60.4. STOPPING TIMES 1959which is exactly the same form as 60.4.9 except m is replaced with m— 1. Now repeat thisprocess till you get the following inequalityi+]| X;rdP > | XrdP+ Xj42dPAN[S=i] jai /ANS=) [T=] AN[S=i)N[T>i+2]The right hand side equalsi+]y | XrdP+ Xj42dPjai /AMS=i)T =i] AN[S=iJ[T<i+1]°itl> XrdP+ Xi4,dPfai JAN[S=iJT =i] AN[S=iJN[T <i+1]¢= | XrdP+ XrdP+ Xj41dPAN[S=iN[T=i] AN[S=iN[T=i+ 1] ANS=A OT <i41]©= | XidP + XiidP + XiidPAn[S=i]N[T =i] AN[S=i)N[T=i+1] An[S=iN[T>i+1]= | XidP + XiidPAn[S=in[T=i] AN[S=iJN[T>i+1]| XidP + XiidPAn|s=in[T=i] AN[S=a)[T<aJ©2 | XjdP + XjdPAn[s=in[T=i] JAn[S=iJn[T <iJ€= | X;dP+ X;dPAn[S=i)n[T =i] AN[S=i[T >i]= | X;dP = X;dP = XsdPAN[S=iN[T >] AN S=i] AN[S=i]In the case where {X,,} is a martingale, you replace every occurance of > in the aboveargument with =. This proves the lemma. §fThis lemma is called the optional sampling theorem. Another version of this theoremis the case where you have an increasing sequence of stopping times, {T;,};_,. Thus if{X,} is a sequence of random variables each -¥,, measurable, the sequence {Xz, } is also asequence of random variables such that X7, is measurable with respect to 7, where -¥7,is an increasing sequence of o fields. In the case where X,, is a submartingale (martingale)it is reasonable to ask whether {X7,} is also a submartingale (martingale). The optionalsampling theorem says this is often the case.Theorem 60.4.4 Let {T,,} be an increasing bounded sequence of stopping times and let{X,,} be a submartingale (martingale) adapted to the increasing sequence of © algebras,{Fy}. Then {Xz,} is a submartingale (martingale) adapted to the increasing sequence ofo algebras {.F¥,r,}.