Download Discrete-Time Markov Chains: Two-Time-Scale Methods and by G. George Yin, Qing Zhang PDF

By G. George Yin, Qing Zhang

This e-book makes a speciality of the idea and functions of discrete-time two-time-scale Markov chains. a lot attempt during this publication is dedicated to designing process versions bobbing up from those functions, reading them through analytic and probabilistic suggestions, and constructing possible computational algorithms that allows you to lessen the inherent complexity.  This publication provides effects together with asymptotic expansions of chance vectors, structural houses of profession measures, exponential bounds, aggregation and decomposition and linked restrict procedures, and interface of discrete-time and continuous-time systems.  one of many salient positive factors is that it incorporates a various diversity of functions on filtering, estimation, regulate, optimization, and Markov choice approaches, and monetary engineering.  This publication can be a big reference for researchers within the parts of utilized likelihood, regulate thought, operations learn, in addition to for practitioners who use optimization techniques.  a part of the e-book is additionally utilized in a graduate process utilized chance, stochastic strategies, and applications.

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Extra resources for Discrete-Time Markov Chains: Two-Time-Scale Methods and Applications (Stochastic Modelling and Applied Probability)

Example text

We also have the following lemma. 1 For all f ∈ F and t ≥ 0, the following assertions hold: (a) P (t, f )P ∗ (f ) = P ∗ (f )P (t, f ) = P ∗ (f )P ∗ (f ) = P ∗ (f ), and P ∗ (f )e = e. (b) Q(f )P ∗ (f ) = P ∗ (f )Q(f ) = 0, and V¯ (f ) = P ∗ (f )r(f ). (c) (P (t, f ) − P ∗ (f ))n = P (nt, f ) − P ∗ (f ) for all integers n ≥ 1. ∞ (d) 0 P (t, f ) − P ∗ (f ) dt < ∞, where D := supi∈S j ∈S |dij | for any matrix D = [dij ]|S|×|S| . 2), P (t + s, f ) = P (t, f )P (s, f ) = P (s, f )P (t, f ). 2) we obtain (a).

Then a∈A(i) x(i, a) > 0 for all i ∈ S. Define a randomized stationary policy π x by π x (a|i) := x(i, a) b∈A(i) x(i, b) ∀a ∈ A(i) and i ∈ S. 68) Then π x is in Π s , and xπ x (i, a) = x(i, a) for all a ∈ A(i) and i ∈ S. 66) we have xπ (x, a) = 1. 67) and q(j |i, π)μπ (i) = 0 ∀j ∈ S. i∈S Hence, xπ is a feasible solution to D-LP. ˆ > 0}. 64) S ′ is not empty. 68) implies that x(i, a) = π x (a|i)u(i). 69) a∈A(i) q(j |i, a)π x (a|i), substituting u(i) ˆ = 1. 12, u(i) ˆ = μπ x (i) > 0 for all i ∈ S, and so S ′ = S.

The following result establishes a relationship between feasible solutions to the D-LP and randomized stationary policies. 66). Then xπ := {xπ (i, a), a ∈ A(i), i ∈ S} is a feasible solution to the D-LP problem. (b) Let x := {x(i, a), a ∈ A(i), i ∈ S} be a feasible solution to the D-LP problem. Then a∈A(i) x(i, a) > 0 for all i ∈ S. Define a randomized stationary policy π x by π x (a|i) := x(i, a) b∈A(i) x(i, b) ∀a ∈ A(i) and i ∈ S. 68) Then π x is in Π s , and xπ x (i, a) = x(i, a) for all a ∈ A(i) and i ∈ S.

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