About some skewed Brownian diffusions: explicit representation of their transition densities and exact simulation

In this talk we first discuss an explicit representation of the transition density of Brownian dynamics undergoing their motion through semipermeable and semireflecting barriers, called skewed Brownian motions.
We use this result to present an exact simulation of these diffusions, and comment some (still) open problems. Eventually we consider the exact simulation of Brownian diffusions whose drift admits finitely many jumps.

Tuan Anh Nguyen (Universität Duisburg-Essen)
The random conductance model under degenerate conditions

The aim of the talk is to briefly introduce some ideas in my PhD thesis. Random motions in random media is an interesting topic that has been studied intensively since several decades. Although these models are relatively simple mathematical objects, they have a wide variety of interesting properties from the theoretical point of view. In the talk I would like to draw your attention to an important branch within this topic, namely reversible random walks moving among nearest neighbour random conductances on $\mathbb{Z}^d$ -- the random conductance model. Reversibility provides the model a variety of interesting connections with other fields in mathematics, for instance, percolation theory and especially stochastic homogenization. Many questions coming from this model have been answered by techniques from partial differential equations and harmonic analysis. As seen in the name of the talk, I would like to consider this model under ''degenerate conditions''. Here, ''degenerate'' has essentially two meanings. First, the conductances are not assumed to be bounded from above and below and stochastically independent. Second, we also consider the case of zero conductances, where the random walk can only move on a subgraph of $\mathbb{Z}^d$. Since there are percolation clusters, where the existence of the infinite cluster does not rely on stochastic independence, it is reasonable to accept the lack of stochastic independence. In the first part of the talk I introduce quenched invariance principles (joint work with Jean-Dominique Deuschel and Martin Slowik). We assume that the positive conductances have some certain moment bounds, however, not bounded from above and below, and give rise to a unique infinite cluster and prove a quenched invariance principle for the continuous-time random walk among random conductances under relatively mild conditions on the structure of the infinite cluster. An essential ingredient of our proof is a new anchored relative isoperimetric inequality. In the second part I would like to talk about Liouville principles. As in the first part, I also assume some moment bounds and prove a first order Liouville property for this model. Using the corrector method introduced by Papanicolaou and Varadhan, the first and the second part are closely related to each other at the technical level. I also introduce a discrete analogue of the Dirichlet-to-Neumann estimate, which compares the tangential and normal derivatives of a harmonic function on the boundary of a domain. Although it is a purely deterministic classical result, it is used in the second part and perhaps useful for numerical analysis.

Clemens Printz (Universität Duisburg-Essen)
Stochastic averaging for multiscale Markov processes applied to a Wright-Fisher model with fluctuating

We present a new result on stochastic averaging for sequences of bivariate Markov processes $((X_t^n,Z_t^n)_{t\in[0,\infty)})_{n\in\mathbb{N}}$ whose components evolve on different time scales. Under suitable conditions, convergence of certain functionals of the fast variables $Z^n$ guarantees convergence of the (not necessarily Markovian on its own) slow variables $X^n$ to a limiting Markov process $(X_t)_{t\in[0,\infty)}$.
With this tool we can generalize the well-known diffusion limit of a Wright-Fisher model with randomly fluctuating selection. Whereas the classical result assumes the selection coefficients to be independent for different generations, we allow the environment to persist with a positive probability. The diffusion limit turns out to depend on this probability.
This talk is based on joint work with Martin Hutzenthaler and Peter Pfaffelhuber.

Gradient flow formulation and longtime behaviour of a constrained Fokker-Planck equation

We consider a Fokker-Planck equation which is coupled to an externally given time-dependent constraint on its first moment. This constraint introduces a Lagrange-multiplier which renders the equation nonlocal and nonlinear. In this talk we exploit an interpretation of this equation as a Wasserstein gradient flow of a free energy ${\mathcal{F}}$ on a time-constrained manifold. First, we prove existence of solutions by passing to the limit in an explicit Euler scheme obtained by minimizing $h {\mathcal{F}}(\varrho)+W_2^2(\varrho^0,\varrho)$ among all $\varrho$ satisfying the constraint for some $\varrho^0$ and time-step $h>0$. Second, we provide quantitative estimates for the rate of convergence to equilibrium when the constraint converges to a constant. The proof is based on the investigation of a suitable relative entropy with respect to minimizers of the free energy chosen according to the constraint. The rate of convergence can be explicitly expressed in terms of constants in suitable logarithmic Sobolev inequalities.

Michael Weinig (Universität Duisburg-Essen)
Nested Monte-Carlo Simulation Estimates of Expected Value of Partial Perfect Information

The expected value of partial perfect information (EVPPI) expresses the value gaining by acquiring further information on certain unknowns in a decision-making process. In this talk, we will show and compare various approaches to estimate the EVPPI like nested Monte-Carlo simulations, Multi-Level Monte-Carlo simulations, and regression-based algorithms.
Further, we introduce an unbiased estimator based on a randomized version multi-level Monte-Carlo algorithm.
The EVPPI estimate is mostly used in medical decision making. We will show some numerical result in evaluating the benefit of further information when deciding between two different treatment options.

Sebastian Mentemeier (TU Dortmund)
Solutions to complex smoothing equations

In many models of Applied Probability and Statistical Physics quantities of interest satisfy distributional limit theorems (i.e. convergence in law) which are nonstandard in the sense that neither normal nor $\alpha$-stable laws appear as limiting distributions. Sometimes, there is even no convergence in law, but periodical fluctuations centered at some limit law occur. Common to all these models is that the limiting distributions satisfy so-called smoothing equations. These are equations of the form
\begin{equation}\label{SFPE} X \stackrel{\text{law}}{=} \sum_{j \ge 1} T_j X_j + C, \end{equation}
where $X_1, X_2, \dots$ are i.i.d.~copies of the (unknown)  random variable $X$ and independent of the (given) random variables $(C, T_1, T_2, \dots)$.
I will start my talk by presenting several examples where this phenomenon occurs. Then I will present a general theory for solving equations of type (1) in the case where $X$ as well as $C, T_1, T_2, \dots$ are complex-valued random variables. These results are then applied to the examples mentioned in the introduction.
This talk is based on joint work with Matthias Meiners, University Innsbruck.

The asymptotic behavior of the ground state energy of the Anderson model on large regular trees

The Anderson model was invented 1985 by Anderson and describes the quantum mechanical motion of a particle in a random potential in $Z^d$. It is related to random walks in a random environment.  The Anderson model on regular tree was introduced 1973 by Abou-Chacra, Thouless and Anderson. We give a detailed description of the ground state energy on large finite symmetric subtrees. This is joint work with Francisco Hoecker-Escuti (TU Hamburg-Harburg).

Dec 12 

Larisa Yaroslavtseva (Universität Passau)
On sub-polynomial lower error bounds for strong approximation of SDEs

We consider the problem of strong approximation of the solution of a stochastic differential equation (SDE) at the final time based on finitely many evaluations of the driving Brownian motion $W$. While the majority of results for this problem deals with equations that have globally Lipschitz continuous coefficients, such assumptions are typically not met for real world applications. In recent years a number of positive results for this problem has been established under substantially weaker assumptions on the coefficients such as global monotonicity conditions: new types of algorithms have been constructed that are easy to implement and still achieve a polynomial rate of convergence under these weaker assumptions.
In our talk we present negative results for this problem. First   we show that there exist  SDEs  with bounded smooth coefficients such that their solutions can not be approximated by means of any kind of adaptive method with a polynomial rate of convergence. Even worse, we show that for any sequence $(a_n)_{n \in \mathbb N}\subset (0, \infty)$, which may converge to zero arbitrarily slowly, there exists an SDE  with bounded smooth coefficients such that no approximation method based on $n$ adaptively chosen evaluations of $W$ on average can achieve a smaller absolute mean error than the given number $a_n$. While the diffusion coefficients of these pathological SDEs are globally Lipschitz continuous, the first order partial derivatives of the drift coefficients are, essentially, of exponential growth. In the second part of the talk we show that sub-polynomial rates of convergence may happen even when the first order partial derivatives of the coefficients have at most polynomial growth, which is one of the typical assumptions in the literature on numerical approximation of SDEs with globally monotone coefficients.
The talk is based on joint work with Arnulf Jentzen (ETH Zürich) and Thomas Müller-Gronbach (University of Passau).

Sung-Soo Byun (Seoul National University/Universität Bielefeld)
CFT and SLE in a doubly connected domain

In this talk, I will present certain implementations of conformal field theory (CFT) in a doubly connected domain. The statistical fields in these implementations are generated by central charge modifications of the Gaussian free field with excursion reflected/Dirichlet boundary conditions. I will explain Ward’s equation in terms of a stress energy tensor, Lie derivative operators and the modular parameter. Combining Ward’s equation with the level 2 degeneracy equation for the boundary condition changing operator, I will outline the relation between CFT and Schramm-Loewner Evolution (SLE) theory. As applications, I will present a version of restriction property and Friedrich-Werner’s formula for annulus SLE and explain how to apply the method of screening to find explicit solutions to the partial differential equations for the annulus SLE partition functions introduced by Lawler and Zhan.
This is based on joint work with Nam-Gyu Kang and Hee-Joon Tak.

Sebastian Hummel (Universität Bielefeld)
Lines of descent in a deterministic model with mutation, selection and pairwise interaction

We consider a classical deterministic model for the evolution of a haploid population with two allelic types which is subject to mutation, selection, and a special form frequency-dependent selection. The deterministic model arises (also) as the large population limit of the Moran model, in which neither parameters nor time are rescaled. Despite the deterministic nature of this limiting process, the ancestry of single individuals in the population is still stochastic. In the case with mutation and selection, we describe it via a killed ancestral selection graph and connect it to the deterministic process via duality; this leads to a stochastic representation of the deterministic solution. In particular, the stationary state obtains a nice probabilistic interpretation. We generalise the construction to the case with frequency-dependent selection.

