arXiv subjects

Alexander Steinicke

Publications and source records attributed to Alexander Steinicke.

At least 19 recordsLinked to original sources

arXiv2026

Deviation frequencies of Brownian path property approximations

This case study proposes a.s.~convergence quantifications of many classical sample path property approximations of Brownian motion in terms of the tradeoff between a.s.~rates and the integrability of the modulus of convergence, as well as the deviation frequencies. This includes Lévy's construction of Brownian motion, the Kolmogorov-Chentsov (and the Kolmogorov-Totoki) continuity theorem, Lévy's modulus of continuity, the Paley-Wiener-Zygmund theorem, the a.s.~approximation of the quadratic variation as well as the laws of the iterated logarithm by Khinchin, Chung and Strassen, among others.

math.PR↗

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

Locally Lipschitz Path Dependent FBSDEs with Unbounded Terminal Conditions in Brownian and L{é}vy Settings

This paper is dedicated to the analysis of forward backward stochastic differential equations driven by a L{é}vy process. We assume that the generator and the terminal condition are path-dependent and satisfy a local Lipschitz condition. We study solvability and Malliavin differentiability of such BSDEs. The proof of the existence and uniqueness is done in three steps. First of all, we truncate and localize the terminal condition and the generator. Then we use an iteration argument to get bounds for the solutions of the truncated BSDE (independent from the level of truncation). Finally, we let the level of truncation tend to infinity. A stability result ends the proof. The Malliavin differentiability result is based on a recent characterisation for the Malliavin Sobolev space D 1,2 by S. Geiss and Zhou.

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↗

arXiv2025

A Skorohod measurable universal functional representation of solutions to semimartingale SDEs

In this paper we show the existence of a universal Skorohod measurable functional representation for a large class of semimartingale-driven stochastic differential equations. For this we prove that paths of the strong solutions of stochastic differential equations can be written as measurable functions of the paths of their driving processes into the space of all càdlàg functions equipped with the Borel sigma-field generated by all open sets with respect to the Skorohod metric. This result can be applied to calculate Malliavin derivatives for SDEs driven by pure-jump Lévy processes with drift.

math.PR↗

arXiv2025

On the tradeoff between almost sure error tolerance and mean deviation frequency in martingale convergence

In this article we quantify almost sure martingale convergence theorems in terms of the tradeoff between asymptotic almost sure rates of convergence (error tolerance) and the respective modulus of convergence. For this purpose we generalize {an} elementary quantitative version of the first Borel-Cantelli lemma on the statistics of the deviation frequencies (error incidence), which was recently established by the authors. First we study martingale convergence in $L^2$, and in the setting of the Azuma-Hoeffding inequality. In a second step we study the strong law of large numbers for martingale differences in two settings: uniformly bounded increments in $L^p$, $p\geq 2$, using the respective Baum-Katz-Stoica theorems, and uniformly bounded exponential moments with the help of the martingale estimates by Lesigne and Volný. We also present applications for the tradeoff for the multicolor generalized Pólya urn process, the Generalized Chinese restaurant process, statistical M-estimators, as well as the a.s.~excursion frequencies of the Galton-Watson branching process. Finally, we relate the tradeoff concept to the convergence in the Ky Fan metric.

math.PR↗

arXiv2024

Worst-Case Optimal Investment in Incomplete Markets

We study and solve the worst-case optimal portfolio problem as pioneered by Korn and Wilmott (2002) of an investor with logarithmic preferences facing the possibility of a market crash with stochastic market coefficients by enhancing the martingale approach developed by Seifried in 2010. With the help of backward stochastic differential equations (BSDEs), we are able to characterize the resulting indifference optimal strategies in a fairly general setting. We also deal with the question of existence of those indifference strategies for market models with an unbounded market price of risk. We therefore solve the corresponding BSDEs via solving their associated PDEs using a utility crash-exposure transformation. Our approach is subsequently demonstrated for Heston's stochastic volatility model, Bates' stochastic volatility model including jumps, and Kim-Omberg's model for a stochastic excess return.

q-fin.MF↗

arXiv2023

Product formulas for multiple stochastic integrals associated with Lévy processes

In the present paper, we obtain an explicit product formula for products of multiple integrals w.r.t. a random measure associated with a Lévy process. As a building block, we use a representation formula for products of martingales from a compensated-covariation stable family. This enables us to consider Lévy processes with both jump and Gaussian part. It is well known that for multiple integrals w.r.t. the Brownian motion such product formulas exist without further integrability conditions on the kernels. However, if a jump part is present, this is, in general, false. Therefore, we provide here sufficient conditions on the kernels which allow us to establish product formulas. As an application, we obtain explicit expressions for the expectation of products of iterated integrals, as well as for the moments and the cumulants for stochastic integrals w.r.t. the random measure. Based on these expressions, we show a central limit theorem for the long time behaviour of a class of stochastic integrals. Finally, we provide methods to calculate the number of summands in the product formula.

