arXiv subjects

Andreas Neuenkirch

Publications and source records attributed to Andreas Neuenkirch.

At least 19 recordsLinked to original sources

arXiv2026

Stochastic nonlocal traffic flow models with Markovian noise

We extend our recently introduced stochastic nonlocal traffic flow model to more general random perturbations, including Markovian noise derived from a discretized Jacobi-type stochastic differential equation. Invoking a deterministic stability estimate, we show that the arising random weak entropy solutions are measurable, ensuring that quantities such as the expectation are well-defined. We show that the proposed Jacobi-type noise is of particular interest as it ensures interpretability, preserves boundedness, and significantly alters the stochastic realizations compared to the previous white noise approach. Moreover, we introduce a local solution operator which provides information on the local effect of the noise and utilize it to derive a mean-value hyperbolic nonlocal PDE, which serves as a proxy for the mean value of the exact solution. The quality of this proxy and the impact of the noise process are analyzed in several simulation studies.

math.NA↗

arXiv2026

Stochastic differential equations driven by fractional Brownian motion: dependence on the Hurst parameter

Stochastic models with fractional Brownian motion as source of randomness have become popular since the early 2000s. Fractional Brownian motion (fBm) is a Gaussian process, whose covariance depends on the so-called Hurst parameter $H\in (0,1)$. Consequently, stochastic models with fBm also depend on the Hurst parameter $H$, and the stability of these models with respect to $H$ is an interesting and important question. In recent years, the continuous (or even smoother) dependence on the Hurst parameter has been studied for several stochastic models, including stochastic integrals with respect to fBm, stochastic differential equations (SDEs) driven by fBm and also stochastic partial differential equations with fractional noise, for different topologies, e.g., in law or almost surely, and for finite and infinite time horizons. In this manuscript, we give an overview of these results with a particular focus on SDE models.

math.PR↗

arXiv2025

Exponential stability of finite-$N$ consensus-based optimization

We study the finite-agent behavior of Consensus-Based Optimization (CBO), a recent metaheuristic for the global minimization of a function, that combines drift toward a consensus estimate with stochastic exploration. While previous analyses focus on asymptotic mean-field limits, we investigate the stability properties of CBO for finite population size $ N $. Following a hierarchical approach, we first analyze a deterministic formulation of the algorithm and then extend our results to the fully stochastic setting governed by a system of stochastic differential equations. Our analysis reveals that essential stability properties, including almost sure and mean square exponential convergence, persist in both regimes and provides sharp quantitative estimates on the rates of convergence.

math.OC↗

arXiv2025

Functional differential equations driven by càdlàg rough paths

The existence of unique solutions is established for rough differential equations (RDEs) with path-dependent coefficients and driven by càdlàg rough paths. Moreover, it is shown that the associated solution map, also known as Itô-Lyons map, is locally Lipschitz continuous. These results are then applied to various classes of rough differential equations, such as controlled RDEs and RDEs with delay, as well as stochastic differential equations with delay. To that end, a joint rough path is constructed for a càdlàg martingale and its delayed version, that corresponds to stochastic Itô integration.

math.PR↗

arXiv2024

Using Low-Discrepancy Points for Data Compression in Machine Learning: An Experimental Comparison

Low-discrepancy points (also called Quasi-Monte Carlo points) are deterministically and cleverly chosen point sets in the unit cube, which provide an approximation of the uniform distribution. We explore two methods based on such low-discrepancy points to reduce large data sets in order to train neural networks. The first one is the method of Dick and Feischl [4], which relies on digital nets and an averaging procedure. Motivated by our experimental findings, we construct a second method, which again uses digital nets, but Voronoi clustering instead of averaging. Both methods are compared to the supercompress approach of [14], which is a variant of the K-means clustering algorithm. The comparison is done in terms of the compression error for different objective functions and the accuracy of the training of a neural network.

stat.ML↗

arXiv2024

A nonlocal traffic flow model with stochastic velocity

In this paper, we investigate a nonlocal traffic flow model based on a scalar conservation law, where a stochastic velocity function is assumed. In addition to the modeling, theoretical properties of the stochastic nonlocal model are provided, also addressing the question of well-posedness. A detailed numerical analysis offers insights how the stochasticity affects the evolution of densities. Finally, numerical examples illustrate the mean behavior of solutions and the influence of parameters for a large number of realizations.

math.NA↗

arXiv2024

On the convergence order of the Euler scheme for scalar SDEs with Hölder-type diffusion coefficients

We study the Euler scheme for scalar non-autonomous stochastic differential equations, whose diffusion coefficient is not globally Lipschitz but a fractional power of a globally Lipschitz function. We analyse the strong error and establish a criterion, which relates the convergence order of the Euler scheme to an inverse moment condition for the diffusion coefficient. Our result in particular applies to Cox-Ingersoll-Ross-, Chan-Karolyi-Longstaff-Sanders- or Wright-Fisher-type stochastic differential equations and thus provides a unifying framework.

math.NA↗

arXiv2023

Sharp $L^1$-Approximation of the log-Heston SDE by Euler-type methods

We study the $L^1$-approximation of the log-Heston SDE at equidistant time points by Euler-type methods. We establish the convergence order $ 1/2-ε$ for $ε>0$ arbitrarily small, if the Feller index $ν$ of the underlying CIR process satisfies $ν> 1$. Thus, we recover the standard convergence order of the Euler scheme for SDEs with globally Lipschitz coefficients. Moreover, we discuss the case $ν\leq 1$ and illustrate our findings by several numerical examples.

math.NA↗

arXiv2023

The order barrier for the $L^1$-approximation of the log-Heston SDE at a single point

