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Perturbed Renewal Equations with Non-Polynomial Perturbations
Mälardalen University, School of Education, Culture and Communication. (Analytical Finance)ORCID iD: 0000-0002-0835-7536
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 [en]
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
ISBN: 978-91-86135-58-4 (print)OAI: oai:DiVA.org:mdh-9354DiVA, id: diva2:302104
##### 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
##### List of papers
1. 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, 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
##### Keywords
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
2. Exponential asymptotic expansions and Monte Carlo studies for ruin probabilities
Open this publication in new window or tab >>Exponential asymptotic expansions and Monte Carlo studies for ruin probabilities
(English)In: Journal of Statistical Planning and Inference, ISSN 0378-3758, E-ISSN 1873-1171Article in journal (Refereed) Submitted
##### Abstract [en]

This paper presents the exponential asymptotic expansions for the ruin probability in a special model of non-linearly perturbed risk processes with non-polynomial perturbations. Monte Carlo studies are performed to investigate the accuracy and other properties of the asymptotic formulas.

##### Keywords
perturbed risk process; renewal equation; ruin probability; nonlinear perturbation; non-polynomial perturbation; Monte Carlo simulation
##### Identifiers
urn:nbn:se:mdh:diva-9352 (URN)
Available from: 2010-03-04 Created: 2010-03-04 Last updated: 2017-12-12Bibliographically approved
3. 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, 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
##### Keywords
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

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Cite
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