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Generalized fractional calculus and some models of generalized counting processes

Khrystyna Buchak Lyudmyla Sakhno

https://doi.org/10.15559/24-VMSTA254

Pub. online: 30 May 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 11, Issue 4 (2024), pp. 439–458

Abstract

Models of generalized counting processes time-changed by a general inverse subordinator are considered, their distributions are characterized, and governing equations for them are presented. The equations are given in terms of the generalized fractional derivatives, namely, convolution-type derivatives with respect to Bernštein functions. Some particular examples are presented.

Sample path properties of multidimensional integral with respect to stochastic measure

Boris Manikin

Vadym Radchenko

https://doi.org/10.15559/24-VMSTA256

Pub. online: 30 May 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 11, Issue 4 (2024), pp. 421–437

Abstract

The integral with respect to a multidimensional stochastic measure, assuming only its σ-additivity in probability, is studied. The continuity and differentiability of realizations of the integral are established.

Properties of the entropic risk measure EVaR in relation to selected distributions

Yuliya Mishura

Kostiantyn Ralchenko

Petro Zelenko

Volodymyr Zubchenko

https://doi.org/10.15559/24-VMSTA255

Pub. online: 30 Apr 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 11, Issue 4 (2024), pp. 373–394

Abstract

Entropic Value-at-Risk (EVaR) measure is a convenient coherent risk measure. Due to certain difficulties in finding its analytical representation, it was previously calculated explicitly only for the normal distribution. We succeeded to overcome these difficulties and to calculate Entropic Value-at-Risk (EVaR) measure for Poisson, compound Poisson, Gamma, Laplace, exponential, chi-squared, inverse Gaussian distribution and normal inverse Gaussian distribution with the help of Lambert function that is a special function, generally speaking, with two branches.

Unsolved problem about stability of stochastic difference equations with continuous time and distributed delay

Leonid Shaikhet

https://doi.org/10.15559/24-VMSTA253

Pub. online: 11 Apr 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 11, Issue 4 (2024), pp. 395–402

Abstract

Despite the fact that the theory of stability of continuous-time difference equations has a long history, is well developed and very popular in research, there is a simple and clearly formulated problem about the stability of stochastic difference equations with continuous time and distributed delay, which has not been solved for more than 13 years. This paper offers to readers some generalization on this unsolved problem in the hope that it will help move closer to its solution.

Combinatorial approach to the calculation of projection coefficients for the simplest Gaussian-Volterra process

Iryna Bodnarchuk

Yuliya Mishura

https://doi.org/10.15559/24-VMSTA252

Pub. online: 9 Apr 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 11, Issue 4 (2024), pp. 403–419

Abstract

The Gaussian-Volterra process with a linear kernel is considered, its properties are established and projection coefficients are explicitly calculated, i.e. one of possible prediction problems related to Gaussian processes is solved.

Consistency of local linear regression estimator for mixtures with varying concentrations

Daniel Horbunov

Rostyslav Maiboroda

https://doi.org/10.15559/24-VMSTA250

Pub. online: 19 Mar 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 11, Issue 3 (2024), pp. 359–372

Abstract

Finite mixtures with different regression models for different mixture components naturally arise in statistical analysis of biological and sociological data. In this paper a model of mixtures with varying concentrations is considered in which the mixing probabilities are different for different observations. A modified local linear estimation (mLLE) technique is developed to estimate the regression functions of the mixture component nonparametrically. Consistency of the mLLE is demonstrated. Performance of mLLE and a modified Nadaraya–Watson estimator (mNWE) is assessed via simulations. The results confirm that the mLLE technique overcomes the boundary effect typical to the NWE.

