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E-grāmata: Nonlinearly Perturbed Semi-Markov Processes

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Nonlinearly Perturbed Semi-Markov ProcessesPalielināt
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The book presents new methods of asymptotic analysis for nonlinearly perturbed semi-Markov processes with a finite phase space. These methods are based on special time-space screening procedures for sequential phase space reduction of semi-Markov processes combined with the systematical use of operational calculus for Laurent asymptotic expansions. Effective recurrent algorithms are composed for getting asymptotic expansions, without and with explicit upper bounds for remainders, for power moments of hitting times, stationary and conditional quasi-stationary distributions for nonlinearly perturbed semi-Markov processes. These results are illustrated by asymptotic expansions for birth-death-type semi-Markov processes, which play an important role in various applications. The book will be a useful contribution to the continuing intensive studies in the area. It is an essential reference for theoretical and applied researchers in the field of stochastic processes and their applications that will contribute to continuing extensive studies in the area and remain relevant for years to come. 


Recenzijas

This book is aimed at studying asymptotic expansions for moment hitting times, stationary and conditional quasi-stationary distributions, and various further functionals, for nonlinearly perturbed semi-Markov processes having finite phase space. The bibliography is broad and relevant, and includes various publications of the authors thanks to their wide expertise on the topics under investigation. book is of interest to researchers working on semi-Markov processes, and provides a useful reference for investigations dealing with asymptotic problems for perturbed stochastic processes. (Antonio Di Crescenzo, Mathematical Reviews, July, 2018)

