia, 13
Pistolato, Francesca, 21
Eisenberg, Julia, 8Salvador, Beatriz, 22Gross, Anmnabel, 14
Sasada, Makiko, 9
Hucker, Laura, 14
Vardanyan, Viktorya, 22
Kerriou, Céline, 15
Véber, Amandine, 5Korfhage, Marilyn, 15

Wang, Yuqiong, 23

Kotelnikova, Valeriya, 16
Koutsimpela, Angeliki, 16

Weinberger, Johanna, 23

Kubatovics, Kata, 17Winter, Anita, 5Langer, Sophie, 8Zhu, Xiaohan, 24

25