Distribution of shifted discrete random walk generated by distinct random variables and applications in ruin theory

Simonas Gervė Andrius Grigutis

https://doi.org/10.15559/24-VMSTA249

Pub. online: 19 Mar 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 11, Issue 3 (2024), pp. 323–357

Abstract

In this paper, the distribution function

\[ \varphi (u)=\mathbb{P}\Bigg(\underset{n\geqslant 1}{\sup }{\sum \limits_{i=1}^{n}}({X_{i}}-\kappa )\lt u\Bigg),\]

and the generating function of $\varphi (u+1)$ are set up. We assume that $u\in \mathbb{N}\cup \{0\}$, $\kappa \in \mathbb{N}$, the random walk $\{{\textstyle\sum _{i=1}^{n}}{X_{i}},\hspace{0.1667em}n\in \mathbb{N}\}$ involves $N\in \mathbb{N}$ periodically occurring distributions, and the integer-valued and nonnegative random variables ${X_{1}},{X_{2}},\dots $ are independent. This research generalizes two recent works where $\{\kappa =1,N\in \mathbb{N}\}$ and $\{\kappa \in \mathbb{N},N=1\}$ were considered respectively. The provided sequence of sums $\{{\textstyle\sum _{i=1}^{n}}({X_{i}}-\kappa ),\hspace{0.1667em}n\in \mathbb{N}\}$ generates the so-called multi-seasonal discrete-time risk model with arbitrary natural premium and its known distribution enables to compute the ultimate time ruin probability $1-\varphi (u)$ or survival probability $\varphi (u)$. The obtained theoretical statements are verified in several computational examples where the values of the survival probability $\varphi (u)$ and its generating function are provided when $\{\kappa =2,\hspace{0.1667em}N=2\}$, $\{\kappa =3,\hspace{0.1667em}N=2\}$, $\{\kappa =5,\hspace{0.1667em}N=10\}$ and ${X_{i}}$ adopts the Poisson and some other distributions. The conjecture on the nonsingularity of certain matrices is posed.

Editorial

Kȩstutis Kubilius Kostiantyn Ralchenko

https://doi.org/10.15559/24-VMSTA251

Pub. online: 15 Mar 2024 Type: Editorial

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 11, Issue 2 (2024), pp. 129–131

Optimal estimation of the local time and the occupation time measure for an α-stable Lévy process

Chiara Amorino

Arturo Jaramillo Mark Podolskij

https://doi.org/10.15559/24-VMSTA243

Pub. online: 6 Feb 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 11, Issue 2 (2024), pp. 149–168

Abstract

A novel theoretical result on estimation of the local time and the occupation time measure of an α-stable Lévy process with $\alpha \in (1,2)$ is presented. The approach is based upon computing the conditional expectation of the desired quantities given high frequency data, which is an ${L^{2}}$-optimal statistic by construction. The corresponding stable central limit theorems are proved and a statistical application is discussed. In particular, this work extends the results of [20], which investigated the case of the Brownian motion.

A law of the iterated logarithm for small counts in Karlin’s occupancy scheme

Alexander Iksanov

Valeriya Kotelnikova

https://doi.org/10.15559/24-VMSTA248

Pub. online: 30 Jan 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 11, Issue 2 (2024), pp. 217–245

Abstract

In the Karlin infinite occupancy scheme, balls are thrown independently into an infinite array of boxes $1,2,\dots $ , with probability ${p_{k}}$ of hitting the box k. For $j,n\in \mathbb{N}$, denote by ${\mathcal{K}_{j}^{\ast }}(n)$ the number of boxes containing exactly j balls provided that n balls have been thrown. Small counts are the variables ${\mathcal{K}_{j}^{\ast }}(n)$, with j fixed. The main result is a law of the iterated logarithm (LIL) for the small counts as the number of balls thrown becomes large. Its proof exploits a Poissonization technique and is based on a new LIL for infinite sums of independent indicators ${\textstyle\sum _{k\ge 1}}{1_{{A_{k}}(t)}}$ as $t\to \infty $, where the family of events ${({A_{k}}(t))_{t\ge 0}}$ is not necessarily monotone in t. The latter LIL is an extension of a LIL obtained recently by Buraczewski, Iksanov and Kotelnikova (2023+) in the situation when ${({A_{k}}(t))_{t\ge 0}}$ forms a nondecreasing family of events.

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Online ISSN: 2351-6054
Print ISSN: 2351-6046

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