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Predictable projections and predictable dual projections of a two parameter stochastic process

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Predictable projections and predictable dual projections of a two parameter stochastic process
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PREDICTABLE PROJECTIONS AND PREDICTABLE
DUAL PROJECTIONS OF A TWO PARAMETER STOCHASTIC PROCESS















By

PETER GRAY


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2006














ACKNOWLEDGMENTS

I am indebted to my supervisor, Dr. Nicolae Dinculeanu, for his infinite patience with me while I slowly learned the material. Also, I am grateful to the many excellent teachers that I have had during my journey at the University of Florida. Finally, for the friendly banter and hearty laughs that I hold in fond memory, I thank Julia, Connie, and Gretchen.














TABLE OF CONTENTS

ACKNOW LEDGMENTS .......................................................................... ii

ABSTRACT .................................................................................................. v

CHAPTER

1 THE CROSS SECTION THEOREM ................................................... 1

Introduction .......................................................................................... 1
Predictable a-Algebras ........................................................................ 2
Stopping Times ................................................................................... 6
Projections 7r[A] ................................................................................. 9
Sets C5 ................................................................................................ 11
The Cross Section Theorem ............................................................... 21

2 PREDICTABLE PROJECTIONS ........................................................ 37

Projections PX ....................................................................................... 37
The Uniqueness of PX ............................................................................ 39
The Existence of PX ............................................................................ 43

3 PREDICTABLE DUAL PROJECTIONS ............................................... 50

Step Filtrations (',) ............................................................................ 50
Predictable Dual Projections XP ............................................................. 52
The Uniqueness of XP ............................................................................ 54
Predictable Dual Projections of Measures ............................................ 55
Processes Associated With Stochastic, R-Valued Measures ............... 58
The Existence of XP ............................................................................ 64

4 VECTOR-VALUED PREDICTABLE DUAL PROJECTIONS ................ 77

Predictable Dual Projections W P ........................................................ 77
The Uniqueness of W P ......................................................................... 81
Processes Associated With Stochastic, E-Valued Measures ............... 82
The Existence of W P .......................................................................... 89









5 AN EXTENSION OF THE RADON-NIKODYM THEOREM
TO MEASURES WITH FINITE SEMIVARIATION .............................. 94

6 SUMMARY AND CONCLUSIONS ........................................................ 106

R EFER ENC E LIST .................................................................................... 107

BIO G RAPHICAL SKETCH ........................................................................ 108














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PREDICTABLE PROJECTIONS AND PREDICTABLE
DUAL PROJECTIONS OF A TWO PARAMETER STOCHASTIC PROCESS

By

PETER GRAY

August 2006

Chair: Nicolae Dinculeanu
Major Department: Mathematics

The framework of this dissertation consists of a probability space (Q,f,P); a fi[ tration (c'Ft)tER, such that if (9s)R, is a filtration satisfying G, = Ts- for every s c then we have G3s = ,s- for every predictable stopping time S for (Gs)sR+; a double filtration (3:s,t)s,ER+ such that sl = , for every s,t > 0; and a Banach space E. We study initially a real-valued, two parameter stochastic process X : Q x + __+ and then we extend some of our results to a vector-valued process Y : n X __ E.

In Chapter 1 we start by defining the predictable a-algebra p of subsets of

x + to be the u-algebra generated by the left continuous processes X that are adapted to the double filtration (f,1,),,R+. Then we prove the main result of the chapter, the cross section theorem for sets in po.

In Chapter 2 we define the predictable projection of a measurable process

x - R to be a predictable process +X x --+ R such that for every








stopping time Z we have E[ Xz Iq 1 z ] = (PX)z 1z<> almost surely. Then, using the cross section theorem, we show that the predictable projection is unique up to an evanescent set. In addition, we demonstrate that every bounded, measurable process X has a predictable projection.

In Chapter 3 we define the predictable dual projection of a right continuous, measurable process X : x 2 --+ R with integrable variation to be a right continuous, predictable process XP : L x R2 --. R with integrable variation such that for each bounded, measurable process (p :�K2x R -- R we have E[f P(p dX] = E[ J dXP Then we show that the predictable dual projection is unique up to an evanescent set. We also establish that the predictable dual projection of the process X exists if the filtration (,Ft)t R+ is a step filtration.

In Chapters 4 and 5 we turn our attention from real-valued processes X to vectorvalued processes Y. In this setting, our formulations are based not on finite variation and integrable variation, but on finite semivariation and integrable semivariation.














CHAPTER 1
THE CROSS SECTION THEOREM

Introduction

The theory surrounding one parameter stochastic processes has applications in many fields. In finance for instance, a filtration (Jt),R contains information that is known up to time t about a market; a martingale (XI)tIR, for (T'F,),.R reflects the price of stock options; a predictable process (Ht),=R. houses the number of shares to be held at time t; and a stopping time S for (,t)tR. indicates when stocks should be sold for optimal profit. Note a discrete (or step) filtration suffices for good results in many markets.

The predictable projection (PX,)tR� and the predictable dual projection (XP')IR, for the process (X,)ER. both play a role in one parameter stochastic theory. For an example of this we retread the finance stage that was set above. The random variable PXs, which is the predictable projection (pX,)t R. evaluated at a (predictable) stopping time S, may be regarded as an updated version of the expected selling price E[Xs] of stock options, given the market information ,%s-.

The goal of this dissertation is to extend the definition and existence of the predictable projection and the predictable dual projection to a two parameter process (XS,1)v,. This extension is difficult because, while the set R+ of positive real numbers is totally ordered, the set R2 of ordered pairs of positive real numbers is not. In order to reduce slightly the complexity that we face, we will retain a one parameter

1






2

flavor: our framework will be built around the double filtration n = YS f , where (,F4)R is a right continuous, complete filtration.

Predictable a-Algebras

The main result of this chapter is a cross section theorem for predictable subsets of , x 1+ relative to the double filtration (,,)st R satisfying Y, = , for s, t > 0. The cross section theorem will be derived with invaluable help from the Monotone Class theorem, which may be found in the text Probabilities and Potential (Dellacherie & Meyer 1975, p. 13-1).

In this section we introduce the predictable u-algebra p of subsets of the space



Notation and Terminology 1.1 The following will be used in the sequel. I.1a (0 ,.T,P) is a probability space.

1.1b R+ is the set of non-negative real numbers. N is the set of natural numbers.

