By G. George Yin, Qing Zhang

This ebook makes a speciality of the speculation and purposes of discrete-time two-time-scale Markov chains. a lot attempt during this booklet is dedicated to designing method versions coming up from those purposes, examining them through analytic and probabilistic strategies, and constructing possible computational algorithms with a purpose to lessen the inherent complexity. This ebook provides effects together with asymptotic expansions of likelihood vectors, structural homes of profession measures, exponential bounds, aggregation and decomposition and linked restrict techniques, and interface of discrete-time and continuous-time structures. one of many salient beneficial properties is that it incorporates a assorted diversity of purposes on filtering, estimation, regulate, optimization, and Markov selection approaches, and monetary engineering. This ebook should be a massive reference for researchers within the components of utilized likelihood, regulate conception, operations learn, in addition to for practitioners who use optimization recommendations. a part of the ebook is also utilized in a graduate process utilized likelihood, stochastic procedures, and purposes.

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**Additional resources for Discrete-Time Markov Chains: Two-Time-Scale Methods and Applications**

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Let αε (γ) be a continuous-time process obtained from ξ(γ) and αkε by ε αε (γ) = αξ(γ) . Then ∞ P (αε (γ) = i) = P (αkε = i)P (ξ(γ) = k). k=0 This implies P (αε (γ) = i) = p(γ) = P (αε (εγ) = i). Concerning the time-scale separation, we note: In a continuous-time setting, we work in a ﬁnite time horizon t ∈ [0, T ], whereas in a discrete-time formulation, the time horizon is of the order O(1/ε), so we work with 0 ≤ k ≤ T /ε , where T /ε denotes the integer part of T /ε. We also note that the time-space separation method considered in the deterministic discrete-time singular perturbation problems by Naidu [117], which mainly dealt with boundary value problems, cannot be carried over to our formulation.

Dγ Denote Qε = Q + εQ, where Q = (q ij ) and let q0 = max(|˜ q ii |) = max i i q˜ij , j=i and ﬁx q > q0 . Deﬁne Pε = 1 I + Q +ε q 1 Q . q Then all entries of P ε are nonnegative and P ε 1lm0 = 1lm0 , for ε small enough. Therefore, P ε is a transition probability matrix, and p(γ) ˙ = p(γ)[q(P ε − I)]. 18 1. Introduction, Overview, and Examples The formal solution of the above forward equation is p(γ) = p0 exp(γq(P ε − I)) ∞ = p0 exp(−γq) j=0 (γq)j ε j (P ) . j! 17) Consider a discrete-time Markov chain αkε having transition matrix P ε .

3. 1 Introduction This chapter provides basic background materials needed in the subsequent chapters of the book. It brieﬂy reviews and summarizes related results of random processes, including Markov chains in both discrete time and continuous time, martingales, Gaussian processes, diﬀusions, and switching diﬀusions. Throughout the book, we work with a probability space (Ω, F, P ). A collection of σ-algebras {Ft }, for t ≥ 0 or t = 1, 2, . , or simply Ft , is called a ﬁltration if Fs ⊂ Ft for s ≤ t.