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Published in:
2015 | OriginalPaper | Chapter
1. Infinitesimal specification of continuous time Markov chains
Authors : David F. Anderson, Thomas G. Kurtz
Published in: Stochastic Analysis of Biochemical Systems
Publisher: Springer International Publishing
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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.