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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

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.

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.

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

István Fazekas Csaba Noszály Noémi Uzonyi

https://doi.org/10.15559/19-MSTA137

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

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.

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

Olena Ragulina

https://doi.org/10.15559/19-MSTA136

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

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.

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