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postgraduate thesis: Construction of nonstandard Markov chain models with applications
Title  Construction of nonstandard Markov chain models with applications 

Authors  
Advisors  
Issue Date  2014 
Publisher  The University of Hong Kong (Pokfulam, Hong Kong) 
Citation  Zhu, D. [朱冬梅]. (2014). Construction of nonstandard Markov chain models with applications. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5295517 
Abstract  In this thesis, the properties of some nonstandard Markov chain models and their corresponding parameter estimation methods are investigated. Several practical applications and extensions are also discussed.
The estimation of model parameters plays a key role in the realworld applications of Markov chain models. Some widely used estimation methods for Markov chain models are based on the existence of stationary vectors. In this thesis, some weaker sufficient conditions for the existence of stationary vectors for highorder Markov chain models, multivariate Markov chain models and highorder multivariate Markov chain models are proposed. Furthermore, for multivariate Markov chain models, a new estimation method based on minimizing the prediction error is proposed. Numerical experiments are conducted to demonstrate the efficiency of the proposed estimation methods with an application in demand prediction.
Hidden Markov Model (HMM) is a bivariate stochastic process such that one of the process is hidden and the other is observable. The distribution of observable sequence depends on the hidden sequence. In a traditional HMM, the hidden states directly affect the observable states but not vice versa. However, in reality, observable sequence may also have effect on the hidden sequence. For this reason, the concept of Interactive Hidden Markov Model (IHMM) is introduced, whose key idea is that the transitions of the hidden states depend on the observable states too. In this thesis, efforts are devoted in building a highorder IHMM where the probability laws governing both observable and hidden states can be written as a pair of highorder stochastic difference equations. We also propose a new model by capturing the effect of observable sequence on the hidden sequence through using the threshold principle. In this case, reference probability methods are adopted in estimating the optimal model parameters, while for unknown threshold parameter, Akaike Information Criterion (AIC) is used. We explore asset allocation problems from both domestic and foreign perspective where asset price dynamics follows autoregressive HMM. The object of an investor is not only to maximize the expected utility of the terminal wealth, but also to ensure that the risk of the portfolio described by the ValueatRisk (VaR) does not exceed a specified level.
In many decision processes, fuzziness is a major source of imprecision. As a perception of usual Markov chains, the definition of fuzzy Markov chains is introduced. Compared to traditional Markov chain models, fuzzy Markov chains are relatively new and many properties of them are still unknown. Due to the potential applications of fuzzy Markov chains, we provide some characterizations to ensure the ergodicity of these chains under both maxmin and maxproduct compositions. 
Degree  Doctor of Philosophy 
Subject  Markov processes 
Dept/Program  Mathematics 
Persistent Identifier  http://hdl.handle.net/10722/202358 
DC Field  Value  Language 

dc.contributor.advisor  Zang, W   
dc.contributor.advisor  Ching, WK   
dc.contributor.author  Zhu, Dongmei   
dc.contributor.author  朱冬梅   
dc.date.accessioned  20140918T02:28:14Z   
dc.date.available  20140918T02:28:14Z   
dc.date.issued  2014   
dc.identifier.citation  Zhu, D. [朱冬梅]. (2014). Construction of nonstandard Markov chain models with applications. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5295517   
dc.identifier.uri  http://hdl.handle.net/10722/202358   
dc.description.abstract  In this thesis, the properties of some nonstandard Markov chain models and their corresponding parameter estimation methods are investigated. Several practical applications and extensions are also discussed. The estimation of model parameters plays a key role in the realworld applications of Markov chain models. Some widely used estimation methods for Markov chain models are based on the existence of stationary vectors. In this thesis, some weaker sufficient conditions for the existence of stationary vectors for highorder Markov chain models, multivariate Markov chain models and highorder multivariate Markov chain models are proposed. Furthermore, for multivariate Markov chain models, a new estimation method based on minimizing the prediction error is proposed. Numerical experiments are conducted to demonstrate the efficiency of the proposed estimation methods with an application in demand prediction. Hidden Markov Model (HMM) is a bivariate stochastic process such that one of the process is hidden and the other is observable. The distribution of observable sequence depends on the hidden sequence. In a traditional HMM, the hidden states directly affect the observable states but not vice versa. However, in reality, observable sequence may also have effect on the hidden sequence. For this reason, the concept of Interactive Hidden Markov Model (IHMM) is introduced, whose key idea is that the transitions of the hidden states depend on the observable states too. In this thesis, efforts are devoted in building a highorder IHMM where the probability laws governing both observable and hidden states can be written as a pair of highorder stochastic difference equations. We also propose a new model by capturing the effect of observable sequence on the hidden sequence through using the threshold principle. In this case, reference probability methods are adopted in estimating the optimal model parameters, while for unknown threshold parameter, Akaike Information Criterion (AIC) is used. We explore asset allocation problems from both domestic and foreign perspective where asset price dynamics follows autoregressive HMM. The object of an investor is not only to maximize the expected utility of the terminal wealth, but also to ensure that the risk of the portfolio described by the ValueatRisk (VaR) does not exceed a specified level. In many decision processes, fuzziness is a major source of imprecision. As a perception of usual Markov chains, the definition of fuzzy Markov chains is introduced. Compared to traditional Markov chain models, fuzzy Markov chains are relatively new and many properties of them are still unknown. Due to the potential applications of fuzzy Markov chains, we provide some characterizations to ensure the ergodicity of these chains under both maxmin and maxproduct compositions.   
dc.language  eng   
dc.publisher  The University of Hong Kong (Pokfulam, Hong Kong)   
dc.relation.ispartof  HKU Theses Online (HKUTO)   
dc.rights  The author retains all proprietary rights, (such as patent rights) and the right to use in future works.   
dc.rights  Creative Commons: Attribution 3.0 Hong Kong License   
dc.subject.lcsh  Markov processes   
dc.title  Construction of nonstandard Markov chain models with applications   
dc.type  PG_Thesis   
dc.identifier.hkul  b5295517   
dc.description.thesisname  Doctor of Philosophy   
dc.description.thesislevel  Doctoral   
dc.description.thesisdiscipline  Mathematics   
dc.description.nature  published_or_final_version   
dc.identifier.doi  10.5353/th_b5295517   