Continuous Semi-Markov Processes

This title considers the special of random processes known as semi-Markov processes. These possess the Markov property with respect to any intrinsic Markov time such as the first exit time from an open set or a finite iteration of these times. The class of semi-Markov processes includes strong Markov processes, L vy and Smith stepped semi-Markov processes, and some other subclasses. Extensive coverage is devoted to non-Markovian semi-Markov processes with continuous trajectories and, in particular, to semi-Markov diffusion processes. Readers looking to enrich their knowledge on Markov processes will find this book a valuable resource.

This title considers the special of random processes known assemi-Markov processes. These possess the Markov property withrespect to any intrinsic Markov time such as the first exit timefrom an open set or a finite iteration of these times.The class of semi-Markov processes includes strong Markovprocesses, Lévy and Smith stepped semi-Markov processes, andsome other subclasses. Extensive coverage is devoted tonon-Markovian semi-Markov processes with continuous trajectoriesand, in particular, to semi-Markov diffusion processes. Readerslooking to enrich their knowledge on Markov processes will findthis book a valuable resource.

Introduction 9Chapter 1. Stepped Semi-Markov Processes 171.1. Random sequence 171.2. Markov chain 201.3. Two-dimensional Markov chain 251.4. Semi-Markov process 291.5. Stationary distributions 32Chapter 2. Sequences of First Exit Times and RegenerationTimes 372.1. Basic maps 382.2. Markov times 422.3. Deducing sequences 462.4. Correct exit and continuity 552.5. Time of regeneration 63Chapter 3. General Semi-Markov Processes 713.1. Definition of a semi-Markov process 723.2. Transition function of a SM process 793.3. Operators and SM walk 823.4. Operators and SM process 913.5. Criterion of Markov property for SM processes 1033.6. Intervals of constancy 110Chapter 4. Construction of Semi-Markov Processes usingSemi-Markov Transition Functions 1154.1. Realization of an innite systemof pairs 1164.2. Extension of a measure 1214.3. Construction of a measure 1244.4. Construction of a projective system of measures 1274.5. Semi-Markov processes 133Chapter 5. Semi-Markov Processes of Diffusion Type1375.1. One-dimensional semi-Markov processes of diffusion type1385.1.1. Differential equation 1385.1.2. Construction SM process 1435.1.3. Some properties of the process 1595.2. Multi-dimensional processes of diffusion type 1685.2.1. Differential equations of elliptic type 1685.2.2. Neighborhood of arbitrary form 1715.2.3. Neighborhood of spherical form 1785.2.4. Characteristic operator 188Chapter 6. Time Change and Semi-Markov Processes 1976.1. Time change and trajectories 1986.2. Intrinsic time and traces 2066.3. Canonical time change 2116.4. Coordination of function and time change 2236.5. Random time changes 2286.6. Additive functionals 2336.7. Distribution of a time run along the trace 2426.8. Random curvilinear integrals 2526.9. Characteristic operator and integral 2646.10. Stochastic integral 2686.10.1. Semi-martingale and martingale 2686.10.2. Stochastic integral 2756.10.3. Ito-Dynkin's formula 277Chapter 7. Limit Theorems for Semi-Markov Processes2817.1. Weak compactness and weak convergence 2817.2. Weak convergence of semi-Markov processes 289Chapter 8. Representation of a Semi-Markov Process as aTransformed Markov Process 2998.1. Construction by operator 3008.2. Comparison of processes 3028.3. Construction by parameters of Lévy formula 3078.4. Stationary distribution 311Chapter 9. Semi-Markov Model of Chromatography 3259.1. Chromatography 3269.2. Model of liquid column chromatography 3289.3. Some monotone Semi-Markov processes 3329.4. Transfer with diffusion 3379.5. Transfer with nal absorption 346Bibliography 361Index 369