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Perturbed Renewal Equations with Non-Polynomial PerturbationsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2010 (English)Licentiate thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### 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

URN: urn:nbn:se:mdh:diva-9354ISBN: 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)
##### Opponent

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#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt472",{id:"formSmash:j_idt472",widgetVar:"widget_formSmash_j_idt472",multiple:true}); Available from: 2010-03-04 Created: 2010-03-04 Last updated: 2015-06-29Bibliographically approved
##### List of papers

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.

1. Nonlinearly Perturbed Renewal Equation with Perturbations of a Non-polynomial Type$(function(){PrimeFaces.cw("OverlayPanel","overlay301946",{id:"formSmash:j_idt521:0:j_idt525",widgetVar:"overlay301946",target:"formSmash:j_idt521:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Exponential asymptotic expansions and Monte Carlo studies for ruin probabilities$(function(){PrimeFaces.cw("OverlayPanel","overlay302084",{id:"formSmash:j_idt521:1:j_idt525",widgetVar:"overlay302084",target:"formSmash:j_idt521:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. Exponential asymptotics for nonlinearly perturbed renewal equation with non-polynomial perturbations$(function(){PrimeFaces.cw("OverlayPanel","overlay160339",{id:"formSmash:j_idt521:2:j_idt525",widgetVar:"overlay160339",target:"formSmash:j_idt521:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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