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Continuous-time Markov chains and applications. A singular perturbation approach. (English) Zbl 0896.60039
Applications of Mathematics. 37. New York, NY: Springer. xv, 349 p. (1998).
The main matter of the book are stepped Markov processes \((\alpha^\varepsilon (t),\;t\geq 0)\) with a finite state space \({\mathcal M}= \{1, \dots, m\}\) and their applications to Markov decision problems. The authors investigate properties of the process connected with its generator \(Q^\varepsilon (t)\) \((\varepsilon >0)\), where \(t\) is a time parameter showing the relation to a one-dimensional probability distribution of the process: \[ {\partial p^\varepsilon(t) \over \partial t}= p^\varepsilon (t)Q^\varepsilon (t), \quad p^\varepsilon (0) =p^0 \quad \left( p^0_i\geq 0,\sum^m_{i=1} p^0_i= 1\right), \] \(\varepsilon\) is a small parameter defining a family of the processes with a singularity. Such a singularity arises in the case \( Q^\varepsilon(t)={1\over\varepsilon}\widetilde Q(t)+ \widehat Q(t)\), where \(\widetilde Q(t)\) and \(\widehat Q(t)\) are generators themselves. The first interesting property of the family is an asymptotic expansion of the probability distribution containing a regular part and a singular one: \[ p^\varepsilon (t)= \sum^n_{i=1} \varepsilon^i \varphi^{(i)} (t)+ \sum^n_{i=1} \varepsilon^i \psi^{(i)} (t) +o(\varepsilon^n), \] where \(\varphi^{(i)}\), \(\psi^{(i)}\) are some functions which can be found in a constructive way and \(o(\varepsilon^n)\) can be estimated. The case \(\widetilde Q(t)= \text{diag} (\widetilde Q^1(t), \dots, \widetilde Q^l (t))\) is most meaningful, where \(\widetilde Q^i (t)\) is a generator of an irreducible Markov process on the state space \({\mathcal M}_i\) \(({\mathcal M}_i \cap {\mathcal M}_j =\emptyset\) \((i\neq j)\), \({\mathcal M} =\bigcup^l_{i=1} {\mathcal M}_i)\). This expansion can take into account sets of absorbing and transient states. It is surprising that there exist variants of the law of large numbers and of the central limit theorem for these non-homogeneous in time processes. They correspond to a difference between a sojourn time at a state \(i\in {\mathcal M}\) and an appropriate one-dimensional probability. In the above block structure of the matrix \(\widetilde Q(t)\) the problem of interaction of quick and slow parts is investigated. In this interaction an aggregated process \(\overline \alpha^\varepsilon (\cdot)\) \((\overline \alpha^\varepsilon (t)=k\) when \(\alpha^\varepsilon (t) \in {\mathcal M}_k)\) plays the main role. It is interesting that \(\overline \alpha^\varepsilon (\cdot)\) tends weakly to a limit process \(\overline \alpha (\cdot)\) in the metric space \({\mathcal D} [0,T]\) \((T>0)\). On the other hand the set of the processes \(\alpha^\varepsilon (\cdot)\) \((\varepsilon >0)\) is not tight in this space.
As generalizations of the finite-dimensional case the following families are discussed: a countable case when \(\widetilde Q(t)= \text{diag} (\widetilde Q^1(t),\widetilde Q^2(t),\dots)\) (irreducible block structure) and the case of a singularly perturbed diffusion process. The main applications of the authors’ results on the singular perturbation relate to a Markov model of a complex control system. In this system it is natural to distinguish slow and quick parts which demand a hierarchical control. The main idea in this application is to construct a Hamiltonian in terms of a singularly perturbed process and to solve the corresponding Hamilton-Jacoby-Bellman equation. The authors propose some numerical methods for that. The book is convenient for a reader to use due to its clear style and accurate preliminaries, notes and appendices, explaining the plan, the history and technical details.

60J27 Continuous-time Markov processes on discrete state spaces
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
93E20 Optimal stochastic control
34E05 Asymptotic expansions of solutions to ordinary differential equations