Thomas Kruse (Universität Duisburg-Essen)
Multilevel Picard approximations for high-dimensional nonlinear parabolic partial differential equations

In this talk we present a family of new approximation methods for high-dimensional PDEs and BSDEs. A key idea of our methods is to combine multilevel approximations with Picard fixed-point approximations. Thereby we obtain a class of multilevel Picard approximations. Our error analysis proves that for semi-linear heat equations, the computational complexity of one of the proposed methods is bounded by $O(d\,\eps^{-(4+\delta)})$ for any $\delta > 0$, where $d$ is the dimensionality of the problem and $\eps\in(0,\infty)$ is the prescribed accuracy. We illustrate the efficiency of one of the proposed approximation methods by means of numerical simulations presenting approximation accuracy against runtime for several nonlinear PDEs from physics (such as the Allen-Cahn equation) and financial engineering (such as derivative pricing incorporating default risks) in the case of $d=100$ space dimensions.
The talk is based on joint work with Weinan E., Martin Hutzenthaler and Arnulf Jentzen.

Lukas Knichel (RUB)
On the Rate of Convergence to a Gamma Distribution on Wiener Space

We consider a sequence $(F_n)$ of random variables living inside a fixed Wiener chaos of order $p$, converging in law to a centred $\chi^2$ distribution with $\nu$ d.f. It can be shown (using the Malliavin-Stein approach) that the rate of convergence is dominated by the differences of the third and fourth cumulants. More precisely, if $G(\nu)$ is a centred Gamma variable with parameter $\nu$, we have $$ d_{W}(F_n,G(\nu)) \leq_C \sqrt{\max ( \vert\kappa_3(F_n) - \kappa_3(G(\nu)) \vert, \vert \kappa_4(F_n) - \kappa_4(G(\nu))\vert ) }. $$ The same result (even in TV-distance) is available if we replace $G(\nu)$ by a standard Gaussian variable $N$. In 2015, I.Nourdin and G. Peccati showed that in this case, the \textit{exact} rate of convergence is given by the above-mentioned expression, if we omit the square root.
In this talk, we will discuss the question if we can also dispose of the square root in the case of a centred Gamma/$\chi^2$ random variable, and we will see by means of an explicit example, that this is not possible using the Malliavin-Stein approach.

Travelling wave solutions to the KPP equation with branching noise

The one-dimensional KPP-equation driven by space-time white noise, $$ \partial_t u = \partial_{xx} u + \theta u - u^2 + u^{\frac{1}{2}} dW, \qquad t>0, x \in \mathbb{R}, \theta>0, \qquad \qquad u(0,x) = u_0(x) \geq 0 $$ is a stochastic partial differential equation (SPDE) that exhibits a phase transition for initial non-negative finite-mass conditions. This SPDE arises for instance as the high density limit of particle systems which undergo branching random walks and allow for extra death due to overcrowding.

If $\theta$ is below a critical value $\theta_c$, solutions die out to $0$ in finite time, almost surely. Above this critical value, the probability of (global) survival is strictly positive. Let $\theta>\theta_c$. For initial conditions that are ‘’uniformly distributed in space’’, a complete convergence result holds, that is, the corresponding solutions are all in the domain of attraction of a unique non-zero stationary distribution. What can be said for solutions with finite initial mass if we condition on their survival?

In this talk I start with an overview of the main probabilistic ideas and arguments of the above mentioned concepts. In a next step, I explain how to obtain a travelling wave solution to this SPDE. Here, the choice of the wave front marker plays an important role. Finally, I outline why an understanding of the behaviour of travelling wave solutions can help to answer questions on convergence of solutions with arbitrary initial conditions.

Metastability in diffusion processes and synchronization

Non-standard time & location: 17:15 @ WSC-S-U-4.02

The phase diagram of the complex branching Brownian motion energy model

We complete the analysis of the phase diagram of the complex branching Brownian motion energy model by studying Phases I, III and boundaries between all three phases (I-III) of this model. For the properly rescaled partition function, in Phase III and on the boundaries I/III and II/III, we prove a central limit theorem with a random variance. In Phase I and on the boundary I/II, we prove an a.s. and $L^1$ martingale convergence. All results are shown for any given correlation between the real and imaginary parts of the random energy. This is joint work with Lisa Hartung (Courant Institute, NYU)

Time-discretization of stochastic 2-D Navier-Stokes equations by a penalty-projection method

In this talk, I will present a time-discretization method of the stochastic incompressible Navier--Stokes problem using a penalty-projection method. Basically, the talk will consist of three parts. A brief introduction of the mathematical problem. Then an overview of the main computational issue that shares the stochastic and the deterministic form of Navier--Stokes. Different algorithm will be introduced: a main algorithm and in order to treat the nonlinear character of the equation two auxiliary algorithms. At the end of the day we will arrive at the convergence with rate in probability and a strong convergence of the main algorithm.

Convex Hulls of Random Walks: Expected Number of Faces

Consider a random walk $S_i= \xi_1+\dots+\xi_i$, $1\leq i\leq n$, starting at $S_0=0$, whose increments $\xi_1,\dots,\xi_n$ are random vectors in $\mathbb R^d$, $d\leq n$. We are interested in the properties of the convex hull $C_n:=\mathrm{Conv}(S_0,S_1,\dots,S_n)$. Assuming that the tuple $(\xi_1,\dots,\xi_n)$ is exchangeable and some general position condition holds, we derive an explicit formula for the expected number of $k$-dimensional faces of $C_n$ in terms of the Stirling numbers of the first and second kind. Generalizing the classical discrete arcsine law for the position of the maximum due to E. Sparre Andersen, we compute explicitly the probability that for given indices $0\leq i_1 < \dots < i_{k+1}\leq n$, the points $S_{i_1},\dots,S_{i_{k+1}}$ form a $k$-dimensional face of $\mathrm{Conv}(S_0,S_1,\dots,S_n)$. This is done in two different settings: for random walks with symmetrically exchangeable increments and for random bridges with exchangeable increments. The main ingredient in the proof is the computation of the probability that the origin is absorbed by a joint convex hull of several random walks and bridges whose increments are invariant with respect to the action of a direct product of finitely many reflection groups of types $A_{n-1}$ and $B_n$. This probability, in turn, is related to the number of Weyl chambers of a product-type reflection group that are intersected by a linear subspace in general position. All formulae are distribution-free, that is do not depend on the distribution of the $\xi_k$'s. (Joint work with Vladislav Vysotsky and Dmitry Zaporozhets.)

Continuous spin models on annealed generalized random graphs

We study spin models where the spins take values in a general compact Polish space and interact via a pair potential along the edges of a generalized random graph with a given asymptotic weight distribution $P$, obtained by annealing over the random graph distribution. We prove a variational formula for the corresponding annealed pressure.

We furthermore study classes of models with second order phase transitions which include models on an interval and rotation-invariant models on spheres, and classify their critical exponents. We find critical exponents which are modified relative to the corresponding mean-field values when $P$ becomes too heavy-tailed, in which case they move continuously with the tail-exponent of $P$. For large classes of models they are the same as for the Ising model, but we provide conditions under which the model is in a different universality class, and construct an explicit example of such a model on the interval.

This is joint work with Christof Külske and Philipp Schriever.

Simple nested coalescents and two-level Fleming Viot processes

Simple nested coalescent has been introduced to model backwards in time the genealogy, of both, species trees and genes trees. In this setting, the Kingman case corresponds to binary coalescences in the species trees or the genes trees, but not simultaneously. On the other hand, two level Fleming Viot with two level selection arises in Dawson (2015) as the limit in distribution of multilevel multitype population undergoing mutation, selection, genetic drift and spatial migration. In this talk, I will establish a duality relation between Kingman nested coalescents and the two level Fleming Viot associated with a two-level multitype population with genetic drift. Using this relation we can read off the genealogy backwards in time of a Kingman nested.

Cutting-down random trees

Imagine that we destroy a finite tree of size n by cutting its edges one after the other and in uniform random order. We then record the genealogy induced by this destruction process in a random rooted binary tree, the so-called cut-tree. The goal of this talk will be to show a general criterion for the convergence of the rescaled cut-tree in the Gromov-Prohorov topology to a real tree, when the underlying tree has a small height. In particular, we consider uniform random recursive trees, binary search trees, scale-free random trees. The approach relies in the introduction of a continuous version of the cutting-down procedure which we allow us to represent the destruction process up to a certain finite time as Bernoulli bond percolation.

Stochastic domination in space-time for the supercritical contact process

The contact process is a classical model for the spread of infections in a population. In this talk, we focus on the contact process in the supercritical regime for which infections may spread forever with positive probability. Our goal is to understand how this process behaves compared with a process having no spatial correlations. In particular, does the contact process stochastically dominate a non-trivial independent (in space) spin-flip process? Such questions were studied by Liggett and Steif (2006) who proved that, for the process on $\mathbb{Z}^d$, the upper invariant measure stochastically dominates a Bernoulli product measure. We present some space-time versions of their results for the contact process on general graphs. From our methods, we furthermore conclude strong (uniform) mixing properties for certain space-time projections of the contact process. Based on joint work with Rob van den Berg (CWI Amsterdam and VU Amsterdam).

Non-asymptotic analysis of non-linear forms in random elements

We review recent results on non-asymptotic analysis for the distributions of quadratic and almost quadratic forms in random elements with values in a Hilbert space. The study of almost quadratic forms is motivated by approximation problems in multidimensional mathematical statistics. A number of results are optimal - they can not be improved without additional assumptions. A unified approach will be proposed for constructing non-asymptotic approximations on the basis of the general result on approximation accuracy for symmetric functions of several variables.