Q+ is the set of positive rational numbers. + is the set 01+ x 1+, and Q+ is the set Q+ x Q+.

1.1 c Ig(R1+) is the Borel u-algebra generated by the intervals (s, t] of 0+. 1+(12) is the Borel a-algebra generated by the rectangles (s, t] x (u, v] of 0+.

1.1d A function X : Q x 12 __, R is called a two parameter process, and is denoted (Xx,,).

1.1e F� I(R1+) is the u-algebra generated by the semiring Yx IR(R1+). ,F� I (01 2) is the u-algebra generated by the semiring Yx 1 (02+).








I.1f (Y),,+ is a filtration and therefore satisfies

" for each t > 0, F1 is a a-algebra contained in Y, and
" , c Yt if s
We will write simply (J,), and we will assume that the filtration satisfies the usual conditions:

" Y0 contains all the negligible sets (that is, (,T,) is complete), and " Y, fl �F, for every t > 0 (that is, (J',) is right continuous).
S>t

See 1.1j below for another assumption about (,).

1.ig A function S : Q - [0 , oo] is a stopping time for the filtration (Y,) if{ S< } te)e for every t > 0.

Let S,T be two stopping times. The stochastic interval (S, 1] is the set {(U, t )_xR+ S() < t < T(tu)}, while [S, T) is the set {(tu, t) (= .Q x R, S(tu) _< t < T(u) } The stochastic intervals (S , T) and [S , T] are defined in a similar fashion. 1.1h The predictable a-algebra P of subsets of Q x R+ is the a-algebra generated by the sets A x (s, t] and B x {0}, where A e Tt and B E YO. A stopping time S is predictable if the stochastic interval [S , 00) is a predictable set.

1.1i Let S be a stopping time.

,Ys denotes the a-algebra { A E Y I A n { S < t ) c- Y for every t 0} while ,s- denotes the a-algebra generated by the sets in f'Fo as well as sets of the form An{S>t}, wheret >Oand Ae Y.

1.1j (S)SA, is the filtration defined by the following rules.








9 go= f0 and
e 9, = -FT- for every s>O.

We will write simply (g,), and we will assume that the filtration (ift) is such that gs = Ys- for every predictable stopping time S for the filtration (g,). For example, this is the case when f, - fLj for every s E R+, where LsJ is the largest integer less than or equal to s.

1.1k ('Fs,1)stR is a double filtration and therefore satisfies

" for each s,t > 0, Fs,, is a a-algebra contained in F, and
, c3 7,, , if s
We will write simply (if,1), and we will assume that TF,,r = , for every s, t > 0.

1.11 Let Bcfnx R+.

B(a) is the set { (s,t) R+ I (ri,s,t) c B }. K[B], called the projection of B, is the set { U C f (t,s,t) - B for some s,t E R,}.

1.1m The point (oo, cc) will be denoted by co. Therefore, the inequality (s,t) < 00 means that s < cc and t < oo. i.1n Let g : Q - [0, co] x [0, co] be a function. [g] denotes the set { (U,s,t) E Q x 2[ g(m) = (s,t) }, called the graph of g.

We now commence our study of the real-valued two parameter process (X,,,). We begin by defining the predictable o-algebra. Definition 1.2 Let X Q x 2- _. R be a two parameter process.

1.2a (X,,) is left continuous if for every so,to E R + and w c f, we have









lim XS,,(c) = XSO,0(u). Note that this limit is a pointwise limit. (s,t)-.(so,to)
O

1.2b (Xs,/) is adapted to the filtration (F,,,) if for each s,t r R, the random variable Xs,t is F,,-measurable.

1.2c The predictable a-algebra of subsets of L2 x R2 is the a-algebra generated by left continuous two parameter processes (X,,) which are adapted to (F,,). We denote this a-algebra by g. Proposition 1.3 g is generated by the sets (S , cc) x [0 , I] and A x {0} x [0, r2], where r1, r2 E R+, S : Q --+ [0 , oo] is a stopping time for (Gs), and A e Go. Proof Let (X,t) be left continuous and adapted to (,,). Set := Xoo lo}�{o}(s,t) + Xo,_L I O}�( k_ L](S,t) k-O

+ ~ o I~, l(m j�Lx{o}(S, t) + X XALI l(n!�m (* ](St)
+~~~ ~ 7X. n - + ~ ,", n+1...,, ,+l(s,t),
m--0 k,m=0


where 1 is the indicator function, and n e N. Since (X,,,) is left continuous, Y" --+ X pointwise as n -- oo. Fix m, n, and k, and consider the process R,,, := Xk1 ],, (S, Since X.,_ is gm-measurable, R is the pointwise limit of processes of the form
pn

Paj 1Aix(_M,__x(k_,k where for every index i we have a, E R and A, E gm.



But for every index i we have Ai x ( m , m] - (S, , TJ],


where S {F on A , which is a stopping time for (9s); and
CO on A'








-on A,
Ti -{ o Ai which is also a stopping time for (G).
cc on A'


Therefore, R is the pointwise limit of processes of the form Z ai 1 (S,,T]x(-L- ,
i-!

where for every index i we have ai c R, and Si, T, are stopping times for (9,). By considering in a similar manner the processes Xo, I k i�k and X -o I m(+], we obtain

V c 'a{(S, , T11 x (r, , si], A x {0} x (r2 s21, (S2, T21 x {0) , where rI,r2,sI,s2 E I+, A E Go, and S1,S2,T1, and T2 are stopping times for (G,); = -a{(S,oo) x [0,rl], Ax{O}x[O, r2] }, where rj,r2 e R, A r g0, and S is a stopping time for (G,).

In factwe have Vo = aa{(S,oo) x [0, r], Ax{0}x [0, r2]} because the sets (S, oo) x [0 , r1] and A x {0} x [0 , r2] are elements of V.

Stopping Times

Definition 1.4 Let Z 0 - [0 , cc] x [0 , oo] be a function.

1.4a The function Z is a stopping time for the filtration (Y,,) if { Z < z } E ,'F for every z +I .

