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

Vitaliy Golomoziy

https://doi.org/10.15559/19-MSTA138

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.

Logarithmic Lévy process directed by Poisson subordinator

Penka Mayster Assen Tchorbadjieff

https://doi.org/10.15559/19-MSTA142

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

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.

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-MSTA141

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

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.

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-MSTA139

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

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.

Maximum likelihood estimation in the non-ergodic fractional Vasicek model

Stanislav Lohvinenko Kostiantyn Ralchenko

https://doi.org/10.15559/19-MSTA140

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

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.

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