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Perturbed discrete time stochastic models
Stockholm University, Faculty of Science, Department of Mathematics. (Mathematical statistics)
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis, nonlinearly perturbed stochastic models in discrete time are considered. We give algorithms for construction of asymptotic expansions with respect to the perturbation parameter for various quantities of interest. In particular, asymptotic expansions are given for solutions of renewal equations, quasi-stationary distributions for semi-Markov processes, and ruin probabilities for risk processes.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2016. , 48 p.
Keyword [en]
Renewal equation, Perturbation, Asymptotic expansion, Regenerative process, Risk process, Semi-Markov process, Markov chain, Quasi-stationary distribution, Ruin probability, First hitting time, Solidarity property
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:su:diva-128979ISBN: 978-91-7649-422-6 (print)OAI: oai:DiVA.org:su-128979DiVA: diva2:918673
Public defence
2016-06-02, Sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 4: Manuscript. Paper 5: Manuscript. Paper 6: Manuscript.

Available from: 2016-05-10 Created: 2016-04-11 Last updated: 2017-11-14Bibliographically approved
List of papers
1. Exponential Expansions for Perturbed Discrete Time Renewal Equations
Open this publication in new window or tab >>Exponential Expansions for Perturbed Discrete Time Renewal Equations
2013 (English)In: Applied Reliability Engineering and Risk Analysis: Probabilistic Models and Statistical Inference / [ed] Ilia B. Frenkel, Alex Karagrigoriou, Anatoly Lisnianski, Andre Kleyner, Chichester: John Wiley & Sons, 2013, 349-362 p.Chapter in book (Refereed)
Abstract [en]

This chapter presents results about the asymptotic behavior of the solution x(φ)(n) of a perturbed discrete time renewal equation as φ--> 0 and n-->? simultaneously. It consider two cases of so-called quasi-stationary and pseudo-stationary asymptotics, where the limiting distribution f (0)(k) may be, respectively, improper or proper. The author improves the asymptotic relation to the much more advanced form of an exponential asymptotic expansion. The chapter illustrates theoretical results by examples related to queuing systems and risk processes. It briefly shows the way of getting the renewal equation. It repeats the method of finding a similar continuous time renewal equation for ruin probabilities, given, for example in Feller (1966) and Grandell (1991).

Place, publisher, year, edition, pages
Chichester: John Wiley & Sons, 2013
Keyword
perturbed discrete time renewal, pseudo-stationary asymptotics, quasi-stationary asymptotics, queuing systems, risk processes
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-95461 (URN)10.1002/9781118701881.ch23 (DOI)9781118539422 (ISBN)9781118701881 (ISBN)
Available from: 2013-10-29 Created: 2013-10-29 Last updated: 2016-04-28Bibliographically approved
2. Quasi-Stationary Distributions for Perturbed Discrete Time Regenerative Processes
Open this publication in new window or tab >>Quasi-Stationary Distributions for Perturbed Discrete Time Regenerative Processes
2014 (English)In: Theory of Probability and Mathematical Statistics, ISSN 0094-9000, Vol. 89, 153-168 p.Article in journal (Refereed) Published
Abstract [en]

Non-linearly perturbed discrete time regenerative processes with regenerative stopping times are considered. We define the quasi-stationary distributions for such processes and present conditions for their convergence. Under some additional assumptions, the quasi-stationary distributions can be expanded in asymptotic power series with respect to the perturbation parameter. We give an explicit recurrence algorithm for calculating the coefficients in these asymptotic expansions. Applications to perturbed alternating regenerative processes with absorption and perturbed risk processes are presented.

Keyword
Regenerative process, Renewal equation, Non-linear perturbation, Quasi-stationary distribution, Asymptotic expansion, Risk process
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-95468 (URN)
Available from: 2013-10-29 Created: 2013-10-29 Last updated: 2016-04-28Bibliographically approved
3. Asymptotics of Ruin Probabilities for Perturbed Discrete Time Risk Processes
Open this publication in new window or tab >>Asymptotics of Ruin Probabilities for Perturbed Discrete Time Risk Processes
2014 (English)In: Modern Problems in Insurance Mathematics / [ed] Dmitrii Silvestrov, Anders Martin-Löf, Springer, 2014, 95-112 p.Chapter in book (Refereed)
Abstract [en]

We consider the problem of approximating the infinite time horizon ruin probabilities for discrete time risk processes. The approach is based on asymptotic results for non-linearly perturbed discrete time renewal equations. Under some moment conditions on the claim distributions, the approximations take the form of exponential asymptotic expansions with respect to the perturbation parameter. We show explicitly how the coefficients of these expansions can be computed as functions of the coefficients of the expansions of local characteristics for perturbed risk processes.