1.4b The stopping time Z = (S,T) is predictable if

9 S is a predictable stopping time for (9,), and
* (S < co} c (T < oo}.

Proposition 1.5 Let Z,, = (S,,T.) be a sequence of stopping times for any double filtration (,,) that is right continuous in t. Assume that (S,) is increasing. Then Z (sup S,,, limsup T,) is a stopping time for (,
n n








Proof Let (s,t) G R . We must show that { Z < (s, t) } ,. For each index n and each (r,u) C we have {Z, < (r, u)} = {S_ r }n { T < u} e r by hypothesis. Set T := V Ti.
i~n

Then limsup T. , lim T,.
n n

We have

{Z<(s,t)} = {(sup Snlim Tn) < (s,t)}
n n

= {sup Sr_ s}n{lim T' n n

= Sn s) n n { T' <{T _t+Ep}
n--I p=N k-I m=k

where N is any natural number, and (Ep) is any sequence of positive numbers decreasing to 0,


Snu {sn p=N k-I Lm=k
=n OU M} n f{T, < t+Ep}fn{Tm1l < t+Ep}n .... p=N k-I n-I m-k


= nun[{ p=N k=I m=k

since (S,) is increasing;


nu n[{Z. __ (s,t+Ep)} n {ZM,1 < (s,t+p) }n ....]
p==N k-I m=k

E '"FSt+CN N e N being arbitrary. Since (cr,,) is right continuous in t, we conclude that { Z < (s,t) } E=






8

Proposition 1.6 Let n e N, and let Zi = (S;, Ti) i = 1 ... n be a finite set of stopping times for , such that each S is a stopping time for (!9,). Set Ai := {inf Sk =Si} n {S * $}S n f{Si# S2} n ... n {S Si}, i=1 ... n,
k!Sn
n1 n
and set S:= Si 1A, and T:= Ti IA,.
il! iil

Denote (S,T) by minf (Z,), and set Z := minf (Z1). Then i i

" S is a stopping time for (9,),
" Z is a stopping time for ( F.,), and
" if Ti(m)-- o = Si(tu) = o for each index i then T(u)=oo => S(s) =0. Proof We first prove that S is a stopping time for (G,). Note that the sets Ai i =

1 ... n are pairwise disjoint. Further, we have A, e Gs, for each i (Metivier 1982, p. 20), and U Ai = 0. It follows that S is a stopping time for (g,).
iI

Next we prove that Z is a stopping time for (f,,). Let (s, t) c R2. We must show that { Z < (s, t) } g;. By hypothesis, { Z _ (s,) }E 9, for each i.

So we have {Z<(s,t)} U {Zi < (s,t) } n A, i=!


U ({S,< s}nA) n {Z, _ (s,t)}
i=!

E 9".

Lastly we prove the third assertion of the proposition. We have T(o) = co => Ti(m) = oo for some index i such that u c Ai, = Si(w) = oo by hypothesis, S(w) = oo since CU E A,.






9

Projections n[A]

In this section we establish a key result concerning the projection Ir[4] of an YF� 1(R2)-measurable set A. It will emerge that 4A] is F-measurable. Proposition 1.7 Let A e � I1(R+) and let g be a function such that K[ [g] ] E 3, and [g]c A on 4[A]n R [ [g] ]. Then [[g]nA]e( YF. Proof Let A be the collection of sets B ,T� II(R2) such that for any function h : Q - [0, oo] x [0, o] satisfying r[ [h] cY and [h] c B on 4[B] n 4[ [g]] we have ;r[ [h] n B E Y.

Let R, be the ring generated by the sets (C x (s, ] x (uv]) n E x R 2 and L2x ,2 where C E Y and s,t, u,v E R. First we show that A contains 'R. Let B E R and let a function h be such that ;[ [h] ] E Y and [h] c B on 4[B] nr[ [h] ].

Without loss in generality we consider B = C x (s,t] x (u,v], where C e F and s
Wehave ,r[[h]nB] = (n[[h]])n C E Y. Hence, B E A.

Next we show that A is a monotone class. Let (B,) be a monotone sequence from A.

Assume first that (B,) is increasing. Set B:=UB,. Let a function h1 satisfy ir[[hY]]e , and [h ]cB on
n l
,r[B] nl ,[ [h1] ]. Let h' : l --- [0, co] x [0, co] be a function that satisfies the con-






10

Mons set out below. There are four conditions, and they involve the function h1. We require that ir[ [h'] ] = ir[ [hi] ],

h' = h1 on (r[B])c,

[h']cB on r[B1] nr[h],

and [h'] c Bi on (4r[Bi] \ 4[Bi-1] ) Oi r[hI] for i = 2,3,4... Such a function h' exists. Note that we have r[ [h'] ] c ,Y, and [h'] c B, on r[Bi] n r[ [h'] ] for i E N. Therefore for each index i we have r[ [h'] n B ] E ,F (since each B, is in A). Hence we have r[ [h n B ] r[ [h'] n B ] from the definition of h', = '[[hl]nu B, ]

= U [h']n B ]
i




Next, assume that (B,) is decreasing. Set B' := nB,,. Let a function h2 satisfy r[ [h2]] EF and [h2] c B' on
n

r[B'] n r[ [h2] ]. Let h" :n -- [0, cc] x [0 , oo] be a function that satisfies the conditions set out below. We require that [ [h"]] = r[ [h2] ],

h"= h2 on (r[Bl])cUr[B'],

and [h"] c Bi on (;r[Bi] \ r[B+]) n r[h2] for i ( N. Such a function h" exists. Note that we have ;r[ [h"] ] E Y, and [h"] c B, on r[B;] n r[ [h" ] for i E N.






11

Therefore for each index i we have x[ [h"] n Bi] Y F (since each Bi is in A). Hence we have [ [h2] n B'] = [ [h"] n B'] (from the definition of h"), = r[ [h"]n B,]
i

= nr[ [h"] n Bi] (since [h"] is a graph),
i

SF.

We have shown that A is a monotone class which contains the ring of generators of F� 1(12). Because this ring contains the whole space Q x +, we conclude from the Monotone Class theorem that A is equal to F� R(R2 ). The statement of the proposition is now seen to be true. Corollary 1.8 Let A e F� 1R(R2). Then K[A] E Y. Proof We are able to find a function h : Q -- [0,cc] x [0, oo] such that 4[ [h]] -Q and [h] c A on 4[A] nx[ [h] ]. From Proposition 1.7 we obtain [ [h] n A] E F. For such a function h we have 4-[ [h] n A ] = 4[A]. Hence we have 4[A] c Y.

Sets /C6

We continue to prepare for the Cross Section theorem by introducing a special collection ICs of predictable sets with compact cross sections. Definition 1.9 Let K c +. x 2.

