7.2 Introductory on Stochastic Processes 113(ii)F 0 contains all null sets ofF∞. This means:A⊂B∈F∞andP[B]=0
implyA∈F 0 (F 0 is saturated).
The natural filtration generated by a processXis defined as follows: we
give the description forT=R+:
(i) for alltwe defineHt=σ(Xu;u≤t)andH∞=
∨∞
t=0Ht=σ(Xt,^0 ≤t).
(ii) Gt=
⋂
s>tHsandG∞=H∞.
(iii)N={A|∃B∈G∞,A⊂BandP[B]=0}.
(iv)Ft=σ(Gt,N).
The filtration (Ft)t≥ 0 is right continuous and satisfies the usual conditions.
The filtration (Ft)t≥ 0 is called the natural filtration of X. The filtration
(Ht)t≥ 0 is sometimes called the internal history ofX.
In the caseT=[0,1] orR+we suppose that the processSis cadlag, i.e.,
for almost everyω∈ΩthemapT→Rd,t→St(ω) is right continuous and
has left limits (where meaningful).
IfX:T→Rdis cadlag we define ∆Xt(ω)=Xt(ω)−lims↗tXs(ω)=
Xt(ω)−Xt−(ω). The processXis called continuous if almost surely,T→Rd,
t→Xt(ω) is continuous. Although the problems forT={ 0 ,...,n},N,[0,1],
orR+are different, there is a possibility to treat many aspects in the same
way. This is done through an embedding ofTinR+. The finite discrete time
caseT={ 0 ,...,n}is treated in the following way. Form∈N,m≥nwe
putSm=SnandFm=Fn, thus embedding the caseT={ 1 ,...,n}into the
caseT=N.Thecase[0,1] is embedded inR+in a similar waySu=S 1 and
Fu=F 1 foru≥1. To embedNinR+we put forn≤t<n+1,St=Snand
Ft=Fn.
In view of this possibility to embed every time set intoR+we will only
work withT=R+.
OnR+×Ω we consider differentσ-algebras. They are the basis to do
stochastic analysis. Theσ-algebra consisting of Borel sets onR+is denoted
byB(R+). Theσ-algebraB(R+)⊗F∞denotes theσ-algebra onR+×Ωof
all measurable subsets. A processX :R+×Ω→R, which is measurable
forB(R+)⊗F∞is simply called measurable. Theσ-algebra generated by all
stochastic intervals of the form [[0,T[[ w h e r eT is a stopping time, is called
theoptionalσ-algebra. It is denoted byO. Under the usual conditions, right
continuous adapted processes are measurable with respect toO.Conversely
Ois generated by the set of all adapted right continuous real-valued processes
(see Dellacherie [D 72]).
Theσ-algebra generated by all stochastic intervals of the form [[0,T]] w h e r e
Tis a stopping time, is called thepredictableσ-algebra. To be precise, when
F 0 is not trivial, we also have to include the sets of the form{ 0 }×A,whereA
runs throughF 0. The predictableσ-algebra is denoted byP. It is generated by
the set of all left continuous adapted real-valued processes. One can even show
thatPis already generated by the set of all continuous adapted real-valued
processes. This implies in particular thatP⊂O(see [D 72]).