By G. George Yin, Qing Zhang

ISBN-10: 1461443458

ISBN-13: 9781461443452

ISBN-10: 1461443466

ISBN-13: 9781461443469

Prologue and Preliminaries: creation and evaluate- Mathematical preliminaries.- Markovian models.- Two-Time-Scale Markov Chains: Asymptotic Expansions of options for ahead Equations.- profession Measures: Asymptotic houses and Ramification.- Asymptotic Expansions of strategies for Backward Equations.- Applications:MDPs, Near-optimal Controls, Numerical equipment, and LQG with Switching: Markov choice Problems.- Stochastic regulate of Dynamical Systems.- Numerical tools for keep an eye on and Optimization.- Hybrid LQG Problems.- References.- Index

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**Additional info for Continuous-time Markov chains and applications : a two-time-scale approach**

**Example text**

5, which are used extensively in the chapters to follow. 6 reviews Gaussian and diﬀusion processes. 7 discusses switching diﬀusion processes. 8 closes the chapter with some postscript notes. 2 Martingales Let (Ω, F , P ) be a probability space. G. Yin and Q. 1007/978-1-4614-4346-9 2, © Springer Science+Business Media, LLC 2013 17 18 2. Mathematical Preliminaries that are right continuous with left-hand limits by D([0, T ]; Rr ). Consider x(·) = {x(t) ∈ Rr : t ≥ 0}. If for each t ≥ 0, x(t) is a random vector (or an Rr -valued random variable), we call x(·) a stochastic process and write it as x(t), t ≥ 0, or simply x(t) if there is no confusion.

They are dpi0 (t) = −λ0 pi0 (t) + μ1 pi1 (t), dt dpij (t) = λj−1 pi,j−1 (t) − (λj + μj )pij (t) dt +μj+1 pi,j+1 (t), j ≥ 1. and dp0j (t) = −λ0 p0j (t) + λ0 p1j (t), dt dpij (t) = μi pi−1,j (t) − (λi + μi )pij (t) dt +λi pi+1,j (t), i ≥ 1. More details of the derivations can be found in Karlin and Taylor [105, pp. 135-137]. There are many possible variations and/or speciﬁcations of the birth and death processes. For instance, if λi = 0 for all i, the process becomes the pure death process. If μi = 0 for all i, the underlying process is the pure birth process.

A. Mitropolskii among others. For detailed survey and historical development, consult the work of Bogoliubov and Mitropolskii [18], Eckhaus [54], Erd´elyi [58], Il’in [92], Kevorkian and Cole [108, 109], Krylov and Bogoliubov [133], Lochak and Meunier [149], Nayfeh [161], O’Malley [163], Smith [199], Vasil’eava and Butuzov [210], Wasow [215, 216], and the references therein. Singular perturbation methods have been extensively used in various branches of physics including statistical mechanics, solid state physics, chemical physics, molecular biophysics (see Gardiner [68], Risken [182], van Kampen [208], Schuss [188], H¨anggi, Talkner, and Borkovec [80] and the over hundreds of references cited there).

### Continuous-time Markov chains and applications : a two-time-scale approach by G. George Yin, Qing Zhang

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