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Nicolas Lanchier - Stochastic Modeling

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Stochastic Modeling
Author:
Nicolas Lanchier
Publisher:
Springer
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2017
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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 gamblers 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, Kingmans 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.

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Part I
Probability theory

Springer International Publishing AG 2017

Nicolas Lanchier Stochastic Modeling Universitext 10.1007/978-3-319-50038-6_1

1. Basics of measure and probability theory

Nicolas Lanchier 1

(1)

School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, USA

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 Mr, 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 []. 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.

Following Kolmogorovs work, this first chapter gives a brief overview of the most basic concepts, definitions, and results of measure theory along with their connections with probability. Since the main topics of this textbook are probability theory and stochastic processes, most of the proofs are omitted and we refer the reader to Rudin [].

We start with a simple probability question showing the limitations of the Riemann integral [] and the reason why measure theory is essential.

Question What is the probability that a real number chosen uniformly at random in the unit interval is a rational number?

Applying concepts seen at the undergraduate level in order to compute this probability leads to the integral of a function that is not Riemann integrable, and therefore not even defined. In particular, to be able to answer this simple question, one needs to define a more general integral.

Motivated by the limitations of the Riemann integral, the second section introduces the first main concepts of measure theory: -algebras, measurable spaces, measurable functions, positive measures and finally the abstract integral with a construction due to Lebesgue [] for detailed solutions to a number of interesting exercises and many historical notes.

In the last section, we turn our attention to some of the earliest and most important results in measure theory. Having a sequence of random variables with a pointwise limit, is the expected value of the limit equal to the limit of the expected values? In other words, do the limit and the expected value commute? We will see that, at least in the following two contexts, the answer is yes.

Monotone convergence Limit and expected value commute when the sequence of random variables is monotone.

Dominated convergence Limit and expected value commute when the sequence is dominated by an integrable random variable.

We also state another useful result, Fubinis theorem [], which gives conditions under which two expected values or integrals commute. Counterexamples are also given showing that the assumptions of the previous theorems cannot be omitted.

1.1 Limitations of the Riemann integral

The Riemann integral was introduced by Bernhard Riemann in his qualification to become an instructor []. This short section shows the limitations of this integral through a simple example and the reason why measure theory is fundamental in probability theory. Consider the following problem: Find the probability that a number chosen uniformly at random in the unit interval is rational. To answer this question using tools from undergraduate probability, we let

where

means that the law of the random variable on the left-hand side is given by the - photo 2

means that the law of the random variable on the left-hand side is given by the right-hand side. Then, the probability to be found is

(1.1)

The function

is known as the Dirichlet function. To find its integral, recall that the Riemann sum of a function is defined from a tagged partition of the domain of the function as illustrated on the left-hand side of Figure ) does not exist. Using the framework of measure theory, however, we can properly define and compute this integral. This is done at the end of the next section.

Fig. 1.1

Partitioning the domain (construction of the Riemann integral) versus partitioning the range (construction of the Lebesgue/abstract integral).

1.2 Construction of the abstract integral

This section is devoted to the construction of the abstract integral, which was introduced by Henri Lebesgue in his Ph.D. dissertation [] in an attempt to generalize the Riemann integral and make it more flexible. Like the Riemann integral, the abstract integral is a linear operator defined on a set of functions, but it is also more powerful for the following two reasons.

The abstract integral is much more general than the Riemann integral because there is in fact one integral associated to each so-called positive measure.
The special case of the Lebesgue measure gives rise to an integral that extends the Riemann integral to a much larger set of functions. This set of functions includes in particular the Dirichlet function.

To construct the abstract integral, we first need a few definitions.

Definition 1.1.

Let be a set. From the point of view of probability theory, we think of this set as a sample space : set of the outcomes of an experiment.

A collection of subsets of is said to be a -algebra whenever
- for all , we have
- for each sequence , we have .
The pair is then called a measurable space .
Members of the -algebra are called measurable sets in measure theory and are interpreted as events in probability theory.

It follows from the definition that the -algebra also contains the empty set and is stable under countable intersections. More generally, any set obtained from elementary set operations involving countably many measurable sets is again measurable. In the context of probability theory, the -algebra or set of events represents the information available: the largest -algebra, which consists of all subsets of , means perfect information, whereas the smallest one, which reduces to the sample space and the empty set, means no information. For any collection

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