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Editorial

K. Kubilius Yu. Mishura L. Sakhno

https://doi.org/10.15559/17-VMSTA81

Pub. online: 3 Jul 2017 Type: Editorial

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 2 (2017), pp. 91–92

Compositions of Poisson and Gamma processes

Khrystyna Buchak Lyudmyla Sakhno

https://doi.org/10.15559/17-VMSTA79

Pub. online: 29 Jun 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 2 (2017), pp. 161–188

Abstract

In the paper we study the models of time-changed Poisson and Skellam-type processes, where the role of time is played by compound Poisson-Gamma subordinators and their inverse (or first passage time) processes. We obtain explicitly the probability distributions of considered time-changed processes and discuss their properties.

Bonus–malus systems with different claim types and varying deductibles

Olena Ragulina

https://doi.org/10.15559/17-VMSTA80

Pub. online: 28 Jun 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 2 (2017), pp. 141–159

Abstract

The paper deals with bonus–malus systems with different claim types and varying deductibles. The premium relativities are softened for the policyholders who are in the malus zone and these policyholders are subject to per claim deductibles depending on their levels in the bonus–malus scale and the types of the reported claims. We introduce such bonus–malus systems and study their basic properties. In particular, we investigate when it is possible to introduce varying deductibles, what restrictions we have and how we can do this. Moreover, we deal with the special case where varying deductibles are applied to the claims reported by policyholders occupying the highest level in the bonus–malus scale and consider two allocation principles for the deductibles. Finally, numerical illustrations are presented.

Multi-state models for evaluating conversion options in life insurance

Guglielmo D’Amico

Montserrat Guillen Raimondo Manca Filippo Petroni

https://doi.org/10.15559/17-VMSTA78

Pub. online: 10 May 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 2 (2017), pp. 127–139

Abstract

In this paper we propose a multi-state model for the evaluation of the conversion option contract. The multi-state model is based on age-indexed semi-Markov chains that are able to reproduce many important aspects that influence the valuation of the option such as the duration problem, the time non-homogeneity and the ageing effect. The value of the conversion option is evaluated after the formal description of this contract.

Quantifying and estimating additive measures of interaction from case-control data

Ola Hössjer Lars Alfredsson Anna Karin Hedström Magnus Lekman Ingrid Kockum Tomas Olsson

https://doi.org/10.15559/17-VMSTA77

Pub. online: 26 Apr 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 2 (2017), pp. 109–125

Abstract

In this paper we develop a general framework for quantifying how binary risk factors jointly influence a binary outcome. Our key result is an additive expansion of odds ratios as a sum of marginal effects and interaction terms of varying order. These odds ratio expansions are used for estimating the excess odds ratio, attributable proportion and synergy index for a case-control dataset by means of maximum likelihood from a logistic regression model. The confidence intervals associated with these estimates of joint effects and interaction of risk factors rely on the delta method. Our methodology is illustrated with a large Nordic meta dataset for multiple sclerosis. It combines four studies, with a total of 6265 cases and 8401 controls. It has three risk factors (smoking and two genetic factors) and a number of other confounding variables.

A functional limit theorem for random processes with immigration in the case of heavy tails

Alexander Marynych Glib Verovkin

https://doi.org/10.15559/17-VMSTA76

Pub. online: 12 Apr 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 2 (2017), pp. 93–108

Abstract

Let $(X_{k},\xi _{k})_{k\in \mathbb{N}}$ be a sequence of independent copies of a pair $(X,\xi )$ where X is a random process with paths in the Skorokhod space $D[0,\infty )$ and ξ is a positive random variable. The random process with immigration $(Y(u))_{u\in \mathbb{R}}$ is defined as the a.s. finite sum $Y(u)=\sum _{k\ge 0}X_{k+1}(u-\xi _{1}-\cdots -\xi _{k})\mathbb{1}_{\{\xi _{1}+\cdots +\xi _{k}\le u\}}$. We obtain a functional limit theorem for the process $(Y(ut))_{u\ge 0}$, as $t\to \infty $, when the law of ξ belongs to the domain of attraction of an α-stable law with $\alpha \in (0,1)$, and the process X oscillates moderately around its mean $\mathbb{E}[X(t)]$. In this situation the process $(Y(ut))_{u\ge 0}$, when scaled appropriately, converges weakly in the Skorokhod space $D(0,\infty )$ to a fractionally integrated inverse stable subordinator.

Asymptotic behaviour of non-isotropic random walks with heavy tails

Mark Kelbert Enzo Orsingher

https://doi.org/10.15559/17-VMSTA75

Pub. online: 6 Apr 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 1 (2017), pp. 79–89

Abstract

A random flight on a plane with non-isotropic displacements at the moments of direction changes is considered. In the case of exponentially distributed flight lengths a Gaussian limit theorem is proved for the position of a particle in the scheme of series when jump lengths and non-isotropic displacements tend to zero. If the flight lengths have a folded Cauchy distribution the limiting distribution of the particle position is a convolution of the circular bivariate Cauchy distribution with a Gaussian law.

Randomly stopped maximum and maximum of sums with consistently varying distributions

Ieva Marija Andrulytė Martynas Manstavičius Jonas Šiaulys

https://doi.org/10.15559/17-VMSTA74

Pub. online: 6 Mar 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 1 (2017), pp. 65–78

Abstract

Let $\{\xi _{1},\xi _{2},\dots \}$ be a sequence of independent random variables, and η be a counting random variable independent of this sequence. In addition, let $S_{0}:=0$ and $S_{n}:=\xi _{1}+\xi _{2}+\cdots +\xi _{n}$ for $n\geqslant 1$. We consider conditions for random variables $\{\xi _{1},\xi _{2},\dots \}$ and η under which the distribution functions of the random maximum $\xi _{(\eta )}:=\max \{0,\xi _{1},\xi _{2},\dots ,\xi _{\eta }\}$ and of the random maximum of sums $S_{(\eta )}:=\max \{S_{0},S_{1},S_{2},\dots ,S_{\eta }\}$ belong to the class of consistently varying distributions. In our consideration the random variables $\{\xi _{1},\xi _{2},\dots \}$ are not necessarily identically distributed.

L^{p} (p \geq 2)

-solutions of generalized BSDEs with jumps and monotone generator in a general filtration

M’hamed Eddahbi Imade Fakhouri Youssef Ouknine

https://doi.org/10.15559/17-VMSTA73

Pub. online: 7 Feb 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 1 (2017), pp. 25–63

Abstract

In this paper, we study multidimensional generalized BSDEs that have a monotone generator in a general filtration supporting a Brownian motion and an independent Poisson random measure. First, we prove the existence and uniqueness of ${\mathbb{L}}^{p}(p\ge 2)$-solutions in the case of a fixed terminal time under suitable p-integrability conditions on the data. Then, we extend these results to the case of a random terminal time. Furthermore, we provide a comparison result in dimension 1.

Generalized fractional Brownian motion

Mounir Zili

https://doi.org/10.15559/16-VMSTA71

Pub. online: 16 Jan 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 1 (2017), pp. 15–24

Abstract

We introduce a new Gaussian process, a generalization of both fractional and subfractional Brownian motions, which could serve as a good model for a larger class of natural phenomena. We study its main stochastic properties and some increments characteristics. As an application, we deduce the properties of nonsemimartingality, Hölder continuity, nondifferentiablity, and existence of a local time.

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

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