114 7 A Primer in Stochastic Integration
IfT:Ω→R+∪{∞}is a stopping time,FTis theσ-algebra of events prior
toTi.e.FT={A|A∈F∞and for allt∈R+we haveA∩{T≤t}∈Ft}.
We also need theσ-algebra of events “strictly” prior toT.Thisσ-algebra,
denoted byFT−, is not defined using a description of its elements. It is defined
as theσ-algebra generated byF 0 and by elements of the formA∩{t<T}
whereA∈Ft. ClearlyFT− ⊂FT. It is easy to see that a stopping time
T:Ω→TisFT−-measurable.
Remark 7.2.1.The difference betweenFT−andFTwill turn out to be crucial.
Typically the following happens. IfTis a stopping time where an optional
cadlag process jumps, thenFT− does not provide any information on the
jump size, whereasFTdoes also contain this information. In insurance terms
we could say that ifT is the stopping time given by the arrival of a letter
announcing a new claim, thenFT− gives all the information prior to this
arrival, including the fact that a letter has arrived. Theσ-algebraFTalso
contains the information on the claim size.
Definition 7.2.2.A stopping time T is called predictable if there is an in-
creasing sequence of stopping times(Tn)∞n=1such thatTn↗Talmost surely
andTn<Ton{ 0 <T}.
Under the usual conditionsT is predictable if and only if the set [[T]] =
[[T,T]] = {(T(ω),ω)|T(ω)<∞}is in the predictableσ-algebraP.One
can show thatPis generated by the stochastic intervals [[0,T[[ w h e r e Tis a
predictable stopping time.
Definition 7.2.3.The stopping timeTis called totally inaccessible if for each
predictable stopping timeτ, we haveP[τ=T<∞]=0.
The following description is proved in the theory of stochastic processes
(see [D 72]). Recall thatπ:Ω×R+→Ω denotes the canonical projection.
Proposition 7.2.4.LetTbe a stopping time.
(i) FT={π(A∩[[T]] )|A∈O},
(ii)FT−={π(A∩[[T]] )|A∈P}.
As a consequence, a functionf :Ω→ R,thatisF∞-measurable is
FT-measurable for a given stopping timeTif and only if there is an optional
processX, i.e., a processX:Ω×R+→Rwhich is measurable with respect
toO, such that on{T<∞}we haveXT=f.ThemapfisFT−-measurable
if and only if there is a predictable processYsuch that on{T<∞}we have
YT=f.
For a stopping timeT, we define the processXTas (XT)t=Xt∧T.We
callXTtheprocessXstopped at timeT.
Definition 7.2.5.If (P) is a property of stochastic processes, then a stochas-
tic processXsatisfies (P) locally if there is an increasing sequence of stopping
times(Tn)∞n=1such thatTn↗∞almost surely, and for eachnthe process
XTnsatisfies (P).