Low-frequency estimation for moving average Lévy processes

This talk is devoted to statistical inference for the stochastic integrals of the type $$ Z_{t}=\int_{R} K(t-s) dL_{s},$$ where $K$ is a deterministic function, and $L$ is a Lévy process. In particular, I will present several new ideas yielding the construction of a consistent estimator for the Lévy density of $L$. Moreover, I intend to discuss the mixing properties, which draw particular interest from the theoretical point of view.

Biased random walk in random environments

We explain two models for biased random walks in random environment (biased random walk on percolation clusters, biased random walk among random conductances) which describe transport in an inhomogenous medium. We give an overview of typical questions and present several results about Einstein relation and monotonicity of the speed.

Approximation techniques for probabilistic inference --
a statistical physics perspective

Probabilistic, Bayesian methods provide important approaches to data modelling in the field of machine learning. Inference on unobserved variables or parameters typically requires the performance of high-dimensional integrals or sums. If the dimensionality of the problem is very large, one often has to apply approximate inference methods which approximate intractable multivariate probability distributions by tractable ones.

In this talk I will focus on the so-called “Expectation Propagation” message passing techniques for approximate inference in latent Gaussian variable models. These are related to (and were, in parts motivated by) the “Thouless-Anderson-Palmer” (TAP) mean field approach to spin-glass models in physics.

I will give an introduction to the basic idea behind this approach and then discuss a combination of ideas from random matrix theory and dynamical functional methods of statistical physics which could be used to improve the efficiency of such methods and to study the convergence of inference algorithms for large systems.

Convergence of Markov processes to the invariant measure with applications to SPDEs and stochastic delay equations

While convergence of finite-dimensional Markov processes (e.g., SDEs) to the invariant measure is quite well understood by now, less is known about the convergence of infinite-dimensional Markov processes (e.g., SPDEs). In the first part of the talk we explain how the classical methods (which are based on the construction of a Lyapunov function) can be extended to study convergence of infinite-dimensional Markov processes in the Wasserstein metric. This generalizes recent results of M. Hairer, J. Mattingly, M. Scheutzow (2011). In the second part of the talk we provide some specific applications to SPDEs and stochastic delay equations and discuss the arising challenges. (Joint work with Alexey Kulik and Michael Scheutzow)

[1] O. Butkovsky (2014). Subgeometric rates of convergence of Markov processes in the Wasserstein metric. Annals of Applied Probability, 24, 526-552.

[2] O. Butkovsky, M. Scheutzow (2017). Invariant measures for stochastic functional differential equations. arXiv:1703.05120.

Exchangeable ultrametrics and tree-valued Fleming-Viot processes

The Donnelly-Kurtz lookdown model contains an evolving genealogy. The genealogical tree of the population at each time can be described by an isomorphy class of a metric measure space to obtain a tree-valued Fleming-Viot process. The states of this process can be viewed as ergodic components of the genealogical distance matrices. In this context, we also discuss a general representation for exchangeable ultrametrics in terms of sampling from marked metric measure spaces.

Stochastische Optimierungsstrategien fur Insider Trading auf einem zeitstetigen Finanzmarkt

Wir legen einen Wahrscheinlichkeitsraum $(\Omega, \mathcal{F},P)$ zugrunde, der die Entwicklung eines Finanzmarkts beschreibt. Auf diesem agiert neben den üblichen Teilnehmern ein Trader, der im Besitz nicht öffentlicher Information ist. Für diesen Insider wird seine optimale Investment Strategie bzgl. einer vorgegebenen Nutzenfunktion ermittelt. Anschließend wird untersucht wie sich der Verlust von Information auf die Strategie und den Nutzen auswirkt. Hierzu werden nicht injektive Funktionen auf die Zufallsvariable, die die Insider Information repräsentiert, angewandt. Es werden Identitäten entwickelt, die zur Gewinnung der neuen aus der alten Strategie dienen konnen.

BM on trees: a cover time bound

Simple symmetric random walk on a finite, connected graph is positive recurrent and the mean time to visit all vertices is finite. Recently, variable speed motions on metric measure trees have been constructed. In this talk we will raise the question what happens if we replace the assumption on the finiteness of the graph by completeness of the tree. (Joint with Omer Angel, Siva Athreya & Manjunath Krishnapur).

Asymptotic growth of supercritical Galton-Watson processes

The Kesten-Stigum theorem describes the growth of supercritical Galton-Watson processes. Its classical proof relies on the analysis of probability generating functions. We look at a conceptual proof due to Lyons, Pemantle and Peres that uses size-biased Galton-Watson trees and basic measure theory. We also discuss its generalization to the case of stationary and ergodic environments.

Nonlinear Schrödinger equations on spheres

We will discuss the Cauchy problem for Schrödinger equations with a nonlinearity of odd order posed on spheres of arbitrary dimension. The methods used are multilinear eigenfunction estimates and a spectral decomposition of the Laplacian, which allows the construction of suitable function spaces. In these spaces we can prove space-time estimates of Strichartz type. As the main result, we will see that the Cauchy problem is locally well-posed for initial data in an appropriate Sobolev space. This extends previous results of Herr and Burq-Gérard-Tzvetkov.

Non-standard weekday (Wednesday), time (16:15) & place (WSC-S-U-4.01)

Structural adaptive dimension reduction for musical signals modelling

In the talk we discuss problem of simplification of musical signals which arise in cochlear implant music transmission. The main peculiarity of musical signals is highly varying temporal and spectral structure. Also the hearing loss and the hearing aid limitations should be used as an a-priori knowledge. The simplification of the music signals may be achieved by an adaptive dimension reduction of spectral data and by the extraction of important musical features. To this end, we develop a novel unsupervised segmentation procedure for music signals which relies on an explained variance criterion in the eigenspace of the constant-Q spectral domain.

Speciation and recombination on fitness landscapes

The number of fitness maxima in the random NK model

Visiting Session @ Rhein-Ruhr SPP Seminar
16:00 @ Ground floor lecture hall, Biozentrum, Universität zu Köln
(Building 304, Zuelpicher Strasse 47b, D-50674 Cologne)

Super-replication with fixed transaction costs

We study super-replication of contingent claims in markets with fixed transaction costs. The first result in this paper reveals that in reasonable continuous time financial market the super–replication price is prohibitively costly and leads to trivial buy–and–hold strategies. Our second result is derives non trivial scaling limits of super–replication prices in the binomial models with small fixed costs. (Joint work with P.Bank)

Non-standard weekday (Thursday), time (10:15), place (WSC-S-U-3.03)

State spaces of (continuum) trees: R-trees versus Algebraic Trees

In order to construct tree-valued stochastic processes, one needs a topological space of trees as state space. While this is not an issue for finite (or countable) graph-theoretic trees, we want to consider global limits as the number of vertices tends to infinity. We call the limiting objects also "tree" but have to make precise what we mean by this. The standard approach is to consider continuum trees to be metric (measure) spaces (so-called $\mathbb{R}$-trees) and equip the space of them with Gromov-Hausdorff or Gromov-weak topology.

We argue that sometimes it is more natural to consider a different type of structure, namely the tree-structure instead of the distance. First, because distances may behave very ``wild'' in certain cases and neglecting them makes proving some limit results more feasible. Second, because sometimes one might want to preserve structural properties such as being binary in the limit.

We present a framework for a space of such (continuum) trees possessing no metric- but only a tree-structure. We call them algebraic trees, because we formalise the tree-structure by a tertiary operation on the tree, namely the branch point map. We construct a natural, topology on spaces of sufficiently nice algebraic trees. In the binary case, the resulting space is compact and intimately related to the set of triangulations of the circle as introduced by Aldous, equipped with the Hausdorff metric.

Visiting Session @ Rhein-Ruhr SPP Seminar
16:30 @ Ground floor lecture hall, Biozentrum, Universität zu Köln
(Building 304, Zuelpicher Strasse 47b, D-50674 Cologne)

Visiting Session @ Rhein-Ruhr SPP Seminar
16:00 @ Ground floor lecture hall, Biozentrum, Universität zu Köln
(Building 304, Zuelpicher Strasse 47b, D-50674 Cologne)

Bootstrap confidence sets for spectral projectors of sample covariance

Alexey Naumov (Skoltech), Vladimir Ulyanov (MSU)

Let $X_1, \ldots ,X_n$ be i.i.d. sample in $\mathbb{R}^p$ with zero mean and the covariance matrix $S$. The problem of recovering the projector onto the eigenspace of $S$ from these observations naturally arises in many applications. Recent technique from [Koltchinskii and Lounici, 2015b] helps to study the asymptotic distribution of the distance in the Frobenius norm between the true projector $P_r$ on the subspace of the $r$-th eigenvalue and its empirical counterpart $\hat{P}_r$ in terms of the effective trace of $S$. This paper offers a bootstrap procedure for building sharp confidence sets for the true projector $P_r$ from the given data. This procedure does not rely on the asymptotic distribution of $\Vert P_r - \hat{P}_r \Vert_2$ and its moments, it applies for small or moderate sample size $n$ and large dimension $p$. The main result states the validity of the proposed procedure for finite samples with an explicit error bound on the error of bootstrap approximation. This bound involves some new sharp results on Gaussian comparison and Gaussian anti-concentration in high dimension. Numeric results confirm a nice performance of the method in realistic examples. These are the joint results with V.Spokoiny.

Non-standard time & location: 12:00 @ WSC-O-4.43

Convergence of Brownian motions on metric measure spaces under the Riemannian Curvature-Dimension condition

The goal of this talk is to characterize the weak convergence of Brownian motions in terms of a geometric convergence of the underlying spaces. As main results, we show several equivalences between the weak convergence of Brownian motions and the pointed measured Gromov convergence of the underlying spaces satisfying the Riemannian Curvature-Dimension (RCD) condition. Here the RCD condition is a generalization of "Ricci curvature $\geq K$" and admits various singular spaces like the Gromov-Hausdorff limit of Riemannian manifolds with Ric $\geq K$, or several infinite-dimensional spaces.

