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2006 | Buch

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Point Process Theory and Applications

Marked Point and Piecewise Deterministic Processes

verfasst von: Martin Jacobsen

Verlag: Birkhäuser Boston

Buchreihe : Probability and its Applications

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The book aims at presenting a detailed and mathematically rigorous exposition of the theory and applications of a class of point processes and piecewise deterministic p- cesses. The framework is suf?ciently general to unify the treatment of several classes of stochastic phenomena: point processes, Markov chains and other Markov processes in continuous time, semi-Markov processes, queueing and storage models, and li- lihood processes. There are applications to ?nance, insurance and risk, population models, survival analysis, and congestion models. A major aim has been to show the versatility of piecewise deterministic Markov processes for applications and to show how they may also become useful in areas where thus far they have not been much in evidence. Originally the plan was to develop a graduate text on marked point processes - dexed by time which would focus on probabilistic structure and be essentially se- contained. However, it soon became apparent that the discussion should naturally include a traditional class of continuous time stochastic processes constructed from certain marked point processes. This class consists of ‘piecewise deterministic p- cesses’; that is, processes with ?nitely many jumps on ?nite time intervals which, roughly speaking, develop deterministically between the random jump times. The - position starts with the point process theory and then uses this to treat the piecewise deterministic processes.

Inhaltsverzeichnis

Frontmatter

Theory

Frontmatter

1. Introduction

2. Simple and Marked Point Processes

Abstracts

This chapter contains the definitions of simple and marked point processes (SPPs and MPPs, respectively) recording the random occurrences over time of random events and shows how to identify the point processes with counting processes (CPs) and random counting measures (RCMs). The canonical spaces K, and K_E of sequences of time-points of events and their marks are introduced together with the space W of counting process paths and the space M of discrete counting measures, counting timepoints and marks. It is demonstrated how SPPs and MPPs may be viewed as random variables with values in the sequence spaces K and K_E respectively, while CPs are W-valued and RCMs are M-valued random variables. The definitions and notation given in the chapter are fundamental for everything that follows.

3. Construction of SPPs and MPPs

Abstracts

This chapter contains the fundamental construction of canonical point processes (i.e., probabilities on the sequence spaces K and K_E), canonical counting processes (probabilities on the space W) and canonical random counting measures (probabilities on the space M). The construction is performed using successive regular conditional distributions. The chapter also has a section on how to view certain types of continuous time stochastic processes as MPPs, an approach examined in detail in Chapters 6 and 7. Finally, a number of basic examples that will reappear at various points in the text are presented.

4. Compensators and Martingales

Abstracts

This chapter contains the basic theory of probability measures on the space W of counting process paths and the space M of discrete counting measures. Compensators and compensating measures are defined using hazard measures, and by exploring the structure of adapted and predictable processes on W and M (a structure that is discussed in detail), it is shown how the various forms of compensators characterize probabilities on canonical spaces. Also, these probabilities are described by the structure of basic martingales. Stochastic integrals (all elementary) are discussed and the martingale representation theorem is established. It is shown (It ô’s formula) how processes adapted to the filtration generated by an RCM may be decomposed into a predictable process and a local martingale. Finally, there is a discussion of compensators and compensating measures for counting processes and random counting measures defined on arbitrary filtered probability spaces.

Much of the material presented in this chapter is essential for what follows and complete proofs are given for the main results. Some of these proofs are quite long and technical and rely on techniques familiar only to readers well acquainted with measure theory and integration. It should be reasonably safe to omit reading the proofs of e.g., the following results: Theorem 4.1.1 (the proof of the last assertion), Proposition 4.2.1, Theorem 4.3.2, Proposition 4.3.5, Proposition 4.5.1 (although it is useful to understand why (4.65) implies that f is constant) and Theorem 4.6.1.

5. Likelihood Processes

Abstracts

In this chapter the structure of the likelihood process (process of Radon-Nikodym derivatives) is derived when considering two probability measures on the canonical spaces W and M, assuming that one of the probabilities is locally absolutely continuous with respect to the other. Also, it is shown how to change measure using martingales or just local martingales.

It is important to understand the contents of Theorem 5.1.1, but not necessary to read the proof in detail. Also, Section 5.2 may be omitted on a first reading.

6. Independence

Abstracts

In the first part of this chapter it is shown how stochastic independence between finitely many marked point processes may be characterized in terms of the structure of the compensating measures. The last part of the chapter is devoted to the study of CPs and RCMs with independent increments (stationary or not), which are characterized as those having deterministic compensating measures, and of other stochastic processes with independent increments, with, in particular, a discussion of compound Poisson processes and more general Lévy processes.

The discussion of general Lévy processes at the end of the chapter is intended for information only and is not required reading. One may also omit reading the technical parts of the proof of Theorem 6.2.1(i) and the proof of Proposition 6.2.3(i).

7. Piecewise Deterministic Markov Processes

Abstracts

This chapter contains the basic theory for piecewise deterministic Markov processes, whether homogeneous or not, based exclusively on the theory of marked point processes from the previous chapters and presented through the device of viewing a PDMP as a process adapted to the filtration generated by an RCM. The strong Markov property is established, various versions of Itô’s formula for PDMPs are given, the socalled full infinitesimal generator for a homogeneous PDMP is discussed, invariant measures are treated, and the chapter concludes with a section on likelihood processes for PDMPs.

At a first reading one may omit Sections 7.5, 7.7, 7.8 and the last part of Section 7.9 (dealing with multiplicative functionals).

Applications

Frontmatter

8. The Basic Models from Survival Analysis

Abstracts

This chapter deals with the problem of estimating an unknown survival distribution based on observation of an iid sample subjected to right-censoring. Also treated is the estimation problem for the Cox regression model.

9. Branching, Ruin, Soccer

Abstracts

In this chapter three quite different models are presented: a branching process for the evolution of a population with age-dependent birth and death intensities; a classical model from risk theory with a discussion of the problem of calculating the probability of ruin and the distribution of the time to ruin; a model for how a soccer game develops over time, using a simple multiplicative Poisson model as starting point.

10. A Model from Finance

Abstracts

A simple specific PDMP model is set up to describe the price of a finite number of risky assets. For this model we discuss self-financing trading strategies, arbitrage and the concept of equivalent martingale measures, and the fair pricing of contingent claims.

11. Examples of Queueing Models

Abstracts

In the first section the classical GI/G/1 model is treated with the emphasis of finding when there is equilibrium (stationarity). For the simplest case, the M/M/1 model, stationarity is discussed in detail, not just for the length of the queue but also involving the time since most the recent arrival and the time since the present service started. The second section deals with the description of some PDMP models of queueing networks that are not just (homogeneous) Markov chains.

Appendices

Frontmatter

A. Differentiation of Cadlag Functions

B. Filtrations, Processes, Martingales

Backmatter

Titel: Point Process Theory and Applications
verfasst von: Martin Jacobsen
Verlag: Birkhäuser Boston
Electronic ISBN: 978-0-8176-4463-5
Print ISBN: 978-0-8176-4215-0
DOI: https://doi.org/10.1007/0-8176-4463-6