Anatoliy Pogorui�s main research interests include probability, stochastic processes, mathematical modeling of an ideal gas using multi-dimensional random motions and the interaction of telegraph particles in semi-Markov environments and the application of random evolutions in the reliability theory of storage systems. Anatoliy Swishchuk is Professor of mathematical finance at the University of Calgary, Canada. His research areas include financial mathematics, random evolutions and their applications, stochastic calculus and biomathematics. Ramón M. Rodríguez-Dagnino has investigated applied probability aimed at modeling systems with stochastic behavior, random motions in wireless networks, video trace modeling and prediction, information source characterization, performance analysis of networks with heavytail traffic, generalized Gaussian estimation and spectral analysis.

• Preface ixAcknowledgments xiiiIntroduction xvPart 1. Basic Methods 1Chapter 1. Preliminary Concepts 31.1. Introduction to random evolutions 31.2. Abstract potential operators 71.3. Markov processes: operator semigroups 111.4. Semi-Markov processes 141.5. Lumped Markov chains 171.6. Switched processes in Markov and semi-Markov media 19Chapter 2. Homogeneous Random Evolutions (HRE) and their Applications 232.1. Homogeneous random evolutions (HRE) 242.1.1. Definition and classification of HRE 242.1.2. Some examples of HRE 252.1.3. Martingale characterization of HRE 282.1.4. Analogue of Dynkin’s formula for HRE 342.1.5. Boundary value problems for HRE 362.2. Limit theorems for HRE 372.2.1. Weak convergence of HRE 372.2.2. Averaging of HRE 392.2.3. Diffusion approximation of HRE 422.2.4. Averaging of REs in reducible phase space: merged HRE 452.2.5. Diffusion approximation of HRE in reducible phase space 482.2.6. Normal deviations of HRE 512.2.7. Rates of convergence in the limit theorems for HRE 53Part 2. Applications to Reliability, Random Motions, and Telegraph Processes 57Chapter 3. Asymptotic Analysis for Distributions of Markov, Semi-Markov and Random Evolutions 593.1. Asymptotic distribution of time to reach a level that is infinitely increasing by a family of semi-Markov processes on the set ℕ 613.2. Asymptotic inequalities for the distribution of the occupation time of a semi-Markov process in an increasing set of states 743.3. Asymptotic analysis of the occupation time distribution of an embedded semi-Markov process (with increasing states) in a diffusion process 773.4. Asymptotic analysis of a semigroup of operators of the singularly perturbed random evolution in semi-Markov media 823.5. Asymptotic expansion for distribution of random motion in Markov media under the Kac condition 903.5.1. The equation for the probability density of the particle position performing a random walk in ℝn 903.5.2. Equation for the probability density of the particle position 913.5.3. Reduction of a singularly perturbed evolution equation to a regularly perturbed equation 933.6. Asymptotic estimation for application of the telegraph process as an alternative to the diffusion process in the Black–Scholes formula 963.6.1. Asymptotic expansion for the singularly perturbed random evolution in Markov media in case of disbalance 963.6.2. Application to an economic model of stock market 100Chapter 4. Random Switched Processes with Delay in Reflecting Boundaries 1034.1. Stationary distribution of evolutionary switched processes in a Markov environment with delay in reflecting boundaries 1044.2. Stationary distribution of switched process in semi-Markov media with delay in reflecting barriers 1094.2.1. Infinitesimal operator of random evolution with semi-Markov switching 1104.2.2. Stationary distribution of random evolution in semi-Markov media with delaying boundaries in balance case 1134.2.3. Stationary distribution of random evolution in semi-Markov media with delaying boundaries 1214.3. Stationary efficiency of a system with two unreliable subsystems in cascade and one buffer: the Markov case 1244.3.1. Introduction 1244.3.2. Stationary