A CROSS SECTION THEOREM. II PETER GRAY Several applications of a one-parameter stochastic process (

PETER GRAY

Several applications of a one-parameter stochastic process (Xt)t∈R+ feature the predictable projection^pXtand the dual predictable projectionX_t^pofXt. Certain results surrounding these projections are derived using the Cross Section Theorem.

(For a statement of the Cross Section Theorem, see [D-M, p. 137].) The treatment of a two-parameter stochastic process (Xs,t)s,t∈R+ is similar: first a cross section theorem is established, after which it is used to derive uniqueness results about the projections ofXs,t. (For other results pertaining to two-parameter processes, see [C-W], [Dz], [L], and [W].) In our previous paper [G] we presented a cross section theorem based on the double filtration Fs,t = Fs− for s, t ≥ 0. The theorem states that for any predictable subsetA ∈ Ω×R²+ and for any > 0 there exists a predictable stopping time Z : Ω →[0,∞]×[0,∞] for the double filtrationFs,t=Fs−such that [Z]⊂AandP(π[A])−P(π[ [Z] ])≤, where [Z]

denotes the graph ofZ,πis the projection onto the set Ω, andP the probability measure on Ω. In this paper we use this result to derive a cross section theorem for the more general double filtration Fs,t = Fs∨t− with s, t≥ 0. Under this double filtration, desirably, information from the second time parametertis not lost. We will prove that for any predictable subsetA∈Ω×R²+and for any >0 there exists a predictable stopping timeZ : Ω →[0,∞]×[0,∞] for the double filtrationFs,t=Fs∨t−such that [Z]⊂AandP(π[A])−P(π[ [Z] ])≤.

AMS 2000 Subject Classification: Primary 60G020; Secondary 60G40.

Key words: two-parameter stochastic process, cross section.

1. PRELIMINARIES The framework for this paper consists of

• a probability space (Ω,F, P);

• a filtration (F_t)t∈R+ of subalgebras ofFsatisfying the usual conditions, and such that for the filtration (G_s)s∈R+ defined byG_s=F_s−for every s∈ R+ we have G_S = F_S− for every predictable stopping time S for (G_s)s∈R+; and

• a double filtration (F_s,t⁰⁰ )s,t∈R+ such thatF_s,t⁰⁰ =G_s∨tfor everys, t∈R+. (See [D, p. 323] for more on double filtrations.)

REV. ROUMAINE MATH. PURES APPL.,54(2009),1, 25–32

2. THE PREDICTABLE σ-ALGEBRA ℘⁰⁰

In this section we characterize theσ-algebra ℘⁰⁰ of subsets of Ω×R²₊ in terms of σ-algebras which have already been studied. This will enable us in ensuing sections to readily provide results pertaining to℘⁰⁰-measurable subsets A of the set Ω×R²+.

Definition 2.1. LetX: Ω×R²+→Rbe a two parameter process.

2.1aX isleft continuous if for each s0, t0 ∈R+ and $∈Ω we have

(s,t)→(slim₀,t0) s≤s0

t≤t0

X_s,t($) =X_s0,t0($).

2.1bXisadaptedto the filtration (F_s,t⁰⁰ ) if for eachs, t∈R+the function Xs,t is F_s,t⁰⁰ -measurable.

2.1c The predictable σ-algebra of subsets of Ω×R²+ is the σ-algebra generated by left continuous, two-parameter processes X which are adapted to (F_s,t⁰⁰ ). We denote thisσ-algebra by ℘⁰⁰.

Remark. LetB : Ω×R+ →Rbe a left continuous, one parameter process that is adapted to the filtration (F_t)t∈R+ (for example, Brownian motion; see [O, p. 72]). Then the two-parameter process X : Ω×R²₊ → R given by Xs,t($) =Bs($)Bt($) for every ($, s, t)∈Ω×R²+ is℘⁰⁰-measurable.

Definition 2.2. The σ-algebra ℘ of subsets of Ω×R²+ is the σ-algebra generated by left continuous, two-parameter processes X which are adapted to the double filtration (F_s,t)s,t∈R+ given byF_s,t =G_sfor everys, t∈R+. The σ-algebra ℘⁰ of subsets of Ω×R²₊ is the σ-algebra generated by left contin- uous, two-parameter processes X which are adapted to the double filtration (F_s,t⁰ )s,t∈R+ given by given by F_s,t⁰ =G_t for everys, t∈R+.

Remark. For results surrounding the σ-algebra ℘ (and by analogy, ℘⁰), see [G].

