Stochastic Analysis of Biochemical Systems

verfasst von: David F. Anderson, Thomas G. Kurtz

Verlag: Springer International Publishing

Buchreihe : Mathematical Biosciences Institute Lecture Series

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

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Über dieses Buch

This book focuses on counting processes and continuous-time Markov chains motivated by examples and applications drawn from chemical networks in systems biology. The book should serve well as a supplement for courses in probability and stochastic processes. While the material is presented in a manner most suitable for students who have studied stochastic processes up to and including martingales in continuous time, much of the necessary background material is summarized in the Appendix. Students and Researchers with a solid understanding of calculus, differential equations and elementary probability and who are well-motivated by the applications will find this book of interest.

David F. Anderson is Associate Professor in the Department of Mathematics at the University of Wisconsin and Thomas G. Kurtz is Emeritus Professor in the Departments of Mathematics and Statistics at that university. Their research is focused on probability and stochastic processes with applications in biology and other areas of science and technology.

These notes are based in part on lectures given by Professor Anderson at the University of Wisconsin – Madison and by Professor Kurtz at Goethe University Frankfurt.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Infinitesimal specification of continuous time Markov chains

Abstract

The basic building blocks of our models are counting processes, that is, nonnegative-integer-valued processes that are constant except for jumps of + 1. In this chapter, we introduce the notion of an intensity for a counting process and show how a counting process model can be specified by specifying a functional form for its intensity. The counting process corresponding to the intensity can be determined either as the solution of a stochastic equation or as the solution of a martingale problem. By writing a lattice-valued (e.g., \(\mathbb{Z}^{d}\)-valued) Markov chain in terms of counting processes that count the number of jumps of each of a countable number of types, we can specify the chain by specifying the intensities of the counting processes as functions of the state of the Markov chain.