The recovery of a recessive allele in a Mendelian diploid model

Optimal hedging and risk contribution in energy markets via coherent risk measures

Let us consider an energy company that sells a fluctuating volume $V$ of energy units at random price $S$ per unit and obtains the income $X=VS$. Today's markets do not provide simple instruments that can be used for hedging volumetric risk, i.e. the risk that appears due to the randomness in $V$. In the current literature hedging of volumetric risk is settled via utility functions. Here we consider this task using coherent risk measures and obtain explicit hedging strategies in some cases. If we extract oil in one country and sell it in another one, then we additionally have currency risk. Motivated by the dependence between oil prices and currency exchange rates we settle a problem of hedging both currency and volumetric risks via multidimensional coherent risk measures, with put options on oil prices as hedging instruments, and present its solution in some situations.

Distributional properties of the stochastic heat equation via Malliavin Calculus

Malliavin Calculus is an infinite dimensional calculus defined on the Wiener space. It was developed in 1976 by Paul Malliavin. It has been used to study regularity properties (conditions for being absolutely continuos and/or having smooth density) of functionals defined on the Wiener space as well as in some applications in finance (via the Clark-Ocone formula). In this talk we consider some properties of the Mallivin derivative and its adjoint operator, known as the Skorokhod integral. We then apply this theory to study distributional properties of the Stochastic Heat Equation. More specifically, we will see how this theory helps us find conditions under which the solution to this equation has a smooth density.

Random processes conditioned to stay positive and occupation times

Random walks provide some of the most rich examples of stochastic processes. Starting with a random walk with real values (zero mean and finite variance), we are interested in finding out how it behaves if we restrict the state space. In particular, what happens when we are only interested in the positive real line. This version of the process is known as random walk conditioned to stay positive. Given the analogies between random walks with finte variance and Brownian motion, we also conditioned the latter to stay positive. The resulting process can be seen as 3-d Bessel process. Once these conditioned processes have been introduced we analyse the asymptotic behaviour of their occupation times; to be more precise, we seek for deterministic functions that describe the limiting behaviour of these processes.

Limiting behavior of super-replication prices with small transaction costs: multi-asset version of Kusuoka results

We consider super-replication with small transaction costs in complete multi-asset multinomial markets. For this setup we prove that the limit of the corresponding super-replication prices equals to a multidimensional $G$-expectation which we find explicitly. These results can be interpreted as a multidimensional extension of Kusuoka (1995). This is a joint work with P.Bank and A.P.Perkkio.

High-dimensional percolation

Percolation is one of the simplest ways to define models in statistical physics and mathematics which displays a non-trivial critical behaviour. This model describes how a graph/lattice/network behaves under random removal of edges. In the last 40+ years, it has been an active field of research due to richness of the behaviour of the model and due to its numerous applications. We introduce the classical version of the model on the lattice $\mathbb{Z}^d$, where percolation has the remarkable feature that it undergoes a sharp phase transition: There exists a critical value $p_c = p_c(d) \in (0,1)$ such that if you randomly retain a fraction of less than $p_c$ edges (i.e., remove a fraction of more than $1-p_c$ of all edges in i.i.d. manner), then the resulting graph consists of finite connected components only. Conversely, if you retain a fraction of more than $p_c$ edges (i.e., randomly remove a smaller than $1-p_c$ fraction of edges), then the resulting graph will contain exactly one infinite connected component. What happens at criticality, i.e., if we randomly retain exactly the critical fraction of $p_c$ edges is only understood for $d = 2$ (where $p_c(2) = 0.5$) and for $d > 10$ (where $p_c(d) \approx 1/(2d-1)$). In the talk, we review known results about percolation and give an idea of how to obtain results for $d > 10$.

Perpetual integrals for Lévy processes

Due to time-change properties of stochastic differential equations, it is important to understand finiteness properties of time-integrals of Brownian motion (or Lévy Processes). We show how to use fluctuation theory and Jeulin’s lemma to prove a 0-1 law for time-integrals of Lévy processes with local times.

Time dependent perturbations of symmetric Markov processes and related evolution problems

A diffusion subject to a deterministic movement and and external supply of particles can be modelled by a Brownian motion with drift and killing. In probability this can be studied by Girsanov and Feynman-Kac transforms, in analysis by a heat equation with first order and potential terms. If the drift depends on time, we can no longer operate with one-parameter semigroups, and if a related Markov process exists, it will not be homogeneous. Combining evolution semigroups, time-dependent Dirichlet forms and space-time processes we deduce probabilistic representations for classical solutions of abstract evolution problems. The result generalizes a formula well known for Brownian motion and applies for instance to Lévy processes, Random walks or diffusions on fractals. The talk is based on joint work with Gerald Trutnau, SNU Seoul.

The allelic spectrum and its applications on genomic data and to an epidemiological model of host-parasite coevolution

Advances in empirical population genetics have made apparent the need for models that simultaneously account for selection and demography. To address this need, I consider the Wright-Fisher diffusion under selection and piecewise constant population sizes. In the case of genic selection, I will sketch the derivation of the transition density. For general diploid selection, I will apply a moment-based approach to devise an efficient and fast algorithm for the computation of the allele frequency spectrum. I will discuss several applications that are of interest for the analysis of whole-genome sequences. In respect of a further application, I will introduce and discuss a system of coupled differential equations describing the coevolution of host and parasite alleles.

Higher order variance reduction for discretised diffusions via regression

A novel approach towards variance reduction for discretised diffusion processes is presented. The proposed approach involves specially constructed control variates and allows for a significant reduction in the variance of the terminal functionals. In this way the complexity order of the plain Monte Carlo algorithm ($\varepsilon^{-3}$ in the case of a first order scheme and $\varepsilon^{-2.5}$ in the case of a second order scheme) can be reduced down to $\varepsilon^{-2+\delta}$ for any $\delta\in [0,0.25)$ with $\varepsilon$ being the precision to be achieved. These theoretical results are illustrated by several numerical examples.

Upper and lower estimates for boundaries in non-linear optimal stopping problems

We consider a sequential testing problem of two hypotheses concerning the drift value of a fractional Brownian motion. We show that it can be reduced to the optimal stopping problem for a standard Brownian motion with non-linear cost of observation. Using standard technique one may characterize optimal stopping boundaries as an unique solution of a system of non-linear integral equations, which can be used for numerical evaluation via backward induction technique. However, for this, one needs to have some end-point T, after which the values of optimal boundaries are known. The main point of the talk is how to obtain some upper and lower estimates for the boundaries which provide the choice of such a T. In contrast to known methods based on the study of corresponding free-boundary problem, our approach is more probabilistic.

Bounds for the expected maximum of a fractional Brownian motion and related processes

We study expected maxima of Gaussian processes that are Holder continuous in L2-norm and/or satisfy the opposite inequality for the L2-norms of their increments. Examples of such processes include the fractional Brownian motion and some of its relatives. We establish upper and lower bounds for the expected maximum of such a process and investigate the rate of convergence to that quantity of its discrete approximation. Further properties of these two maxima are established in the special case of the fractional Brownian motion. This is a joint work with Konstantin Borovkov, Yulia Mishura and Alexander Novikov.

Maximising functions of the average of a martingale with given terminal law

In this talk, we consider the problem of finding the martingale which maximises the expected value of some function of the average of the martingale, subject to a given terminal law of the process. This is a problem which arises in model-independent pricing of financial derivatives, and is commonly known as a Martingale Optimal Transport problem. Our methods differ from most approaches to these problems in that we consider the problem as a dynamic programming problem, where the controlled process is the conditional distribution of the martingale at the final time. By formulating the problem in this manner, we are able to determine the maximal price through a PDE formulation. Notably, this approach does not require specific constraints on the function (e.g. convexity), and would appear to be generalisable to many related problems. (Joint work with S. Källblad).

On trading American put options with interactive volatility

We introduce a simple stochastic volatility model, which takes into account hitting times of the asset price, and study the optimal stopping problem corresponding to a put option whose time horizon (after the asset price hits a certain level) is exponentially distributed. We obtain explicit optimal stopping rules in various cases one of which is interestingly complex because of an unexpectedly disconnected continuation region. Finally, we discuss in detail how these stopping rules could be used for trading an American put when the trader expects a market drop in the near future.

Asymptotic lower bounds for optimal tracking: a linear programming approach

We consider the problem of tracking a target whose dynamics is modeled by a continuous Ito semimartingale. The aim is to minimize both deviation from the target and tracking efforts. We establish the existence of asymptotic lower bounds for this problem, depending on the cost structure. These lower bounds can be related to the time-average control of Brownian motion, which is characterized as a deterministic linear programming problem. A comprehensive list of examples with explicit expressions for the lower bounds is provided.

Nash equilibria of threshold type for two-player nonzero-sum games of stopping

In this talk I consider two-player nonzero-sum games of optimal stopping on a class of regular diffusions with singular boundary behaviour (in the sense of Itô and McKean, p. 108). I show that Nash equilibria are realised by stopping the diffusion at the first exit time from suitable intervals whose boundaries solve a system of algebraic equations. Under mild additional assumptions we also prove uniqueness of the equilibrium. The talk is based on a joint work with Tiziano De Angelis and John Moriarty.

Dynamic contracting under moral hazard

We consider a contracting problem that a firm faces in the presence of managerial moral hazard and stochastic cashflows. We first develop a general contracting setting. We then restrict attention to contracts that admit appropriate state space representations. In the latter context, we establish the link between the optimal contract and the solution to a suitable stochastic control problem.