distribution of Markov stochastic evolutions 1254.3.3. Stationary efficiency of a system with two unreliable subsystems in cascade and one buffer 1294.3.4. Mathematical model 1314.3.5. Main mathematical results 1334.3.6. Numerical results for the symmetric case 1384.4. Application of random evolutions with delaying barriers to modelling control of supply systems with feedback: the semi-Markov switching process 1414.4.1. Estimation of stationary efficiency of one-phase system with a reservoir 1414.4.2. Estimation of stationary efficiency of a production system with two unreliable supply lines 149Chapter 5. One-dimensional Random Motions in Markov and Semi-Markov Media 1595.1. One-dimensional semi-Markov evolutions with general Erlang sojourn times 1605.1.1. Mathematical model 1605.1.2. Solution of PDEs with constant coefficients and derivability of functions ranged in commutative algebras 1685.1.3. Infinite-dimensional case 1715.1.4. The distribution of one-dimensional random evolutions in Erlang media 1725.2. Distribution of limiting position of fading evolution 1815.2.1. Distribution of random power series in cases of uniform and Erlang distributions 1825.2.2. The distribution of the limiting position 1905.3. Differential and integral equations for jump random motions 1915.3.1. The Erlang jump telegraph process on a line 1925.3.2. Examples 1985.4. Estimation of the number of level crossings by the telegraph process 1995.4.1. Estimation of the number of level crossings for the telegraph process in Kac’s condition 202References 205Index 219Summary of Volume 2 221

Anatoliy Pogorui�s main research interests include probability, stochastic processes, mathematical modeling of an ideal gas using multi-dimensional random motions and the interaction of telegraph particles in semi-Markov environments and the application of random evolutions in the reliability theory of storage systems. Anatoliy Swishchuk is Professor of mathematical finance at the University of Calgary, Canada. His research areas include financial mathematics, random evolutions and their applications, stochastic calculus and biomathematics. Ramón M. Rodríguez-Dagnino has investigated applied probability aimed at modeling systems with stochastic behavior, random motions in wireless networks, video trace modeling and prediction, information source characterization, performance analysis of networks with heavytail traffic, generalized Gaussian estimation and spectral analysis.

• Preface ixAcknowledgments xiiiIntroduction xvPart 1. Basic Methods 1Chapter 1. Preliminary Concepts 31.1. Introduction to random evolutions 31.2. Abstract potential operators 71.3. Markov processes: operator semigroups 111.4. Semi-Markov processes 141.5. Lumped Markov chains 171.6. Switched processes in Markov and semi-Markov media 19Chapter 2. Homogeneous Random Evolutions (HRE) and their Applications 232.1. Homogeneous random evolutions (HRE) 242.1.1. Definition and classification of HRE 242.1.2. Some examples of HRE 252.1.3. Martingale characterization of HRE 282.1.4. Analogue of Dynkin’s formula for HRE 342.1.5. Boundary value problems for HRE 362.2. Limit theorems for HRE 372.2.1. Weak convergence of HRE 372.2.2. Averaging of HRE 392.2.3. Diffusion approximation of HRE 422.2.4. Averaging of REs in reducible phase space: merged HRE 452.2.5. Diffusion approximation of HRE in reducible phase space 482.2.6. Normal deviations of HRE 512.2.7. Rates of convergence in the limit theorems for HRE 53Part 2. Applications to Reliability, Random Motions, and Telegraph Processes 57Chapter 3. Asymptotic Analysis for Distributions of Markov, Semi-Markov and Random Evolutions 593.1. Asymptotic distribution of time to reach a level that is infinitely increasing by a family of semi-Markov processes on the set ℕ 613.2. Asymptotic inequalities for the distribution of the occupation time of a semi-Markov process in an increasing set of states 743.3. Asymptotic analysis of the occupation time distribution of an embedded semi-Markov process (with increasing states) in a diffusion