Poisson Processes as Particular Markov Processes
We first introduce some practical and theoretical issues of modeling by means of Markov processes. Point processes are introduced in order to model jump instants. The Poisson process is then characterized as a point process without memory. The rest of the
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Poisson Processes as Particular Markov Processes
Abstract We first introduce some practical and theoretical issues of modeling by means of Markov processes. Point processes are introduced in order to model jump instants. The Poisson process is then characterized as a point process without memory. The rest of the chapter consists in its rather detailed study, including various results concerning its simulation and approximation. This study is essential to understand the abstract constructions and the simulation methods for jump Markov processes developed in the following chapters.
4.1 Quick Introduction to Markov Processes Markovian modeling of randomly evolving systems raises several issues. After giving some examples, we discuss a reasonably simple mathematical framework in which results can actually be proved.
4.1.1 Some Issues in Markovian Modeling Many application fields feature random phenomena which evolve in continuous time, and in a continuous fashion between sudden state changes. The latter are called jumps, occur at jump instants, and may for instance be due to: • • • •
arrivals, ends of service, and transfers of customers in communication networks, contagion, recovery, and deaths of individuals in epidemiology models, births, deaths, mutations, and transfers of individuals in ecological models, interactions or reactions between particles in physical or chemical models.
The natural state space may be discrete, as is the case when counting customers at the various network resources, or infected individuals in an epidemic. The evolution is then necessarily constant between the jumps. The state space may also be continuous, as is the case when measuring phenotypes such as size or speed of individuals in ecology, or the energy in certain models of physics. The evolution may then be purely continuous, or mix jumps with continuous (possibly constant) stretches. C. Graham, D. Talay, Stochastic Simulation and Monte Carlo Methods, Stochastic Modelling and Applied Probability 68, DOI 10.1007/978-3-642-39363-1_4, © Springer-Verlag Berlin Heidelberg 2013
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Poisson Processes as Particular Markov Processes
The goal here will be to model random phenomena having a property of lack of memory of the past conditional on the present. A Markov process will be defined as a random evolution such that at any instant the law of the future evolution only depends on the past through the present state. The main case considered is when this law does not depend on the value of the present instant; else, the process will be said to be inhomogeneous (in time). An important element of the study of Markov process with jumps is their set of discontinuities. In this chapter, point processes will be introduced as a representation for a sequence of jump instants without accumulation point. This is followed by detailed study of Poisson processes, which are point processes having a certain property of lack of memory. This study will be fundamental in the sequel, for abstract comprehension and construction as well as for effecti
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