Notation 2.3. LetU denote the set Ω× {(s, t)∈R²+ |s≤t}, letLdenote the set Ω× {(s, t) ∈ R²+ | s > t}, and for each n ∈ N let U_n denote the set Ω× {(s, t)∈R²+|s, t= 0 or (1−¹_n)s < t}.

Notation 2.4. Let A and B be collections of sets. We denote by A ∨ B the collection {A∪B |A∈ AandB ∈ B} .

Theorem2.5. We have℘⁰⁰= (℘∩L)∨(℘⁰∩U).

Proof. Set Q := (℘∩L)∨(℘⁰∩U). First, we show that Q ⊂℘⁰⁰. The generators of the σ-algebra ℘∩L are the processes of the form X1_L, where X : Ω×R²₊ → R is left continuous and adapted to the double filtration

(F_s,t)s,t∈R+. Each such generatorX1L is left continuous and adapted to the double filtration (F_s,t⁰⁰ )s,t∈R+. In fact,X and1_L are both left continuous, and for any s, t∈R+ we have

X_s,t,(1_L)_s,t ∈ G_s⊂ G_s∨t=F_s,t⁰⁰ . We conclude that ℘∩L⊂℘⁰⁰.

Next, the generators of the σ-algebra ℘⁰ ∩U are processes of the form Y1_U, where Y : Ω×R²+ → R is left continuous and adapted to the double filtration (F_s,t⁰ )s,t∈R+. Let the processY1_U be such a generator. We have

Y1_U= lim

n Y1_Un,

and for each index n we have Y1_Un ∈ ℘⁰⁰, because both Y and 1_Un are left continuous, and for any s, t∈R+ we have

Y_s,t,(1_Un)_s,t∈ G_t⊂ G_s∨t=F_s,t⁰⁰ .

We conclude that the process Y1_Uis ℘⁰⁰-measurable. Therefore, we have℘⁰∩ U⊂℘⁰⁰. It follows that Q ⊂℘⁰⁰.

Lastly, we show that ℘⁰⁰ ⊂ Q. Let X : Ω×R²+ → R be a generator of

℘⁰⁰; that is, let the process X be left continuous and adapted to the double filtration (F_s,t⁰⁰ )s,t∈R+. Note that the processX1L is℘∩L-measurable. In fact, X and 1_L are both left continuous, and so X1_L is left continuous. Further, for any s, t∈R+ we have

(X1L)s,t =

( 0∈ F_s,t ifs≤t, Xs,t∈ F_s,t⁰⁰ =G_s∨t=G_s=F_s,t ifs > t.

So, X1_L ∈ ℘. Then we have (X1_L)1_L ∈ ℘∩ L; that is, X1_L is ℘∩ L- measurable. Further, the process X1U is ℘⁰ ∩U-measurable. In fact, let Y : Ω×R²+→Rbe the process given by

Ys,t($) =

( Xs,t($) if$∈Ω and 0≤s≤t, X_t,t($) if$∈Ω ands > t.

Then Y is left continuous, and for each s, t∈R+ the random variable Y_s,t is G_t-measurable. Hence Y is ℘⁰-measurable, and so the process X1_U = Y1_U is ℘⁰ ∩U-measurable. We conclude that the process X = X1_L +X1_U is Q-measurable. It follows that℘⁰⁰⊂ Q.

Corollary2.6. LetX : Ω×R²+→Rbe a two-parameter process. Then X is ℘⁰⁰-measurable iff X1L is℘-measurable and X1U is ℘⁰-measurable.

Proof. Assume first that X is ℘⁰⁰-measurable. Then the process X1L is

℘⁰⁰∩L-measurable. But we have

℘⁰⁰∩L= ((℘∩L)∨(℘⁰∩U))∩L by Theorem 2.5

=℘∩L sinceU∩L=∅;⊂℘ sinceL∈℘.

Hence the process X1_L is ℘-measurable. Also, the process X1_U is ℘⁰⁰∩U- measurable. But we have

℘⁰⁰∩U = ((℘∩L)∨(℘⁰∩U))∩U =℘⁰∩U ⊂℘⁰ sinceU ∈℘⁰

(because 1U = limn1Un, and each process 1Un is ℘⁰-measurable). Therefore, the process X1_U is ℘⁰-measurable. Next, assume that X1_L is ℘-measurable and X1U is ℘⁰-measurable. Then the process X1L = (X1L)1L is ℘ ∩ L- measurable, and the process X1_U = (X1_U)1_U is ℘⁰∩U-measurable. Hence the processX1_L+X1_U is (℘∩L)∨(℘⁰∩U)-measurable; that is, the process X is℘⁰⁰-measurable.

