Markov Renewal and Piecewise Deterministic Processes

Preface
Contents
Notations
1 Introduction
2 Markov Renewal Processes and Related Processes
	2.1 Kernels and General Markov Chains
	2.2 Renewal Kernels and Convolution
	2.3 Markov Renewal Processes
	2.4 Changing the Observation Time
	2.5 Semi-Markov Processes and CSMPs
	2.6 Other Classical Processes Associated with Markov Renewal Processes
	2.7 Semiregenerative Processes and Markov Renewal Equations
3 First Steps with PDMPs
	3.1 General PDMPs
	3.2 Parametrized CSMPs and PDMPs
	3.3 Simulation
	3.4 Some Examples
4 Hitting Time Distribution
	4.1 Killed and Stopped PDMPs
	4.2 Process Decomposition
5 Intensity of Some Marked Point Processes
	5.1 Two Intensities Related to Markov Renewal Processes
	5.2 Intensity Decomposition
	5.3 Intensities Related to PDMP
6 Generalized Kolmogorov Equations
	6.1 Kolmogorov Equations for CSMPs
	6.2 Kolmogorov Equations for PDMPs
	6.3 Some Semigroup Properties in the Absence of Dirac Measures
7 A Martingale Approach
	7.1 Martingales Related to Markov Renewal Processes
	7.2 Martingales Related to Simple CSMPs
	7.3 Martingales Related to Parametrized PDMPs
	7.4 Generators
8 Stability
	8.1 Connections Between Invariant Measures of a CSMP and those of Its Driving Chain
	8.2 CSMP Recurrence from That of Its Driving Chain
	8.3 Intensities of Stable CSMPs
	8.4 Invariant Measure for PDMPs
	8.5 Intensities of Stable PDMPs
	8.6 Asymptotic Results
9 Numerical Methods
	9.1 Method
	9.2 An Explicit Scheme for PDMPs Without Boundary Associated with Ordinary Differential Equations
	9.3 An Implicit Scheme for PDMPs Without Boundary Associated with Ordinary Differential Equations
	9.4 An Implicit Scheme for PDMPs with Boundary
	9.5 Numerical Experiments
10 Switching Processes
	10.1 The General Model
	10.2 Vector of Independent Parametrized Switching Processes
	10.3 Process Decomposition
	10.4 Semiregenerative Property, Link with the CSMP
	10.5 Conditions to Get a Markov Process
	10.6 Case of a Semimartingale Intrinsic Process
	10.7 Case of an Inhomogeneous Intrinsic Process
Appendix A Tools
A.1  Conditional Expectation, Conditional Probability Distribution
A.2  Functions of Bounded Variation
A.3  Hazard Rate, Hazard Measure
A.4  Compensator of a Marked Point Process
Appendix B Interarrival Distributions with Several Dirac Measures
Appendix C Proof of Convergence of the Scheme of Sect. 9.4摥映數爠eflinkparCEGR9.49
C.1  Uniqueness
C.2  Tightness
C.3  Convergence
Appendix  References
Index