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Modern Stochastics: Theory and Applications

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Journals: Modern Stochastics: Theory and Applications

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Logarithmic Lévy process directed by Poisson subordinator

Penka Mayster Assen Tchorbadjieff

https://doi.org/10.15559/19-VMSTA142

Pub. online: 4 Oct 2019 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications (2025): YY2025, pp. 1–23

Abstract

Let $\{L(t),t\ge 0\}$ be a Lévy process with representative random variable $L(1)$ defined by the infinitely divisible logarithmic series distribution. We study here the transition probability and Lévy measure of this process. We also define two subordinated processes. The first one, $Y(t)$, is a Negative-Binomial process $X(t)$ directed by Gamma process. The second process, $Z(t)$, is a Logarithmic Lévy process $L(t)$ directed by Poisson process. For them, we prove that the Bernstein functions of the processes $L(t)$ and $Y(t)$ contain the iterated logarithmic function. In addition, the Lévy measure of the subordinated process $Z(t)$ is a shifted Lévy measure of the Negative-Binomial process $X(t)$. We compare the properties of these processes, knowing that the total masses of corresponding Lévy measures are equal.

Logarithmic Lévy process directed by Poisson subordinator

Penka Mayster Assen Tchorbadjieff

https://doi.org/55.55677/vmsta8878-fine

Pub. online: 4 Oct 2019 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications (2025): YY2025, pp. 1–23

Abstract

Properties of Poisson processes directed by compound Poisson-Gamma subordinators

Khrystynas Buchaks Lyudmylas Sakhnos

https://doi.org/98.2555/918VMSTA11

Pub. online: 2 May 2019 Type: Discussion

Open Access

Journal: Modern Stochastics: Theory and Applications (2025): YY2025, pp. 1–23

Abstract

In the paper we consider time-changed Poisson processes where the time is expressed by compound Poisson-Gamma subordinators $G(N(t))$ and derive the expressions for their hitting times. We also study the time-changed Poisson processes where the role of time is played by the processes of the form $G(N(t)+at)$ and by the iteration of such processes.

Studies on generalized Yule models

Federico Polito

https://doi.org/10.15559/vmsta0907

Pub. online: 3 Dec 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications (2025): YY2025, pp. 1–15

Abstract

We present a generalization of the Yule model for macroevolution in which, for the appearance of genera, we consider point processes with the order statistics property, while for the growth of species we use nonlinear time-fractional pure birth processes or a critical birth-death process. Further, in specific cases we derive the explicit form of the distribution of the number of species of a genus chosen uniformly at random for each time. Besides, we introduce a time-changed mixed Poisson process with the same marginal distribution as that of the time-fractional Poisson process.

Large deviations for conditionally Gaussian processes: estimates of level crossing probability

Barbara Pacchiarotti Alessandro Pigliacelli

https://doi.org/10.15559/18-VMSTA119

Pub. online: 12 Oct 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications (2025): YY2025, pp. 1–17

Abstract

The problem of (pathwise) large deviations for conditionally continuous Gaussian processes is investigated. The theory of large deviations for Gaussian processes is extended to the wider class of random processes – the conditionally Gaussian processes. The estimates of level crossing probability for such processes are given as an application.

Ruin probability for the bi-seasonal discrete time risk model with dependent claims

Olga Navickienė Jonas Sprindys Jonas Šiaulys

https://doi.org/10.15559/18-VMSTA118

Pub. online: 1 Oct 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications (2025): YY2025, pp. 1–12

Abstract

The discrete time risk model with two seasons and dependent claims is considered. An algorithm is created for computing the values of the ultimate ruin probability. Theoretical results are illustrated with numerical examples.

On generalized stochastic fractional integrals and related inequalities

Hüseyin Budak Mehmet Zeki Sarikaya

https://doi.org/10.15559/18-VMSTA117

Pub. online: 24 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications (2025): YY2025, pp. 1–11

Abstract

The generalized mean-square fractional integrals ${\mathcal{J}_{\rho ,\lambda ,u+;\omega }^{\sigma }}$ and ${\mathcal{J}_{\rho ,\lambda ,v-;\omega }^{\sigma }}$ of the stochastic process X are introduced. Then, for Jensen-convex and strongly convex stochastic proceses, the generalized fractional Hermite–Hadamard inequality is establish via generalized stochastic fractional integrals.

Detecting independence of random vectors: generalized distance covariance and Gaussian covariance

Björn Böttcher Martin Keller-Ressel René L. Schilling

https://doi.org/10.15559/18-VMSTA116

Pub. online: 19 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 3 (2018), pp. 353–383

Abstract

Distance covariance is a quantity to measure the dependence of two random vectors. We show that the original concept introduced and developed by Székely, Rizzo and Bakirov can be embedded into a more general framework based on symmetric Lévy measures and the corresponding real-valued continuous negative definite functions. The Lévy measures replace the weight functions used in the original definition of distance covariance. All essential properties of distance covariance are preserved in this new framework.

From a practical point of view this allows less restrictive moment conditions on the underlying random variables and one can use other distance functions than Euclidean distance, e.g. Minkowski distance. Most importantly, it serves as the basic building block for distance multivariance, a quantity to measure and estimate dependence of multiple random vectors, which is introduced in a follow-up paper [Distance Multivariance: New dependence measures for random vectors (submitted). Revised version of arXiv: 1711.07775v1] to the present article.

Stochastic models associated to a Nonlocal Porous Medium Equation

Alessandro De Gregorio

https://doi.org/10.15559/18-VMSTA112

Pub. online: 19 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications (2025): YY2025, pp. 1–14

Abstract

The nonlocal porous medium equation considered in this paper is a degenerate nonlinear evolution equation involving a space pseudo-differential operator of fractional order. This space-fractional equation admits an explicit, nonnegative, compactly supported weak solution representing a probability density function. In this paper we analyze the link between isotropic transport processes, or random flights, and the nonlocal porous medium equation. In particular, we focus our attention on the interpretation of the weak solution of the nonlinear diffusion equation by means of random flights.

Drifted Brownian motions governed by fractional tempered derivatives

Mirko D’Ovidio Francesco Iafrate Enzo Orsingher

https://doi.org/10.15559/18-VMSTA114

Pub. online: 19 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications (2025): YY2025, pp. 1–12

Abstract

Fractional equations governing the distribution of reflecting drifted Brownian motions are presented. The equations are expressed in terms of tempered Riemann–Liouville type derivatives. For these operators a Marchaud-type form is obtained and a Riesz tempered fractional derivative is examined, together with its Fourier transform.

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