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On a linear functional for infinitely divisible moving average random fields

Stefan Roth

https://doi.org/10.15559/19-VMSTA143

Pub. online: 22 Oct 2019 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 4 (2019), pp. 443–478

Abstract

Given a low-frequency sample of the infinitely divisible moving average random field $\{{\textstyle\int _{{\mathbb{R}^{d}}}}f(t-x)\Lambda (dx),\hspace{2.5pt}t\in {\mathbb{R}^{d}}\}$, in [13] we proposed an estimator $\widehat{u{v_{0}}}$ for the function $\mathbb{R}\ni x\mapsto u(x){v_{0}}(x)=(u{v_{0}})(x)$, with $u(x)=x$ and ${v_{0}}$ being the Lévy density of the integrator random measure Λ. In this paper, we study asymptotic properties of the linear functional ${L^{2}}(\mathbb{R})\ni v\mapsto {\left\langle v,\widehat{u{v_{0}}}\right\rangle _{{L^{2}}(\mathbb{R})}}$, if the (known) kernel function f has a compact support. We provide conditions that ensure consistency (in mean) and prove a central limit theorem for it.

On estimation of expectation of simultaneous renewal time of time-inhomogeneous Markov chains using dominating sequence

Vitaliy Golomoziy

https://doi.org/10.15559/19-VMSTA138

Pub. online: 14 Oct 2019 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 3 (2019), pp. 333–343

Abstract

The main subject of the study in this paper is the simultaneous renewal time for two time-inhomogeneous Markov chains which start with arbitrary initial distributions. By a simultaneous renewal we mean the first time of joint hitting the specific set C by both processes. Under the condition of existence a dominating sequence for both renewal sequences generated by the chains and non-lattice condition for renewal probabilities an upper bound for the expectation of the simultaneous renewal time is obtained.

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 Volume 6, Issue 4 (2019), pp. 419–441

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.

Estimation of the drift parameter for the fractional stochastic heat equation via power variation

Zeina Mahdi Khalil Ciprian Tudor

https://doi.org/10.15559/19-VMSTA141

Pub. online: 3 Oct 2019 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 4 (2019), pp. 397–417

Abstract

We define power variation estimators for the drift parameter of the stochastic heat equation with the fractional Laplacian and an additive Gaussian noise which is white in time and white or correlated in space. We prove that these estimators are consistent and asymptotically normal and we derive their rate of convergence under the Wasserstein metric.

Spatial quadratic variations for the solution to a stochastic partial differential equation with elliptic divergence form operator

Mounir Zili

Eya Zougar

https://doi.org/10.15559/19-VMSTA139

Pub. online: 3 Oct 2019 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 3 (2019), pp. 345–375

Abstract

We introduce a stochastic partial differential equation (SPDE) with elliptic operator in divergence form, with measurable and bounded coefficients and driven by space-time white noise. Such SPDEs could be used in mathematical modelling of diffusion phenomena in medium consisting of different kinds of materials and undergoing stochastic perturbations. We characterize the solution and, using the Stein–Malliavin calculus, we prove that the sequence of its recentered and renormalized spatial quadratic variations satisfies an almost sure central limit theorem. Particular focus is given to the interesting case where the coefficients of the operator are piecewise constant.

Maximum likelihood estimation in the non-ergodic fractional Vasicek model

Stanislav Lohvinenko Kostiantyn Ralchenko

https://doi.org/10.15559/19-VMSTA140

Pub. online: 23 Sep 2019 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 3 (2019), pp. 377–395

Abstract

We investigate the fractional Vasicek model described by the stochastic differential equation $d{X_{t}}=(\alpha -\beta {X_{t}})\hspace{0.1667em}dt+\gamma \hspace{0.1667em}d{B_{t}^{H}}$, ${X_{0}}={x_{0}}$, driven by the fractional Brownian motion ${B^{H}}$ with the known Hurst parameter $H\in (1/2,1)$. We study the maximum likelihood estimators for unknown parameters α and β in the non-ergodic case (when $\beta <0$) for arbitrary ${x_{0}}\in \mathbb{R}$, generalizing the result of Tanaka, Xiao and Yu (2019) for particular ${x_{0}}=\alpha /\beta $, derive their asymptotic distributions and prove their asymptotic independence.

Taylor’s power law for the N-stars network evolution model

István Fazekas Csaba Noszály Noémi Uzonyi

https://doi.org/10.15559/19-VMSTA137

Pub. online: 16 Sep 2019 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 3 (2019), pp. 311–331

Abstract

Taylor’s power law states that the variance function decays as a power law. It is observed for population densities of species in ecology. For random networks another power law, that is, the power law degree distribution is widely studied. In this paper the original Taylor’s power law is considered for random networks. A precise mathematical proof is presented that Taylor’s power law is asymptotically true for the N-stars network evolution model.

The risk model with stochastic premiums and a multi-layer dividend strategy

Olena Ragulina

https://doi.org/10.15559/19-VMSTA136

Pub. online: 28 Aug 2019 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 3 (2019), pp. 285–309

Abstract

The paper deals with a generalization of the risk model with stochastic premiums where dividends are paid according to a multi-layer dividend strategy. First of all, we derive piecewise integro-differential equations for the Gerber–Shiu function and the expected discounted dividend payments until ruin. In addition, we concentrate on the detailed investigation of the model in the case of exponentially distributed claim and premium sizes and find explicit formulas for the ruin probability as well as for the expected discounted dividend payments. Lastly, numerical illustrations for some multi-layer dividend strategies are presented.

Arithmetic of (independent) sigma-fields on probability spaces

Matija Vidmar

https://doi.org/10.15559/19-VMSTA135

Pub. online: 20 Jun 2019 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 3 (2019), pp. 269–284

Abstract

This note gathers what is known about, and provides some new results concerning the operations of intersection, of “generated σ-field”, and of “complementation” for (independent) complete σ-fields on probability spaces.

Moderate deviations for a stochastic Burgers equation

Rachid Belfadli Lahcen Boulanba Mohamed Mellouk

https://doi.org/10.15559/19-VMSTA134

Pub. online: 16 May 2019 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 2 (2019), pp. 167–193

Abstract

A moderate deviations principle for the law of a stochastic Burgers equation is proved via the weak convergence approach. In addition, some useful estimates toward a central limit theorem are established.

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