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On singularity of distribution of random variables with independent symbols of Oppenheim expansions

Liliia Sydoruk Grygoriy Torbin

https://doi.org/10.15559/17-VMSTA87

Pub. online: 26 Oct 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 3 (2017), pp. 273–283

Abstract

The paper is devoted to the restricted Oppenheim expansion of real numbers ($\mathit{ROE}$), which includes already known Engel, Sylvester and Lüroth expansions as partial cases. We find conditions under which for almost all (with respect to Lebesgue measure) real numbers from the unit interval their $\mathit{ROE}$-expansion contain arbitrary digit i only finitely many times. Main results of the paper state the singularity (w.r.t. the Lebesgue measure) of the distribution of a random variable with i.i.d. increments of symbols of the restricted Oppenheim expansion. General non-i.i.d. case is also studied and sufficient conditions for the singularity of the corresponding probability distributions are found.

Random iterations of homeomorphisms on the circle

Katrin Gelfert Örjan Stenflo

https://doi.org/10.15559/17-VMSTA86

Pub. online: 5 Oct 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 3 (2017), pp. 253–271

Abstract

We study random independent and identically distributed iterations of functions from an iterated function system of homeomorphisms on the circle which is minimal. We show how such systems can be analyzed in terms of iterated function systems with probabilities which are non-expansive on average.

Weighted entropy: basic inequalities

Mark Kelbert Izabella Stuhl Yuri Suhov

https://doi.org/10.15559/17-VMSTA85

Pub. online: 2 Oct 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 3 (2017), pp. 233–252

Abstract

This paper represents an extended version of an earlier note [10]. The concept of weighted entropy takes into account values of different outcomes, i.e., makes entropy context-dependent, through the weight function. We analyse analogs of the Fisher information inequality and entropy power inequality for the weighted entropy and discuss connections with weighted Lieb’s splitting inequality. The concepts of rates of the weighted entropy and information are also discussed.

Quantifying non-monotonicity of functions and the lack of positivity in signed measures

Youri Davydov Ričardas Zitikis

https://doi.org/10.15559/17-VMSTA84

Pub. online: 28 Sep 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 3 (2017), pp. 219–231

Abstract

In various research areas related to decision making, problems and their solutions frequently rely on certain functions being monotonic. In the case of non-monotonic functions, one would then wish to quantify their lack of monotonicity. In this paper we develop a method designed specifically for this task, including quantification of the lack of positivity, negativity, or sign-constancy in signed measures. We note relevant applications in Insurance, Finance, and Economics, and discuss some of them in detail.

Asymptotic behavior of functionals of the solutions to inhomogeneous Itô stochastic differential equations with nonregular dependence on parameter

Grigorij Kulinich Svitlana Kushnirenko

https://doi.org/10.15559/17-VMSTA83

Pub. online: 22 Sep 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 3 (2017), pp. 199–217

Abstract

The asymptotic behavior, as $T\to \infty $, of some functionals of the form $I_{T}(t)=F_{T}(\xi _{T}(t))+{\int _{0}^{t}}g_{T}(\xi _{T}(s))\hspace{0.1667em}dW_{T}(s)$, $t\ge 0$ is studied. Here $\xi _{T}(t)$ is the solution to the time-inhomogeneous Itô stochastic differential equation

\[d\xi _{T}(t)=a_{T}\big(t,\xi _{T}(t)\big)\hspace{0.1667em}dt+dW_{T}(t),\hspace{1em}t\ge 0,\hspace{2.5pt}\xi _{T}(0)=x_{0},\]

$T>0$ is a parameter, $a_{T}(t,x),x\in \mathbb{R}$ are measurable functions, $|a_{T}(t,x)|\le C_{T}$ for all $x\in \mathbb{R}$ and $t\ge 0$, $W_{T}(t)$ are standard Wiener processes, $F_{T}(x),x\in \mathbb{R}$ are continuous functions, $g_{T}(x),x\in \mathbb{R}$ are measurable locally bounded functions, and everything is real-valued. The explicit form of the limiting processes for $I_{T}(t)$ is established under nonregular dependence of $a_{T}(t,x)$ and $g_{T}(x)$ on the parameter T.

The self-normalized Donsker theorem revisited

Peter Parczewski

https://doi.org/10.15559/17-VMSTA82

Pub. online: 18 Sep 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 3 (2017), pp. 189–198

Abstract

We extend the Poincaré–Borel lemma to a weak approximation of a Brownian motion via simple functionals of uniform distributions on n-spheres in the Skorokhod space $D([0,1])$. This approach is used to simplify the proof of the self-normalized Donsker theorem in Csörgő et al. (2003). Some notes on spheres with respect to $\ell _{p}$-norms are given.

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.

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

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