Filtrations

Continuing with the heuristics, suppose that we are interested in a random experiment taking place over an infinite expanse of time. Let T = R+ or T =Nbe the time set. For each time t, letFt be the information gathered during [0, t] by an observer of the experiment. For s < t, we must have Fs⊂Ft. The familyF={Ft: t∈T}, then, depicts the flow of information as the experiment progresses over time. The following definition formalizes this concept.

4.8 Definition. LetT be a subset ofR. For eachtinT, letFtbe a sub- σ-algebra ofH. The family F={Ft: t∈T} is called a filtration provided thatFs⊂Ft fors < t.

In other words, a filtration is an increasing family of sub-σ-algebras ofH. The simplest examples are the filtrations generated by stochastic processes:

IfX ={Xt: t∈T}is a stochastic process, then puttingFt=σ{X_s: s≤t, s∈T}yields a filtrationF={Ft: t∈T}. The reader is invited to ponder the meaning of the next proposition for such a filtration. Of course, the aim is to approximate eternal variables by random variables that become known in finite time.

4.9 Proposition. Let F={Fn : n∈ N} be a filtration and put F_∞ =

n∈NFn. For each bounded random variable V in F_∞ there are bounded variables Vn inFn,n∈N, such that

limn E|V_n−V| = 0.

Remark. Note thatE|Vn−V|=V_n−V1in the notation of section 3;

thus, the approximation here is in the sense ofL¹-space. Also, we may add to the conclusion that EV_n → EV; this follows from the observation that

|EV_n−EV| ≤E|V_n−V|. Proof. LetC=

nFn. By definition,F∞=σC. ObviouslyCis a p-system.

To complete the proof via the monotone class theorem, we start by lettingMb

be the collection of all bounded variables inF_∞ having the approximation property described. It is easy to see thatMbincludes constants and is a vector space overRand includes the indicators of events inC. Thus,Mbwill include all boundedV in F_∞once we check the remaining monotonicity condition.

Let (Uk)⊂Mbbe positive and increasing to a bounded variableV inF_∞. Then, for eachk≥1 there areUk,ninFn,n∈N, such thatE|Uk,n−Uk| →0 as n → ∞. Put n₀ = 0, and for eachk ≥ 1 choose nk > nk−1 such that Uˆk=Uk,n_k satisfies

E|Uˆk−Uk|< 1 k.

Moreover, since (Uk) is bounded and converges to V, the bounded conver- gence implies thatE|Uk−V| →0. Hence,

E|Uˆk−V| ≤ E|Uˆk−Uk| + E|Uk−V| → 0 4.10

ask→ ∞. Withn₀= 0 chooseV₀= 0 and putVn= ˆUk for all integersnin (nk, nk+1]; then, Vn ∈Fn_k ⊂Fn, andE|Vn−V| →0 as n→ ∞ in view of 4.10. This is what we need to show thatV ∈Mb. In the preceding proposition, theVnare shown to exist but are unspecified.

A very specific version will appear later employing totally new tools; see the martingale convergence theorems of ChapterVand, in particular, Corollary V.3.30 there.

Exercises and complements

4.11 p-systems forσX. Let T be an arbitrary index set. LetX = (X_t)_t∈T, where X_t takes values in (E_t,Et) for each t in T. For each t, let Ct be a p-system that generatesEt. LetG0 be the collection of allG⊂Ω having the form

G=

t∈S

{X_t∈A_t}

for some finiteS⊂TandAtinCtfor everytinS. Show thatG0is a p-system that generatesG=σX.

4.12 Monotone class theorem.This is a generalization of the monotone class theorem I.2.19. We keep the setting and notations of the preceding exercise.

Sec. 4 Information and Determinability 81 Let M be a monotone class of mappings from Ω into ¯R. Suppose that M includes everyV : Ω→[0,1] having the form

V =

t∈S

1A_t◦Xt, S finite, At∈Ct for everytin S.

Then, every positiveV inσX belongs toM. Prove.

4.13 Special case.In the setting of the exercises above, supposeEt=Rand Et =BR for all t. Let M be a monotone class of mappings from Ω into ¯R. Suppose thatMincludes everyV of the form

V =f₁◦Xt₁· · ·fn◦Xt_n

withn≥1 andt₁, . . . , tn in T andf₁, . . . , fn bounded continuous functions from R into R. Then, M contains all positive V in σX. Prove. Hint: Start by showing that, if A is an open interval of R, then 1A is the limit of an increasing sequence of bounded continuous functions.

4.14 Determinability. If X and Y are random variables taking values in (E,E) and (D,D), then we say thatX determines Y ifY =f◦X for some f : E→D measurable with respect toEandD. Then,σX ⊃σY obviously.

Heuristically,XdeterminesY if knowingX(ω) is sufficient for knowingY(ω), this being true for every possibility ω. To illustrate the notion in a simple setting, let T be a positive random variable and define a stochastic process X= (Xt)t∈R+ by setting, for eachω

Xt(ω) =

0 ift < T(ω), 1 ift≥T(ω).

Show thatX andT determine each other. IfT represents the time of failure for a device, then X is the process that indicates whether the device has failed or not. ThatX andT determine each other is intuitively obvious, but the measurability issues cannot be ignored altogether.

4.15 Warning. A slight change in the preceding exercise shows that one must guard against raw intuition. LetT have a distribution that is absolutely continuous with respect to the Lebesgue measure onR+; in fact, all we need is thatP{T =t}= 0 for everytin R+. Define

X_t(ω) =

1 ift=T(ω) 0 otherwise.

Show that, for each t in R₊, the random variable Xt is determined by T. But, contrary to raw intuition,T is not determined byX= (Xt)t∈R+. Show this by following the steps below:

a) For each t, we have Xt = 0 almost surely. Therefore, for every sequence (tn) inR+,Xt₁ =Xt₂ =. . .= 0 almost surely.

b) IfV ∈σX, thenV =calmost surely for some constantc. It follows thatT is not inσX.

4.16 Arrival processes. Let T = (T₁, T₂, . . .) be an increasing sequence of R+-valued variables. Define a stochastic process X = (Xt)t∈R+ with state spaceNby

Xt= ∞ n=1

1_(0,t]◦Tn, t∈R₊.

Show that X andT determine each other. IfTn represents then-th arrival time at a store, thenXtis the number of customers who arrived during (0, t].

So,X andT are the same phenomena viewed from different angles.

5 Independence

This section is about independence, a truly probabilistic concept. For random variables, the concept reduces to the earlier definition: they are inde- pendent if and only if their joint distribution is the product of their marginal distributions.

Throughout, (Ω,H,P) is a probability space. As usual, if G is a sub-σ- algebra ofH, we regard it both as a collection of events and as the collection of all numerical random variables that are measurable with respect to it.

Recall thatσX is theσ-algebra on Ω generated byX, andX here can be a random variable or a collection of random variables. Finally, we writeFI for

i∈IFi as in I.1.8and refer to it as theσ-algebra generated by the collection ofσ-algebrasFi,i∈I.