arXiv subjects

Gunther Leobacher

Publications and source records attributed to Gunther Leobacher.

At least 19 recordsLinked to original sources

arXiv2026

A dice game, a multinomial walk, and the inverted Dirichlet distribution

We consider a simple dice game, which leads to an intriguing study of multinomial walks, with surprising and seemingly paradoxical properties. The winning and losing probabilities of a general version of the game are investigated via conjugacy relations between Gamma and Poisson distributions, as well as between negative multinomial and inverted Dirichlet distributions. We show a monotonicity property of the regularized beta function, which implies a monotonicity property of the winning probability. Furthermore, the asymptotic behavior of the game for one or several parameters of the game tending to infinity is analyzed, as well as the probability of being last in the game.

math.PR↗

arXiv2025

Measure and dimension theory of permeable sets and its applications to fractals

We study {\it permeable} sets. These are sets $Θ\subset \mathbb{R}^d$ which have the property that each two points $x,y\in \mathbb{R}^d$ can be connected by a short path $γ$ which has small (or even empty, apart from the end points of $γ$) intersection with $Θ$. We investigate relations between permeability and Lebesgue measure and establish theorems on the relation of permeability with several notions of dimension. It turns out that for most notions of dimension each subset of $\mathbb{R}^d$ of dimension less than $d-1$ is permeable. We use our permeability result on the Nagata dimension to characterize permeability properties of self-similar sets with certain finiteness properties.

math.GN↗

arXiv2022

Tractability of $L_2$-approximation and integration in weighted Hermite spaces of finite smoothness

In this paper we consider integration and $L_2$-approximation for functions over $\RR^s$ from weighted Hermite spaces. The first part of the paper is devoted to a comparison of several weighted Hermite spaces that appear in literature, which is interesting on its own. Then we study tractability of the integration and $L_2$-approximation problem for the introduced Hermite spaces, which describes the growth rate of the information complexity when the error threshold $\varepsilon$ tends to 0 and the problem dimension $s$ grows to infinity. Our main results are characterizations of tractability in terms of the involved weights, which model the importance of the successive coordinate directions for functions from the weighted Hermite spaces.

math.NA↗

arXiv2022

Well-posedness and numerical schemes for one-dimensional McKean-Vlasov equations and interacting particle systems with discontinuous drift

In this paper, we first establish well-posedness results for one-dimensional McKean-Vlasov stochastic differential equations (SDEs) and related particle systems with a measure-dependent drift coefficient that is discontinuous in the spatial component, and a diffusion coefficient which is a Lipschitz function of the state only. We only require a fairly mild condition on the diffusion coefficient, namely to be non-zero in a point of discontinuity of the drift, while we need to impose certain structural assumptions on the measure-dependence of the drift. Second, we study Euler-Maruyama type schemes for the particle system to approximate the solution of the one-dimensional McKean-Vlasov SDE. Here, we will prove strong convergence results in terms of the number of time-steps and number of particles. Due to the discontinuity of the drift, the convergence analysis is non-standard and the usual strong convergence order $1/2$ known for the Lipschitz case cannot be recovered for all schemes.

math.PR↗

arXiv2022

Continuous functions with impermeable graphs

We construct a Hölder continuous function on the unit interval which coincides in uncountably (in fact continuum) many points with every function of total variation smaller than 1 passing through the origin. We say that a function with this property has impermeable graph, and we present further examples of functions both with permeable and impermeable graphs. The first example function is subsequently used to construct an example of a continuous function on the plane which is intrinsically Lipschitz continuous on the complement of the graph of a Hölder continuous function with impermeable graph, but which is not Lipschitz continuous on the plane. As another main result we construct a continuous function on the unit interval which coincides in a set of Hausdorff dimension 1 with every function of total variation smaller than 1 which passes through the origin.

math.CA↗

arXiv2021

Exception sets of intrinsic and piecewise Lipschitz functions

We consider a class of functions defined on metric spaces which generalizes the concept of piecewise Lipschitz continuous functions on an interval or on polyhedral structures. The study of such functions requires the investigation of their exception sets where the Lipschitz property fails. The newly introduced notion of permeability describes sets which are natural exceptions for Lipschitz continuity in a well-defined sense. One of the main results states that continuous functions which are intrinsically Lipschitz continuous outside a permeable set are Lipschitz continuous on the whole domain with respect to the intrinsic metric. We provide examples of permeable sets in $\mathbb{R}^d$, which include Lipschitz submanifolds.

math.GN↗

arXiv2021

On the Existence of Solutions of a Class of SDEs with Discontinuous Drift and Singular Diffusion

The classical result by Itô on the existence of strong solutions of stochastic differential equations (SDEs) with Lipschitz coefficients can be extended to the case where the drift is only measurable and bounded. These generalizations are based on techniques presented by Zvonkin and Veretennikov, which rely on the uniform ellipticity of the diffusion coefficient. In this paper we study the case of degenerate ellipticity and give sufficient conditions for the existence of a solution. The conditions on the diffusion coefficient are more general than previous results and we gain fundamental insight into the geometric properties of the discontinuity of the drift on the one hand and the diffusion vector field on the other hand. Besides presenting existence results for the degenerate elliptic situation, we give an example illustrating the difficulties in obtaining more general results than those given. The particular types of SDEs considered arise naturally in the framework of combined optimal control and filtering problems.