process 773.4. Asymptotic analysis of a semigroup of operators of the singularly perturbed random evolution in semi-Markov media 823.5. Asymptotic expansion for distribution of random motion in Markov media under the Kac condition 903.5.1. The equation for the probability density of the particle position performing a random walk in ℝn 903.5.2. Equation for the probability density of the particle position 913.5.3. Reduction of a singularly perturbed evolution equation to a regularly perturbed equation 933.6. Asymptotic estimation for application of the telegraph process as an alternative to the diffusion process in the Black–Scholes formula 963.6.1. Asymptotic expansion for the singularly perturbed random evolution in Markov media in case of disbalance 963.6.2. Application to an economic model of stock market 100Chapter 4. Random Switched Processes with Delay in Reflecting Boundaries 1034.1. Stationary distribution of evolutionary switched processes in a Markov environment with delay in reflecting boundaries 1044.2. Stationary distribution of switched process in semi-Markov media with delay in reflecting barriers 1094.2.1. Infinitesimal operator of random evolution with semi-Markov switching 1104.2.2. Stationary distribution of random evolution in semi-Markov media with delaying boundaries in balance case 1134.2.3. Stationary distribution of random evolution in semi-Markov media with delaying boundaries 1214.3. Stationary efficiency of a system with two unreliable subsystems in cascade and one buffer: the Markov case 1244.3.1. Introduction 1244.3.2. Stationary distribution of Markov stochastic evolutions 1254.3.3. Stationary efficiency of a system with two unreliable subsystems in cascade and one buffer 1294.3.4. Mathematical model 1314.3.5. Main mathematical results 1334.3.6. Numerical results for the symmetric case 1384.4. Application of random evolutions with delaying barriers to modelling control of supply systems with feedback: the semi-Markov switching process 1414.4.1. Estimation of stationary efficiency of one-phase system with a reservoir 1414.4.2. Estimation of stationary efficiency of a production system with two unreliable supply lines 149Chapter 5. One-dimensional Random Motions in Markov and Semi-Markov Media 1595.1. One-dimensional semi-Markov evolutions with general Erlang sojourn times 1605.1.1. Mathematical model 1605.1.2. Solution of PDEs with constant coefficients and derivability of functions ranged in commutative algebras 1685.1.3. Infinite-dimensional case 1715.1.4. The distribution of one-dimensional random evolutions in Erlang media 1725.2. Distribution of limiting position of fading evolution 1815.2.1. Distribution of random power series in cases of uniform and Erlang distributions 1825.2.2. The distribution of the limiting position 1905.3. Differential and integral equations for jump random motions 1915.3.1. The Erlang jump telegraph process on a line 1925.3.2. Examples 1985.4. Estimation of the number of level crossings by the telegraph process 1995.4.1. Estimation of the number of level crossings for the telegraph process in Kac’s condition 202References 205Index 219Summary of Volume 2 221

Anatoliy Pogorui�s main research interests include probability, stochastic processes, mathematical modeling of an ideal gas using multi-dimensional random motions and the interaction of telegraph particles in semi-Markov environments and the application of random evolutions in the reliability theory of storage systems. Anatoliy Swishchuk is Professor of mathematical finance at the University of Calgary, Canada. His research areas include financial mathematics, random evolutions and their applications, stochastic calculus and biomathematics. Ramón M. Rodríguez-Dagnino has investigated applied probability aimed at modeling systems with stochastic behavior, random motions in wireless networks, video trace modeling and prediction, information source characterization, performance analysis of networks with heavytail traffic, generalized Gaussian estimation and spectral analysis.