Preface v
List of Symbols
xiii
1 Introduction
1(16)
1.1 Examples
1(9)
1.1.1 Linear and Nonlinear Perturbations and High-Order Asymptotic Expansions
1(2)
1.1.2 Laurent Asymptotic Expansions
3(2)
1.1.3 Recurrent Algorithms of Phase Space Reduction
5(4)
1.1.4 Rates of Convergence and Explicit Upper Bounds for Remainders of Asymptotic Expansions
9(1)
1.2 Contents of the Book
10(4)
1.2.1
Chapter 2
11(1)
1.2.2
Chapter 3
11(2)
1.2.3
Chapter 4
13(1)
1.2.4
Chapter 5
13(1)
1.2.5
Chapter 6
14(1)
1.2.6 Appendix A
14(1)
1.3 Conclusion
14(3)
2 Laurent Asymptotic Expansions
17(20)
2.1 Laurent Asymptotic Expansions with Remainders Given in the Standard Form
17(5)
2.1.1 Definition of Laurent Asymptotic Expansions with Remainders Given in the Standard Form
17(2)
2.1.2 Operational Rules for Laurent Asymptotic Expansion with Remainders Given in the Standard Form
19(3)
2.1.3 Algebraic Properties of Operational Rules for Laurent Asymptotic Expansions with Remainders Given in the Standard Form
22(1)
2.2 Laurent Asymptotic Expansions with Explicit Upper Bounds for Remainders
22(6)
2.2.1 Definition of Laurent Asymptotic Expansions with Explicit Upper Bounds for Remainders
22(2)
2.2.2 Operational Rules for Laurent Asymptotic Expansion with Explicit Upper Bounds for Remainders
24(3)
2.2.3 Algebraic Properties of Operational Rules for Laurent Asymptotic Expansions with Explicit Upper Bounds for Remainders
27(1)
2.3 Proofs of Lemmas 2.1-2.8
28(9)
2.3.1 Lemmas 2.1 and 2.5
28(1)
2.3.2 Lemmas 2.2 and 2.6
29(5)
2.3.3 Lemmas 2.3 and 2.7
34(1)
2.3.4 Lemmas 2.4 and 2.8
34(3)
3 Asymptotic Expansions for Moments of Hitting Times for Nonlinearly Perturbed Semi-Markov Processes
37(30)
3.1 Perturbed Semi-Markov Processes
37(9)
3.1.1 Perturbed Semi-Markov Processes
37(2)
3.1.2 Hitting Times for Semi-Markov Processes
39(2)
3.1.3 Stationary Distributions for Semi-Markov Processes
41(1)
3.1.4 Perturbation Conditions for Semi-Markov Processes
42(4)
3.2 Reduction of Phase Spaces for Semi-Markov Processes
46(6)
3.2.1 Semi-Markov Processes with Reduced Phase Spaces
47(3)
3.2.2 Basic Model Conditions for Reduced Semi-Markov Processes
50(1)
3.2.3 Hitting Times for Reduced Semi-Markov Processes
51(1)
3.3 Asymptotic Expansions for Moments of Hitting Times for Perturbed Semi-Markov Processes
52(15)
3.3.1 Asymptotic Expansions with Remainders Given in the Standard Form for Transition Characteristics of Reduced Semi-Markov Processes
52(5)
3.3.2 Asymptotic Expansions with Remainders Given in the Standard Form for Moments of Hitting Times for Reduced Semi-Markov Processes
57(4)
3.3.3 Asymptotic Expansions with Explicit Upper Bounds for Remainders for Transition Characteristics of Reduced Semi-Markov Processes
61(4)
3.3.4 Asymptotic Expansions with Explicit Upper Bounds for Remainders for Moments of Hitting Times for Reduced Semi-Markov Processes
65(2)
4 Asymptotic Expansions for Stationary Distributions of Nonlinearly Perturbed Semi-Markov Processes
67(14)
4.1 Asymptotic Expansions with Remainders Given in the Standard Form for Stationary Distributions of Perturbed Semi-Markov Processes
67(4)
4.1.1 Asymptotic Expansions with Remainders Given in the Standard Form for Expectations of Return Times
68(2)
4.1.2 Asymptotic Expansions with Remainders Given in the Standard Form for Stationary Distributions
70(1)
4.2 Asymptotic Expansions with Explicit Upper Bounds for Remainders for Stationary Distributions of Perturbed Semi-Markov Processes
71(4)
4.2.1 Asymptotic Expansions with Explicit Upper Bounds for Remainders for Expectations of Return Times
72(1)
4.2.2 Asymptotic Expansions with Explicit Upper Bounds for Remainders for Stationary Distributions
73(2)
4.3 Asymptotic Expansions for Stationary Functionals for Perturbed Semi-Markov Processes
75(6)
4.3.1 Asymptotic Expansions for Conditional Quasi-Stationary Distributions
75(3)
4.3.2 Asymptotic Expansions for Stationary Limits of Additive Functionals
78(3)
5 Nonlinearly Perturbed Birth-Death-Type Semi-Markov Processes
81(26)
5.1 Stationary and Quasi-Stationary Distributions for Perturbed Birth-Death-Type Semi-Markov Processes
82(10)
5.1.1 Perturbed Birth-Death-Type Semi-Markov Processes
82(3)
5.1.2 Stationary and Quasi-Stationary Distributions for Perturbed Birth-Death-Type Semi-Markov Processes
85(2)
5.1.3 Reduced Birth-Death-Type Semi-Markov Processes
87(2)
5.1.4 Explicit Formulas for Expectations of Return Times and Stationary Probabilities for Birth-Death-Type Semi-Markov Processes
89(3)
5.2 Asymptotic Expansions with Remainders Given the Standard Form for Perturbed Birth-Death-Type Semi-Markov Processes
92(10)
5.2.1 Asymptotic Expansions with Remainders Given in the Standard Form for Expectations of Return Times and Related Functionals
92(6)
5.2.2 Asymptotic Expansions with Remainders Given in the Standard Form for Stationary and Conditional Quasi-Stationary Distributions
98(3)
5.2.3 Asymptotic Expansions with Remainders Given in the Standard Form and the Algorithm of Sequential Phase Space Reduction
101(1)
5.3 Asymptotic Expansions with Explicit Upper Bounds for Remainders for Perturbed Birth-Death-Type Semi-Markov Processes
102(5)
5.3.1 Asymptotic Expansions with Explicit Upper Bounds for Remainders for Expectations of Return Times and Related Functionals
102(3)
5.3.2 Asymptotic Expansions with Explicit Upper Bounds for Remainders for Stationary and Conditional Quasi-Stationary Distributions
105(1)
5.3.3 Asymptotic Expansions with Explicit Upper Bounds for Remainders and the Algorithm of Sequential Phase Space Reduction
106(1)
6 Examples and Survey of Applied Perturbed Stochastic Models
107(16)
6.1 A Background for Numerical Examples
107(4)
6.1.1 Perturbation Conditions for Transition Probabilities of Embedded Markov Chains
107(3)
6.1.2 Perturbation Conditions for Expectations of Transition Times
110(1)
6.2 Numerical Examples
111(4)
6.2.1 Asymptotic Expansions with Remainders Given in the Standard Form
111(3)
6.2.2 Asymptotic Expansions with Explicit Upper Bounds for Remainders
114(1)
6.3 A Survey of Perturbed Stochastic Models
115(8)
6.3.1 Perturbed Queuing Systems
116(1)
6.3.2 Perturbed Stochastic Networks
117(2)
6.3.3 Perturbed Bio-Stochastic Systems
119(1)
6.3.4 Other Classes of Perturbed Stochastic Systems
120(3)
A Methodological and Bibliographical Remarks
123(8)
A.1 Methodological Remarks
123(2)
A.2 New Results Presented in the Book
125(3)
A.3 New Problems
128(3)
References 131(10)
Index 141
Dmitrii Silvestrov: Candidate of Science [ eq. PhD], (1969, Kiev University), Doctor of Science (1972, Kiev University), Professor at Kiev University (1974-1992), Mälardalen University (1999-2009) and Stockholm University from 2009. The main areas of research are stochastic processes and actuarial and financial mathematics. Author of 10 books and more than 140 research papers, co-editor of 15 collective books and proceedings. Supervised 22 doctoral students who subsequently obtained eq. PhD and PhD degrees. Sergei Silvestrov: PhD (1996, Umeå University), Docent at Lund University (2002-2011), Professor at Mälardalen University from 2011. The main areas of research are non-commutative analysis and geometry and applied matrix analysis. Author of 1 book and more than 120 research papers, co-editor of 15 collective books and proceedings. Supervised 6 doctoral students who subsequently obtained PhDdegrees.
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