Front Matter

Published:2021
Crispin Gardiner, "Front Matter", Elements of Stochastic Methods, Crispin Gardiner
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This book provides a compact practical exposition of the stochastic methods and their applications for a wide range of fields, including population dynamics, infectious diseases, financial modeling, hydrology, physics, quantum optics, chemistry, and many others. The aim is to give practitioners an idea of how to formulate their current problem stochastically, and how to get an idea of what the predictions of such modeling are.
Elements of Stochastic Methods features:
A treatment based on Markov processes, including an outline of their theoretical foundations, and a description of basic algorithms for simulating Markov processes
A straightforward and practical formulation of stochastic differential equations and Ito calculus, including simulation methods
Heavytailed processes, Mandelbrot's fractional Brownian motion, as well as the more traditional diffusion and jump processes
This book is an essential resource for researchers, professionals, and graduate students in physics and in any field where random processes are relevant.
Preface
When AIP Publishing invited me to contribute a book to their new eBook series, I knew at once that I should write a brief book on stochastic methods. The fields in which stochastic methods are relevant are very diverse—population dynamics, infectious diseases, genetic diversity, financial modeling, hydrology, psychology, physics, chemistry, and many others. The compact format of the AIP series suited my belief that a short book would be appreciated, something in which the main aim is to give practitioners an idea of how to formulate their current problem stochastically, and how to get an idea of what are the predictions of such modeling. And all of this should be contained in one compact volume.
I expect typical readers of this book to be people with a need to model practical problems in their own field. They are not mathematicians, and while they are interested in why things are true, they have little interest in rigorous proof or abstract mathematical formulations. They want answers.
The result is this book. It should be viewed as both a distillation of my earlier book Stochastic Methods,^{1} keeping only the essentials, and as an extension, widening the scope to include examples from many fields. Mostly, readers who want more detail or rigor will find explicit references to the appropriate sections of Stochastic Methods. As much as possible, notation and terminology are the same in both books, so that it should be easy to use both books simultaneously without undue strain.
To achieve compactness, I have had to be quite selective. There are no rigorous mathematical proofs, although I do try to explain why results are true. But I do cover most of the basic techniques used in practice. I restrict the coverage to Markov processes, and processes expressible in terms of Markov processes. This is in practice not a serious restriction.
Since it is now recognized that they play a major role in many natural phenomena, I have included heavytailed distributions and processes, not as an afterthought, but as an integral part of the development of the book. This means that my discussion of probability distributions distinguishes between:
Those which have a welldefined variance, for which the central limit theorem is therefore valid,
Heavytailed distributions, which do not have a welldefined variance, and for which the central limit theorem is not valid. For these, the central limit theorem is replaced by the concept of stable distributions as the appropriate limit distributions of sums of independent variables.
As well, I have also included Mandelbrot's fractional Brownian motion. This is not a Markov process, but is Gaussian, and easily expressed in terms of Markov processes. Its significance in the understanding of the longtime correlations that appear to be an essential part of many natural phenomena is unchallenged, even though it does not provide all of the answers.
The influence of my late colleagues, Dan Walls (1942–1999) and Nico van Kampen (1921–2013) on this book and much of my other work has been profound, and I would like to dedicate this book to their memory.
C. W. Gardiner, Stochastic Methods (Springer, Heidelberg, Berlin; 1st ed. 1985, 2nd ed. 1997, 3rd ed. 2004, 4th ed. 2009)—I want to thank Springer Verlag for permission to include somematerial from Stochastic Methods in this book.