1.9a We say that K has compact cross sections if for each u G ;[K], the cross section K(w) c R2 is compact.

1.9b The cross sectional closure of K, denoted K*, is given by K*() = (K(w) )* for every tu c K2, where (K(w) )* denotes the closure of the






12

cross section K(m) c R+. (The closure includes all adherent points.) Proposition 1.10 Let N E N, and let (K,) be a sequence of subsets of nx .+ with compact cross sections. Assume that each set K, has the following properties: ;r[B* n K,.] e for every .T� 1g(R2)-measurable set B; and there is a stopping time Z,, = (S.,T.) such that

9 S, is a predictable stopping time for (Os),
* [Z,] c K,
" {Z < oo} = ;[K.],
" Tn(i) = oo Sn(w ) = oo for every wE.K, and
" (s,t) E K.(tu) s> S,(uy) for every 0 E K.

N 00
Set K:=UK, and K':=AlK,.
n-I ?I-]

Then K and K' both have compact cross sections, and K has all the above properties of the sets K,. Further, if the sequence (Kn) is decreasing, then the set K' also has all the above properties.

Proof The finite union and the countable intersection of compact subsets of R+ is compact.

Therefore, K and K' have compact cross sections. Next, let B C F� (R+).

N
We have ;r[B* n K] = ;r[B* n U K,]

N
= U i[B* n K,,] n=I




and ir[B* N K'] = ir[B* n n K,] n-I








CO
= r ,[B* K,] if (K,) is decreasing,
n=I

since each set B* fl K, has compact cross sections;
E ,..

We now show that K has the second property set out in the proposition. Set Z := minf (Z, ). (See Proposition 1.6 for the definition of "minf".)
n-I...N

From Proposition 1.6 we know that Z is a stopping time, and that if we write Z = (S,T) then S is a stopping time for (9,). Note that the stopping time S is a predictable stopping time because S is the minimum of finitely many predictable stopping times S,, n = I.. .N. Next, let u c [Z]. Then Z(u) < oo. There is an index n such that Z(ti) = Z,(tu). From this equality we deduce that Z"(Cf) < X.

By hypothesis we have (tu,Z(w)) c K. Hence we have (w,Z(w)) = (tu,Z.(w))



cK.

We now show that { Z < oo} = ,[K].

N
We have [K] = 7r[UK] n-I

N
n=I








N
U {Z, < oo} by a property of the set K., n-I
N
SU ({s, < ao} n {Tn < 00)

N
U {S. < oo} by a property of the function Z,, n-I

N
U {S, ff= nt n
Penultimately, note that by Proposition 1.6 we have T(w) = OD =* S() =0 for every W E .

Lastly we show that (s, t) c K(u) implies that s > S(tu) for every w E- Q. We have (s,t) e K(u) = (ay,s,t) E K (ay,s,t) c Kn for some index n, Ss > Sn() by hypothesis, > S(n) since S = inf S..
n=I.N

To complete the proof, assume that the sequence (K,) is decreasing. We verify that the set K' has all five properties that were listed. Let Z := (sup S , limsup Tn). Note that Z is a stopping time.
n n

To see this, it is enough to show (by Proposition 1.5) that the sequence (S,) is increasing.

Let GY E L2 and no E N, and assume first that Zn0+I(1) < Then (t, So,,,+(), Tn0+I( U)) E Kn0+, by a property of the set K,+.






15

But we have the containment Ko+] c K,.. So we have (tu, So+I(m), To+I(ti)) c Kno(ay). By a property of the set K,, we have S+1 (tu) _ So(r). Next, assume that Zo+,(tu) - oo. Then S0+1(t) = oo or T,+](W) = X. We have So+] (w) - co by a property of the set K,0+1. Hence So+(u) > S~o(c), and we have shown that Z is a stopping time. Now note, the stopping time sup S, is a predictable stopping time for (9,).
n

This follows since each S, is a predictable stopping time for (G), and since (S,) is increasing.

We now prove that [Z] c K. Let Z(w) < oo. We must show that (tu, Z(w)) c K'. Since Z(tg) < co, we have sup S(r) < oo.
n

Therefore S.(tu) < oo for every index n. It follows from a property of K. that Z.(m) < oo for every n. So (ay, Z.(tu)) c K, for every n, since [Z.] c K, for all n. There is a subsequence (T.,((U))k of (T.(zu)). (that depends on iv) such that T,,(t) - limsup Tn(M) as k - oo.
n

Since (S.(G)) is increasing, we have S.k('U) -. sup S.(w) as k-- oo.
n

Thus, we have (U. Z k(O)) = (0, Sk(tu), Tf,(tu)) -. (w, Z(u)) as k-- co. For each k we have (u, Z,(tu)) c Knk. Since by assumption the sequence (K.) is decreasing, then (0, Z.k(u))k is a sequence that eventually belongs to each of the sets Knk, k e N.






16

By the compactness of the cross sections involved, it follows that (tu, Z(w)) = lim (w, Znk(u)) belongs to each of the sets Kk, k e N.
k

So we have (a, Z(w)) n fl Kk
k

K'.

We now show that { Z < oo} = K[ K']. First we establish that { Z < oo {Z. < oo}.
n

In fact, let v E { Z < x }. Suppose there is a number no E N such that Z,.(u) < 00. Then either S2o(tu) = cc or T.o(r) = oo. Since T1o(u) = co = S.o() = o, we obtain So(tu) = cc. Since (Sn) is increasing, we have S,(tu) = oo for each m > no. It follows that sup S.([U) = c.
n

Therefore Z(tu) = (sup S1(tu) , limsup T.(G)) < oo, and we have reached a
1 n

contradiction. Because of this contradiction, we conclude that Z1(w) < oo for every n e N.

Hence, we have {Z < oo c n (Z, < o}.
n

Next let tue E { Z. < cc }. Then for each n we have Z,(tu) < cc.
n

Note for each n we have [Z2] c K,,. Therefore for each n we have (&r,Z2(aT)) c Kn c K1.






17

Since K, has compact cross sections, there is a number M in R2 (that depends on tu) such that (s,t) < M V (st)c K (E). Thus, for each n we have Z,(tu) < M. Consequently we have Z(w) - (sup S.(uy), limsup T()) < M.
n n

Accordingly, we have u E {Z < oo}. Hence, we have nl{Z, < oo} c {Z n

Therefore we have {Z < oo} = n { Zn < 0o ) (by the preceding lines),
n

= nir[K.] since {Z. < oo}- r[K.] for all n,
n

= 7r[ f K, ] since all sets K, have compact cross sections,
n

= ir[ K'].

