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Nonlinearly Perturbed Renewal Equations: asymptotic Results and Applications
Mälardalen University, School of Education, Culture and Communication. (Analytical FInance)ORCID iD: 0000-0002-0835-7536
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we investigate a model of nonlinearly perturbed continuous-time renewal equation. Some characteristics of the renewal equation are assumed to have non-polynomial perturbations, more specifically they can be expanded with respect to a non-polynomial asymptotic scale. The main result of the present study is exponential asymptotic expansions for the solution of the perturbed renewal equation. These asymptotic results are also applied to various applied probability models like perturbed risk processes, perturbed M/G/1 queues and perturbed dam/storage processes.

The thesis is based on five papers where the model described above is successively studied.

Place, publisher, year, edition, pages
Västerås: Mälardalen University , 2011. , 33 p.
Series
Mälardalen University Press Dissertations, ISSN 1651-4238 ; 106
Keyword [en]
Nonlinearly perturbed renewal equation, perturbed renewal equation, nonlinear perturbation, non-polynomial perturbation, perturbed risk process, perturbed storage process
National Category
Mathematics
Research subject
Mathematics/Applied Mathematics
Identifiers
URN: urn:nbn:se:mdh:diva-12953ISBN: 978-91-7485-032-1 (print)OAI: oai:DiVA.org:mdh-12953DiVA: diva2:438488
Public defence
2011-10-28, Gamma, Högskoleplan 1, Mälardalens Högskola, Västerås, 13:15 (English)
Opponent
Supervisors
Available from: 2011-09-05 Created: 2011-09-02 Last updated: 2015-06-29Bibliographically approved
List of papers
1. Exponential asymptotics for nonlinearly perturbed renewal equation with non-polynomial perturbations
Open this publication in new window or tab >>Exponential asymptotics for nonlinearly perturbed renewal equation with non-polynomial perturbations
2008 (English)In: Journal of Numerical and Applied Mathematics, ISSN 0868-6912, Vol. 96, no 1, 173-197 p.Article in journal (Refereed) Published
Abstract [en]

The model of nonlinearly perturbedcontinuous-time renewal equation is studied in this paper.The perturbation conditions considered involve asymptoticalexpansions with respect to asymptotic scale$\{\varphi_{n,m}(\varepsilon) = \varepsilon^{n +m\omega}\}$,with $n, m$ being non-negative integers and $\omega >1$ beingirrational number. Such asymptotical scale results in non-polynomialtype of asymptotic expansions for solutions for perturbed renewalequations. An example of risk processes with perturbations describedabove and asymptotic expansions in diffusion approximation for ruinprobabilities in this model are given.

Place, publisher, year, edition, pages
Kiev: TBiMC, 2008
Keyword
Renewal equation, nonlinear perturbation, non-polynomial perturbation, exponential asymptotic expansion, risk process, ruin probability
National Category
Probability Theory and Statistics Mathematics
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-5285 (URN)
Available from: 2009-02-13 Created: 2009-02-13 Last updated: 2015-06-29Bibliographically approved
2. Nonlinearly Perturbed Renewal Equation with Perturbations of a Non-polynomial Type
Open this publication in new window or tab >>Nonlinearly Perturbed Renewal Equation with Perturbations of a Non-polynomial Type
2010 (English)In: Proceedings of the International Symposium on Stochastic Models in Reliability Engineering, Life Science and Operations Management, Beer Sheva, 2010. / [ed] Frenkel, I., Gertsbakh, I., Khvatskin L., Laslo Z. Lisnianski, A., Beer Sheva: SCE - Shamoon College of Engineering , 2010, 754-763 p.Conference paper, Published paper (Refereed)
Abstract [en]

The object of study is a model of nonlinearly perturbed continuous-time renewal equation with multivariate non-polynomial perturbations. The characteristics of the distribution generating the renewal equation are assumed to have expansions in the perturbation parameter with respect to a non-polynomial asymptotic scale which can be considered as a generalization of the standard polynomial scale. Exponential asymptotics for such a model are obtained and applications are given.

