arXiv subjects

Manfred Marvin Marchione

Publications and source records attributed to Manfred Marvin Marchione.

9 recordsLinked to original sources

arXiv2026

Dirichlet problems and exit distributions for the telegraph process and its planar extensions

In this paper, we study boundary-value problems describing the exit distribution of finite-velocity random motions from prescribed domains. For the standard telegraph process, with and without drift, we derive the Dirichlet problems governing the exit point and mean exit time from a closed interval. We then extend the analysis to a planar finite-velocity model with orthogonal directions, for which we obtain the associated Laplace and Poisson-type equations for the exit distribution and mean exit time. In the special case of an infinite strip, explicit solutions are obtained. In all cases, we show that our equations and results converge, in the hydrodynamic limit, to the corresponding ones for Brownian motion.

math.PR↗

arXiv2026

One-dimensional and planar random motions with variable propagation speeds

In this paper, we study univariate and planar random motions with variable propagation speeds. We first consider motions with space-varying velocity, which can be reduced to constant-velocity motions by means of suitable nonlinear transformations. We examine a special case of a motion which is confined within the unit interval. To provide a general expression of the moments of this process, we introduce a new family of polynomials which generalize the classical Euler polynomials. We then examine a planar extension of this process which moves along orthogonal directions. A process with velocity depending on the direction is also examined, and its mean conditional on the initial direction and the number of direction changes is given in terms of confluent hypergeometric functions. We conjecture that, in the hydrodynamic limit, this process is absorbed at a point in an arbitrarily small time. We finally study a motion with time-dependent velocity. We prove that this process can be represented as an integral with respect to a standard telegraph process, and we obtain its covariance function explicitly. Moreover, we show that this process behaves as a Itô integral with respect to Brownian motion in the hydrodynamic limit.

math.PR↗

arXiv2024

On a planar random motion with asymptotically correlated components

We study a planar random motion $\big(X(t),\,Y(t)\big)$ with orthogonal directions, where the direction switches are governed by a homogeneous Poisson process. At each Poisson event, the moving particle turns clockwise or counterclockwise according to a rule which depends on the current direction. We prove that the components of the vector $\big(X(t),\,Y(t)\big)$ can be represented as linear combinations of two independent telegraph processes with different intensities. The exact distribution of $\big(X(t),\,Y(t)\big)$ is then obtained both in the interior of the support and on its boundary, where a singular component is present. We show that, in the hydrodynamic limit, the process behaves as a planar Brownian motion with correlated components. The distribution of the time spent by the process moving vertically is then studied. We obtain its exact distribution and discuss its hydrodynamic limit. In particular, in the limiting case, the process $\big(X(t),\,Y(t)\big)$ spends half of the time moving vertically.

math.PR↗

arXiv2024

Planar random motions in a vortex

We study a planar random motion $\big(X(t),Y(t)\big)$ with orthogonal directions which can turn clockwise, counterclockwise and reverse its direction each with a different probability. The support of the process is given by a time-varying square and the singular distributions on the boundary and the diagonals of the square are obtained. In the interior of the support, we study the hydrodynamic limit of the distribution. We then investigate the time $T(t)$ spent by the process moving vertically and the joint distribution of $\big(T(t),Y(t)\big)$. We prove that, in the hydrodynamic limit, the process $\big(X(t),Y(t)\big)$ spends half the time moving vertically.

math.PR↗

arXiv2024

Non-monotone dependence modeling with copulas: an application to the volume-return relationship

This paper introduces an innovative method for constructing copula models capable of describing arbitrary non-monotone dependence structures. The proposed method enables the creation of such copulas in parametric form, thus allowing the resulting models to adapt to diverse and intricate real-world data patterns. We apply this novel methodology to analyze the relationship between returns and trading volumes in financial markets, a domain where the existence of non-monotone dependencies is well-documented in the existing literature. Our approach exhibits superior adaptability compared to other models which have previously been proposed in the literature, enabling a deeper understanding of the dependence structure among the considered variables.

stat.ME↗

arXiv2023

GEMAct: a Python package for non-life (re)insurance modeling

This paper introduces $\textbf{gemact}$, a $\textbf{Python}$ package for actuarial modelling based on the collective risk model. The library supports applications to risk costing and risk transfer, loss aggregation, and loss reserving. We add new probability distributions to those available in $\textbf{scipy}$, including the (a, b, 0) and (a, b, 1) discrete distributions, copulas of the Archimedean family, the Gaussian, the Student t and the Fundamental copulas. We provide an implementation of the AEP algorithm for calculating the cumulative distribution function of the sum of dependent, non-negative random variables, given their dependency structure specified with a copula. The theoretical framework is introduced at the beginning of each section to give the reader with a sufficient understanding of the underlying actuarial models.

stat.AP↗

arXiv2022

Pseudoprocesses related to higher-order equations of vibrations of rods

In this paper we study Fresnel pseudoprocesses whose signed measure density is a solution to a higher-order extension of the equation of vibrations of rods. We also investigate space-fractional extensions of the pseudoprocesses related to the Riesz operator. The measure density is represented in terms of generalized Airy functions which include the classical Airy function as a particular case. We prove that the Fresnel pseudoprocess time-changed with an independent stable subordinator produces genuine stochastic processes. In particular, if the exponent of the subordinator is chosen in a suitable way, the time-changed pseudoprocess is identical in distribution to a mixture of stable processes. The case of a mixture of Cauchy distributions is discussed and we show that the symmetric mixture can be either unimodal or bimodal, while the probability density function of an asymmetric mixture can possibly have an inflection point.

math.PR↗

arXiv2022

Stable distributions and pseudo-processes related to fractional Airy functions

In this paper we study pseudo-processes related to odd-order heat-type equations composed with Lévy stable subordinators. The aim of the article is twofold. We first show that the pseudo-density of the subordinated pseudo-process can be represented as an expectation of damped oscillations with generalized gamma distributed parameters. This stochastic representation also arises as the solution to a fractional diffusion equation, involving a higher-order Riesz-Feller operator, which generalizes the odd-order heat-type equation. We then prove that, if the stable subordinator has a suitable exponent, the time-changed pseudo-process becomes a genuine Lévy stable process. This result permits us to obtain a power series representation for the probability density function of an arbitrary asymmetric stable process of exponent $ν>1$ and skewness parameter $β$, with $0<\lvertβ\lvert<1$. The methods we use in order to carry out our analysis are based on the study of a fractional Airy function which emerges in the investigation of the higher-order Riesz-Feller operator.

math.PR↗

arXiv2022

Hitting distribution of a correlated planar Brownian motion in a disk

In this paper we study the hitting probability of a circumference $C_R$ for a correlated Brownian motion $\underline{B}(t)=\left(B_1(t), B_2(t)\right)$, $ρ$ being the correlation coefficient. The analysis starts by first mapping the circle $C_R$ into an ellipse $E$ with semiaxes depending on $ρ$ and transforming the differential operator governing the hitting distribution into the classical Laplace operator. By means of two different approaches (one obtained by applying elliptic coordinates) we obtain the desired distribution as a series of Poisson kernels.

math.PR↗