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Nonlinearly Perturbed Renewal Equation with Perturbations of a Non-polynomial Type
Mälardalen University, School of Education, Culture and Communication. (Analytical Finance)ORCID iD: 0000-0002-0835-7536
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, p. 754-763Conference 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. p. 754-763
##### Keywords [en]
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
OAI: oai:DiVA.org:mdh-9347DiVA, id: diva2:301946
##### 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
##### 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)
##### 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
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)
##### Supervisors
Available from: 2011-09-05 Created: 2011-09-02 Last updated: 2015-06-29Bibliographically approved

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Cite
Citation style
• apa
• ieee
• modern-language-association-8th-edition
• vancouver
• Other style
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Language
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