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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.

The asymptotic error of chaos expansion approximations for stochastic differential equations

Tony Huschto Mark Podolskij Sebastian Sager

https://doi.org/10.15559/19-VMSTA133

Pub. online: 23 Apr 2019 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 2 (2019), pp. 145–165

Abstract

In this paper we present a numerical scheme for stochastic differential equations based upon the Wiener chaos expansion. The approximation of a square integrable stochastic differential equation is obtained by cutting off the infinite chaos expansion in chaos order and in number of basis elements. We derive an explicit upper bound for the ${L^{2}}$ approximation error associated with our method. The proofs are based upon an application of Malliavin calculus.

A copula-based bivariate integer-valued autoregressive process with application

Andrius Buteikis Remigijus Leipus

https://doi.org/10.15559/19-VMSTA130

Pub. online: 12 Mar 2019 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 2 (2019), pp. 227–249

Abstract

A bivariate integer-valued autoregressive process of order 1 (BINAR(1)) with copula-joint innovations is studied. Different parameter estimation methods are analyzed and compared via Monte Carlo simulations with emphasis on estimation of the copula dependence parameter. An empirical application on defaulted and non-defaulted loan data is carried out using different combinations of copula functions and marginal distribution functions covering the cases where both marginal distributions are from the same family, as well as the case where they are from different distribution families.

Note on AR(1)-characterisation of stationary processes and model fitting

Marko Voutilainen

Lauri Viitasaari Pauliina Ilmonen

https://doi.org/10.15559/19-VMSTA132

Pub. online: 8 Mar 2019 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 2 (2019), pp. 195–207

Abstract

It was recently proved that any strictly stationary stochastic process can be viewed as an autoregressive process of order one with coloured noise. Furthermore, it was proved that, using this characterisation, one can define closed form estimators for the model parameter based on autocovariance estimators for several different lags. However, this estimation procedure may fail in some special cases. In this article, a detailed analysis of these special cases is provided. In particular, it is proved that these cases correspond to degenerate processes.

Editorial

K. Kubilius Yu. Mishura L. Sakhno

https://doi.org/10.15559/19-VMSTA131

Pub. online: 19 Feb 2019 Type: Editorial

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 1 (2019), pp. 1–2

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

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