3. STOPPING TIMES

In this section we define what is meant by a stopping time and a predict- able stopping time, for each of the three double filtrations of interest.

Definition 2.7. LetZ : Ω→[0,∞]×[0,∞] be a function.

2.7a We say that Z is a stopping time for the filtration (F_s,t⁰⁰ )s,t∈R+

(respectively (F_s,t)s,t∈R+, (F_s,t⁰ )s,t∈R+) if the set{Z ≤z} is an element of F_z⁰⁰ (respectively F_z,F_z⁰) for everyz∈R²+.

2.7b LetZ be a stopping time for (F_s,t)s,t∈R+. We define theσ-algebra F_Z ⊂Ω byF_Z ={A∈ F |A∩ {Z ≤z} ∈ F_z ∀z∈R²+}. Let Z be a stopping time for (F_s,t⁰ )s,t∈R+. We define the σ-algebra F_Z⁰ ⊂ Ω by F_Z⁰ = {A ∈ F | A∩ {Z ≤z} ∈ F_z⁰ ∀z∈R²+}. Let Z be a stopping time for (F_s,t⁰⁰)s,t∈R+. We define the σ-algebra F_Z⁰⁰ ⊂Ω byF_Z⁰⁰ ={A∈ F |A∩ {Z ≤z} ∈ F_z⁰⁰ ∀z∈R²+}.

2.7c We say that the stopping time Z = (S, T) for (F_s,t)s,t∈R+ is a predictable stopping time for (F_s,t)s,t∈R+ if

• S is a predictable stopping time for the filtration (G_s)s∈R+, and

• {S <∞} ⊂ {T <∞}.

Note, for any predictable stopping time Z = (S, T) for (F_s,t)s,t∈R+ we have F_Z =F_S−.

2.7d We say that the stopping time Z = (S, T) for (F_s,t⁰ )s,t∈R+ is a predictable stopping time for (F_s,t⁰ )s,t∈R+ if

• T is a predictable stopping time for the filtration (G_s)s∈R+, and

• {T <∞} ⊂ {S <∞}.

Note, for any predictable stopping time Z = (S, T) for (F_s,t⁰ )s,t∈R+ we have F_Z⁰ =F_T−.

2.7eWe say that the stopping time Z = (S, T) for (F_s,t⁰⁰ )s,t∈R+ is apre- dictable stopping time for (F_s,t⁰⁰)s,t∈R+ if the following three conditions hold.

• Z_{S>T} := (S_{S>T}, T_{S>T}) is a predictable stopping time for (F_s,t)s,t∈R+.

• Z_{S≤T} := (S_{S≤T}, T_{S≤T}) is a predictable stopping time for (F_s,t⁰ )s,t∈R+.

• The supremum S∨T of S and T, is a predictable stopping time for (G_s)s∈R+.

Note, it can be shown that for any predictable stopping time Z = (S, T) for (F_s,t⁰⁰ )s,t∈R+ we have F_Z⁰⁰ = G_S∨T. Accordingly, for any predictable stopping time Z = (S, T) for (F_s,t⁰⁰ )s,t∈R+ we have F_Z⁰⁰ =F_S∨T−.

Here are three examples of predictable stopping times for (F_s,t⁰⁰ )s,t∈R+.

•Z = (S, T) such that S, T are predictable stopping times for (G_s)s∈R+. In fact, the stopping time S_{S>T} for (G_s)s∈R+ is predictable because {S > T} ∈ F_S−, and further we have

{S_{S>T} <∞}={S <∞} ∩ {S > T}={S <∞} ∩ {S > T} ∩ {T <∞}

⊂ {T <∞} ∩ {S > T}={T_{S>T} <∞},

thereby confirming thatZ_{S>T} is a predictable stopping time for (F_s,t)s,t∈R+. Similarly, Z_{S≤T} is a predictable stopping time for (F_s,t⁰ )s,t∈R+. Finally, note that S∨T is a predictable stopping time for (G_s)s∈R+ since bothS andT are predictable stopping times for (G_s)s∈R+.

