By G. George Yin, Qing Zhang
This booklet makes a speciality of two-time-scale Markov chains in discrete time. Our motivation stems from present and rising purposes in optimization and keep an eye on of complicated platforms in production, instant conversation, and ?nancial engineering. a lot of our e?ort during this ebook is dedicated to designing method versions bobbing up from a variety of purposes, examining them through analytic and probabilistic options, and constructing possible compu- tionalschemes. Ourmainconcernistoreducetheinherentsystemcompl- ity. even though all the purposes has its personal particular features, them all are heavily comparable in the course of the modeling of uncertainty as a result of leap or switching random techniques. Oneofthesalientfeaturesofthisbookistheuseofmulti-timescalesin Markovprocessesandtheirapplications. Intuitively,notallpartsorcom- nents of a large-scale method evolve on the similar expense. a few of them swap speedily and others fluctuate slowly. The di?erent premiums of diversifications let us lessen complexity through decomposition and aggregation. it might be perfect if lets divide a wide approach into its smallest irreducible subsystems thoroughly separable from each other and deal with each one subsystem indep- dently. even though, this is infeasible actually because of numerous actual constraints and different issues. therefore, we need to take care of events within which the platforms are just approximately decomposable within the feel that there are susceptible hyperlinks one of the irreducible subsystems, which dictate the oc- sional regime alterations of the method. An e?ective technique to deal with such close to decomposability is time-scale separation. that's, we organize the platforms as though there have been time scales, quick vs. gradual. xii Preface Followingthetime-scaleseparation,weusesingularperturbationmeth- ology to regard the underlying structures.
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Extra resources for Discrete-time Markov Chains: Two-time-scale Methods and Applications
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12 Ch. 3. Relationship and dependence among chapters random regime switching is delineated by a finite-state Markov chain, based on which a discrete-time Markov modulated portfolio selection model is formulated. Under broad conditions, we derive the limit problem that turns out to be a Markov modulated mean-variance control problem in continuous time, which is a LQG problem with regime switching, and in which the control weight is indefinite. We then proceed to obtain its near-optimal control based on the limit problem.
12) such that P = 1lν, a matrix having identical rows, with each row being the stationary distribution associated with the transition matrix P . Moreover, |Ψi (k)| ≤ Kλk0 for some K > 0 and i = 0, 1. In the above and hereafter, K > 0 represents a generic positive constant; its values may change for different uses. It is understood that the convention K + K = K and KK = K is used. The λ0 given above is related to the absolute value of the “largest” non-unity eigenvalues of P . Note that any transition matrix always has an eigenvalue 1.
Let pεk denote the probability vector pεk = (P (αkε = 1), . . , P (αkε = m0 )) ∈ R1×m0 . , pε0 = i 0 p0 = (p10 , . . , pm 0 ) such that p0 ≥ 0 for i = 1, . . , m0 and p0 1lm0 = m0 i ε i=1 p0 = 1. It is well known that pk is a solution of the vector-valued difference equation pεk+1 = pεk Pkε , k = 0, 1, . . , T /ε , pε0 = p0 . 2) with 0 ≤ k ≤ T /ε ? We are equally interested in finding limit properties of the associated equations for k-step transition probability matrices ε Pk+1 = Pkε Pkε , k = 0, 1, .