• Preface ixAcknowledgments xiiiIntroduction xvPart 1. Basic Methods 1Chapter 1. Preliminary Concepts 31.1. Introduction to random evolutions 31.2. Abstract potential operators 71.3. Markov processes: operator semigroups 111.4. Semi-Markov processes 141.5. Lumped Markov chains 171.6. Switched processes in Markov and semi-Markov media 19Chapter 2. Homogeneous Random Evolutions (HRE) and their Applications 232.1. Homogeneous random evolutions (HRE) 242.1.1. Definition and classification of HRE 242.1.2. Some examples of HRE 252.1.3. Martingale characterization of HRE 282.1.4. Analogue of Dynkin’s formula for HRE 342.1.5. Boundary value problems for HRE 362.2. Limit theorems for HRE 372.2.1. Weak convergence of HRE 372.2.2. Averaging of HRE 392.2.3. Diffusion approximation of HRE 422.2.4. Averaging of REs in reducible phase space: merged HRE 452.2.5. Diffusion approximation of HRE in reducible phase space 482.2.6. Normal deviations of HRE 512.2.7. Rates of convergence in the limit theorems for HRE 53Part 2. Applications to Reliability, Random Motions, and Telegraph Processes 57Chapter 3. Asymptotic Analysis for Distributions of Markov, Semi-Markov and Random Evolutions 593.1. Asymptotic distribution of time to reach a level that is infinitely increasing by a family of semi-Markov processes on the set ℕ 613.2. Asymptotic inequalities for the distribution of the occupation time of a semi-Markov process in an increasing set of states 743.3. Asymptotic analysis of the occupation time distribution of an embedded semi-Markov process (with increasing states) in a diffusion process 773.4. Asymptotic analysis of a semigroup of operators of the singularly perturbed random evolution in semi-Markov media 823.5. Asymptotic expansion for distribution of random motion in Markov media under the Kac condition 903.5.1. The equation for the probability density of the particle position performing a random walk in ℝn 903.5.2. Equation for the probability density of the particle position 913.5.3. Reduction of a singularly perturbed evolution equation to a regularly perturbed equation 933.6. Asymptotic estimation for application of the telegraph process as an alternative to the diffusion process in the Black–Scholes formula 963.6.1. Asymptotic expansion for the singularly perturbed random evolution in Markov media in case of disbalance 963.6.2. Application to an economic model of stock market 100Chapter 4. Random Switched Processes with Delay in Reflecting Boundaries 1034.1. Stationary distribution of evolutionary switched processes in a Markov environment with delay in reflecting boundaries 1044.2. Stationary distribution of switched process in semi-Markov media with delay in reflecting barriers 1094.2.1. Infinitesimal operator of random evolution with semi-Markov switching 1104.2.2. Stationary distribution of random evolution in semi-Markov media with delaying boundaries in balance case 1134.2.3. Stationary distribution of random evolution in semi-Markov media with delaying boundaries 1214.3. Stationary efficiency of a system with two unreliable subsystems in cascade and one buffer: the Markov case 1244.3.1. Introduction 1244.3.2. Stationary distribution of Markov stochastic evolutions 1254.3.3. Stationary efficiency of a system with two unreliable subsystems in cascade and one buffer 1294.3.4. Mathematical model 1314.3.5. Main mathematical results 1334.3.6. Numerical results for the symmetric case 1384.4. Application of random evolutions with delaying barriers to modelling control of supply systems with feedback: the semi-Markov switching process 1414.4.1. Estimation of stationary efficiency of one-phase system with a reservoir 1414.4.2. Estimation of stationary efficiency of a production system with two unreliable supply lines 149Chapter 5. One-dimensional Random Motions in Markov and Semi-Markov Media 1595.1. One-dimensional semi-Markov evolutions with general Erlang sojourn times 1605.1.1. Mathematical model 1605.1.2. Solution of PDEs with constant coefficients and derivability of functions ranged in commutative algebras 1685.1.3. Infinite-dimensional case 1715.1.4. The distribution of one-dimensional random evolutions in Erlang media 1725.2. Distribution of limiting position of fading evolution 1815.2.1. Distribution of random power series in cases of uniform and Erlang distributions 1825.2.2. The distribution of the limiting position 1905.3. Differential and integral equations for jump random motions 1915.3.1. The Erlang jump telegraph process on a line 1925.3.2. Examples 1985.4. Estimation of the number of level crossings by the telegraph process 1995.4.1. Estimation of the number of level crossings for the telegraph process in Kac’s condition 202References 205Index 219Summary of Volume 2 221