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Semi-Markov Models for Insurance and Option Rewards
Mälardalen University, Department of Mathematics and Physics.
2007 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis presents studies of semi-Markov models for insurance and option rewards. The thesis consists of the introduction and six papers. The introduction presents the results of the thesis in an informal way.

In paper A, a general semi-Markov reward model is presented. Recurrence relations for evaluation of higher moments of the reward process are given, as well as a backward semi-Markov reward processes are applied to insurance problems for the first time.

In paper B, models for disability insurance given in paper A are further extended. Statistical evidences of relevance of semi-Markov setting are given. Applications to profit-risk analysis for contracts are considered.

In paper C, a more detailed explanation of the algorithmic for the non-homogenous backward semi-Markov reward process is given. Two algorithmic approaches to solve the problem in an iterative manner are given. One of the algorithms is presented in a pseudo-code.

In paper D, the geometrical Brownian motion with drift and volatility controlled by a semi-Markov processes is considered as a price process in option valuation. The discrete version is examined and limit theorems describing the transition from discrete to continuous time are given. Monte-Carlo algorithms are described.

In paper E, a general price process represented by a two-component Markov process is considered. American options with pay-off functions, which admit power type upper bounds are studied. Both the transition characteristics of the price processes and the pay-off functions are assumed to depend on a perturbation parameter and to converge to the corresponding limits. Results about the convergence of reward functionals for American options are presented.

In paper F, convergence for option rewards when the price processes are perturbed exponential Lévy type process controlled by semi-Markov indices is studied. Both European and American type options with pay-off functions which admit power type upper bounds are considered. The paper continues research started in paper D and gives a key example for paper E.

Place, publisher, year, edition, pages
Institutionen för matematik och fysik , 2007.
Series
Mälardalen University Press Dissertations, ISSN 1651-4238 ; 38
Keyword [en]
semi-Markov process, discrete time, insurance, actuarial, higher order reward, disability, variance, skewness, kurtosis, reward process, stochastic volatility, controlling semi-Markov process, Monte Carlo algorithm, convergence, optimal stopping, skeleton approximation, regime switching, semi-Markov modulated, European option, American option, Lévy process.
National Category
Mathematics
Research subject
Matematik/tillämpad matematik
Identifiers
URN: urn:nbn:se:mdh:diva-170ISBN: 978-91-85485-33-8 (print)OAI: oai:DiVA.org:mdh-170DiVA: diva2:120597
Public defence
2007-01-26, Gamma, Hus U, Högskoleplan , Mälardalens högskola, Västerås, 13:15
Opponent
Supervisors
Available from: 2006-12-12 Created: 2006-12-12
List of papers
1. Discrete Time Backward Semi-Markov Reward Processes and an Application to Disability Insurance Problems
Open this publication in new window or tab >>Discrete Time Backward Semi-Markov Reward Processes and an Application to Disability Insurance Problems
(English)Manuscript (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:mdh:diva-4050 (URN)
Available from: 2006-12-12 Created: 2006-12-12 Last updated: 2016-02-11Bibliographically approved
2. SEMI-MARKOV REWARD MODELS FOR DISABILITY INSURANCE
Open this publication in new window or tab >>SEMI-MARKOV REWARD MODELS FOR DISABILITY INSURANCE
2006 (English)In: Theory of Stochastic Processes, ISSN 0321-3900, Vol. 12(28), no 3-4, 239-254 p.Article in journal (Refereed) Published
Abstract [en]

A semi-Markov model for disability insurance is described. Statisticalevidences of relevance semi-Markov setting are given. High order semiMarkov backward reward models are invented. Applications of these models to profit-risk analysis of disability insurance contracts are considered.

National Category
Natural Sciences
Identifiers
urn:nbn:se:mdh:diva-4051 (URN)
Available from: 2006-12-12 Created: 2006-12-12 Last updated: 2017-02-15Bibliographically approved
3. An algorithmic approach to discrete time non-homogeneous backward semi-Markov reward processes with an application to disability insurance
Open this publication in new window or tab >>An algorithmic approach to discrete time non-homogeneous backward semi-Markov reward processes with an application to disability insurance
2007 (English)In: Methodology and Computing in Applied Probability, ISSN 1387-5841, E-ISSN 1573-7713, Vol. 9, no 4, 497-519 p.Article in journal (Refereed) Published
Abstract [en]

In this paper semi-Markov reward models are presented. Higher moments of the reward process is presented for the first time applied to in timenon-homogeneous semi-Markov insurance problems. Also an example is presented based on real disability data. Different algorithmic approaches to solve the problem is described.

National Category
Natural Sciences
Identifiers
urn:nbn:se:mdh:diva-4052 (URN)10.1007/s11009-006-9012-4 (DOI)000249951800003 ()2-s2.0-35148839909 (Scopus ID)
Available from: 2006-12-12 Created: 2006-12-12 Last updated: 2016-01-13Bibliographically approved
4. A pricing process with stochastic volatility controlled by a semi-Markov process
Open this publication in new window or tab >>A pricing process with stochastic volatility controlled by a semi-Markov process
2004 (English)In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 33, no 3, 591-608 p.Article in journal (Refereed) Published
Abstract [sv]

This paper is devoted to the investigation of the geometrical Brownian motion as a price process where the drift and volatility are controlled by a semi-Markov process. Conditions of risk-neutral measure are given as well as a formula for the risk-neutral price for European options. The discrete version, the binomial model controlled by a semi-Markov chain, is examined and limit theorems describing the transition from discrete time binomial to continuous time model are given. A system of partial differential equations for distribution functions of average volatility is given. Related Monte Carlo algorithms are described.

National Category
Mathematics
Identifiers
urn:nbn:se:mdh:diva-4053 (URN)10.1081/STA-120028686 (DOI)000220104900012 ()
Available from: 2006-12-12 Created: 2006-12-12 Last updated: 2015-01-30Bibliographically approved
5. Convergence of Option Rewards for Markov Type Price Processes Controlled by Stochastic Indices. 1.
Open this publication in new window or tab >>Convergence of Option Rewards for Markov Type Price Processes Controlled by Stochastic Indices. 1.
(English)Manuscript (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:mdh:diva-4054 (URN)
Available from: 2006-12-12 Created: 2006-12-12 Last updated: 2017-02-15Bibliographically approved
6. Convergence of Option Rewards for Exponential Levy Type Price Processes Controlled by Semi-Markov Indices
Open this publication in new window or tab >>Convergence of Option Rewards for Exponential Levy Type Price Processes Controlled by Semi-Markov Indices
(English)Manuscript (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:mdh:diva-4055 (URN)
Available from: 2006-12-12 Created: 2006-12-12 Last updated: 2017-02-22Bibliographically approved

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