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2017 | Buch

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Stochastic Modeling

verfasst von: Nicolas Lanchier

Verlag: Springer International Publishing

Buchreihe : Universitext

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

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Über dieses Buch

Three coherent parts form the material covered in this text, portions of which have not been widely covered in traditional textbooks. In this coverage the reader is quickly introduced to several different topics enriched with 175 exercises which focus on real-world problems. Exercises range from the classics of probability theory to more exotic research-oriented problems based on numerical simulations. Intended for graduate students in mathematics and applied sciences, the text provides the tools and training needed to write and use programs for research purposes.

The first part of the text begins with a brief review of measure theory and revisits the main concepts of probability theory, from random variables to the standard limit theorems. The second part covers traditional material on stochastic processes, including martingales, discrete-time Markov chains, Poisson processes, and continuous-time Markov chains. The theory developed is illustrated by a variety of examples surrounding applications such as the gambler’s ruin chain, branching processes, symmetric random walks, and queueing systems. The third, more research-oriented part of the text, discusses special stochastic processes of interest in physics, biology, and sociology. Additional emphasis is placed on minimal models that have been used historically to develop new mathematical techniques in the field of stochastic processes: the logistic growth process, the Wright –Fisher model, Kingman’s coalescent, percolation models, the contact process, and the voter model. Further treatment of the material explains how these special processes are connected to each other from a modeling perspective as well as their simulation capabilities in C and Matlab™.

Inhaltsverzeichnis

Frontmatter

Probability theory

Frontmatter

Chapter 1. Basics of measure and probability theory

Abstract

The first use of mathematics to solve probability problems goes back to 1654 with the works of Fermat and Pascal. Their joint effort was motivated by questions raised by Antoine Gombaud, Chevalier de Méré, who was interested in betting strategies in the context of dice games. One of his main questions was: Is the probability of getting at least one double six when rolling two fair dice 24 times larger or smaller than one-half? A few years later, Huygens wrote the first book in probability theory [46] while, since the beginning of the 18th century, a number of mathematicians, including Bernoulli, de Moivre, Laplace, Poisson and Chebyshev, have made major discoveries in this field. But the most important turn in the history of probability since Fermat and Pascal is certainly the publication of Kolmogorov’s monograph [56] in 1933 which defines the axiomatic foundations of probability theory and marks the beginning of modern probability. For an English translation, see [58]. Though this point of view might be simplistic, the main idea of his work was to redefine probability concepts from their analog in measure theory.

Nicolas Lanchier

Chapter 2. Distribution and conditional expectation

Abstract

Following the same spirit as in the previous chapter, we continue to explore the implications of applying the results from measure theory to the field of probability theory. This chapter introduces in particular concepts that will be key later to studying various stochastic processes.

Nicolas Lanchier

Chapter 3. Limit theorems

Abstract

In this chapter, we move one more step toward the world of stochastic processes by considering sequences of random variables, the convergence of these sequences, and limit theorems in the special case when these random variables are independent and identically distributed.

Nicolas Lanchier

Stochastic processes

Frontmatter

Chapter 4. Stochastic processes: general definition

Abstract

This short chapter introduces the main topic of this textbook, namely, stochastic processes. As was true for the law of large numbers and the central limit theorem, stochastic processes are collections of random variables defined on the same probability space. These random variables, however, are not usually assumed to be independent. Instead, they are connected by some dependency relationships that can typically be expressed using conditional expectation or probability. In this chapter, we give the general definition of a stochastic process and define martingales and Markov chains along with a couple of examples.

Nicolas Lanchier

Chapter 5. Martingales

Abstract

Martingales are extensively used in physics, biology, sociology and economics, among other fields. In particular, we will later use martingales in a biological context as models of fair competition involving species that have the same fitness. However, to explain the theory, it is more convenient to think of a martingale as the process that keeps track of the fortune of a gambler playing a fair game. Similarly, we think of submartingales and supermartingales as the processes that, respectively, keep track of the fortune of a gambler playing a favorable and an unfavorable game.

Nicolas Lanchier

Chapter 6. Branching processes

Abstract

Branching processes, also called Galton–Watson processes after the name of their inventors (Francis Galton and Henry William Walton), are some of the simplest stochastic processes describing single-species dynamics. Such processes ignore the presence of a spatial structure but include general offspring distributions. The first reference to these processes is [39].

The first section gives a precise description of the model. The main assumption is that the number of offspring produced by each individual is independent and identically distributed across individuals and generations, which implies that the number of individuals evolves according to a discrete-time Markov chain.

The second section exhibits some useful connections with martingales suggesting that the process experiences a phase transition when the expected value μ of the offspring distribution is equal to one. From these connections, we prove exponentially fast extinction when μ < 1 and, more interestingly, almost sure extinction in the critical case μ = 1 using the martingale convergence theorem.