math.PR↗

arXiv2020

Change of drift in one-dimensional diffusions

It is generally understood that a given one-dimensional diffusion may be transformed by Cameron-Martin-Girsanov measure change into another one-dimensional diffusion with the same volatility but a different drift. But to achieve this we have to know that the change-of-measure local martingale that we write down is a true martingale; we provide a complete characterization of when this happens. This is then used to discuss absence of arbitrage in a generalized Heston model including the case where the Feller condition for the volatility process is violated.

q-fin.MF↗

arXiv2020

Zur Irrationalität in der Schule

Irrational numbers are introduced usually already introduced in lower secondary level schools. But typically, maybe with the exception of $\sqrt{2}$, no mathematical proof of irrationality is provided. In particular it is not proven that famous Euler's number $e$ as well as the number $π$ are irrational. In this article we want to show how this can be done with very elementary methods from calculus. In addition, we offer geometrical variants for many of the analytical statements, which in particular create variability in the level of requirements. ----- Irrationale Zahlen werden in der Schule bereits in der Sekundarstufe I eingeführt. Allerdings wird typischerweise, mit Ausnahme vielleicht für $\sqrt{2}$, kein mathematischer Beweis zur Irrationalität geführt. Insbesondere wird nicht bewiesen, dass die berühmte Eulersche Zahl $e$ sowie die Kreiszahl $π$ irrationale Zahlen sind. In diesem Artikel wollen wir aufzeigen, wie dies mit recht elementaren Methoden der Analysis möglich ist. Darüber hinaus bieten wir für viele der analytischen Aussagen geometrische Varianten zur Veranschaulichung, die insbesondere Variabilität im Anspruchsniveau schaffen.

math.HO↗

arXiv2020

Existence, Uniqueness and Regularity of the Projection onto Differentiable Manifolds

We investigate the maximal open domain $\mathscr{E}(M)$ on which the orthogonal projection map $p$ onto a subset $M\subseteq \mathbb{R}^d$ can be defined and study essential properties of $p$. We prove that if $M$ is a $C^1$ submanifold of $\mathbb{R}^d$ satisfying a Lipschitz condition on the tangent spaces, then $\mathscr{E}(M)$ can be described by a lower semi-continuous frontier function. We show that this frontier function is continuous if $M$ is $C^2$ or if the topological skeleton of $M^c$ is closed and we provide an example showing that the frontier function need not be continuous in general. We demonstrate that, for a $C^k$-submanifold $M$ with $k\ge 2$, the projection map is $C^{k-1}$ on $\mathscr{E}(M)$, and we obtain a differentiation formula for the projection map which is used to discuss boundedness of its higher order derivatives on tubular neighborhoods. A sufficient condition for the inclusion $M\subseteq\mathscr{E}(M)$ is that $M$ is a $C^1$ submanifold whose tangent spaces satisfy a local Lipschitz condition. We prove in a new way that this condition is also necessary. More precisely, if $M$ is a topological submanifold with $M\subseteq\mathscr{E}(M)$, then $M$ must be $C^1$ and its tangent spaces satisfy the same local Lipschitz condition. A final section is devoted to highlighting some relations between $\mathscr{E}(M)$ and the topological skeleton of $M^c$.

math.DG↗

arXiv2019

Independence in Mathematics -- the key to a Gaussian law

In this manuscript we discuss the notion of (statistical) independence embedded in its historical context. We focus in particular on its appearance and role in number theory, concomitantly exploring the intimate connection of independence and the famous Gaussian law of errors. As we shall see, this at times requires us to go adrift from the celebrated Kolmogorov axioms, which give the appearance of being ultimate ever since they have been introduced in the $1930$s. While these insights are known to many a mathematician, we feel it is time for both a reminder and renewed awareness. We present the independence of the coefficients in a binary expansion, the independence of divisibility by primes, and the resulting, famous central limit theorem of Paul Erdős and Mark Kac on the number of different prime factors of a number $n\in\mathbb N$. We shall also present some of the (modern) developments in the framework of lacunary series that have its origin in a work of Raphaël Salem and Antoni Zygmund.

math.PR↗

arXiv2019

Convergence of the Euler-Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient

We prove strong convergence of order $1/4-ε$ for arbitrarily small $ε>0$ of the Euler-Maruyama method for multidimensional stochastic differential equations (SDEs) with discontinuous drift and degenerate diffusion coefficient. The proof is based on estimating the difference between the Euler-Maruyama scheme and another numerical method, which is constructed by applying the Euler-Maruyama scheme to a transformation of the SDE we aim to solve.

math.NA↗

arXiv2019

Approximation methods for piecewise deterministic Markov processes and their costs