Place, publisher, year, edition, pages
Springer, 2014
Series
EAA Series, ISSN 1869-6929
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-95469 (URN)10.1007/978-3-319-06653-0_7 (DOI)978-3-319-06652-3 (ISBN)978-3-319-06653-0 (ISBN)
Available from: 2013-10-29 Created: 2013-10-29 Last updated: 2016-04-28Bibliographically approved
4. Quasi-Stationary Asymptotics for Perturbed Semi-Markov Processes in Discrete Time
Open this publication in new window or tab >>Quasi-Stationary Asymptotics for Perturbed Semi-Markov Processes in Discrete Time
2017 (English)In: Methodology and Computing in Applied Probability, ISSN 1387-5841, E-ISSN 1573-7713, Vol. 19, no 4, 1047-1074 p.Article in journal (Refereed) Published
Abstract [en]

We consider a discrete time semi-Markov process where the characteristics defining the process depend on a small perturbation parameter. It is assumed that the state space consists of one finite communicating class of states and, in addition, one absorbing state. Our main object of interest is the asymptotic behavior of the joint probabilities of the position of the semi-Markov process and the event of non-absorption as time tends to infinity and the perturbation parameter tends to zero. The main result gives exponential expansions of these probabilities together with a recursive algorithm for computing the coefficients in the expansions. An application to perturbed epidemic SIS models is discussed.

Keyword
Semi-Markov process, Perturbation, Asymptotic expansion, Regenerative process, Renewal equation, Solidarity property, First hitting time
National Category
Mathematics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-148811 (URN)10.1007/s11009-016-9530-7 (DOI)000413792200004 ()
Conference
15th Applied Stochastic Models and Data Analysis International Conference (ASMDA), Piraeus, Greece June 30-July 04, 2015
Available from: 2017-11-14 Created: 2017-11-14 Last updated: 2017-11-14Bibliographically approved
5. Asymptotic expansions for moment functionals of perturbed discrete time semi-Markov processes
Open this publication in new window or tab >>Asymptotic expansions for moment functionals of perturbed discrete time semi-Markov processes
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper, we study mixed power-exponential moment functionals of nonlinearly perturbed semi-Markov processes in discrete time. Conditions under which the moment functionals of interest can be expanded in asymptotic power series with respect to the perturbation parameter are given. We show how the coefficients in these expansions can be computed from explicit recursive formulas. In particular, the results of the present paper have applications for studies of quasi-stationary distributions.

Keyword
Semi-Markov process, Perturbation, Asymptotic expansion, Renewal equation, Solidarity property, First hitting time
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-128972 (URN)
Available from: 2016-04-11 Created: 2016-04-11 Last updated: 2016-04-28
6. Asymptotics for quasi-stationary distributions of perturbed discrete time semi-Markov processes
Open this publication in new window or tab >>Asymptotics for quasi-stationary distributions of perturbed discrete time semi-Markov processes
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper, we study quasi-stationary distributions of nonlinearly perturbed semi-Markov processes in discrete time. This type of distributions is of interest for the analysis of stochastic systems which have finite lifetimes, but are expected to persist for a long time. We obtain asymptotic power series expansions for quasi-stationary distributions and it is shown how the coefficients in these expansions can be computed from a recursive algorithm. As an illustration of this algorithm, we present a numerical example for a discrete time Markov chain.

Keyword
Semi-Markov process, Perturbation, Quasi-stationary distribution, Asymptotic expansion, Renewal equation, Markov chain
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-128976 (URN)
Available from: 2016-04-11 Created: 2016-04-11 Last updated: 2016-04-28

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