We now show that limsup T,(m) = oo => sup S.(t) = cc for every 0 e 9.
n n

In fact for each zu e Q we have limsup Tn(tff) = 00 Z(t) < o
n

z Zlo(w) n

=> either S,.(uy) = -o or T.o(tg) = O S.o(tu) = oo (as Tno(rz) = x =* Sno(U) = 0), sup S.(W) = 00.






18

Lastly we prove that (s,t) c K'(tu) => s > sup S,(w) for every W E 0.
n

In fact, for each tu c Q we have


(st) c K'(w) = (,s,t) E K'= n K,
n-I

(t,s,t) r K, for every n, S(s,t) e K,(a) for every n,

= s ? S"(tu) for every n by a property of K.,

Ss > sup S,((U).
n

Definition 1.11 We define the set C to be the collection of finite unions

N
U [Si+1i,T1Ani]x[ri ,r1, where ni, N N, i >0, ri, iE R+, and Si,T are stopping times for (9,), for every index i. We define the set C, to be the collection of countable intersections of sets from K. Proposition 1.12 The set K3 is closed under finite union and countable intersection. Further, the elements of K6 have all the properties that were presented in Proposition

1.10.

Proof It is evident that K6 is closed under countable intersection. Next we show that K6 is closed under finite union. Let ME N and let (K,)=1 ..M be a finite family from )C6. For each index i we may write Ki = fl Kj, where each set Kj is an element of 1.
j=l

M Mo0
Then we have U K; = UnKi
z I i=1 j-=








00
n Kj, U K2j2 U ... U KM,,
jhj2,... jM =1

which is a countable intersection of elements of C, since each set Ku, U K2j. U ... U KMJM is an element of 1C.

M
Accordingly, the set U Ki is an element of C,5.
i-I

We now prove that the elements of C,5 possess all the properties that were presented in Proposition 1.10.

Let S,T be stopping times for (g,), let E > 0, let n c N, and let r,r' E R. In view of Proposition 1.10, it is enough to show that the set K0 := [S + E , TAn ] x [r, r'] possesses all the aforementioned properties. Without loss in generality, we assume that S + c < TAn and r < r'. Let B c ,f� I?(R2). We will show that ir[B* n K0] e X. Let (6m) be a sequence of positive numbers decreasing to 0. For each index m, denote by Km the set [S + E - Sm , TAn + 6,] x [r - mr' + Sm], and by K, the set (S + c - ,., TAn + 3,) x (r - ,r' +3). We have 4[B* n K0] = fl [B n Kin].


In fact, we have n[B n Km] c flr[B* n K.] since B c B*,
m=1 or=I

- R[B* n n Kn] since (Kn) is decreasing, and since each set B* n Km has comm,1

pact cross sections.






20

This last set is equal to the projection [B* n Ko]. On the other hand, for each index m we have ,[B* n Ko] c ;[B* nl K] since K0 c K ,

Sr[B n K'] since every cross section of the set K, is an open ball when S+E < oo,

C r[BfNK,.] since K, c K,. Therefore, 4rB* n Ko] c n 4[ n Kn].
m=!


So we have r[B* n Ko] n n B n K,]
M,- I

E F by Corollary 1.8, since K,. e Y� I(R+) for every index m.

We now reveal the remaining properties of the set Ko. Set Z := (S + c , r). Note that Z is a stopping time. In fact, let (s,t) E 1. Then {Z < (s,)} = {S < s) if r < t 0 otherwise


In either case we have {Z _ (st)} 9 ;s. We complete the proof by making the following five observations. The stopping time S + E is a predictable stopping time since E > 0. The graph [Z] is a subset of K0, since by assumption S + E <_ TAn and r < r'. We have {Z < co} = {S+E < co} = 4[[S+E,TAn]] since S+E < TAn, = 7r[Ko].






21

For each 0iEQ we have r(w) = oo = S(r) +E = oo, since r < oo. For each a r.n we have

(s,t) c Ko(ai) =* sc [S(m )+E , T(ti)An] and t E [r, r']

> >_ (S + )t.

The Cross Section Theorem We begin with some important precursors to the cross section theorem for predictable subsets of +1 � R2.

Definition 1.13 Let 91 be the ring of subsets of R2 generated by the sets (z, z'] such that z,z' E R+, with z < z. The measure y is the set function from 9 into R+ that is given by p((z, z']) = the area of the rectangle (z, z'] for every z < z', and which is additively extended to R1.

Remark The measure p is the Lebesgue measure on 91, and can be extended to a sigma-additive measure on (), with values in [0 , oo]. We still denote by p this sigma-additive extension.

Lemma 1.14 Let B c X � 1I(R2) and UY E Q. Then B(w) E1(R2). Proof Let F c �, z,z' E R2+, and (B,) c Q x R+. Then (F x (z, z'])(tg) ( while (BI - B2)(0) = BI(w) - B2(w) and (UB,)(w) = UB,(t).
n n

Notation 1.15 Let A,B E � I(R2), A3. E R+, and z e R2. We denote by ABA;L the set

{ w A B I an( (A n n x [o, zt)( ra) ) > mhi( (B n se x [as, zb)(e)in When A c B and 0 < A < 1, the reader might think of the set AB z as being the






22

projection of the portion of A n n x [0 , z] that fills some Borel cross section of B n Q x [0 , z] by a factor of 100 percent or more. Proposition 1.16 Let A, B E JF � (12(R), and let z E R2. Then for every A E R+ we have ABA.; c .
n
Proof Let R be the ring generated by the sets U Fi x (zi , z'], where n E N, i=1

F, r JF, and zi, z' E R2 for every index i.

no
Fix A0 r R. Write A0 n n x [0 , z] = U Fo,, x B0,,, where the sets F0,1 E Y are mutually disjoint, and where each set B0, is an element of 1(R+). Let A be the collection of sets C in Y� B( 2) such that for every X. C R +, the set AoCA; is an element of F. We first show that A contains the ring R. Let C1 e R and let E R+.

nI
Write Cl n n x [0, z] = U F 1, x B 1j, where the sets F 1,, E 3F are mutually disjoint, and where each set B1,i is an element of +). Then AoC1 = U Fo, nF j, where the union is taken over all pairs (ij) such that
(41)
/t(Boj) > X/y(Bij).

