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A Course in Applied Stochastic Processes By A. Goswami, B. V. Rao
English | PDF | 2006 | 226 Pages | ISBN : 8185931690 | 20.86 MB
In terms of pre-requisites, the book does not demand much. Although an exposure to elementary Probability Theory would help, it is by no means es- sential. Chapter 0 is meant to supply the necessary background in Probability. The only real pre-requisite is an exposure to undergraduate Linear Algebra and Calculus, and of course, the necessary motivation. We would like to highlight one interesting feature of the present book. A substantial amount of basic Martingale Theory and Theory of Diffusion Processes have been presented in this book and we have been able to do this without resorting to any measure- theoretic framework. We have not just 'conveyed the idea without rigour' - in most cases, we have given completely rigorous proofs. Here is a brief summary of what the reader is in for. Chapter 0 gives a brief introduction to the necessary background in Probability. It almost starts from scratch and takes the reader through to Martingale Theory, Markov Chains, and a little of Diffusion Processes. Chapter 1 discusses the elementary theory of Discrete Time one-dimensional Branching Processes a la Galton-Watson. Much of the material covered here is available in the books of Harris, Feller and Karlin, as referred to at the end of the chapter. Chapter 2 is preparatory to Chapter 3. It contains the necessary introduction to Mathematical Genetics and the relevent Probability Models. An important topic here is the Hardy- Wienberg Laws. Some of the basic concepts in Population Genetics like SeHing, Sibmating, Gene Identity are elaborately discussed in this chapter and some related mathematical analyses are presented. Chapter 3 contains one of the most important and interesting topics of this book. We mainly discuss various Markov Chain models in Population Genetics. Of course the classical Wright- Fisher Model is the starting point. But many other models, not easily available in standard texts, are discussed at length. Towards the end, some nice Diffusion approximations to these Markov Chains are also discussed. Chapter 4 discusses Stochastic models in the spread of Epidemics. Some non-Stochastic models are discussed first to create motivation for their Stochastic counterparts. It is only in this chapter that we use some Continuous Time Markov Chain models. The most important topic in this chapter - at least in our opinion - consists of the Threshold Theorems. These theorems are believed to depict, in a nut shell, the temporal spread of Epidemics. At the end of each chapter, we have given a list of references as suggested supplementary readings. This is primarily aimed at readers who might take an active interest in pursuing further studies in these areas. Each of the chapters contain a fairly large number of exercises and except in Chapter 0, these exercises are given at the end of the chapters. In Chapter 0, the exercises are spread out over various sections. Sometimes the exercises are accompanied by enough hints, whenever deemed necessary. It is advisable for a serious reader to attempt the exercises as far as possible. Many of the exercises are taken from the various sources referred to throughout the book. For the sake of brevity we refrain from specific acknowledgements. The index at the end should be of help for a quick reference to important concepts and definitions. With a few exceptions, items are listed only according to their first appearance.
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