| Preface |
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ix | |
| Acknowledgments |
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xiii | |
| Introduction |
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xv | |
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1 | (56) |
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Chapter 1 Preliminary Concepts |
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3 | (20) |
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1.1 Introduction to random evolutions |
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3 | (4) |
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1.2 Abstract potential operators |
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7 | (4) |
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1.3 Markov processes: operator semigroups |
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11 | (3) |
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1.4 Semi-Markov processes |
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14 | (3) |
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17 | (2) |
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1.6 Switched processes in Markov and semi-Markov media |
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19 | (4) |
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Chapter 2 Homogeneous Random Evolutions (HRE) and their Applications |
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23 | (34) |
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2.1 Homogeneous random evolutions (HRE) |
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24 | (13) |
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2.1.1 Definition and classification of HRE |
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24 | (1) |
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2.1.2 Some examples of HRE |
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25 | (3) |
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2.1.3 Martingale characterization of HRE |
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28 | (6) |
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2.1.4 Analogue of Dynkin's formula for HRE |
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34 | (2) |
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2.1.5 Boundary value problems for HRE |
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36 | (1) |
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2.2 Limit theorems for HRE |
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37 | (20) |
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2.2.1 Weak convergence of HRE |
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37 | (2) |
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39 | (3) |
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2.2.3 Diffusion approximation of HRE |
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42 | (3) |
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2.2.4 Averaging of REs in reducible phase space: merged HRE |
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45 | (3) |
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2.2.5 Diffusion approximation of HRE in reducible phase space |
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48 | (3) |
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2.2.6 Normal deviations of HRE |
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51 | (2) |
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2.2.7 Rates of convergence in the limit theorems for HRE |
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53 | (4) |
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Part 2 Applications to Reliability, Random Motions, and Telegraph Processes |
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57 | (148) |
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Chapter 3 Asymptotic Analysis for Distributions of Markov, Semi-Markov and Random Evolutions |
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59 | (44) |
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3.1 Asymptotic distribution of time to reach a level that is infinitely increasing by a family of semi-Markov processes on the set N |
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61 | (13) |
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3.2 Asymptotic inequalities for the distribution of the occupation time of a semi-Markov process in an increasing set of states |
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74 | (3) |
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3.3 Asymptotic analysis of the occupation time distribution of an embedded semi-Markov process (with increasing states) in a diffusion process |
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77 | (5) |
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3.4 Asymptotic analysis of a semigroup of operators of the singularly perturbed random evolution in semi-Markov media |
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82 | (8) |
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3.5 Asymptotic expansion for distribution of random motion in Markov media under the Kac condition |
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90 | (6) |
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3.5.1 The equation for the probability density of the particle position performing a random walk in R™ |
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90 | (1) |
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3.5.2 Equation for the probability density of the particle position |
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91 | (2) |
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3.5.3 Reduction of a singularly perturbed evolution equation to a regularly perturbed equation |
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93 | (3) |
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3.6 Asymptotic estimation for application of the telegraph process as an alternative to the diffusion process in the Black--Scholes formula |
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96 | (7) |
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3.6.1 Asymptotic expansion for the singularly perturbed random evolution in Markov media in case of disbalance |
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96 | (4) |
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3.6.2 Application to an economic model of stock market |
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100 | (3) |
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Chapter 4 Random Switched Processes with Delay in Reflecting Boundaries |
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103 | (56) |
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4.1 Stationary distribution of evolutionary switched processes in a Markov environment with delay in reflecting boundaries |
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104 | (5) |
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4.2 Stationary distribution of switched process in semi-Markov media with delay in reflecting barriers |
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109 | (15) |
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4.2.1 Infinitesimal operator of random evolution with semi-Markov switching |
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110 | (3) |
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4.2.2 Stationary distribution of random evolution in semi-Markov media with delaying boundaries in balance case |
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113 | (8) |
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4.2.3 Stationary distribution of random evolution in semi-Markov media with delaying boundaries |
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121 | (3) |
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4.3 Stationary efficiency of a system with two unreliable subsystems in cascade and one buffer: the Markov case |
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124 | (17) |
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124 | (1) |
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4.3.2 Stationary distribution of Markov stochastic evolutions |
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125 | (4) |
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4.3.3 Stationary efficiency of a system with two unreliable subsystems in cascade and one buffer |
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129 | (2) |
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131 | (2) |
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4.3.5 Main mathematical results |
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133 | (5) |
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4.3.6 Numerical results for the symmetric case |
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138 | (3) |
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4.4 Application of random evolutions with delaying barriers to modeling control of supply systems with feedback: the semi-Markov switching process |
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141 | (18) |
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4.4.1 Estimation of stationary efficiency of one-phase system with a reservoir |
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141 | (8) |
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4.4.2 Estimation of stationary efficiency of a production system with two unreliable supply lines |
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149 | (10) |
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Chapter 5 One-dimensional Random Motions in Markov and Semi-Markov Media |
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159 | (46) |
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5.1 One-dimensional semi-Markov evolutions with general Erlang sojourn times |
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160 | (21) |
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160 | (8) |
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5.1.2 Solution of PDEs with constant coefficients and derivability of functions ranged in commutative algebras |
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168 | (3) |
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5.1.3 Infinite-dimensional case |
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171 | (1) |
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5.1.4 The distribution of one-dimensional random evolutions in Erlang media |
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172 | (9) |
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5.2 Distribution of limiting position of fading evolution |
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181 | (10) |
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5.2.1 Distribution of random power series in cases of uniform and Erlang distributions |
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182 | (8) |
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5.2.2 The distribution of the limiting position |
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190 | (1) |
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5.3 Differential and integral equations for jump random motions |
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191 | (8) |
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5.3.1 The Erlang jump telegraph process on a line |
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192 | (6) |
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198 | (1) |
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5.4 Estimation of the number of level crossings by the telegraph process |
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199 | (6) |
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5.4.1 Estimation of the number of level crossings for the telegraph process in Kac's condition |
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202 | (3) |
| References |
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205 | (14) |
| Index |
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219 | (2) |
| Summary of Volume 2 |
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221 | |
Lisainfo e-raamatute kohta