Therefore we have AoCI ' E F. Accordingly we have C1 E A, since ;L E R+ was arbitrary.

Next, let (C,) be an increasing sequence from A.









Set C := c.
"=

We will show that C e A. Note for each tu E D we have ( (C n Q x [0 , z])(t)) = lim., p( (C. n Q x [0 , z])(a) ), since y is sigma-additive. Let A E R,, and let (6,) be a sequence of positive numbers decreasing to 0, with A- 31 > 0.

We have AOC;L = J fl AoC,, x-b, (since u is finite on [0, z]),
m=I n-Il

E Y.

Since A E R+ was arbitrary, we conclude that C E A. Now let (Cn') be a decreasing sequence from A.
'0
Set C' fC
n=1

We will show that C' rc A. Note for each u e K we have #( (C' n Q x [0 , z])(x) ) = lim,,-. p( (C' n Q x [0 , z])(w) ), since p is sigma-additive and since ( (C, n n x [0, z])(w) ) < oo for every w i D. Let A, e R+.

We have AoC .= U AoC' . Since ;L E R+ was arbitrary, we conclude that
n-l

C' c A.

By applying the Monotone Class theorem we deduce that for every A0 ER, C e F� f(R2), and A, e R+, we have AoC ; e T. Now let A' be the collection of sets A' in ,Y� I(R2) such that for every A, e I+, the set A'B I is an element of F.








By the above we have R c A'. Next, let (A') be an increasing sequence from A'. Set A':= UA.
n-I

We will show that A' e A'. Let ;Le +. Since /p is sigma-additive we have A'B ;L= U AnBx.,
n-I

J (F.

Since A e R+ was arbitrary, we conclude that A' e A. Lastly, let (An) be a decreasing sequence from A'.

{0
Set A" l.
n -I

We will show that A" - A'. Let ). E R, and let (5,) be a sequence of positive numbers decreasing to 0.

0o cc
Since y is sigma-additive we have A"B,, = UnA B ,




Since k E- R+ was arbitrary, we conclude that A"e A'. We have shown that A' is a monotone class containing R. By applying the Monotone Class theorem we deduce that for every ,t e R+, the set ABA;L is an element of YF. Proposition 1.17 Let z e RI, let (An) be an increasing sequence from F� a(R2), and let A = U A. Let B be a measurable subset of A, let c > 0,
n-I






25

and let 0 :s A < 1. There is an index N c N such that P([ B n a x [o, z]]) P( (B n AN)B;L ) < E.

Proof Since y is sigma-additive, for each w E f we have /((BNA. n o x [o, z])(w)) j( (BNAf K2 x [, z])(t)) = j( (B n x [, z])(t)).

Therefore, since A < I and since z E t2 is finite, for each W E K there is an index N(tu) such that /j( (B N AN(,) n L2 x [0, zj)(t)) > I.u( (B 1 2 x [0 z])( ) Hence we have (BnA,)B;L, / (BnA)B;..=. Since each set (B n A,)B;Lz is an element of F (see Proposition 1.16), we have P( (B n A,)B;,) 7 P( (B N A)BI-) = P( BBA )

= P(ir[ B n x [0, z] 1).

The statement of the proposition now follows. Proposition 1.18 Let z G R2, Ao E p, and Eo > 0. Let Bo be a measurable subset of Ao. Then there is an element KO of K5 such that Ko c Ao n 1 x [0, z] and P(r[Bonfux[0,z]]) - P([BonKonfx[0,z]]) _< Eo. Proof Let A be the collection of sets A E p such that for every E > 0, 0 < I < 1, and B cA, there is an element K of /C5 such that K c An Lx [0,z] and P(;[BnL2x[0,z]]) - P((BnK)BA,) < E. Let S denote the ring generated by the sets (S , T] x (s , t] and D x (0) x (u, v], such that S,T are stopping times for (Ge), D E Go, and s,t,u,v E R+. Note that we have p = aa(S).






26

It is our goal to show that A is a monotone class containing S. It will follow from the Monotone Class theorem that A Let A c S.

n mI
We may write A U (Si, T,] x (r, , si] U U [0Aj , Uj] x (vj, wj] where S,, T,, and Uj are stopping times for (g,), Aj E g0, and ri,si,vj,wj are real numbers, for every index i and every index j.


Without loss in generality, we will consider A = U (Si , T1J x (ri , s,], and we will asI-I

sume that the sets A' := (S , TJ x (r, , si] are pairwise disjoint. Let E > 0, 0 < IL < 1, and let B be a measurable subset of A. Let (E,) be a sequence of positive numbers decreasing to 0. For each L E N and each index i denote by KL,, the set [Si + EL , Ti A L] x [r, + EL , S] E K.,6. For each index i we have KL,, / Ai as L - oo. By Proposition 1.17, for each index i there is a number Li E N such that P([ B n A' n x [0, z]]) - P( (B n A' KL,)(B n A)AL) < 6i. Set K := U KL,..
i_ I

Wehave Knnx[0,z] E )Cb, Knix[0,z] c Anfx[0,z], and P([BAn x[0,z]]) - P((BfnKn0x[0,z])Bk_) =P(Kr[BN x[0,z]] \ (BnK)BI, )

since (B n K n x [0, z])B.;L = (B n K)BIz. c B n n x [0, z],









= PQII[BfUA'flix [O,z]] \ (BflUK l(nA))


UP U[ BlA'f L x [O,z]] \ U(BnKL)(BnA)'hA)
( i =I /-1I



since the sets Ai are disjoint,


nP(r[ BnA'nf2x[o,]] \ (BnKL,,)(BnA)a,)
n
< EP([ BnA'I.Q x [0,z]] (BnKL,.,)(BnIA')A,,


<
'
-Il

< E.

Thus, A E A. Since A e S was arbitrary, we have S c A. Next, let (A') be an increasing sequence from A. Set A' U A'. We will show that A' c A.

Let B' be a measurable subset of A', and let E > 0 and 0 < L < 1. There are numbers A 1, 2L2 such that 0 < Al < 1 and 111 A2 = 1. By Proposition 1.17 there is an index N e N such that P(4[ B' f L2 x [0, z]]) - P( (B' nA ,)B>.) < Since A, e A, then for the measurable subset C:= [ ((B'-I A%)B',.) x 1 ] i (B'" A' ) n n x [0, z] of AN there is a set KeK:C such that K c AN nE2x[o ,z] c An f2x[0 ,z], and P([ C fn L x [0, z] ]) - P( (C iK)C;,.z ) < E But [cn x[o,z]] = (B'n A,)B'L.-.