Fine integral representations through small deviations of quadratic variation

The talk is based on a joint paper with Taras Shalaiko (Mannheim University). It will be devoted to stochastic integral representations of the form $$\xi = \int_0^T \phi(s) dB^H(s),$$ where $\phi$ is an adapted process, and $B^H$ is a fractional Brownian motion with Hurst parameter $H>1/2$. Studying such representations is motived by applications in financial mathematics, where $\phi$ plays a role of the risky component of a self-financing portfolio. I will discuss how the estimates for probabilities of small deviations of realised quadratic variation of $B^H$ can be used to construct the above integral representations under some mild assumptions on $\xi$. To this end, we define a generalisation of the fractional integral introduced by Martina Zähle.

Sweeps in the ancestral selection graph

We will see, why and how the ancestral selection graph can be used to study fixation of beneficial alleles in structured populations.

Invariance of closed convex cones for stochastic partial differential equations

In this talk, we provide necessary and sufficient conditions for stochastic invariance of closed convex cones in Hilbert spaces for semilinear stochastic partial differential equations driven by Wiener processes and Poisson random measures. Several examples accompany our results.

Perturbation theory for Markov chains via Wasserstein distance

Perturbation theory for Markov chains addresses the question how small differences in the transitions of Markov chains are reflected in differences between their distributions. We prove powerful and flexible bounds on the distance of the n-th step distributions of two Markov chains when one of them satisfies a Wasserstein contractivity condition.

Our work is motivated by the recent interest in approximate Markov chain Monte Carlo (MCMC) methods in the analysis of big data sets. By using an approach based on Lyapunov functions, we provide estimates for geometrically ergodic Markov chains under weak conditions. In an autoregressive model, our bounds cannot be improved in general. We illustrate our theory by showing quantitative estimates for approximate versions of two prominent MCMC algorithms, the Metropolis-Hastings and stochastic Langevin algorithms.

This is joint work with Daniel Rudolf (University of Jena)

Hedging price risk in commodity markets in presence of volumetric and currency risk

Let us consider an energy company that sells fluctuating volume $V$ of energy units at random price~$S$ per unit and obtains the income $X=VS$. Markets today do not provide simple instruments that can be used for hedging volumetric risk (See~\cite{OOD06}). Traditional measure of risk which is widely used in practice is V@R. We consider the problem of V@R minimization in incomplete markets using put options. We try to find the optimal strategy in $h\in {\mathbb R}$ that solves the following problem: $$ V@R\Bigl(V * S + h ((K-S)^+ - EP(K))\Bigr) \rightarrow \min_{h\in{\mathbb R},K\in {\mathbb R}_{+}}, $$ (due to incompleetness of commodity markets we could not make a perfect hedge). Here we introduce the solution of finding optimal number of put options bought with strike $K$ to minimize V@R in some cases (also the case of ''natural hedging'' is also introduced). To solve it we use the following function: $$q(K)(a,h)={\sf P}(VS+h((K-S)^{+}-P(K))\leq a)$$ and investigate its properties. So we try to find optimal hedge in presence of volumetric risk. However when you extract oil in one country and sell it in another you also have currency risk. So you need to solve the same task in presence of currency risk where usually only price risk is traded. Here we provide some fundamental facts which allow to model currency exchange rate and oil price together (correlation does not provide good results in this case). Using this fact we provide some examples of optimal hedging in presence of currency and volume risk using put options on oil prices.

Optimal stopping problems for a Brownian motion with drift and disorder: Applications to financial mathematics.

We consider solutions of the several sequential problems for a Brownian motion. For example, we show how to solve Chernoff's problem of testing statistical hypotheses about the sign of drift. We show how to obtain a solution to the problem of sequential testing of three statistical hypotheses about the drift of a Brownian motion. We consider financial problems of the type "When to sell Apple?" and some other problems.

Implied Volatilities from Strict Local Martingales

Several authors have proposed to model price bubbles in stock markets by specifying a strict local martingale for the risk-neutral stock price process. Such models are consistent with absence of arbitrage (in the NFLVR sense) while allowing fundamental prices to diverge from actual prices and thus modeling investors’ exuberance during the appearance of a bubble. We show that the strict local martingale property as well as the “distance to a true martingale” can be detected from the asymptotic behavior of implied option volatilities for large strikes, thus providing a model-free asymptotic test for the strict local martingale property of the underlying. This talk is based on joint work with Antoine Jacquier.

Regularity of Cox-Ingersoll-Ross processes in the case of an accessible boundary point.

Cox-Ingersoll-Ross (CIR) processes are widely used in financial modeling such as in Heston's stochastic volatility model as the variance process for the pricing of financial derivatives. In Heston's stochastic volatility model, it is commonly assumed that the parameters of the variance process satisfy Feller's boundary condition. As a consequence, the variance process is strictly positive. On the other side, if Feller's boundary condition is not satisfied, we can show for half of the parameter regime in which the boundary point 0 is accessible that the variance process spends $\mathbb{P}$-almost surely only Lebesgue time $0$ at the boundary point $0$. Since calibrations of the Heston model frequently result in parameters such that the boundary is accessible, we focus on this interesting case. In the literature, positive strong convergence rates for numerical approximations of CIR processes have been established in the case of an inaccessible boundary point. Our main result shows for every $p\in(0,\infty)$ that the drift-implicit square-root Euler approximations proposed in Alfonsi (2005) converge in the strong $ L^p $-distance with a positive rate for half of the parameter regime in which the boundary point is accessible. Moreover, we prove local Lipschitz continuity in the initial value for CIR processes in the case where the boundary point $0$ is accessible.

SPDEs, criticality and renormalisation

In this colloquium I will report on recent progress in the theory of stochastic PDEs and their connections with statistical mechanics. My main examples will be the KPZ equation and the dynamic $\Phi^4$ model. Both of these equations arise as scaling limits of interacting particle systems: the KPZ equation as a scaling limit for surface growth models and the $\Phi^4$ model as a scaling limit of an Ising-type model of a ferromagnet. The mathematical treatment of these equations is challenging due to the low regularity of the random ``noise term'' present. In particular, solutions are too irregular to apply the usual ``deterministic'' solution techniques. Even worse sometimes ``infinite counterterms'' have to be subtracted to produce non-trivial solutions. I will discuss this renormalisation procedure in some detail for the $\Phi^4$ model. I will first explain, why this procedure is necessary and how it can be implemented mathematically. Then I will discuss the interpretation on the level of particle approximations.

Galerkin approximations for stochastic partial differential equations

In this talk I give an introduction to Finite Element approximations for parabolic and hyperbolic stochastic partial differential equations. I demonstrate that in the parabolic case, due to the nice properties of the parabolic operator, Galerkin approximations give the expected convergence results (expected as for deterministic equations). In the case of an hyperbolic equation, however, some artificial smoothing has to be employed, since the driving noise process and therefore the solution is too irregular for a (standard) Galerkin approximation. For an approximation of these problems, I introduce a (stochastic) version of a Streamline Diffusion method.

Controlling the occupation time of a geometric martingale

We consider control problems in which the volatility of a geometric martingale can be chosen to take any value between a minimal and a maximal value. Special attention is paid to the strategy that consists in choosing the minimal volatility if the martingale is above a given threshold and to choose the maximal volatility else (play safe if ahead, take risks if behind).

Semiparametric estimation in the normal variance-mean mixture model

In this talk, I intend to present some fresh ideas concerning the estimation in the normal variance-mean mixture models. These models are closely related to the class of time-changed L{\'e}vy processes, and naturally appear in some interesting problems like modelling of the sizes of diamonds in marine deposits of South West Afrika. The focus of our research is the simultaneous estimation of all finite-dimensional parameters of the model and the mixing distribution.

Numerical approximations of stochastic partial differential equations with superlinearly growing nonlinearities

In this talk, we study numerical approximations of stochastic partial differential equations (SPDEs) with superlinearly growing nonlinearities. We first give examples of some of such SPDEs, including stochastic Burgers equations, stochastic 2-D Navier-Stokes equations, Cahn-Hilliard-Cook type equations, and stochastic Kuramoto-Sivashinsky equations. We then show that the fully discrete exponential Euler method converges in the almost sure sense with some rate for all of the aforementioned examples of SPDEs. Finally, we will discuss about a work in progress on how to use this result to show that a newly proposed scheme, called the fully discrete nonlinearity-stopped exponential Euler method, converges strongly in the $L^p$-norm for all $p\in(0,2)$ for all of the aforementioned examples of SPDEs.

Random orthogonal projections, universality, and the smallest singular values of random matrices

We start with a particular problem that arises in econometrics and deals with random orthogonal projection matrices. We show that this problem is intimately related to the Marchenko-Pastur theorem and, in general, the universality phenomenon in the random matrix theory. We also discuss some lower bounds on the smallest singular values of random matrices that are needed to solve the above problem, but can be of independent interest in a much more general context.

Genetic Variability in the Mendelian Diploid Model

Adaptive dynamics are well studied for the haploid reproduction model. In this talk we are interested in the genetic evolution of a diploid hermaphroditic population, which is modeled by a three-type nonlinear birth-and-death process with competition and Mendelian reproduction. In a recent paper, Collet, Méléard and Metz (2013) have shown that on the mutation time-scale the process converges to the Trait-Substitution Sequence of adaptive dynamics, stepping from one homozygotic state to another with higher fitness.

In the first part of the talk, we prove that, under the assumption that the dominant allele is also the fittest one, the recessive allele survives for a time of order at least $K^{1/6}$, where $K$ is the size of the population.

In the second part we give a sufficient condition on the competition such that the recessive a- allele can recover after the occurrence of a new fitter mutant. This leads to genetic variability in the populations.

Dr. Fabian Dickmann (University of Duisburg-Essen)

Pricing Bermudan options via multi-level approximation methods

This talk is about an approach to reduce the computational complexity of various approximation methods for pricing Bermudan options. Given a sequence of continuation value estimates corresponding to different levels of spatial approximation, we propose a multi-level low biased estimate of the price. It turns out that the resulting complexity gain can be of order $1 / \epsilon$ where $\epsilon$ denotes the desired precision measured in terms of the root-mean-squared error.

