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1、CHAPTER1Introduction1.WhatintheWorldisaStochasticProcess?2.HowtoCharacterizeaStochasticProcess?3.WhattoDowithaStochasticProcess?TheGrandPlanCharacterizationTransientBehaviorLimitingBehaviorFirstPassageTimesCosts/Rewards34INTRODUCTION“Allbusinessproceedsonbeliefs,onjudgemento
2、fprobabilities,andnotoncer-tainties”CharlesW.Eliot1.1WhatintheWorldisaStochasticProcess?Considerasystemthatevolvesrandomlyintime,forexample,thestockmarketin-dex,theinventoryinawarehouse,thequeueofcustomersataservicestation,water-levelinareservoir,thestateofthemachinesinafact
3、ory,etc.Supposeweobservethissystematdiscretetimepointsn=0,1,2,···,say,everyhour,everyday,everyweek,etc.LetXnbethestateofthesystemattimen.Forexample,XncanbetheDow-Jonesindexattheendofthen-thworkingday;thenumberofunsoldcarsonadealer’slotatthebeginningofdayn;theintensityofthen-
4、thearthquake(measuredontheRichterscale)tohittheContinentalUnitedstatesinthiscentury;orthenumberofrobberiesinacityondayn,tonameafew.Wesaythat{Xn,n≥0}isadiscrete-timestochasticprocessdescribingthesystem.Ifthesystemisobservedcontinuouslyintime,withX(t)beingitsstateattimet,theni
5、tisdescribedbyacontinuous-timestochasticprocess{X(t),t≥0}.Forexample,X(t)mayrepresentthenumberoffailedmachinesinamachine-shopattimet,thepositionofahurricaneattimet,orthemountofmoneyinabankaccountattimet,etc.Moreformally,astochasticprocessisacollectionrandomvariables{X(τ),τ∈T
6、},indexedbytheparameterτtakingvaluesintheparametersetT.TherandomvariabletakesvaluesinthesetS,calledthestate-spaceofthestochasticprocess.Inmanyapplicationstheparametertrepresentstime.Throughoutthisbookweshallencountertwocases:1.T={0,1,2,···}.Inthiscasewewrite{Xn,n≥0}insteadof
7、{X(τ),τ∈T}.2.T=[0,∞).Inthiscasewewrite{X(t),t≥0}insteadof{X(τ),τ∈T}.Also,weshallalmostalwaysencounterS⊆{0,1,2,···}orS⊆(−∞,∞).Weshallrefertotheformercaseasthediscretestate-spacecase,andthelattercaseasthecontinuousstate-spacecase.Let{X(τ),τ∈T}beastochasticprocesswithstate-spac
8、eS,andletx:T→Sbeafunction.Onecanthinkof{x(τ),τ∈T}asapossibleevolution(traje