Nicolas Lanchier

Chapter 7. Discrete-time Markov chains

Abstract

The first mathematical results for discrete-time Markov chains with a finite state space were generated by Andrey Markov [71] whose motivation was to extend the law of large numbers to sequences of nonindependent random variables. The first important results in the more difficult context of countably infinite Markov chains were established 30 years later by Andrey Kolmogorov

Nicolas Lanchier

Chapter 8. Symmetric simple random walks

Abstract

Random walks denote a general class of stochastic processes for which the definition significantly varies across the literature. Since the ultimate target of this textbook is spatial stochastic processes, the random walks we are interested in are the symmetric simple random walks on graphs as defined in Example 7.2. As previously mentioned, the random walk on the integers can be generated from the consecutive flips of a fair coin and is therefore a natural mathematical object that was already studied in the 17th century by Bernoulli and de Moivre. However, it was only as recently as 1905 that the term random walk was introduced in [81].

Nicolas Lanchier

Chapter 9. Poisson point and Poisson processes

Abstract

This chapter starts with a general description of Poisson point processes. These processes are defined from four natural axioms describing the spatial distribution of so-called Poisson points scattered homogeneously in a random manner across the d-dimensional Euclidean space. They are used for instance as a model for the distribution of stars and galaxies in three dimensions or the distribution of plants and trees in two dimensions. These processes are characterized by a single parameter called the intensity that can be seen as a measure of the density of Poisson points. It is proved that the number of Poisson points in every bounded Borel set is Poisson distributed with parameter proportional to the Lebesgue measure of this set, which explains the name of these processes.

Nicolas Lanchier

Chapter 10. Continuous-time Markov chains

Abstract

As in discrete time, continuous-time Markov chains are stochastic processes in which the future depends on the past only through the present, or equivalently, given the present, past, and future are independent. Since there is no next time when time is continuous, the process is now characterized by transition rates instead of transition probabilities. Also, the analog of the transition matrix in continuous time is the so-called intensity matrix.

Nicolas Lanchier

Special models

Frontmatter

Chapter 11. Logistic growth process

Abstract

Starting from this chapter, we focus on special stochastic processes that are important in physics, biology, and sociology. The models are not chosen at random. They form a coherent structure and tell a story. All the models are minimal in term of their number of parameters and can be classified as either invasion models or as competition models. They mostly differ in their level of realism from the point of view of the inclusion or not of a temporal structure and/or an explicit spatial structure, but are otherwise closely related.

Nicolas Lanchier

Chapter 12. Wright–Fisher and Moran models

Abstract

The first references introducing the Wright–Fisher model and the Moran model discussed in this chapter are [97] and [73], respectively. Contrary to the logistic growth process, these stochastic processes are competition models. In particular, the relevant mathematical tools to study these models are quite different. However, all three models are closely related from the point of view of the level of complexity they take into account, in that all three stochastic models are spatially implicit dynamical systems. In particular, as in the logistic growth process, the models can be described starting from a set of sites that we think of as spatial locations and the dynamics is dictated by global interactions. Rather than being either empty or occupied, these sites are now occupied by one of two possible types of individuals that we call type 0 and type 1.

Nicolas Lanchier

Chapter 13. Percolation models

Abstract

We now make a U-turn going from dynamical spatially implicit processes in the last two chapters to static spatially explicit percolation models in this chapter. In particular, the actual spatial distance between individuals is no longer ignored and is in fact the key to constructing the models.

Nicolas Lanchier

Chapter 14. Interacting particle systems

Abstract

Most mathematical models introduced in the life and social sciences literature that describe inherently spatial phenomena of interacting populations consist of systems of ordinary differential equations or stochastic counterparts assuming global interactions such as the logistic growth process and the Moran model. These models, however, leave out any spatial structure, while past research has identified the spatial component as an important factor in how communities are shaped, and spatial models can result in predictions that differ from nonspatial models.

Nicolas Lanchier

Chapter 15. The contact process

Abstract

In this chapter, we prove that, similarly to the other invasion models introduced in this textbook, the contact process exhibits a phase transition: There is a critical birth parameter above which the process starting with one individual survives with positive probability but below which it goes extinct eventually with probability one.

Nicolas Lanchier

Chapter 16. The voter model

Abstract

As in the Wright–Fisher and Moran models, the two configurations in which all the individuals have the same type or share the same opinion are absorbing states for the voter model. However, at least starting with infinitely many individuals of each type, the time to fixation to one of these absorbing states is almost surely infinite, which allows for the possibility of interesting transient or long-term behaviors. In fact, even though it does not depend on any parameter, the voter model exhibits very rich dynamics with various behaviors depending on the spatial dimension. This contrasts with the contact process that has behavior similar in any dimension.

Nicolas Lanchier

Chapter 17. Numerical simulations in C and Matlab

Abstract

This chapter is devoted to numerical simulations, focusing on the ten most popular stochastic models presented in this textbook. Following the chronology of the textbook, we start with the three classical models: the gambler’s ruin chain, the branching process, and the simple birth and death process. Then, we look at the spatially implicit stochastic models and finally at the spatially explicit ones: percolation models and interacting particle systems.

Nicolas Lanchier

Backmatter

Titel: Stochastic Modeling
verfasst von: Nicolas Lanchier
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-50038-6
Print ISBN: 978-3-319-50037-9
DOI: https://doi.org/10.1007/978-3-319-50038-6