Place, publisher, year, edition, pages
Beer Sheva: SCE - Shamoon College of Engineering, 2010
Keyword
Renewal equation, nonlinear perturbation, non-polynomial perturbation, exponential asymptotic expansion, risk process, ruin probability
National Category
Mathematics Computational Mathematics
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-9347 (URN)
Conference
The International Symposium on Stochastic Models in Reliability Engineering, Life Sciences and Operations Management. February 8-11, 2010. Beer Sheva, Israel.
Available from: 2010-03-03 Created: 2010-03-03 Last updated: 2015-08-06Bibliographically approved
3. Analytical and Numerical Studies of Perturbed Renewal Equations with Multivariate Non-Polynomial Perturbations
Open this publication in new window or tab >>Analytical and Numerical Studies of Perturbed Renewal Equations with Multivariate Non-Polynomial Perturbations
2010 (English)In: Journal of Applied Quantitative Methods, ISSN 1842-4562, Vol. 5, no 3, 411-428 p.Article in journal (Refereed) Published
Abstract [en]

The object of study is a model of nonlinearly perturbed continuous-time renewal equation with multivariate non-polynomial perturbations. The characteristics of the distribution generating the renewal equation are assumed to have expansions in a perturbation parameter with respect to a non-polynomial asymptotic. Exponential asymptotics for such a model as well as their applications are given. Numerical studies are performed to gain insights into the asymptotical results.

Place, publisher, year, edition, pages
Association for Development through Science and Education, Romania (ADSE)., 2010
Keyword
perturbed renewal equation, nonlinear perturbation, non-polynomial perturbation, perturbed risk process, ruin probability
National Category
Computational Mathematics
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-11698 (URN)
Available from: 2011-02-02 Created: 2011-02-02 Last updated: 2015-06-29Bibliographically approved
4. Asymptotically Improper Perturbed Renewal Equations: Asymptotic Results and Their Applications
Open this publication in new window or tab >>Asymptotically Improper Perturbed Renewal Equations: Asymptotic Results and Their Applications
2011 (English)Report (Other academic)
Abstract [en]

We consider a family of asymptotically improper perturbed renewal equations where the characteristics of the distribution functions generating the perturbed renewal equations are perturbed in a particular way. More specifically, those characteristics are nonlinear functions of the perturbation parameter such that they can be expanded into asymptotic expansions of a non-polynomial type with respect to the perturbation parameter. We give asymptotic results, namely the exponential asymptotic expansions, for the solutions of the perturbed renewal equations. An application to perturbed storage processes is also presented.

Place, publisher, year, edition, pages
Västerås: Mälardalens University, 2011. 21 p.
Series
School of Education, Culture and Communication, Division of Applied Mathematics, ISSN 1404-4978 ; 2011-1
Keyword
Perturbed renewal equations, nonlinear perturbations, non-polynomial perturbations, perturbed storage processes
National Category
Probability Theory and Statistics
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-12917 (URN)
Available from: 2011-08-25 Created: 2011-08-25 Last updated: 2015-06-29Bibliographically approved
5. NONLINEARLY PERTURBED RENEWAL EQUATIONS: THE NON-POLYNOMIAL CASE
Open this publication in new window or tab >>NONLINEARLY PERTURBED RENEWAL EQUATIONS: THE NON-POLYNOMIAL CASE
2012 (English)In: Theory of Probability and Mathematical Statistics, ISSN 0094-9000, Vol. 84, 117-129 p.Article in journal (Refereed) Published
Abstract [en]

Models of nonlinearly perturbed renewal equations with non-polynomial perturbations are studied. Exponential asymptotic expansions are given for the solutions to the perturbed renewal equations under consideration. An application to perturbed M/G/1/ queues is presented.

Place, publisher, year, edition, pages
Kiev, Ukraine: Kiev University (English translation by American Mathematical Society), 2012
Keyword
Perturbed renewal equation, non-polynomial perturbation
National Category
Probability Theory and Statistics
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-12913 (URN)10.1090/S0094-9000-2012-00865-X (DOI)2-s2.0-84865087490 (Scopus ID)
Available from: 2011-08-25 Created: 2011-08-25 Last updated: 2015-06-29Bibliographically approved

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