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Large deviations for conditionally Gaussian processes: estimates of level crossing probability

Barbara Pacchiarotti Alessandro Pigliacelli

https://doi.org/10.15559/18-MSTA119

Pub. online: 12 Oct 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 5, Issue 4 (2018), pp. 483–499

Abstract

The problem of (pathwise) large deviations for conditionally continuous Gaussian processes is investigated. The theory of large deviations for Gaussian processes is extended to the wider class of random processes – the conditionally Gaussian processes. The estimates of level crossing probability for such processes are given as an application.

Ruin probability for the bi-seasonal discrete time risk model with dependent claims

Olga Navickienė Jonas Sprindys Jonas Šiaulys

https://doi.org/10.15559/18-VMSTA118

Pub. online: 1 Oct 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 1 (2019), pp. 133–144

Abstract

The discrete time risk model with two seasons and dependent claims is considered. An algorithm is created for computing the values of the ultimate ruin probability. Theoretical results are illustrated with numerical examples.

Ruin probability for the bi-seasonal discrete time risk model with dependent claims

Olga Navickienė Jonas Sprindys Jonas Šiaulys

https://doi.org/10.15559/18-MSTA118

Pub. online: 1 Oct 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 6, Issue 1 (2019), pp. 133–144

Abstract

On generalized stochastic fractional integrals and related inequalities

Hüseyin Budak Mehmet Zeki Sarikaya

https://doi.org/10.15559/18-VMSTA117

Pub. online: 24 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 4 (2018), pp. 471–481

Abstract

The generalized mean-square fractional integrals ${\mathcal{J}_{\rho ,\lambda ,u+;\omega }^{\sigma }}$ and ${\mathcal{J}_{\rho ,\lambda ,v-;\omega }^{\sigma }}$ of the stochastic process X are introduced. Then, for Jensen-convex and strongly convex stochastic proceses, the generalized fractional Hermite–Hadamard inequality is establish via generalized stochastic fractional integrals.

On generalized stochastic fractional integrals and related inequalities

Hüseyin Budak Mehmet Zeki Sarikaya

https://doi.org/10.15559/18-MSTA117

Pub. online: 24 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 5, Issue 4 (2018), pp. 471–481

Abstract

Stochastic models associated to a Nonlocal Porous Medium Equation

Alessandro De Gregorio

https://doi.org/10.15559/18-VMSTA112

Pub. online: 19 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 4 (2018), pp. 457–470

Abstract

The nonlocal porous medium equation considered in this paper is a degenerate nonlinear evolution equation involving a space pseudo-differential operator of fractional order. This space-fractional equation admits an explicit, nonnegative, compactly supported weak solution representing a probability density function. In this paper we analyze the link between isotropic transport processes, or random flights, and the nonlocal porous medium equation. In particular, we focus our attention on the interpretation of the weak solution of the nonlinear diffusion equation by means of random flights.

Stochastic models associated to a Nonlocal Porous Medium Equation

Alessandro De Gregorio

https://doi.org/10.15559/18-MSTA112

Pub. online: 19 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 5, Issue 4 (2018), pp. 457–470

Abstract

Drifted Brownian motions governed by fractional tempered derivatives

Mirko D’Ovidio Francesco Iafrate Enzo Orsingher

https://doi.org/10.15559/18-VMSTA114

Pub. online: 19 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 4 (2018), pp. 445–456

Abstract

Fractional equations governing the distribution of reflecting drifted Brownian motions are presented. The equations are expressed in terms of tempered Riemann–Liouville type derivatives. For these operators a Marchaud-type form is obtained and a Riesz tempered fractional derivative is examined, together with its Fourier transform.

Drifted Brownian motions governed by fractional tempered derivatives

Mirko D’Ovidio Francesco Iafrate Enzo Orsingher

https://doi.org/10.15559/18-MSTA114

Pub. online: 19 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 5, Issue 4 (2018), pp. 445–456

Abstract

Approximation of solutions of the stochastic wave equation by using the Fourier series

Vadym Radchenko

Nelia Stefans’ka

https://doi.org/10.15559/18-VMSTA115

Pub. online: 19 Sep 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 4 (2018), pp. 429–444

Abstract

A one-dimensional stochastic wave equation driven by a general stochastic measure is studied in this paper. The Fourier series expansion of stochastic measures is considered. It is proved that changing the integrator by the corresponding partial sums or by Fejèr sums we obtain the approximations of mild solution of the equation.

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