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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.

Existence of density function for the running maximum of SDEs driven by nontruncated pure-jump Lévy processes

Takuya Nakagawa

Ryoichi Suzuki

https://doi.org/10.15559/24-VMSTA245

Pub. online: 23 Jan 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 11, Issue 3 (2024), pp. 303–321

Abstract

The existence of density function of the running maximum of a stochastic differential equation (SDE) driven by a Brownian motion and a nontruncated pure-jump process is verified. This is proved by the existence of density function of the running maximum of the Wiener–Poisson functionals resulting from Bismut’s approach to the Malliavin calculus for jump processes.

Asymptotic normality of the LSE for chirp signal parameters

Alexander Ivanov

Viktor Hladun

https://doi.org/10.15559/24-VMSTA247

Pub. online: 23 Jan 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 11, Issue 2 (2024), pp. 195–216

Abstract

A time continuous statistical model of chirp signal observed against the background of stationary Gaussian noise is considered in the paper. Asymptotic normality of the LSE for parameters of such a sinusoidal regression model is obtained.

Power law in Sandwiched Volterra Volatility model

Giulia Di Nunno Anton Yurchenko-Tytarenko

https://doi.org/10.15559/24-VMSTA246

Pub. online: 23 Jan 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 11, Issue 2 (2024), pp. 169–194

Abstract

The paper presents an analytical proof demonstrating that the Sandwiched Volterra Volatility (SVV) model is able to reproduce the power-law behavior of the at-the-money implied volatility skew, provided the correct choice of the Volterra kernel. To obtain this result, the second-order Malliavin differentiability of the volatility process is assessed and the conditions that lead to explosive behavior in the Malliavin derivative are investigated. As a supplementary result, a general Malliavin product rule is proved.

Almost everywhere continuity of conditional expectations

Alberto Alonso

Fernando Brambila-Paz

https://doi.org/10.15559/23-VMSTA240

Pub. online: 11 Jan 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 11, Issue 3 (2024), pp. 247–263

Abstract

A necessary and sufficient condition on a sequence ${\{{\mathcal{A}_{n}}\}_{n\in \mathbb{N}}}$ of σ-subalgebras which assures convergence almost everywhere of conditional expectations for functions in ${L^{\infty }}$ is given. It is proven that for $f\in {L^{\infty }}(\mathcal{A})$

\[ \mathsf{E}(f|{\mathcal{A}_{n}})\stackrel{a.e.}{\longrightarrow }\mathsf{E}(f|{\mathcal{A}_{\mu a.e.}}).\]

Stochastic Lotka–Volterra mutualism model with jumps

Olga Borysenko

Oleksandr Borysenko

https://doi.org/10.15559/23-VMSTA242

Pub. online: 9 Jan 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 11, Issue 3 (2024), pp. 289–301

Abstract

The existence and uniqueness of the global positive solution are proved for the system of stochastic differential equations describing a two-species Lotka–Volterra mutualism model disturbed by white noise, centered and noncentered Poisson noises. For the considered system, sufficient conditions of stochastic ultimate boundedness, stochastic permanence, nonpersistence and strong persistence in the mean are obtained.

On min- and max-Kies families: distributional properties and saturation in Hausdorff sense

Tsvetelin Zaevski

Nikolay Kyurkchiev

https://doi.org/10.15559/24-VMSTA244

Pub. online: 9 Jan 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 11, Issue 3 (2024), pp. 265–288

Abstract

The purpose of this paper is to explore two probability distributions originating from the Kies distribution defined on an arbitrary domain. The first one describes the minimum of several Kies random variables whereas the second one is for their maximum – they are named min- and max-Kies, respectively. The properties of the min-Kies distribution are studied in details, and later some duality arguments are used to examine the max variant. Also the saturations in the Hausdorff sense are investigated. Some numerical experiments are provided.

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

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