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Preface

Markov and Semi-Markov Processes

(34)

Preliminaries

(1)

Markov Processes

(17)

Markov Chains

(4)

Continuous-Time Markov Processes

(4)

Diffusion Processes

(1)

Processes with Independent Increments

(3)

Processes with Locally Independent Increments

(1)

Martingale Characterization of Markov Processes

(4)

Semi-Markov Processes

(6)

Markov Renewal Processes

(2)

Markov Renewal Equation and Theorem

(2)

Auxiliary Processes

(1)

Compensating Operators

(1)

Martingale Characterization of Markov Renewal Processes

(1)

Semimartingales

(3)

Counting Markov Renewal Processes

(3)

Reducible-Invertible Operators

(4)

Stochastic Systems with Switching

(32)

Introduction

(1)

Stochastic Integral Functionals

(4)

Increment Processes

(3)

Stochastic Evolutionary Systems

(3)

Markov Additive Processes

(1)

Stochastic Additive Functionals

(3)

Random Evolutions

(9)

Continuous Random Evolutions

(4)

Jump Random Evolutions

(2)

Semi-Markov Random Evolutions

(3)

Extended Compensating Operators

(2)

Markov Additive Semimartingales

(6)

Impulsive Processes

(2)

Continuous Predictable Characteristics

(4)

Stochastic Systems in the Series Scheme

(36)

Introduction

(1)

Random Evolutions in the Series Scheme

(6)

Continuous Random Evolutions

(4)

Jump Random Evolutions

(2)

Average Approximation

(7)

Stochastic Additive Functionals

(5)

Increment Processes

(2)

Diffusion Approximation

(9)

Stochastic Integral Functionals

(3)

Stochastic Additive Functionals

(4)

Stochastic Evolutionary Systems

(1)

Increment Processes

(1)

Diffusion Approximation with Equilibrium

(13)

Locally Independent Increment Processes

(3)

Stochastic Additive Functionals with Equilibrium

(4)

Stochastic Evolutionary Systems with Semi-Markov Switching

(6)

Stochastic Systems with Split and Merging

103

(36)

Introduction

103

(1)

Phase Merging Scheme

104

(12)

Ergodic Merging

104

(6)

Merging with Absorption

110

(2)

Ergodic Double Merging

112

(4)

Average with Merging

116

(6)

Ergodic Average

117

(2)

Average with Absorption

119

(1)

Ergodic Average with Double Merging

120

(1)

Double Average with Absorption

121

(1)

Diffusion Approximation with Split and Merging

122

(12)

Ergodic Split and Merging

123

(3)

Split and Merging with Absorption

126

(2)

Ergodic Split and Double Merging

128

(2)

Double Split and Merging

130

(2)

Double Split and Double Merging

132

(2)

Integral Functionals in Split Phase Space

134

(5)

Ergodic Split

134

(3)

Double Split and Merging

137

(1)

Triple Split and Merging

138

(1)

Phase Merging Principles

139

(54)

Introduction

139

(1)

Perturbation of Reducible-Invertible Operators

140

(10)

Preliminaries

140

(1)

Solution of Singular Perturbation Problems

141

(9)

Average Merging Principle

150

(10)

Stochastic Evolutionary Systems

151

(1)

Stochastic Additive Functionals

152

(2)

Increment Processes

154

(2)

Continuous Random Evolutions

156

(1)

Jump Random Evolutions

157

(2)

Random Evolutions with Markov Switching

159

(1)

Diffusion Approximation Principle

160

(13)

Stochastic Integral Functionals

161

(4)

Continuous Random Evolutions

165

(4)

Jump Random Evolutions

169

(3)

Random Evolutions with Markov Switching

172

(1)

Diffusion Approximation with Equilibrium

173

(9)

Locally Independent Increment Processes

174

(1)

Stochastic Additive Functionals

175

(1)

Stochastic Evolutionary Systems with Semi-Markov Switching

176

(6)

Merging and Averaging in Split State Space

182

(6)

Preliminaries

182

(2)

Semi-Markov Processes in Split State Space

184

(2)

Average Stochastic Systems

186

(2)

Diffusion Approximation with Split and Merging

188

(5)

Ergodic Split and Merging

188

(1)

Split and Double Merging

189

(1)

Double Split and Merging

190

(1)

Double Split and Double Merging

191

(2)

Weak Convergence

193

(26)

Introduction

193

(1)

Preliminaries

193

(3)

Pattern Limit Theorems

196

(13)

Stochastic Systems with Markov Switching

196

(5)

Stochastic Systems with Semi-Markov Switching

201

(4)

Embedded Markov Renewal Processes

205

(4)

Relative Compactness

209

(7)

Stochastic Systems with Markov Switching

209

(3)

Stochastic Systems with Semi-Markov Switching

212

(1)

Compact Containment Condition

213

(3)

Verification of Convergence

216

(3)

Poisson Approximation

219

(24)

Introduction

219

(1)

Stochastic Systems in Poisson Approximation Scheme

220

(11)

Impulsive Processes with Markov Switching

220

(5)

Impulsive Processes in an Asymptotic Split Phase Space

225

(3)

Stochastic Additive Functionals with Semi-Markov Switching

228

(3)

Semimartingale Characterization

231

(12)

Impulsive Processes as Semimartingales

232

(5)

Stochastic Additive Functionals

237

(6)

Applications I

243

(26)

Absorption Times

243

(6)

Stationary Phase Merging

249

(4)

Superposition of Two Renewal Processes

253

(5)

Semi-Markov Random Walks

258

(11)

Introduction

258

(1)

The algorithms of approximation for SMRW

259

(3)

Compensating Operators

262

(3)

The singular perturbation problem

265

(2)

Stationary Phase Merging Scheme

267

(2)

Applications II

269

(18)

Birth and Death Processes and Repairable Systems

269

(7)

Introduction

269

(1)

Diffusion Approximation

270

(2)

Proofs of the Theorems

272

(4)

Levy Approximation of Impulsive Processes

276

(11)

Introduction

276

(2)

Levy Approximation Scheme

278

(4)

Proof of Theorems

282

(5)

Problems to Solve

287

(14)

Appendix A Weak Convergence of Probability Measures

301

(4)

A.1 Weak Convergence

301

(2)

A.2 Relative Compactness

303

(2)

Appendix B Some Limit Theorems for Stochastic Processes

305

(6)

B.1 Two Limit Theorems for Semimartingales

305

(3)

B.2 A Limit Theorem for Composed Processes

308

(3)

Appendix C Some Auxiliary Results

311

(4)

C.1 Backward Recurrence Time Negligibility

311

(1)

C.2 Positiveness of Diffusion Coefficients

312

(3)

Bibliography

315

(10)

Notation

325

(4)

Index

329

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Stochastic Systems in Merging Phase Space

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Stochastic Systems in Merging Phase Space

9789812565914

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Table of Contents

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