The relatively young theory of structured dependence between stochastic processes has many real-life applications in areas including finance, insurance, seismology, neuroscience, and genetics. With this monograph, the first to be devoted to the modeling of structured dependence between random processes, the authors not only meet the demand for a solid theoretical account but also develop a stochastic processes counterpart of the classical copula theory that exists for finite-dimensional random variables. Presenting both the technical aspects and the applications of the theory, this is a valuable reference for researchers and practitioners in the field, as well as for graduate students in pure and applied mathematics programs. Numerous theoretical examples are included, alongside examples of both current and potential applications, aimed at helping those who need to model structured dependence between dynamic random phenomena.
Author(s): Tomasz R. Bielecki, Jacek Jakubowski, Mariusz Niewȩgłowski
Series: Encyclopedia of Mathematics and its Applications
Publisher: Cambridge University Press
Year: 2020
Language: English
Pages: 278
City: Cambridge
Copyright
Contents
1 Introduction
PART I: CONSISTENCIES
2 Strong Markov Consistency of Multivariate Markov Families and Processes
2.1 Definition of Strong Markov Consistency and an Introduction to its Properties
2.1.1 Strong Markov Consistency of Multivariate Markov Families
2.1.2 Strong Markov Consistency of Multivariate Markov Processes
2.2 Sufficient Conditions for Strong Markov Consistency
2.2.1 Sufficient Conditions for Strong Markov Consistency of Rn-Feller–Markov Evolution Families
2.2.2 Strong Markov Consistency of Feller–Markov Families
3 Consistency of Finite Multivariate Markov Chains
3.1 Definition and Characterization of Strong Markov Consistency
3.1.1 Necessary and Sufficient Conditions for Strong Markov Consistency
3.1.2 Operator Interpretation of Condition (M)
3.2 Definition and Characterization of Weak Markov Consistency
3.2.1 Necessary and Sufficient Conditions for Weak Markov Consistency
3.2.2 Operator Interpretation of Necessary Conditions for Weak Markov Consistency
3.3 When does Weak Markov Consistency Imply Strong Markov Consistency?
4 Consistency of Finite Multivariate Conditional Markov Chains
4.1 Strong Markov Consistency of Conditional Markov Chains
4.1.1 Necessary and Sufficient Conditions for Strong Markov Consistency of a Multivariate (F,FX )-CDMC
4.1.2 Algebraic Conditions for Strong Markov Consistency
4.2 Weak Markovian Consistency of Conditional Markov Chains
4.2.1 Necessary and Sufficient Conditions for Weak Markov Consistency of a Multivariate (F,FX )-CDMC
4.2.2 When Does Weak Markov Consistency Imply Strong Markov Consistency?
5 Consistency of Multivariate Special Semimartingales
5.1 Semimartingale Consistency and Semi-Strong Semimartingale Consistency
5.1.1 Computation of the Fj-Characteristics of X j
5.2 Strong Semimartingale Consistency
5.2.1 Examples of Strongly Semimartingale-Consistent Multivariate Special Semimartingales
5.2.2 Examples of Multivariate Special Semimartingales That Are Not Strongly Semimartingale Consistent
PART II: STRUCTURES
6 Strong Markov Family Structures
6.1 Strong Markov Structures for Nice Feller–Markov Families and Nice Feller–Markov Processes
6.2 Building Strong Markov Structures
6.3 Examples
7 Markov Chain Structures
7.1 Strong Markov Chain Structures for Finite Markov Chains
7.2 Weak Markov Chain Structures for Finite Markov Chains
7.3 Examples
8 Conditional Markov Chain Structures
8.1 Strong CMC Structures
8.2 Weak CMC Structures
9 Special Semimartingale Structures
9.1 Semimartingale Structures: Definition
9.2 Semimartingale Structures: Examples
PART III: FURTHER DEVELOPMENTS
10 Archimedean Survival Processes, Markov Consistency, ASP Structures
10.1 Archimedean Survival Processes and ASP Structures
10.1.1 Preliminaries
10.1.2 Archimedean Survival Processes
10.1.3 ASP Structures
10.2 Weak and Strong Markov Consistency for ASPs
11 Generalized Multivariate Hawkes Processes
11.1 Generalized G-Hawkes Process
11.2 Generalized Multivariate Hawkes Processes
11.2.1 Definition of a Generalized Multivariate Hawkes Process
11.2.2 Existence of Generalized Multivariate Hawkes Process: The Canonical Probability Space Approach
11.2.3 Examples
11.2.4 Construction of a Generalized Multivariate Hawkes Process via Poisson Thinning
11.3 Markovian Aspects of a Generalized Multivariate Hawkes Process
11.4 Hawkes Structures
11.5 Hawkes Consistency of Generalized Multivariate Hawkes Processes
PART IV: APPLICATIONS OF STOCHASTIC STRUCTURES
12 Applications of Stochastic Structures
12.1 Markov and Conditionally Markov Structures
12.1.1 Portfolio Credit Risk Valuation and the Hedging and Valuation of Step-Up Bonds
12.1.2 Counterparty Risk Valuation and Hedging
12.1.3 Premium Evaluation for Unemployment Insurance Products
12.1.4 Multiple Ion Channels
12.1.5 Dynamic Risk Management in Central Clearing Parties
12.2 Special Semimartingale Structures
12.2.1 Poisson Structure and Ruin Probabilities
12.3 Archimedean Survival Process (ASP) Structures
12.3.1 Decision Functionals
12.3.2 Explicit Formulae for Decision Functionals for ASPs
12.3.3 Application of ASP Structures in Maintenance Management
12.4 Hawkes Structures
12.4.1 Seismology
12.4.2 Criminology and Terrorism
12.4.3 Neurology, Neuroscience, and DNA Sequencing
12.4.4 Finance
APPENDICES
Appendix A Stochastic Analysis: Selected Concepts and Results Used in this Book
Appendix B Markov Processes and Markov Families
Appendix C Finite Markov Chains: Auxiliary Technical Framework
Appendix D Crash Course on Conditional Markov Chains and on Doubly Stochastic Markov Chains
Appendix E Evolution Systems and Semigroups of Linear Operators
Appendix F Martingale Problem: Some New Results Needed in this Book
Appendix G Function Spaces and Pseudo-Differential Operators
References
Notation Index
Subject Index
