TV1979 - 156 D.S. SIL’VESTROV [13] J.R. BLUM A N D J. R....

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Unformatted text preview: 156 D.S. SIL’VESTROV [13] J.R. BLUM A N D J. R. ROSENBLATT, On the central limittheorem for thesum of a random number of independentrandom variables,Z.WahrscheinlichkeitstheorieandVerw.Gebiete, 1,4 (1962-1963), pp. 389-393. [14] W.RICHTER,Limittheorems for sequences of random variableswithsequences of random indices, TheoryProb.Applications, 10 (1965),pp.74-89. [15] J.MOGYORODI, A remarkon thestablesequences of random variables,ActaMath. Acad. Sci. Hungar.,17,3 (1966),pp.401-409. [16] H.SREEHARI, An invarianceprinciple for random partialsums,Sankhy-Ser. A,30,4 (1968), pp.432-442. [17] R.PRAKASA, Random centrallimittheorems]:ormartingales,Acta.Math. Acad. Sci.Hungar., 20, 1-2 (1969),pp.217-222. ON HOMOGENEOUS MARKOV MODELS WITH CONTINUOUS TIME AND FINITE OR COUNTABLE STATE SPACE A. A. YUSHKEVIC AND E. A. FAINBERG (TranslatedbyE.Lukacs) Without assuming the homogeneity in time, processes indicated in the titleof this paper were investigated in [13]. In our note, the results of [13] are applied to the homogeneous case,andtheconceptofcanonicalstrategyiscarriedoverfrom 16],[17]to models with continuous time. Incontrast to papers [1],[3]-[11],itisassumed that the controls depend on the whole history. We note that, aside from [13] and [14], in- homogeneous models with more generalcontrolswere studiedby Gikhman and Skorok- hod underadditionalconditionsofcompactnessandcontinuity([12],Chapter2, 5).The physicalaspectsofdescriptionsdependingon thepastofthestrategywere firstdiscussedin [2]. 1. The homogeneous modelZ isgiven by the following elements: (i)thestatespace X--a finiteor denumerable set;(ii)thespace of controlsA--a Borelspace (in 4,itisan arbitrarymeasurablespace);(iii)theprojection/’mameasurablemappingof A toX,where A(x)=]-l(x) is the set of controls admitting the states x; (iv) the transition density q(a,F)ma measure depending measurably on the state a and satisfyingthe conditions q(a,.i(a))=O, q(a,X)<=K< ; (v)therate of payoffr(a)ma measurable functionon A; (vi)thecoefficientofdiscontinuitya--a non-negative constant. The estimateofthepolicyzrand theestimateofthemodel on thetimeinterval[s,u] aregivenby theformulas (1) w( r,R)=E e-r(r[x])dt+e-"R(x,), v(x,R)=supw2(x,r,R), wherethefinalpayoffR isalunctionon X (equalto0foru eo), x isarandom trajectory ofthecontrolledprocess, x’ isthesegmentofthistrajectoryinthetimeinterval[s,t], E" is an integralwith respectto the probabilitymeasure P inthespace oftrajectorieswhich corresponds to theinitialstate x and thepolicy-. The formal definitionsofthespace of trajectories, of the arbitrarypolicy- and the measure P are given in [13]. For s 0, u oo,R 0,thesearguments are omittedinthenotationoftheestimates.The model is summable (.fromabove orJ’rom below)providedtheestimate w(x,-)isfiniteforallx andr ifinthemodelwiththefunctionreitheritspositiveoritsnegativepartissubstitutedforr....
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TV1979 - 156 D.S. SIL’VESTROV [13] J.R. BLUM A N D J. R....

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