The paper investigates properties of mean-square solutions to the Airy equation with random initial data given by stationary processes. The result on the modulus of continiuty of the solution is stated and properties of the covariance function are described. Bounds for the distributions of the suprema of solutions under $\varphi$-sub-Gaussian initial conditions are presented. Several examples are provided to illustrate the results. Extension of the results to the case of fractional Airy equation is given.

Investigation of Airy equations with random initial conditions

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.

Pseudoprocesses Related to Higher-Order Equations of Vibrations of Rods

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 < | β | < 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.

Stable distributions and pseudo-processes related to fractional Airy functions

This paper introduces gemact, a Python package for actuarial modeling based on the collective risk model. The library supports applications to costing and risk transfers, risks aggregation, and claims reserving. We add new probability distributions to those available in scipy, including the (a,b,0) and (a,b,1) discrete distributions, the Archimedean and the Elliptical classes of copulas. We provide a software 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 in brief at the beginning of each section to provide the reader with a sufficient understanding of the underlying actuarial models.

Manfred Marvin Marchione

Papers

Stable distributions and pseudo-processes related to fractional Airy functions

Pseudoprocesses Related to Higher-Order Equations of Vibrations of Rods

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