This book is aimed at researchers, graduate students and engineers who would like to be initiated to Piecewise Deterministic Markov Processes (PDMPs). A PDMP models a deterministic mechanism modified by jumps that occur at random times. The fields of applications are numerous : insurance and risk, biology, communication networks, dependability, supply management, etc.
Indeed, the PDMPs studied so far are in fact deterministic functions of CSMPs (Completed Semi-Markov Processes), i.e. semi-Markov processes completed to become Markov processes. This remark leads to considerably broaden the definition of PDMPs and allows their properties to be deduced from those of CSMPs, which are easier to grasp. Stability is studied within a very general framework. In the other chapters, the results become more accurate as the assumptions become more precise. Generalized Chapman-Kolmogorov equations lead to numerical schemes. The last chapter is an opening on processes for which the deterministic flow of the PDMP is replaced with a Markov process.
Marked point processes play a key role throughout this book.
Author(s): Christiane Cocozza-Thivent
Series: Probability Theory and Stochastic Modelling 100
Edition: 1
Publisher: Springer Nature Switzerland AG
Year: 2021
Language: English
Pages: 252
Tags: Renewal Processes, Hitting Time, Point Processes, Kolmogorov Equations, Switching 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