Dr. Thomas Kruse (Université d'Evry Val d'Essonne)

A generalized Donsker theorem and approximating SDEs with irregular coefficients

We provide a new method for approximating the law of a diffusion $M$ solving a stochastic differential equation with coefficients satisfying the Engelbert-Schmidt conditions. To this end we construct Markov chains whose law can be embedded into the diffusion M with a sequence of stopping times that have expectation $1/N$, where $N \in \mathbb{N}$ is a discretization parameter. The transition probabilities of the Markov chains are determined by a reference probability measure $\mu$, scaled with a factor depending on $N$ and the state. We show that the Markov chains converge in distribution to the diffusion $M$, as $N \to \infty$, thus refining the Donsker-Prokhorov invariance principle. Finally, we illustrate our results with several examples.

The talk is based on joint work with Stefan Ankirchner and Mikhail Urusov.

Tigran Nagapetyan (Wias Berlin)

Weak Multilevel Monte Carlo algorithm

We discuss the possibility of using multilevel Monte Carlo (MLMC) methods for weak approximation schemes. It turns out that by means of a simple coupling between consecutive time discretisation levels, one can achieve the same complexity gain as under the presence of a strong convergence. This general approach can be used in a wide range of problems, while we exemplify this idea in the case of weak Euler scheme for Lévy driven stochastic differential . The numerical performance of the new "weak" MLMC method is illustrated by several examples.
The talk is based on joint work with Denis Belomestny.

Lisa Hartung (University of Bonn)

Extended convergence of the extremal process of branching Brownian motion

Branching Brownian motion is a popular example for a Gaussian process indexed by a tree. A property of key interest is the structure of the process near its maximum for large times. In this talk I recall the construction of the extremal process of branching Brownian motion obtained by Arguin, Bovier and Kistler. Then I explain how this convergence can be extended by adding an extra dimension that encodes the "location" of the particle in the underlying Galton-Watson tree. (joint work with A. Bovier)

Paul Krühner (TU Dortmund)

Optimal density bounds for SDE's with discontinuous drift coefficients

In this talk we study the regularity of solutions to the SDE $dX(t) = b(t,X(t))dt + a(t,X(t))dW(t)$ in a finite dimensional space where $b$ is only assumed to be measurable and bounded. Malliavin invented a great method to study the density properties -- like boundedness of the density -- of $X(t)$ under smoothness assumptions. His approach has been generalised in various directions and to various different applications. We find sharp upper and lower bounds for the density without using any type of variational calculus. This talk is based on joint work with David Baños.

Anselm Hudde (University of Freiburg)

Malliavin Calculus and the Calculation of Greeks

In this talk we introduce the Malliavin Calculus for Isonormal Gaussian processes. This allows us to differentiate and integrate random variables with respect to the white noise that is generated by an Isonormal Gaussian process. We present some of its main properties, including an integration by parts formula. With this formula we can find stochastic weights such that option sensitivities in the Black Scholes model can be easily calculated without differentiating the payoff function and without knowledge of the density of the underlying asset.

Felix Jordan (LMU München)

Evolution of Altruistic Defense Traits in Structured Populations

An altruist is an individual that helps others at having more offspring while paying the cost of having less offspring itself. This would intuitively suggest that bearers of genes for altruism have a disadvantage compared to non-altruists and, therefore, become extinct. Nevertheless, it is known that in a structured population, altruists can prevail, if the benefit to other altruists outweighs the cost to the individual. In the case of altruism taking the form of defense against a parasite, there is a negative feedback for altruists. When altruists become abundant, they reduce the number of parasites, which in turn reduces the benefit that each altruist brings. It is not a priori clear how this mechanism affects the evolution of the defense trait. We analyze this scenario in an island model using diffusion theory. In the limit of infinitely large populations on infinitely many islands, we obtain a condition determining whether a gene for altruism becomes extinct or fixates in the host population.

Dr. Sonja Cox (University of Amsterdam)

Burkholder-Davis-Gundy inequalities for vector-valued stochastic integrals

Stochastic integrals of processes taking values in a Banach space are used to study the solutions to stochastic partial differential equations and obtain regularity results that are not accessible using Hilbert space theory only. A key ingredient for defining these stochastic integrals are the Burkholder-Davis-Gundy inequalities. These inequalities however do not hold in all Banach spaces. In my talk I will discuss different types of decoupling inequalities for random sums in a Banach space, by means of which one can characterize the spaces in which the BDG inequalities hold. Some characterizations are well-known, and some were established in recent work by Stefan Geiss and myself.

Dr. Martin Sauer (TU Berlin)

Analysis and Approximation of Stochastic Nerve Axon Equations

This talk concerns spatially extended conductance based neuronal models with noise described by certain stochastic reaction diffusion equations with only locally Lipschitz continuous nonlinearities. We study the wellposedness of such equations, the approximation in space using the finite difference method and derive explicit error estimates, in particular pathwise and strong convergence rates.

PD Dr. Volker Krätschmer (University of Duisburg-Essen)

Optimal stopping under model uncertainty

In the talk we consider optimal stopping problems with conditional convex risk measures of the form $ \rho^{\Phi}_t(X)=\sup_{\mathbb{Q}\in {\cal Q}_{t}}\left(\mathbb{E}_{\mathbb{Q}}[X|\mathcal{F}_t]- \mathbb{E}\left[\Phi\left(\frac{d\mathbb{Q}}{d\mathbb{P}}\right)\big|{\cal F}_{t}\right]\right),$ where $\Phi: [0,\infty[\rightarrow [0,\infty]$ is a lower semicontinuous convex mapping and ${\cal Q}_{t}$ stands for the set of all probability measures ${\cal Q}$ which are absolutely continuous w.r.t. a given measure $\mathbb{P}$ and $\mathbb{Q}= \mathbb{P}$ on \({\cal F}_{t}.\) Here the model uncertainty risk depends on a (random) divergence $\mathbb{E}\left[\Phi\left(\frac{d\mathbb{Q}}{d\mathbb{P}}\right)\big|{\cal F}_{t}\right]$ measuring the distance between a hypothetical probability measure we are uncertain about and a reference one at time $t.$ Let \((Y_t)_{t\in [0,T]}\) be an adapted nonnegative, right-continuous stochastic process fulfilling some proper integrability condition and let \({\cal T}\) be the set of stopping times on \([0,T]\), then without assuming any kind of time-consistency for the family \((\rho_t^{\Phi}),\) we derive a novel representation \begin{eqnarray*} \sup_{\tau\in {\cal T}}\rho^{\Phi}_0(Y_\tau)= \inf_{x\in\RRR}\left\{\sup\limits_{\tau\in{\cal T}}\mathbb{E}\bigl[\Phi^*(x + Y_{\tau}) - x\bigr]\right\}, \end{eqnarray*} which makes the application of the standard dynamic programming based approaches possible. In particular, we generalize the additive dual representation of Rogers (2002) to the case of optimal stopping under uncertainty. The talk is based on a joint work with Denis Belomestny.

Non-parametric Bayesian inference for multi-dimensional compound Poisson processes

Given a sample from a discretely observed multi-dimensional compound Poisson process, we study the problem of non-parametric estimation of its jump size density $r_0$ and intensity $\lambda_0.$ We take a non-parametric Bayesian approach to the problem and determine posterior contraction rates in this context, which, under some assumptions, we argue to be optimal posterior contraction rates. In particular, our results imply existence of Bayesian point estimates that converge to the true parameter pair $(r_0,\lambda_0)$ at these rates.

Extensions of Transition Semigroups onto $L^p$-spaces and Applications to SDE's

Given both, a Markov semigroup $(P_t)_{t\geq0}$, a priori defined on the bounded and Borel measurable functions, $B_b(E)$, over a metric space $E$, with corresponding Markov kernels $(p_t(x,\cdot))_{t\geq0,x\in E}$, and a semigroup $(R_t)_{t\geq0}$ of linear Operators, defined on $B_b(E)$, of the shape $$R_t\varphi(x)=\int_E\varphi(y)r_t(x,dy)\quad x\in E,\varphi\in B_b(E), t\geq0,$$ with corresponding finite transition kernels $(r_t(x,\cdot))_{t\geq0,x\in E}$, satisfying $r_t(x,\Gamma)\leq e^{at}p_t(x,\Gamma)$ for all $t\geq0$, $x\in E$, $\Gamma\in\mathscr{B}(E)$ and some constant $a\in\mathbb{R}$, we ask under which conditions the semigroup $(R_t)_{t\geq0}$ can be extended from $B_b(E)\cap L^p(E,\mathscr E,\mu)$ onto $L^p(E,\mathscr E,\mu)$, $p\in[1,\infty)$, such that the extension is even strongly continuous. In the talk we give sufficient conditions, slightly generalizing existing results by Da Prato and Zabczyk, under which we can answer in the affirmative. To illustrate the results we will apply them to Lévy-semigroups and some stochastic differential equations.

An equilibrium model for commodity spot and forward prices

The aim of this project is to determine the forward price of a consumption commodity via the interaction of agents in the spot and forward commodity market. We consider a market model that consists of three agents: producers of the commodity, consumers and financial investors (sometimes also called speculators). Producers produce a fixed amount of the commodity at each time point, but can choose how much they offer in the spot market and store the rest for selling at the next time period. They also have a position in forward contracts in order to hedge the commodity price uncertainty. Consumers are setting the spot price of the commodity at each time point by their demand. Finally, investors are investing in the financial markets and, in order to diversify their portfolios, also in the forward commodity market. The equilibrium prices for the commodity are the ones that clear out the spot and forward markets. We assume that producers and investors are utility maximizers and have exponential preferences, while the consumers' demand function is linear. Moreover, the exogenously priced financial market and the demand function are driven by Lévy processes. We solve the maximization problem for each agent and prove the existence of an equilibrium. This setting allows to derive explicit solutions for the equilibrium prices and to analyze the dependence of prices on the model parameters and the agent's risk aversion. This is joint work with Michail Anthropelos and Michael Kupper.