•Z = (S, T) such that Z is a predictable stopping time for (F_s,t)s,t∈R+, andS > T orS =T =∞. In fact, we haveZ{S>T} =Z, which is a predictable stopping time for (F_s,t)s,t∈R+;Z_{S≤T} = (∞,∞), which is a predictable stop- ping time for (F_s,t⁰ )s,t∈R+; andS∨T =S, which is a predictable stopping time for (G_s)s∈R+.

•Z = (S, T) such that Z is a predictable stopping time for (F_s,t⁰ )s,t∈R+, and S ≤T.

4. THE CROSS SECTION THEOREM

We now present the main result of this paper, a Cross Section theorem for ℘⁰⁰-measurable subsetsA of Ω×R²+.

Theorem2.8. Let A⊂Ω×R²+ be ℘⁰⁰-measurable, and let >0. There exists a predictable stopping time Z : Ω → [0,∞]×[0,∞] for the double filtration (F_s,t⁰⁰ )s,t∈R+ such that

2.8a[Z]⊂Aand

2.8b P(π[A])−P(π[ [Z] ])≤.

Proof. By Corollary 2.6, the setA∩Lis℘-measurable, and the setA∩U is ℘⁰-measurable. By [G, Theorem 1.21], we have the following results.

• There is a predictable stopping timeZ1= (S1, T1) for the double filtra- tion (F_s,t)s,t∈R+ such that [Z₁]⊂A∩LandP(π[A∩L])−P(π[ [Z₁] ])≤₂.

• There is a predictable stopping timeZ₂ = (S₂, T₂) for the double filtra- tion (F_s,t⁰ )s,t∈R+ such that [Z2]⊂A∩U andP(π[A∩U])−P(π[ [Z2] ])≤₂. Define the functionZ : Ω→[0,∞]×[0,∞] by

Z =Z11{T₂>S1}+Z21{T₂≤S₁}

= (S11{T₂>S1}+S21{T₂≤S₁}, T11{T₂>S1}+T21{T₂≤S₁}) =: (S, T).

First, we show thatZ is a stopping time for (F_s,t⁰⁰ )s,t∈R+. Letz= (s, t)∈ R²+. We must show that{Z ≤z} ∈ F_z⁰⁰. We have

{Z≤z}={Z₁ ≤z} ∩ {T₂> S₁}S{Z₂≤z} ∩ {T₂ ≤S₁}

={Z₁≤z} ∩ {S₁ ≤s} ∩ {T₂ > S₁}S

{Z₂ ≤z} ∩ {T₂ ≤t} ∩ {T₂ ≤S₁}

={Z₁ ≤z} ∩ {S₁ ≤s} ∩({T₂ > s} ∪({T₂ ≤s} ∩ {T₂ > S1})) S{Z₂≤z} ∩ {T₂ ≤t} ∩({S₁> t} ∪({S₁ ≤t} ∩ {T₂ > S₁}))∈ G_s∨t=F_z⁰⁰, sinceZ1is a stopping time for (F_s,t)s,t∈R+,Z2is a stopping time for (F_s,t⁰ )s,t∈R+, and S1, T2 are stopping times for (G_s)s∈R+.

Next, we show that Z is a predictable stopping time for (F_s,t⁰⁰ )s,t∈R+. To do this, we must show the following:

2.8cThe functionZ_{S>T} is a predictable stopping time for (F_s,t)s,t∈R+. 2.8dThe functionZ{S≤T}is a predictable stopping time for (F_s,t⁰ )s,t∈R+. 2.8eThe function S∨T is a predictable stopping time for (G_s)s∈R+. First, we establish requirement2.8c. We have

{S > T}={S₁ > T1} ∩ {T₂> S1}S

{S₂> T2} ∩ {T₂ ≤S1}

={S₁<∞} ∩ {T₂>S1}S

∅∩ {T₂≤S1} since [Z₁]⊂Land [Z₂]⊂U; ={T₂> S₁}. So, we have

Z_{S>T} =Z_{T2>S1} =Z_1{T2>S1}. We must show that

2.8f The functionZ_1{T2>S1} is a stopping time for (F_s,t)s,t∈R+.

2.8gThe functionS_1{T2>S1} is a predictable stopping time for (G_s)s∈R+, and

2.8h {S_1{T2>S1} <∞} ⊂ {T_1{T2>S1} <∞}.

