arXiv subjects

Nicole Hufnagel

Publications and source records attributed to Nicole Hufnagel.

6 recordsLinked to original sources

arXiv2026

Collision types and times in interacting particle systems

We consider a system of stochastic interacting particles with general diffusion coefficient and drift functions and we study the types of collisions that arise in them. In particular, interactions between particles are inversely proportional to their separation, and the coupling function of interaction is also considered in great generality. Our main result shows that, under positivity and the stated variance compatibility conditions, no two distinct positive-root hyperplanes are reached simultaneously at a positive time. In type $A_{N-1}$, this excludes both collisions involving three or more particles and simultaneous collisions of disjoint pairs. In order to obtain our results we make use of symmetric polynomials in squared root projections; the degree of these polynomials indicates the type of collision, and by a locality argument we show that polynomials indicating a non-simple collision almost surely do not cancel. We use this result to obtain upper and lower bounds for the Hausdorff dimension of the set of collision times in terms of interaction-to-variance ratios. These bounds coincide in particular constant-ratio cases. Our results cover many of the most well-known particle systems, such as the Dyson model and Wishart processes and their extensions to non-constant diffusion coefficients and background drifts.

math.PR↗

arXiv2025

Monte Carlo on a single sample

In this paper, we consider a Monte Carlo simulation method (MinMC) that approximates prices and risk measures for a range $Γ$ of model parameters at once. The simulation method that we study has recently gained popularity [HS20, FPP22, BDG24], and we provide a theoretical framework and convergence rates for it. In particular, we show that sample-based approximations to $\mathbb{E}_θ[X]$, where $θ$ denotes the model and $\mathbb{E}_θ$ the expectation with respect to the distribution $P_θ$ of the model $θ$, can be obtained across all $θ\in Γ$ by minimizing a map $V:H\rightarrow \mathbb{R}$ with $H$ a suitable function space. The minimization can be achieved easily by fitting a standard feedforward neural network with stochastic gradient descent. We show that MinMC, which uses only one sample for each model, significantly outperforms a traditional Monte Carlo method performed for multiple values of $θ$, which are subsequently interpolated. Our case study suggests that MinMC might serve as a new benchmark for parameter-dependent Monte Carlo simulations, which appear not only in quantitative finance but also in many other areas of scientific computing.

math.ST↗

arXiv2023

Collision times of multivariate Bessel processes with their Weyl chambers' boundaries and their Hausdorff dimension

Multivariate Bessel processes, otherwise known as radial Dunkl processes, are stochastic processes defined in a Weyl chamber that are repelled from the latter's boundary by a singular drift with a strength given by the multiplicity function $k$. It is a well-known fact that when $k$ is sufficiently small, these processes hit the Weyl chamber's boundary almost surely, and it was recently shown by the authors that the collision times for the process of type $A$, also known as the Dyson model (a one-dimensional multiple-particle stochastic system), have a Hausdorff dimension that depends on $k$. In this paper, we use the square of the alternating polynomial, which corresponds to the reflection group of the process, to extend this result to all multivariate Bessel processes of rational type, and we show that the Hausdorff dimension of collision times is a piecewise-linear function of the minimum of $k$, but is independent of the dimension of the space where the process lives. This implies that the Hausdorff dimension is independent of the particle number for processes with a particle system representation.

math.PR↗

arXiv2023

Hausdorff dimension of collision times in one-dimensional log-gases

We consider systems of multiple Brownian particles in one dimension that repel mutually via a logarithmic potential on the real line, more specifically the Dyson model. These systems are characterized by a parameter that controls the strength of the interaction, $k>0$. In spite of being a one-dimensional system, this system is interesting due to the properties that arise from the long-range interaction between particles. It is a well-known fact that when $k$ is small enough, particle collisions occur almost surely, while when $k$ is large, collisions never occur. However, aside from this fact there was no characterization of the collision times until now. In this paper, we derive the fractal (Hausdorff) dimension of the set of collision times by generalizing techniques introduced by Liu and Xiao to study the return times to the origin of self-similar stochastic processes. In our case, we consider the return times to configurations where at least one collision occurs, which is a condition that defines unbounded sets, as opposed to a single point, namely, the origin. We find that the fractal dimension characterizes the collision behavior of these systems, and establishes a clear delimitation between the colliding and the non-colliding regions in a way similar to that of an order parameter.

math.PR↗

arXiv2022

Estimation of ergodic square-root diffusion under high-frequency sampling

Gaussian quasi-likelihood estimation of the parameter $θ$ in the square-root diffusion process is studied under high frequency sampling. Different from the previous study of Overbeck and Rydén(1998) under low-frequency sampling, high-frequency of data provides very simple form of the asymptotic covariance matrix. Through easy-to-compute preliminary contrast functions, a practical two-stage manner without numerical optimization is formulated in order to conduct not only an asymptotically efficient estimation of the drift parameters, but also high-precision estimator of the diffusion parameter. Simulation experiments are given to illustrate the results.

math.ST↗

arXiv2020

Martingale estimation functions for Bessel processes

In this paper we derive martingale estimating functions for the dimensionality parameter of a Bessel process based on the eigenfunctions of the diffusion operator. Since a Bessel process is non-ergodic and the theory of martingale estimating functions is developed for ergodic diffusions, we use the space-time transformation of the Bessel process and formulate our results for a modified Bessel process. We deduce consistency, asymptotic normality and discuss optimality. It turns out that the martingale estimating function based of the first eigenfunction of the modified Bessel process coincides with the linear martingale estimating function for the Cox Ingersoll Ross process. Furthermore, our results may also be applied to estimating the multiplicity parameter of a one-dimensional Dunkl process.

math.PR↗