Dr. Mikhail Zhitlukhin (Steklov Mathematical Institute Moskow)

Testing complex hypotheses about the drift of a Brownian motion

We consider two problems of sequential testing of the hypotheses whether the drift of a Brownian motion is positive or negative. The sequential setting assumes that a decision rule consists of a stopping time when the observation is terminated and a decision function choosing the hypothesis to accept. An observer is penalized for the duration of observation and a wrong decision. The goal is to find the optimal trade-off between the duration and the chance of a wrong decision.

The first criterion of optimality we consider was proposed by H. Chernoff and assumes the unknown drift is a priori normally distributed and one needs to minimize the sum of the observation time and the penalty for a wrong decision proportional to the absolute value of the drift. The second criterion, proposed by J. Kiefer and L. Weiss, minimizes the maximum mean observation time for all possible values of the drift parameter among criteria that have fixed probability of a wrong decision.

In the both problems, asymptotically optimal decision rules were obtained long ago. The new result presented in this talk concerns exact optimal decision rules. We show that they can be found from solving some optimal stopping problems and obtain them in an explicit form.

Dr. Alexey Muravlev (Steklov Mathematical Institute Moskow)

On the methods of sequential hypothesis testing for a fractional Brownian motion in the Bayesian setting

The problem of sequential hypothesis testing lies at the origins of sequential analysis and was considered by many authors. Among the most well-known setting are Wald's criteria, Bayesian testing of two simple hypothesis, H. Chernoff's (Bayesian) problem and Kiefer-Weiss criteria. The greatest number of results refers to the case when we observe i.i.d. sequence of random variables or a Brownian motion with unknown drift.

Recently, U.Çetin, A. Novikov and A. Shiryaev considered the Bayesian problem of sequential estimation of the drift of a fractional Brownian motion (that is not Markovian). In this talk we consider the sequential hypothesis testing problems for the same process, and present some general methods how to deal with them. In particular:

1) How to reduce the sequential testing problem for fractional Brownian motion to the similar problem for a standard one (with Hurst index equal to 1/2).

2) How to reduce it to the standard optimal stopping problem of a special form.

3) How to describe continuation and stopping regions.

Richard Bernstein (Otto von Guericke Universität Magdeburg)

Der Satz von Girsanow

Der Satz von Girsanow ist von großer Bedeutung für die Finanzmathematik. Zu seiner Formulierung ist es notwenig, einige Begriffe wie "Adaptiertheit", "Brownsche Bewegung" oder "quadratische Variation" zu erklären. Auch soll eine Möglichkeit zur Konstruktion stochastischer Integrale über elementare Prozesse vorgestellt werden. Der Schlüssel zum Beweis ist Levys Charakterisierung der Brownschen Bewegung.

Henry Wegener (Leibniz Universität Hannover)

Resultate über fast überall Konvergenz (u.a. Ergodentheorie)

Dr. Johannes Rauh (Max-Planck-Institut Leipzig)

Polytopes from Subgraph Statistics

For two graphs \(H\) and \(G\) let \(s(H,G)\) be the subgraph density of \(H\) in \(G\). For a family of graphs \(H_1,\dots,H_d\) we obtain a vector of subgraph densities for each graph \(G\). For fixed \(N\) I want to study the convex hull of all these vectors for all graphs on \(N\) nodes. This polytope plays an important role in the study of random graphs; for example, it equals the convex support of the corresponding exponential random graph model. In my talk I will present relations to the theory of graph limits (graphons), and I will discuss the example of the k-star polytope.

Ph.D. Stephen Tate (Ruhr-Universität Bochum)

The Combinatorics of Mayer's Theory of Cluster and Virial Expansions

In this talk, I introduce the cluster and virial expansions of Mayer in Statistical Mechanics. These are interpreted as weighted generating functions of connected and two-connected graphs, respectively, and arise as the functions describing the relationship between pressure and activity, respectively, density. In two particular models of Statistical Mechanics, one may obtain these two expansions independently of the generating function interpretation. When one compares the two methods of derivation, one finds some interesting combinatorial identities. This talk indicates an explanation of these identities, giving how the cancellations occur in the weighted generating functions.

Prof. Vladimir Panov (Higher School of Economics Moscow)

Maximal deviation distribution for projection estimates of Levy densities

This talk is devoted to projection estimates for Levy densities in both high and low frequency setup. After a short introduction to the theory of statistical inference for Levy processes, I will present new results concerning the convergence rates of the projection estimates. These results are based on some fresh ideas, which allow to reformulate the problem in terms of Gaussian processes.

Prof. Dr. Christoph Thäle (Ruhr-Universität Bochum)

Process-level Poisson approximation

I present a quantitative version of a process-level Poisson limit theorem, where distance is measured by an optimal transportation metric. To illustrate the flexibility of our result, we show applications to a number of limit theorems for classical U-statistics and also apply it to functionals arising in the context of stochastic geometry.

Prof. Dr. Christof Külske (Ruhr-Universität Bochum)

On nonergodic stochastic lattice systems with unique invariant measure

Is there an interacting particle system (IPS) on the 3-dimensional lattice which has a unique time-invariant measure but which does not attract all starting measures in the long time limit?

We provide a construction of such a system showing that the answer to this classical question is yes. The construction draws on properties of an underlying equilibrium measure of continuous spins showing phase transitions and is based on a heuristics of Maes and Shlosman. We also present results on similarities and differences between the non-ergodic continuous-time IPS and related versions of weak probabilistic cellular automata (PCA) which have simultaneous updates in discrete time with quasilocal rules.

M. Sc. Stephan Gufler (Goethe Universität Frankfurt am Main)

Lookdown representation for tree-valued Fleming-Viot processes

We construct the tree-valued Fleming-Viot process from the Lookdown model. The tree-valued Fleming-Viot process was introduced by Greven, Pfaffelhuber, and Winter [Probab. Theory Related Fields 2013]. We generalize this process to include the cases with multiple and simultaneous multiple reproduction events. In case that the associated coalescent comes down from infinity, the construction from the Lookdown model allows to read off a process with values in the space of measure-preserving isometry classes of compact metric measure spaces, endowed with the Gromov-Hausdorff-Prohorov metric. This process has a.s. càdlàg paths with additional jumps at the times when old families become extinct.

Prof. Dr. Alexander Gushchin (Steklov Mathematical Institute Moskow)

Trajectorial counterparts of Doob's maximal inequalities and their applications

We present trajectorial counterparts of Doob's maximal inequalities for supermartingales and submartingales. From these inequalties, we obtain trajectorial counterparts of Doob's \(L^p\) and \(L \log L\) inequalities obtained by Acciaio, Beiglböck, Penkner, Schachermayer and Temme (2013). Another application of our inequalities is connected with the Azéma--Yor solution of the Skorokhod embedding problem. We show that this solution could be guessed from our inequalities. Moreover, we study the problem considered by Hobson (1998): to find an upper bound with respect to stochastic ordering for the maximum of a martingale with given initial and terminal distributions.

Dr. Paolo Di Tella (Humboldt-Universität zu Berlin)

The Chaotic Representation Property of Certain Families of Martingales

We investigate the chaotic representation property of certain families of square integrable martingales, which we call compensated-covariation stable families. First, we introduce the multiple integrals with respect to elements of a compensated-covariation stable family of martingales. The main result is that any compensated-covariation stable family of martingales which satisfies some further conditions possesses the chaotic representation property. As first examples, we consider continuous Gaussian families of martingales and independent families of compensated Poisson processes. Then we apply the result to the case of Lévy processes. We shall construct families of martingales relative to a Lévy filtration which possess the chaotic representation property. We give several examples including Teugels martingales.

Ph.D. Sandra Kliem (Universität Duisburg-Essen)

Modeling evolving phylogenies by means of marked metric measure spaces

In this talk, a model for evolving phylogenies, incorporating branching, mutation and competition is introduced. The state-space consists of marked tree-like metric measure (mmm)-spaces. The model arises as the limit of approximating finite population models with rates dependent on the individuals' traits and their genealogical distances.

The main focus of the talk will be on presenting the notion of mmm-spaces and to highlight their advantages in the given context. In particular, necessary and sufficient conditions for relative compactness of sets in mmm-spaces are explained. The route to verify these conditions to conclude the tightness of the approximating models from above is given.

A similar approximating model and its limit is treated in [Méléard and Tran, 2012] in the framework of nonlinear historical superprocess approximations. In the framework of mmm-spaces, work of [Depperschmidt, Greven, Pfaffelhuber and Winter, 2012--2013] introduces and studies tree-valued Fleming-Viot dynamics. During this talk, new ideas and challenges that arise from working with mmm-spaces in the context of evolving phylogenies are put into context of the above.

Dr. Wolfgang Löhr (Universität Duisburg-Essen)

Invariance principle for variable speed random walks on trees

Everyone knows that simple random walks on \(\mathbb{Z}\) converge, suitably rescaled, in path-space to Brownian motion on \(\mathbb{R}\). This has been generalised by Stone in 1963 to processes in "natural scale" on \(\mathbb{R}\), which are characterised by a speed measure \(\nu\). We call these processes speed-\(\nu\) motions and generalise Stone's observation that they depend continuously on \(\nu\) to the case of trees. More precisely, we show that whenever a locally compact \(\mathbb{R}\)-tree \(T_n\), together with a locally finite speed-measure \(\nu_n\) on \(T_n\), converges in the Gromov-vague topology to a limiting \(\mathbb{R}\)-tree \(T\) with measure \(\nu\), the speed-\(\nu_n\) motions converge to the speed-\(\nu\) motion under a mild lower mass-bound assumption. (joint work with Siva Athreya and Anita Winter)

Lisa Beck (Augsburg University)

Regularization by noise for the stochastic transport equation

We discuss several aspects of weak (\(L^\infty\)-) solutions to the stochastic transport equation \[ du = b
\cdot Du dt + \sigma Du \circ dW_t \] with Stratonovich multiplicative noise. Here, \(b\) is a time dependent vector field (the drift), \(u\) is the unknown, \(\sigma\) a real number, and \(W_t\) a Brownian motion.

