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Exponential asymptotics for nonlinearly perturbed renewal equation with non-polynomial perturbations
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. (Mathematics and Applied Mathematics)ORCID iD: 0000-0002-0835-7536
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. (Mathematics and Applied Mathematics)ORCID iD: 0000-0002-2626-5598
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. (Mathematics and Applied Mathematics)ORCID iD: 0000-0002-0139-0747
2008 (English)In: Journal of Numerical and Applied Mathematics, ISSN 0868-6912, Vol. 96, no 1, p. 173-197Article 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. Vol. 96, no 1, p. 173-197
Keywords [en]
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: urn:nbn:se:mdh:diva-5285OAI: oai:DiVA.org:mdh-5285DiVA, id: diva2:160339
Available from: 2009-02-13 Created: 2009-02-13 Last updated: 2015-06-29Bibliographically approved
In thesis
1. Perturbed Renewal Equations with Non-Polynomial Perturbations
Open this publication in new window or tab >>Perturbed Renewal Equations with Non-Polynomial Perturbations
2010 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis deals with a model of nonlinearly perturbed continuous-time renewal equation with nonpolynomial perturbations. The characteristics, namely the defect and moments, of the distribution function generating the renewal equation are assumed to have expansions with respect to a non-polynomial asymptotic scale: $\{\varphi_{\nn} (\varepsilon) =\varepsilon^{\nn \cdot \w}, \nn \in \mathbf{N}_0^k\}$  as $\varepsilon \to 0$, where $\mathbf{N}_0$ is the set of non-negative integers, $\mathbf{N}_0^k \equiv \mathbf{N}_0 \times \cdots \times \mathbf{N}_0, 1\leq k <\infty$ with the product being taken $k$ times and $\w$ is a $k$ dimensional parameter vector that satisfies certain properties. For the one-dimensional case, i.e., $k=1$, this model reduces to the model of nonlinearly perturbed renewal equation with polynomial perturbations which is well studied in the literature.  The goal of the present study is to obtain the exponential asymptotics for the solution to the perturbed renewal equation in the form of exponential asymptotic expansions and present possible applications.

The thesis is based on three papers which study successively the model stated above. Paper A investigates the two-dimensional case, i.e. where $k=2$. The corresponding asymptotic exponential expansion for the solution to the perturbed renewal equation is given. The asymptotic results are applied to an example of the perturbed risk process, which leads to diffusion approximation type asymptotics for the ruin probability.  Numerical experimental studies on this example of perturbed risk process are conducted in paper B, where Monte Carlo simulation are used to study the accuracy and properties of the asymptotic formulas. Paper C presents the asymptotic results for the more general case where the dimension $k$ satisfies $1\leq k <\infty$, which are applied to the asymptotic analysis of the ruin probability in an example of perturbed risk processes with this general type of non-polynomial perturbations.  All the proofs of the theorems stated in paper C are collected in its supplement: paper D.

Place, publisher, year, edition, pages
Västerås: Mälardalen University, 2010. p. 98
Series
Mälardalen University Press Licentiate Theses, ISSN 1651-9256 ; 116
Keywords
Renewal equation, perturbed renewal equation, non-polynomial perturbation, exponential asymptotic expansion, risk process, ruin probability
National Category
Probability Theory and Statistics
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-9354 (URN)978-91-86135-58-4 (ISBN)
Presentation
2010-05-07, Kappa, Hus U, Högskoleplan 1, Mälardalen University, 13:15 (English)
Opponent
Supervisors
Available from: 2010-03-04 Created: 2010-03-04 Last updated: 2015-06-29Bibliographically approved
2. Nonlinearly Perturbed Renewal Equations: asymptotic Results and Applications
Open this publication in new window or tab >>Nonlinearly Perturbed Renewal Equations: asymptotic Results and Applications
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. p. 33
Series
Mälardalen University Press Dissertations, ISSN 1651-4238 ; 106
Keywords
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:nbn:se:mdh:diva-12953 (URN)978-91-7485-032-1 (ISBN)
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

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Ni, YingSilvestrov, DmitriiMalyarenko, Anatoliy

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