Contents -- Preface -- 1. Markov and Semi-Markov Processes -- 1.1 Preliminaries -- 1.2 Markov Processes -- 1.2.1 Markov Chains -- 1.2.2 Continuous-Time Markov Processes -- 1.2.3 Diffusion Processes -- 1.2.4 Processes with Independent Increments -- 1.2.5 Processes with Locally Independent Increments -- 1.2.6 Martingale Characterization of Markov Processes -- 1.3 Semi-Markov Processes -- 1.3.1 Markov Renewal Processes -- 1.3.2 Markov Renewal Equation and Theorem -- 1.3.3 Auxiliary Processes -- 1.3.4 Compensating Operators -- 1.3.5 Martingale Characterization of Markov Renewal Processes -- 1.4 Semimartingales -- 1.5 Counting Markov Renewal Processes -- 1.6 Reducible-Invertible Operators -- 2. Stochastic Systems with Switching -- 2.1 Introduction -- 2.2 Stochastic Integral Functionals -- 2.3 Increment Processes -- 2.4 Stochastic Evolutionary Systems -- 2.5 Markov Additive Processes -- 2.6 Stochastic Additive Functionals -- 2.7 Random Evolutions -- 2.7.1 Continuous Random Evolutions -- 2.7.2 Jump Random Evolutions -- 2.7.3 Semi-Markov Random Evolutions -- 2.8 Extended Compensating Operators -- 2.9 Markov Additive Semimartingales -- 2.9.1 Impulsive Processes -- 2.9.2 Continuous Predictable Characteristics -- 3. Stochastic Systems in the Series Scheme -- 3.1 Introduction -- 3.2 Random Evolutions in the Series Scheme -- 3.2.1 Continuous Random Evolutions -- 3.2.2 Jump Random Evolutions -- 3.3 Average Approximation -- 3.3.1 Stochastic Additive Functionals -- 3.3.2 Increment Processes -- 3.4 Diffusion Approximation -- 3.4.1 Stochastic Integral Functionals -- 3.4.2 Stochastic Additive Functionals -- 3.4.3 Stochastic Evolutionary Systems -- 3.4.4 Increment Processes -- 3.5 Diffusion Approximation with Equilibrium -- 3.5.1 Locally Independent Increment Processes -- 3.5.2 Stochastic Additive Functionals with Equilibrium.

3.5.3 Stochastic Evolutionary Systems with Semi-Markov Switching -- 4. Stochastic Systems with Split and Merging -- 4.1 Introduction -- 4.2 Phase Merging Scheme -- 4.2.1 Ergodic Merging -- 4.2.2 Merging with Absorption -- 4.2.3 Ergodic Double Merging -- 4.3 Average with Merging -- 4.3.1 Ergodic Average -- 4.3.2 Average with Absorption -- 4.3.3 Ergodic Average with Double Merging -- 4.3.4 Double Average with Absorption -- 4.4 Diffusion Approximation with Split and Merging -- 4.4.1 Ergodic Split and Merging -- 4.4.2 Split and Merging with Absorption -- 4.4.3 Ergodic Split and Double Merging -- 4.4.4 Double Split and Merging -- 4.4.5 Double Split and Double Merging -- 4.5 Integral F'unctionals in Split Phase Space -- 4.5.1 Ergodic Split -- 4.5.2 Double Split and Merging -- 4.5.3 Triple Split and Merging -- 5. Phase Merging Principles -- 5.1 Introduction -- 5.2 Perturbation of Reducible-Invertible Operators -- 5.2.1 Preliminaries -- 5.2.2 Solution of Singular Perturbation Problems -- 5.3 Average Merging Principle -- 5.3.1 Stochastic Evolutionary Systems -- 5.3.2 Stochastic Additive F'unctionals -- 5.3.3 Increment Processes -- 5.3.4 Continuous Random Evolutions -- 5.3.5 Jump Random Evolutions -- 5.3.6 Random Evolutions with Markov Switching -- 5.4 Diffusion Approximation Principle -- 5.4.1 Stochastic Integral F'unctionals -- 5.4.2 Continuous Random Evolutions -- 5.4.3 Jump Random Evolutions -- 5.4.4 Random Evolutions with Markov Switching -- 5.5 Diffusion Approximation with Equilibrium -- 5.5.1 Locally Independent Increment Processes -- 5.5.2 Stochastic Additive F'unctionals -- 5.5.3 Stochastic Evolutionary Systems with Semi-Markov Switching -- 5.6 Merging and Averaging in Split State Space -- 5.6.1 Preliminaries -- 5.6.2 Semi-Markov Processes in Split State Space -- 5.6.3 Average Stochastic Systems -- 5.7 Diffusion Approximation with Split and Merging.

5.7.1 Ergodic Split and Merging -- 5.7.2 Split and Double Merging -- 5.7.3 Double Split and Merging -- 5.7.4 Double Split and Double Merging -- 6. Weak Convergence -- 6.1 Introduction -- 6.2 Preliminaries -- 6.3 Pattern Limit Theorems -- 6.3.1 Stochastic Systems with Markov Switching -- 6.3.2 Stochastic Systems with Semi-Markov Switching -- 6.3.3 Embedded Markov Renewal Processes -- 6.4 Relative Compactness -- 6.4.1 Stochastic Systems with Markov Switching -- 6.4.2 Stochastic Systems with Semi-Markov Switching -- 6.4.3 Compact Containment Condition -- 6.5 Verification of Convergence -- 7. Poisson Approximation -- 7.1 Introduction -- 7.2 Stochastic Systems in Poisson Approximation Scheme -- 7.2.1 Impulsive Processes with Markov Switching -- 7.2.2 Impulsive Processes in an Asymptotic Split Phase Space -- 7.2.3 Stochastic Additive F'unctionals with Semi-Markov Switching -- 7.3 Semimartingale Characterization -- 7.3.1 Impulsive Processes as Semimartingales -- 7.3.2 Stochastic Additive F'unctionals -- 8. Applications I -- 8.1 Absorption Times -- 8.2 Stationary Phase Merging -- 8.3 Superposition of Two Renewal Processes -- 8.4 Semi-Markov Random Walks -- 8.4.1 Introduction -- 8.4.2 The algorithms of approximation for SMRW -- 8.4.3 Compensating Operators -- 8.4.4 The singular perturbation problem -- 8.4.5 Stationary Phase Merging Scheme -- 9. Applications II -- 9.1 Birth and Death Processes and Repairable Systems -- 9.1.1 Introduction . -- 9.1.2 Diffusion Approximation -- 9.1.3 Proofs of the Theorems -- 9.2 Levy Approximation of Impulsive Processes -- 9.2.1 Introduction -- 9.2.2 L6vy Approximation Scheme -- 9.2.3 Proof of Theorems -- Problems to Solve -- Appendix A Weak Convergence of Probability Measures -- A.l Weak Convergence -- A.2 Relative Compactness -- Appendix B Some Limit Theorems for Stochastic Processes.

B.l Two Limit Theorems for Semimartingales -- B.2 A Limit Theorem for Composed Processes -- Appendix C Some Auxiliary Results -- C.l Backward Recurrence Time Negligibility -- C.2 Positiveness of Diffusion Coefficients -- Bibliography -- Notation -- Index.