For the deterministic equation (\(\sigma = 0\)) it is well-known that multiple solutions may exist and that solutions may blow up from smooth initial data in finite time provided that the drift is not regular (basically less than Lipschitz in space). For the stochastic equation (\(\sigma \neq 0\)) instead, a suitable integrability condition on the drift is sufficient to prevent the formation of non-uniqueness and of singularities.

After a short review of some techniques for the deterministic equation, I will present in my talk two recent results for the stochastic equation, which concern the phenomenon of regularization by noise and which are part of a joint project with F. Flandoli, M. Gubinelli and M. Maurelli. More precisely, assuming merely the aforementioned integrability condition, we obtain in a first step conservation of Sobolev regularity of initial data by means of PDE techniques (and not via stochastic characteristics). In a second step this regularity is used to prove path-by-path uniqueness of solutions via a duality argument.

Warning: Non-standard time (Wednesday, 14:15) but usual venue (WSC-S-U-3.03).

Fabian Gerle (Ruhr-Universität Bochum)

Applications of Lindeberg's method in the theory of random matrices

Lindeberg's method for the proof of the central limit theorem is easily generalized and can be applied to elegantly proof limiting theorems for the empirical spectral distribution of certain types of random matrices.

In my talk about the subjects of my master thesis I will first give a sketch of Lindeberg's original proof of the central limit theorem from 1912. Following a work by Sourav Chatterjee we will see, how this argument can be extended to more general functions and other sequences of random variables and how this generalized Lindeberg principle can be applied in the theory of random matrices.

Using this generalized Lindeberg principle we will derive Lindebergesque conditions for the convergence of the empirical spectral distribution of Wigner and Wishart matrices in different setups.

Albert N. Shiryaev (Steklov Institute of Mathematics)

The concept of randomness: from von Mises' collective to Kolmogorov's complexity

Mareike Esser (Universität Bielefeld)

Single-crossover recombination and ancestral recombination trees

We consider the dynamics of the genetic composition of a population evolving under the evolutionary force of recombination. Recombination, which can be briefly described via an exchange of genetic material from maternal and paternal gene-sequences during sexual reproduction, gives rise to dependence problems and creates large state spaces. In continuous time the resulting system of non-linear differential equations could be solved in closed form. Inspired by this, we now investigate the stochastic counterpart. To this end, we trace back the ancestry of a single individual that has evolved in accordance to the Moran model with recombination. For \(N \to \infty\) the ancestry of a single individual is represented by a random binary tree which we call ancestral recombination tree. The probability of such trees can be derived via an inclusion-exclusion principle that leads to a decomposition of the trees into subtrees.

Alexander Kulikov (Moscow Institute of Physics and Technology)

Hedging volumetric risks using put options in commodity markets

Abstract: [pdf]

Loren Coquille (Universität Bonn)

Gaussian free field with disordered pinning on \(\mathbb{Z}^d\), \(d \geq 2\)

Angelika Rohde (Ruhr-Universität Bochum)

Accuracy of empirical projections of high-dimensional Gaussian matrices

Let \(X=C+\mathrm{E}\) with a deterministic matrix \(C\in\mathbb{R}^{M\times M}\) and \(\mathrm{E}\) some centered Gaussian \(M\times M\)-matrix whose entries are independent with variance \(\sigma^2\). In the present work, the accuracy of reduced-rank projections of \(X\) is studied. Non-asymptotic upper and lower bounds are derived, and favorable and unfavorable prototypes of matrices \(C\) in terms of the accuracy of approximation are characterized. The approach does not involve analytic perturbation theory of linear operators and allows for multiplicities in the singular value spectrum. Our main result is some general non-asymptotic upper bound on the accuracy of approximation which involves explicitly the singular values of \(C\), and which is shown to be sharp in various regimes of \(C\). The results are accompanied by lower bounds under diverse assumptions. Consequences on statistical estimation problems, in particular in the recent area of low-rank matrix recovery, are discussed.

Mikhail Urusov (Universität Duisburg-Essen)

On the processes that can be embedded in a geometric Brownian motion

A process is equivalent to a time-change of a geometric Brownian motion if and only if it is a nonnegative supermartingale.

Volker Krätschmer (Universität Duisburg-Essen)

Quasi-Hadamard differentiability of general risk functionals and its application to statistical inference

No seminar.

Patricia Alonso-Ruiz (Universität Ulm)

Dirichlet form and Laplacian on fractal quantum graphs via resistance forms

Resistance forms have turn out to be very useful in the study of analysis on fractals from an intrinsic point of view. Under a suitable choice of a measure, they provide a Dirichlet form on the space and thus a Laplacian on it.

In this talk we introduce fractal quantum graphs, which generalize classical quantum graphs and also include fractal sets like Hanoi attractors. We present in detail the construction of a Dirichlet form on any Hanoi attractor by means of resistance forms. We will also discuss the spectral asymptotics of the Laplacian associated to this Dirichlet form and if time allows it, we will point out some questions concerning the Einstein relation.

This is joint work with D.Kelleher and A.Teplyaev from the University of Connecticut.

Jordan Stoyanov (Newcastle University)

Moment (in)determinacy of probability distributions: recent progress

The discussion will be on heavy tailed probability distributions, one- or multi-dimensional, with finite all moments. One of the questions in the classical moment problem is about the uniqueness. Either such a distribution is uniquely determined by its moments (M-determinate), or it is nonunique (M-indeterminate). Our goal is to describe the current state of art in this area. A brief summary of known and widely used classical criteria (Cramer, Hausdorff, Carleman, Krein, ...) will go in parallel with the following very recent developments:

1. Stieltjes classes for M-indeterminate distributions. Index of dissimilarity.
2. New Hardy’s criterion for uniqueness. Multidimensional moment problem.
3. Nonlinear transformations of random data and their moment (in)determinacy.

There will be several new results, hints for their proof, examples and counterexamples, and also open questions and conjectures.

Warning: Non-standard time (Friday, 14:15) but usual venue (WSC-S-U-3.03).

Johannes Ruf (University of Oxford)

Supermartingales as Radon-Nikodym densities, Novikov's and Kazamaki's criteria, and the distribution of explosion times

I will show how certain countably and finitely additive measures can be associated to a given nonnegative supermartingale. I will review existence and (non-)uniqueness results for such measures. In the second part of my talk, I will give a new proof for the famous criteria by Novikov and Kazamaki, which provide sufficient conditions for the martingale property of a nonnegative local martingale. I will focus on the situation when jumps are present. If time permits, I will then illustrate how a generalized Girsanov formula can be used to compute the distribution of the explosion time of a weak solution to a stochastic differential equation.

This presentation is based on joint papers with Nicolas Perkowski, Martin Larsson, and Ioannis Karatzas.

Anton Klimovsky (Universität Duisburg-Essen)

From Exchangeability to Ultrametricity (II)

I will review some representation results for exchangeable random arrays. As an application, I will sketch how these can be used to derive ultrametricity.

Vladimir Panov (Universität Duisburg-Essen)

Exponential functionals of Lévy processes

The talk is devoted to the exponential functional \(\int_0^t e^{-\xi_s}ds\) of a Lévy process \(\xi_s\). This object is related to the generalized Ornstein-Uhlenbeck process and naturally arises in a broad class of applications. After the introduction to this research area, we will focus on the statistical aspects of the topic. In particular, we will present a new approach, which allows to infer on the characteristics of the Lévy process from the distribution of its exponential functional.

Eyal Neuman (Technion)

Pathwise Uniqueness of the Stochastic Heat Equations with Spatially Inhomogeneous White Noise

We study the solutions of the stochastic heat equation with spatially inhomogeneous white noise. This equation has the form \[ (*) \quad \frac{\partial}{\partial t} u(t,x) = \frac{1}{2}\Delta u(t,x) + \sigma(t,x,u(t,x))\dot{W} , \ \ t \geq 0, \ \ x\in \mathbb{R}. \] Here \( \Delta \) denotes the Laplacian and \( \sigma(t,x,u): \mathbb{R}_{+}\times\mathbb{R}^2\to\mathbb{R} \) is a continuous function with at most a linear growth in the \( u \) variable. We assume that the noise \( \dot{W} \) is a spatially inhomogeneous white noise on \( \mathbb{R}_{+}\times\mathbb{R} \). When \( \sigma(t,x,u)=\sqrt{u} \) such equations arise as scaling limits of critical branching particle systems which are known as catalytic super Brownian motion. In particular we prove pathwise uniqueness for solutions of \( (*) \) if \( \sigma \) is Hölder continuous of index \( \gamma>1-\frac{\eta}{2(\eta+1)} \) in \( u \). Here \( \eta\in(0,1) \) is a constant that defines the spatial regularity of the noise.

An extensive introduction to stochastic partial differential equations will be given in this talk.

Tal Orenshtein (Technische Universität München)

0-1 law for directional transience of one-dimensional excited random walks

We will discuss the following theorem which solves a problem posed by Kosygina and Zerner. For a one dimensional excited random walk in a stationary ergodic and elliptic cookie environment, the probability of being transient to the right (left) is either zero or one. (Joint work with Gideon Amir and Noam Berger.)