Optional sampling

《Optional sampling》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、CHAPTER2OptionalsamplingbyJanvanNeerven1.Optionalsampling:discretetimeInthissectionwediscusstheoptionalsamplingtheoremfordiscretesupermartingales.Throughout,welet(;F;P)beafixedprobabilityspace.1.1.Stoppingtimes.Letusfixafiltration(Fn)n2N,whereN=f0;1;2;:::g.DEFINITION2.1.

2、AstoppingtimeisarandomvariableT:!N[f1gwiththepropertythatfT6ng2Fnforalln2N.Forn2N,n>1,wehavefT=ng=fT6ngnfT6n1g.ClearlythisimpliesthatTisastoppingtimeifandonlyiffT=ng2Fnforalln2N[f1g.IfSandTarestoppingtimes,thenS^T:=minfS;TgandS_T:=maxfS;Tgarestoppingtimesaswell.Indee

3、d,thisfollowsfromfS^T6ng=fS6ng[fT6ngandfS_T6ng=fS6ngfT6ng.Inparticular,takingS=k,weseethatT^kisastoppingtimewheneverTisastoppingtimeandk2N.DEFINITION2.2.LetTbeastoppingtime.The-algebraFTisconsistsofallsetsA2FwiththepropertythatAfT6ng2Fnforalln2N.REMARK2.3.LetTbeast

4、oppingtime.IfweputF1:=FthentriviallywehavefT61g2F1,andforallA2FTwehaveAfT61g2F1.Thuswemayalwaysenlargethefiltration(Fn)n2Ntoafiltration(Fn)n2N[f1gandassumethatthedefiningpropertiesinDefinitions2.1and2.2holdforalln2N[f1g.Foralln>1wehaveAfT=ng=(AfT6ng)n(AfT6n1g).Clearl

5、ythisimpliesthatA2FTifandonlyifAfT=ng2Fnforalln2N.FortheconstantstoppingtimeT=k,withk2N,wehaveFT=Fk.Thisrelationextendstok=1,providedwetakeF1:=FasinRemark2.3.PROPOSITION2.4.IfSandTarestoppingtimessatisfyingS6T,thenFSFT.PROOF.LetA2FSandfixn2N.FromS6TwehavefT6ngfS6nga

6、ndthereforeAfT6ng=AfS6ngfT6ng:SinceAfS6ngFnandfT6ng2Fn,itfollowsthatAfT6ng2Fn.Asaparticularcase,notethatFT^jFT^kwheneverTisastoppingtimeandj6kinN.PROPOSITION2.5.IfTisastoppingtime,thenforallF2FTandk2NwehaveFfT6kg2FT^k:122.OPTIONALSAMPLINGPROOF.FirstnotethatF

7、fT6kg2FksinceTisastoppingtime.Henceforalln>kwehave(FfT6kg)fT^k6ng=FfT6kg2FkFn:Ontheotherhand,forn

8、FT)=E(XjFn)almostsurelyonthesetfT=ng,foralln2N;(2)E(XjFT)=XalmostsurelyonthesetfT=1g.PROOF.Weprove(1)and(2)simultaneouslybyp