To prove2.8f, letz= (s, t)∈R²₊. We have

{Z_1{T2>S1} ≤z}={Z₁≤z} ∩ {T₂> S₁}={Z₁≤z} ∩ {S₁≤s} ∩ {T₂> S₁}

={Z₁≤z} ∩ {S₁≤s} ∩({T₂> s} ∪({T₂≤s} ∩ {T₂ > S₁})∈ G_s=F_z. Next, we verify that condition2.8gis satisfied. Note, since bothS₁ andT₂are predictable stopping times for (G_s)s∈R+, the set{T₂> S₁}is an element ofG_S1−

(see [D-M, p. 128]). Therefore, the functionS_1{T2>S1} is a predictable stopping time for (G_s)s∈R+. Lastly, we show that condition2.8h is met. We have

{S_1{T2>S1} <∞}={S₁ <∞} ∩ {T₂> S1} ⊂ {T₁<∞} ∩ {T₂ > S1} since Z1 is a predictable stopping time for (F_s,t)s,t∈R+, and so {S₁ < ∞} ⊂ {T₁ <∞}; ={T_1{T2>S1}<∞}.

This concludes our verification that requirement 2.8c holds true. Using a similar proof, we are able to establish requirement 2.8d.

Lastly, we establish requirement2.8e. We have

S∨T =S₁∨T₁1_{T2>S1}+S₂∨T₂1_{T2≤S1} =S₁1_{T2>S1}+T₂1_{T2≤S1} sinceS₁ ≥T₁ andS₂≤T₂ because [Z₁]⊂L and [Z₂]⊂U; =S₁∧T₂, which is a predictable stopping time for (G_s)s∈R+, since bothS1 andT2are predictable stopping times for (G_s)s∈R+.

To complete the proof of the theorem, we must show that assertions2.8a and 2.8b are true. We have [Z]⊂ [Z₁]∪[Z₂]⊂ A, and so assertion 2.8a is true. Next, we verify that assertion 2.8b is true. As preparation, we first show the following:

2.8i{T₂ > S₁} ⊂π[Z₁]⊂π[A].

2.8j {T₂<∞}=π[Z2]⊂π[A].

2.8k{T₂ =S₁ =∞} ⊂(π[Z₁]∪π[Z₂])^c. Regarding2.8i, we have

{T₂> S₁} ⊂ {S₁<∞}={S₁ <∞} ∩ {T₁<∞}sinceS₁> T₁ on Ω;

=π[Z₁]⊂π[A∩L] ⊂π[A].

Next, regarding 2.8j, we have

{T₂<∞}={T₂<∞} ∩ {S₂ <∞}sinceS₂ ≤T₂ on Ω; =π[Z₂]⊂π[A].

Lastly, with respect to 2.8k, we have

{T₂=S₁=∞}={T₂=∞} ∩ {S₁=∞} ⊂(π[Z₂])^c∩(π[Z₁])^c = (π[Z₁]∪π[Z₂])^c. We are now ready to prove assertion2.8b. We have

P(π[A])−P(π[Z]) =P(π[A]\π[Z]) since π[Z]⊂π[A];

=P





(π[A]∩ {T₂ > S1}S

π[A]∩ {T₂ ≤S1} ∩ {T₂<∞}

Sπ[A]∩ {T₂ ≤S₁} ∩ {T₂=∞})\(π[Z₁]∩ {T₂ > S₁} Sπ[Z₂]∩ {T₂≤S₁})





≤P





(π[A]∩ {T₂ > S1} \π[Z1]∩ {T₂ > S1})

S(π[A]∩ {T₂ ≤S₁} ∩ {T₂ <∞} \(π[Z₂]∩ {T₂ ≤S₁}) Sπ[A]∩ {T₂ ≤S₁} ∩ {T₂ =∞}





=P





{T₂ > S1} \ {T₂ > S1}

S(π[A]∩ {T₂ <∞} \π[Z₂])∩ {T₂≤S₁} Sπ[A]∩ {T₂ =S1 =∞}



 by 2.8i;

=P(∅S

({T₂ <∞} \ {T₂<∞})S

π[A]∩ {T₂=S₁=∞}) by 2.8j;

=P(∅S

∅S

π[A]∩ {T₂ =S1 =∞})≤P(π[A]∩(π[Z1]∪π[Z2])^c) by2.8k;

=P((π[A∩L]∪π[A∩U])T

(π[Z1]∪π[Z2])^c)

≤P((π[A∩L]\π[Z₁])S

(π[A∩U]\π[Z₂]))

≤P(π[A∩L]\π[Z₁]) +P(π[A∩U]\π[Z₂])≤ 2 +

2 =.

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Received 7 December 2007 University of Florida

College of Journalism and Communication Gainesville, FL 32611, USA

pgray@ufl.edu