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Coles A One-Semester Course on Markov Processes: Foundations For Probabilistic Modeling at Village Green Shopping Centre in Vernon, BC
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The book begins with a fairly comprehensive description of the relatively simple theory of Markov chains. It describes various types of random walks with simple research and proof techniques based on algebra and matrix calculus, enriched by Poly's theorem, the voting theorem, descriptions of various games of chance (including the problem of player bankruptcy), and numerous application examples.Then we move on to continuous-time chains, particularly the simple, conditional, and complex Poisson processes, the pure birth process, and the birth-death process. We describe the possibility of a population explosion, the problem of population extinction, and the application of the complex Poisson process in modeling the risk function for an insurance company. We then move on to a description of continuous-time Markov processes with a state space that is an interval (possibly unbounded) contained within a line. The prime example of such a process is Brownian motion. We describe and prove the properties of this process: the Reflection Principle, the continuity and non-differentiability of trajectories, the Donsker Invariance Principle, and the Laws of the Iterated Logarithm. This process is widely used in stochastic modeling.In the final chapter, we discuss the general theory of Markov processes, with particular emphasis on heavy-tailed processes. This is important from an application perspective, as we increasingly encounter phenomena whose description cannot be based on variables of finite variance. A new feature here is the inclusion of examples of Markov processes defined by transition probabilities given by generalized convolutions.
The book begins with a fairly comprehensive description of the relatively simple theory of Markov chains. It describes various types of random walks with simple research and proof techniques based on algebra and matrix calculus, enriched by Poly's theorem, the voting theorem, descriptions of various games of chance (including the problem of player bankruptcy), and numerous application examples.Then we move on to continuous-time chains, particularly the simple, conditional, and complex Poisson processes, the pure birth process, and the birth-death process. We describe the possibility of a population explosion, the problem of population extinction, and the application of the complex Poisson process in modeling the risk function for an insurance company. We then move on to a description of continuous-time Markov processes with a state space that is an interval (possibly unbounded) contained within a line. The prime example of such a process is Brownian motion. We describe and prove the properties of this process: the Reflection Principle, the continuity and non-differentiability of trajectories, the Donsker Invariance Principle, and the Laws of the Iterated Logarithm. This process is widely used in stochastic modeling.In the final chapter, we discuss the general theory of Markov processes, with particular emphasis on heavy-tailed processes. This is important from an application perspective, as we increasingly encounter phenomena whose description cannot be based on variables of finite variance. A new feature here is the inclusion of examples of Markov processes defined by transition probabilities given by generalized convolutions.
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