28

Hence we have P( (B' n A)B 1.) - P( (C n K)C=,z ) < Note that we have (B' n K)B, (C n K)CkA,. Hence we have P(r[ B' N K x [0, z] ]) - P( (B' n K)Bz)

< P([ 1B' n n x [0, z] ]) - P( (C n K)C =,;)

P([B 'nnx[o,z]]) - P( (B'n A'v)B',z) +

P( (B' n A)B,.) - P( (C n K)C;,,,)

< C + f- 2 2


= C.

We conclude that A' e A. Lastly, let (An) be a decreasing sequence from A. Set A" " .A We will n-I

show that A" E A. Let B" be a measurable subset of A", and let E > 0 and

0 < L < 1. Let ;L' ~ + satisfy I < A'< 1. There is a sequence (;.,) from [0, 1) such that fj 2.L = ;L'.
n-I

Note that the set B" is a measurable subset of A;. So there is a set K1 E K2' suchthat K c Al'nflx [O,z] and P(r[ B" n n x [0, z]]) - P( (B" n K)B ,. )

-- 2 �
The set C :=[ (13" n KI)B", ) X R2 ]NB These C n B"n K n Q x[0, z] isameasurable subset of A". So there is a set K2 E /C6 such that K2 c A" n fn x [0 , z] and P(Cr[ClnLx[0,z]]) - P((CN K2)C1,) _< 22 Note that [Cil x[0,z]] = (B" n K)BA.:-






29

The set C2:- [((C n K2)C1 A2-) xR InC n K2Rox [o, z] is a measurable subset of A". So there is a set K3 e ICS such that K3 c A" n n x [0 , z] and P([ 02fn x [0, z] ]) - P((C2n K3)C2 ) <13-Z-Note that r[ C2fnlx[o,z]] = (CiN K2)C1 , Continuing inductively we obtain a sequence (K,) from /C5 such that for each index n we have K.+ c A"1 fNx[o,z] and P([ C. n x [o, z] ]) - P( (C, n K.+)C., ) < f+ where C,= [((C,-1 n K.)C,,1 ) x+ ] n C, n K. n n x [0, z] and ir[ C. f x [0, z] ] = (C,1 n K.)C,1 j., n = 2,3,4... So for each index n we have P(z[ B" n t x [o, z ]I) - P( (Cn n K,,+)C. n1-,i) =P([ B" n nx [o, z] ]) - P((B" n KI)B",..) + P([Cin L2x[0,z]]) - P((CAn K2)C, X,z) + + P(i[C. n L2x [o, z]]) - P( (Cn K,,)C L,+, ) ++ + f
- 2 22 2+

< E.

The sequence ((C, n K,,+)C. X.-. ). is decreasing, and so we may write P(;r[ B" n n x [0 , z] ]) - P( lim._,n(C. n K,+I)C. A,+,- ) =P([ B" n n x [o, z] I) - lim, P( (C. n K.+m)C. LI) < E.
Now for each index n we have






30

(C, n K,,,+)C. A.,z c (B" n K1 n K2 n n K,+)B" +1 Hence, we have


n -- , , i , z i+l


But we have B ' (B"n Kfi )B.
S n Az)B
n=]

In fact, let y c B".

( "+I n+1
Then p(B" nfn Ki n n x [0,z])()) > Fj[-[ y((B" n 2 x [o,z)()) for every index n. So we have p (B" K n n x [o,z])(tu)) _ f-[n p ((B" n n x


- A'p((B" n o2 x[0z)r) AjU ((B" n L2 x [O,z])() This means that E- (B" n Kn)BI.
n A,

We remark here that if we had considered only A = A' = 0, then all we would have been guaranteed at this point is zu e 7r[B"* fi Kn], which would have been unhelpn-i

ful.
OD F
We now have P(4[B" n (Q x [0, z]]) - P((B" n N)B>) < E.
n-I

Set K:= Kn.
n-I






31

We have K e KQ, K c A" n12x[o ,z] (since Kc A, nftx[o,z] for every n), and P(r[ B" n Q x [0, z] ]) P( (B" n K)B" ) < E. Accordingly, we have A" E A. We have shown that A is a monotone class that contains the ring S of generators of p. Since S generates the whole space f x REI, we have A = p. Then for the sets Ao e p and Bo c Ao, the value Eo > 0, and any number A E [0, 1), we may find an element Ko c- C so that we have Ko c Ao n 1 x [0, z] and P([ BonLx [0, z]]) - P(r[ Bon KoNflx [0, z]]) < P([ Bo n Lx [0, z]]) - P((o n Ko)Bo A. ) since (BonKo)BoA, c r[BonKonQ x[0,z]], < Eo.

The statement of the theorem is now seen to be true. Corollary 1.19 Let A E p, and let E > 0. There is an ,,-measurable function f: Q - [0 , oo] x [0, ao] such that

" [ cA and
" POr[A]) - P( [A]) < E.

Proof Let (z,) be a sequence from R2 such that z, 7 co. For each index n there is a set K. E C, such that K. c A n f2 x [0 , z.] and P(i[ A n 12 x [0, z] ]) - P(4r[K.]) < -E- (take Bo = Ao = A in Theorem 1.18). Set D:= UK. We have Dc A (since z. 7oc), and

oOcoD
P(K[A]) - P(ir[D]) = P~r[ U A n n x [0, z,,] ])-P(;[U K.])


= P( Ui[ A n 9x [0,z]] "I U r[K,])
n=I n= I








< P(7r[ A n n x [,z.] ] \ i[K])




n=F


r,=I



For each index n there is an F-measurable function Z, : .Q -. , o] x [0, 00] such that [Zn] c Kn and 7[ [Zn]] = [K.] (see Proposition 1.12). Set f:= ZI,,If[z,] + E ZMn=2 lrZ.]] ", U g[zi]]
'-I

Then f is an Y-measurable function such that [f] c D and [ [I]] = ,[D]. Accordingly, we have [/ c A and P(4[A]) - P(r[ [M ]) < E. Theorem 1.20 Let Ao e p, co > 0, and let f: .Q - [0, oo] x [0, oD] be an Fmeasurable function such that [/] c A0. There is an element Ko e K, such that

" Ko cAo and
* P(4[1]) - P(4 [ n Ko ) < Eo.

