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Double barrier reflected BSDEs with stochastic Lipschitz coefficient

Mohamed Marzougue Mohamed El Otmani

https://doi.org/10.15559/17-MSTA90

Pub. online: 8 Dec 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 4, Issue 4 (2017), pp. 353–379

Abstract

This paper proves the existence and uniqueness of a solution to doubly reflected backward stochastic differential equations where the coefficient is stochastic Lipschitz, by means of the penalization method.

The risk model with stochastic premiums, dependence and a threshold dividend strategy

Olena Ragulina

https://doi.org/10.15559/17-VMSTA89

Pub. online: 8 Dec 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 4 (2017), pp. 315–351

Abstract

The paper deals with a generalization of the risk model with stochastic premiums where dependence structures between claim sizes and inter-claim times as well as premium sizes and inter-premium times are modeled by Farlie–Gumbel–Morgenstern copulas. In addition, dividends are paid to its shareholders according to a threshold dividend strategy. We derive integral and integro-differential equations for the Gerber–Shiu function and the expected discounted dividend payments until ruin. Next, we concentrate on the detailed investigation of the model in the case of exponentially distributed claim and premium sizes. In particular, we find explicit formulas for the ruin probability in the model without either dividend payments or dependence as well as for the expected discounted dividend payments in the model without dependence. Finally, numerical illustrations are presented.

The risk model with stochastic premiums, dependence and a threshold dividend strategy

Olena Ragulina

https://doi.org/10.15559/17-MSTA89

Pub. online: 8 Dec 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 4, Issue 4 (2017), pp. 315–351

Abstract

Integrated quantile functions: properties and applications

Alexander A. Gushchin

Dmitriy A. Borzykh

https://doi.org/10.15559/17-VMSTA88

Pub. online: 8 Dec 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 4 (2017), pp. 285–314

Abstract

In this paper we provide a systematic exposition of basic properties of integrated distribution and quantile functions. We define these transforms in such a way that they characterize any probability distribution on the real line and are Fenchel conjugates of each other. We show that uniform integrability, weak convergence and tightness admit a convenient characterization in terms of integrated quantile functions. As an application we demonstrate how some basic results of the theory of comparison of binary statistical experiments can be deduced using integrated quantile functions. Finally, we extend the area of application of the Chacon–Walsh construction in the Skorokhod embedding problem.

Integrated quantile functions: properties and applications

Alexander A. Gushchin

Dmitriy A. Borzykh

https://doi.org/10.15559/17-MSTA88

Pub. online: 8 Dec 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 4, Issue 4 (2017), pp. 285–314

Abstract

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.

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

Liliia Sydoruk Grygoriy Torbin

https://doi.org/10.15559/17-MSTA87

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

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.

Random iterations of homeomorphisms on the circle

Katrin Gelfert Örjan Stenflo

https://doi.org/10.15559/17-MSTA86

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

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.

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