In this paper, we analyse piecewise deterministic Markov processes, as introduced in Davis (1984). Many models in insurance mathematics can be formulated in terms of the general concept of piecewise deterministic Markov processes. In this context, one is interested in computing certain quantities of interest such as the probability of ruin of an insurance company, or the insurance company's value, defined as the expected discounted future dividend payments until the time of ruin. Instead of explicitly solving the integro-(partial) differential equation related to the quantity of interest considered (an approach which can only be used in few special cases), we adapt the problem in a manner that allows us to apply deterministic numerical integration algorithms such as quasi-Monte Carlo rules; this is in contrast to applying random integration algorithms such as Monte Carlo. To this end, we reformulate a general cost functional as a fixed point of a particular integral operator, which allows for iterative approximation of the functional. Furthermore, we introduce a smoothing technique which is applied to the integrands involved, in order to use error bounds for deterministic cubature rules. On the analytical side, we prove a convergence result for our PDMP approximation, which is of independent interest as it justifies phase-type approximations on the process level. We illustrate the smoothing technique for a risk-theoretic example, and provide a comparative study of deterministic and Monte Carlo integration.

math.PR↗

arXiv2018

A Strong Order 1/2 Method for Multidimensional SDEs with Discontinuous Drift

In this paper we consider multidimensional stochastic differential equations (SDEs) with discontinuous drift and possibly degenerate diffusion coefficient. We prove an existence and uniqueness result for this class of SDEs and we present a numerical method that converges with strong order 1/2. Our result is the first one that shows strong convergence for such a general class of SDEs. The proof is based on a transformation technique that removes the discontinuity from the drift such that the coefficients of the transformed SDE are Lipschitz continuous. Thus the Euler-Maruyama method can be applied to this transformed SDE. The approximation can be transformed back, giving an approximation to the solution of the original SDE. As an illustration, we apply our result to an SDE the drift of which has a discontinuity along the unit circle.

math.NA↗

arXiv2018

Numerical methods for SDEs with drift discontinuous on a set of positive reach

For time-homogeneous stochastic differential equations (SDEs) it is enough to know that the coefficients are Lipschitz to conclude existence and uniqueness of a solution, as well as the existence of a strongly convergent numerical method for its approximation. Here we introduce a notion of piecewise Lipschitz functions and study SDEs with a drift coefficient satisfying only this weaker regularity condition. For these SDEs we can construct a strongly convergent approximation scheme, if the set of discontinuities is a sufficiently smooth hypersurface satisfying the geometrical property of being of positive reach. We then arrive at similar conclusions as in the Lipschitz case. We will see that, although SDEs are in the center of our interest, we will talk surprisingly little about probability theory here.

math.NA↗

arXiv2018

On the length of arcs in labyrinth fractals

Labyrinth fractals are self-similar dendrites in the unit square that are defined with the help of a labyrinth set or a labyrinth pattern. In the case when the fractal is generated by a horizontally and vertically blocked pattern, the arc between any two points in the fractal has infinite length [Cristea\&Steinsky 2009,Cristea\&Steinsky 2011]. In the case of mixed labyrinth fractals a sequence of labyrinth patterns is used in order to construct the dendrite. In the present article we focus on the length of the arcs between points of mixed labyrinth fractals. We show that, depending on the choice of the patterns in the sequence, both situations can occur: the arc between any two points of the fractal has finite length, or the arc between any two points of the fractal has infinite length. This is in stark contrast to the self-similar case.

math.DS↗

arXiv2018

Supermixed labyrinth fractals

Labyrinth fractals are dendrites in the unit square. They were introduced and studied in the last decade first in the self-similar case [Cristea & Steinsky (2009,2011)], then in the mixed case [Cristea & Steinsky (2017), Cristea & Leobacher (2017)]. Supermixed fractals constitute a significant generalisation of mixed labyrinth fractals: each step of the iterative construction is done according to not just one labyrinth pattern, but possibly to several different patterns. In this paper we introduce and study supermixed labyrinth fractals and the corresponding prefractals, called supermixed labyrinth sets, with focus on the aspects that were previously studied for the self-similar and mixed case: topological properties and properties of the arcs between points in the fractal. The facts and formulae found here extend results proven in the above mentioned cases. One of the main results is a sufficient condition for infinite length of arcs in mixed labyrinth fractals.

math.GT↗

arXiv2017

On the optimal order of integration in Hermite spaces with finite smoothness

We study the numerical approximation of integrals over $\mathbb{R}^s$ with respect to the standard Gaussian measure for integrands which lie in certain Hermite spaces of functions. The decay rate of the associated sequence is specified by a single integer parameter which determines the smoothness classes and the inner product can be expressed via $L_2$ norms of the derivatives of the function. We map higher order digital nets from the unit cube to a suitable subcube of $\mathbb{R}^s$ via a linear transformation and show that such rules achieve, apart from powers of $\log N$, the optimal rate of convergence of the integration error.

math.NA↗