math.PR↗

arXiv2022

Complements and Improvements Regarding Distributivity of the Product for $σ$-Algebras with Respect to the Intersection

We present a variety of refined conditions for $σ$ algebras $\mathcal{A}$ (on a set $X$), $\mathcal{F}, \mathcal{G}$ (on a set $U$) such that the distributivity equation $$(\mathcal{A}\otimes\mathcal{F})\cap(\mathcal{A}\otimes\mathcal{G})=\mathcal{A}\otimes\left(\mathcal{F}\cap\mathcal{G}\right),$$ holds -- or is violated. \\ The article generalizes the results in arXiv:2007.06095 and includes a positive result for $σ$ algebras generated by at most countable partitions, was not covered before. We also present a proof that counterexamples may be constructed whenever $X$ is uncountable and there exist two $σ$-algebras on $X$ which are both countably separated, but their intersection is not. We present examples of such structures. In the last section, we extend Theorem 3.3 of arXiv:2007.06095 from analytic to the setting of Blackwell spaces.

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

Two Approaches for a Dividend Maximization Problem under an Ornstein-Uhlenbeck Interest Rate

We investigate a dividend maximization problem under stochastic interest rates with Ornstein-Uhlenbeck dynamics. This setup also takes negative rates into account. First a deterministic time is considered, where an explicit separating curve $α(t)$ can be found to determine the optimal strategy at time $t$. In a second setting we introduce a strategy-independent stopping time. The properties and behavior of these optimal control problems in both settings are analyzed in an analytical HJB-driven approach as well as using backward stochastic differential equations.

math.OC↗

arXiv2021

On the Configurations of Closed Kinematic Chains in three-dimensional Space

A kinematic chain in three-dimensional Euclidean space consists of $n$ links that are connected by spherical joints. Such a chain is said to be within a closed configuration when its link lengths form a closed polygonal chain in three dimensions. We investigate the space of configurations, described in terms of joint angles of its spherical joints, that satisfy the the loop closure constraint, meaning that the kinematic chain is closed. In special cases, we can find a new set of parameters that describe the diagonal lengths (the distance of the joints from the origin) of the configuration space by a simple domain, namely a cube of dimension $n-3$. We expect that the new findings can be applied to various problems such as motion planning for closed kinematic chains or singularity analysis of their configuration spaces. To demonstrate the practical feasibility of the new method, we present numerical examples.

cs.RO↗

arXiv2021

(Non-)Distributivity of the Product for $σ$-Algebras with Respect to the Intersection

We study the validity of the distributivity equation $$(\mathcal{A}\otimes\mathcal{F})\cap(\mathcal{A}\otimes\mathcal{G})=\mathcal{A}\otimes\left(\mathcal{F}\cap\mathcal{G}\right),$$ where $\mathcal{A}$ is a $σ$-algebra on a set $X$, and $\mathcal{F}, \mathcal{G}$ are $σ$-algebras on a set $U$. We present a counterexample for the general case and in the case of countably generated subspaces of analytic measurable spaces we give an equivalent condition in terms of the $σ$-algebras' atoms. Using this, we give a sufficient condition under which distributivity holds.

math.PR↗

arXiv2020

A deep neural network algorithm for semilinear elliptic PDEs with applications in insurance mathematics

In insurance mathematics optimal control problems over an infinite time horizon arise when computing risk measures. Their solutions correspond to solutions of deterministic semilinear (degenerate) elliptic partial differential equations. In this paper we propose a deep neural network algorithm for solving such partial differential equations in high dimensions. The algorithm is based on the correspondence of elliptic partial differential equations to backward stochastic differential equations with random terminal time.

q-fin.MF↗

arXiv2020

$L^p$-Solutions and Comparison Results for Lévy Driven BSDEs in a Monotonic, General Growth Setting

We present a unified approach to $L^p$-solutions ($p > 1$) of multidimensional backward stochastic differential equations (BSDEs) driven by Lévy processes and more general filtrations. New existence, uniqueness and comparison results are obtained. The generator functions obey a time-dependent extended monotonicity (Osgood) condition in the $y$-variable and have general growth in $y$. Within this setting, the results generalize those of Royer (2006), Yin and Mao (2008), Yao (2017), Kruse and Popier (2016/2017) and Geiss and Steinicke (2018).

math.PR↗

arXiv2020

On a combinatorial problem in the Secret Santa ritual

The Secret Santa ritual, where in a group of people every member presents a gift to a randomly assigned partner, poses a combinatorial problem when considering the probabilities involved in the formation of pairs, where two persons exchange gifts mutually. We give different possible derivations for such probabilities by counting fixed-point-free permutations with certain numbers of 2-cycles.

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↗