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On model fitting and estimation of strictly stationary processes

Marko Voutilainen

Lauri Viitasaari Pauliina Ilmonen

https://doi.org/10.15559/17-VMSTA91

Pub. online: 22 Dec 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 4 (2017), pp. 381–406

Abstract

Stationary processes have been extensively studied in the literature. Their applications include modeling and forecasting numerous real life phenomena such as natural disasters, sales and market movements. When stationary processes are considered, modeling is traditionally based on fitting an autoregressive moving average (ARMA) process. However, we challenge this conventional approach. Instead of fitting an ARMA model, we apply an AR(1) characterization in modeling any strictly stationary processes. Moreover, we derive consistent and asymptotically normal estimators of the corresponding model parameter.

Double barrier reflected BSDEs with stochastic Lipschitz coefficient

Mohamed Marzougue Mohamed El Otmani

https://doi.org/10.15559/17-VMSTA90

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.

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.

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.

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Online ISSN: 2351-6054
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