We study the $L^1$-approximation of the log-Heston SDE at the terminal time point by arbitrary methods that use an equidistant discretization of the driving Brownian motion. We show that such methods can achieve at most order $ \min \{ ν, \tfrac{1}{2} \}$, where $ν$ is the Feller index of the underlying CIR process. As a consequence Euler-type schemes are optimal for $ν\geq 1$, since they have convergence order $\tfrac{1}{2}-ε$ for $ε>0$ arbitrarily small in this regime.

math.NA↗

arXiv2022

The weak convergence order of two Euler-type discretization schemes for the log-Heston model

We study the weak convergence order of two Euler-type discretizations of the log-Heston Model where we use symmetrization and absorption, respectively, to prevent the discretization of the underlying CIR process from becoming negative. If the Feller index $ν$ of the CIR process satisfies $ν>1$, we establish weak convergence order one, while for $ν\leq 1$, we obtain weak convergence order $ν-ε$ for $ε>0$ arbitrarily small. We illustrate our theoretical findings by several numerical examples.

math.NA↗

arXiv2020

The Euler-Maruyama Scheme for SDEs with Irregular Drift: Convergence Rates via Reduction to a Quadrature Problem

We study the strong convergence order of the Euler-Maruyama scheme for scalar stochastic differential equations with additive noise and irregular drift. We provide a general framework for the error analysis by reducing it to a weighted quadrature problem for irregular functions of Brownian motion. Assuming Sobolev-Slobodeckij-type regularity of order $κ\in (0,1)$ for the non-smooth part of the drift, our analysis of the quadrature problem yields the convergence order $\min\{3/4,(1+κ)/2\}-ε$ for the equidistant Euler-Maruyama scheme (for arbitrarily small $ε>0$). The cut-off of the convergence order at $3/4$ can be overcome by using a suitable non-equidistant discretization, which yields the strong convergence order of $(1+κ)/2-ε$ for the corresponding Euler-Maruyama scheme.

math.PR↗

arXiv2019

An adaptive Euler-Maruyama scheme for stochastic differential equations with discontinuous drift and its convergence analysis

We study the strong approximation of stochastic differential equations with discontinuous drift coefficients and (possibly) degenerate diffusion coefficients. To account for the discontinuity of the drift coefficient we construct an adaptive step sizing strategy for the explicit Euler-Maruyama scheme. As a result, we obtain a numerical method which has -- up to logarithmic terms -- strong convergence order $1/2$ with respect to the average computational cost. We support our theoretical findings with several numerical examples.

math.NA↗

arXiv2017

Exponential stability of stochastic evolution equations driven by small fractional Brownian motion with Hurst parameter in $(1/2,1)$

This paper addresses the exponential stability of the trivial solution of some types of evolution equations driven by Hölder continuous functions with Hölder index greater than $1/2$. The results can be applied to the case of equations whose noisy inputs are given by a fractional Brownian motion $B^H$ with covariance operator $Q$, provided that $H\in (1/2,1)$ and ${\rm tr}(Q)$ is sufficiently small.

math.AP↗

arXiv2016

Optimal approximation of Skorohod integrals

In this manuscript, we determine the optimal approximation rate for Skorohod integrals of sufficiently regular integrands. This generalizes the optimal approximation results for Itô integrals. However, without adaptedness and the Itô isometry, new proof techniques are required. The main tools are a characterization via S-transform and a reformulation of the Wiener chaos decomposition in terms of Wick-analytic functionals.

math.PR↗

arXiv2016

The Order Barrier for Strong Approximation of Rough Volatility Models

We study the strong approximation of a rough volatility model, in which the log-volatility is given by a fractional Ornstein-Uhlenbeck process with Hurst parameter $H<1/2$. Our methods are based on an equidistant discretization of the volatility process and of the driving Brownian motions, respectively. For the root mean-square error at a single point the optimal rate of convergence that can be achieved by such methods is $n^{-H}$, where $n$ denotes the number of subintervals of the discretization. This rate is in particular obtained by the Euler method and an Euler-trapezoidal type scheme.

math.PR↗

arXiv2016

Asymptotical stability of differential equations driven by Hölder--continuous paths

In this manuscript, we establish asymptotic local exponential stability of the trivial solution of differential equations driven by Hölder--continuous paths with Hölder exponent greater than $1/2$. This applies in particular to stochastic differential equations driven by fractional Brownian motion with Hurst parameter greater than $1/2$. We motivate the study of local stability by giving a particular example of a scalar equation, where global stability of the trivial solution can be obtained.

math.DS↗

arXiv2016

Discretizing the Heston Model: An Analysis of the Weak Convergence Rate

In this manuscript we analyze the weak convergence rate of a discretization scheme for the Heston model. Under mild assumptions on the smoothness of the payoff and on the Feller index of the volatility process, respectively, we establish a weak convergence rate of order one. Moreover, under almost minimal assumptions we obtain weak convergence without a rate. These results are accompanied by several numerical examples. Our error analysis relies on a classical technique from Talay & Tubaro, a recent regularity estimate for the Heston PDE by Feehan & Pop and Malliavin calculus.

math.NA↗

arXiv2015

The relation between mixed and rough SDEs and its application to numerical methods

We study the relationship between mixed stochastic differential equations and the corresponding rough path equations driven by standard Brownian motion and fractional Brownian motion with Hurst parameter $H>1/2$. We establish a correction formula, which relates both types of equations, analogously to the Itō-Stratonovich correction formula. This correction formula allows to transfer properties, which are established for one type of equation to the other, and we will illustrate this by considering numerical methods for mixed and rough SDEs

math.PR↗