Proof Let A be the collection of predictable sets A such that for any E > 0 and any T-measurable function g : 0 - [0, co] x [0, co] with [g] c A there is an element KeC, such that K c A and P([[g]]) - P(4K[n[g]]) < E. Let S,T be stopping times for (g,), and let s,i e R . First we show that the predictable set A := (S , T] x (s , t] is in A. Let E > 0 and let g be an f-measurable function such that [g] c A. Let (E,) be






33

a sequence of positive numbers decreasing to 0. For each index n, let K,[S+E,, TAn]x[s+E, t] E /C. We have K, /A. Hence we have P(r[ [g] ]) = P([ [g] n A]) = P([ [g] n U K])
n=I

= P(U r[ [g] l K,,)
n-I

= lim,, P(;r[ [g] n K,,]) since P is sigma-additive and since each set r[ [g] n K,, ] is an element of ,; see Corollary 1.8, <_ P(r[ [g] n KN ]) + E

for an existing index N. Note that we have KN e K6, KN c A, and P(ir[ [g]]) - P( [g]r l KN]) 5 E. Hence, we have A e A. We may similarly show that the set B x {0} x (u , v] is an element of A, where B E go and u,v E R+.

Next we show that A is closed under countable unions. It will follow that a monotone increasing sequence from A has its limit in A, and also that A contains the ring S generated by the sets (S , T] x (s, t] and B x {0} x (u, v] such that S,T are stopping times for (g), B e g0, and s,u,v e +. Let (A,) be a sequence from A. Set A' UAn
n-I
Let E > 0 and let g be an ,-measurable function such that [g] c A�.






34

For each index n, the set [g] n An c A, is the graph of the J-measurable function gl, [g].nA] + ccl (,r[[g]n.])c. Hence for each index n there is an element Kn G AC6 such that K. c A, and P(;r[ wg n An ]) - P(r[ [g n Kn ]) < 2We have P(r[ [g] ]) Pr[ [g] n A' ])


= P(I[ [g n UA.])
n-I


P(U r[ [g]n An )
n=1
NI
SP( ~l~~)+



for an existing index N1 E N. N,
Set K:= U K. c K5. Then K c A', and we have
n-I

NI NI P[ [g] ]) - Pr[ [g] n K]) < P(U r[ [g] n An ]) - P(U 7r[ [g] n K,) +
n=1 n=l

NI
< P(U(r[[g]nA.I\ r[[g]nK,])) + 2
n-I

NI
< P(r[[g]nA.]\r[[g]nKn]) +
n=l

NI
< + T
n<

< E.


We conclude that A' E A.








OO
Lastly, let (A,) be a decreasing sequence from A. Let A" :fn A. We will show
n-I

that A" - A. Since A contains S, and since aa(S) = i, we will then conclude that A = Vo. The statement of the theorem will follow as a consequence. Let E > 0, and let g be an ,F-measurable function such that [g] c A". Then [g] c A1, and so there is an element KI = KC6 such that KI c A and Por[ [g] ]) - P(;4[ [g] nK]) < The measurable graph [g] n K, is contained in A2, and so there is an element K2 e AC6 such that K2c A2 and P(r[[gi] n ]) - P(;r[][gnKnK2]) C2 * The measurable graph [g] n K, n K2 is contained in A3, and so there is an element K3 GC6 such that K3 c A3 and P(i[ [g] n K1 nK2 ]) - P(r[ [g] n KinK2nK3]) < 6

Continuing inductively we obtain a sequence (Ku) from /C6 such that for each index n-I n
n we have K. c A, and P(ir[[g]NNK1]) - P(tr[[gl nKJ) <2



Set K := nK,. We have K EC6, K c A", and
n-I

P(r[ [g] ]) - P(r[ [gn K]) = P(r[ [g]]) - P(G[ [g]N nKn ]) n-I
n
= P(,r[ [g]]) - P(N r[ [g] n K, ]) since [g] is a graph,
n-I

n
= P(ir[ [g]]) - lim_. P(7r[ [g] nf K ]) since P is sigma-additive,
i=I






36


-P(K[ [g]]) - P([ [g] n K, ]) + E P(7r[[g]fnn Ki]) - P(r[ [g]nfKi])
n=-2 (Pi--i=



n-l



We conclude that A" c A. We close Chapter 1 with the main result of the chapter, the cross section theorem. Theorem 1.21 Let A E k, and let c > 0. There is a stopping time Z such that

" [Z] c A and
" P(ir[A]) - P(;r[ [Z]]) < E.

Proof By Corollary 1.19 there is an Y-measurable function f satisfying

1.21a [] c A and

1.21bP(r[A]) - P(,r[W]) < By Theorem 1.20 there is an element K c AC, so that

1.21c K c A,

1.21d P(ir[ ]) - P(-[ RKI) n K 2, and

1.21e there is a stopping time Z such that [Z] c K and ,r[K] - r[ [Z]]. We have [Z] c K by 1.21e,

c A by 1.21c;

and P(r[A]) < P(r[ [A) + 2 by 1.21b,
2

< P(ir[[f]n K]) + E by 1.21d,

_ P(;r[K]) + E = P(r[ [Z]]) + E by 1.21e.














CHAPTER 2
PREDICTABLE PROJECTIONS In this chapter we define the predictable projection PX for a measurable two parameter process (X51). In addition, we demonstrate that the projection PX exists, and is unique.

Projections PX In the paragraphs that immediately follow, we motivate the definition of the predictable projection. Definition 2.1 Let Z be a stopping time for (F,,). The a-algebra YFz is defined by Fz = {AcFIAn {Z< (s,t)} cF,, V (s,t) E R'}. Proposition 2.2 Let Z = (S,T) be a stopping time such that { S < 00 } c { T
We have {S< s} = {S < s}l{T < oo} since {S< oo) c {T fl-I


U{z < (S,,n)}




Next we show that Gs c Fz.






38

Let Ae 9s. Then AcY and An{S < s} e q forevery selR+. Let s,t c R,.

We have An {Z <(s,t)} = An{S
So A ,'z.

Lastly we show that YFz c gs. Let A c Fz, and let s E R, and n e N. Then Ae Y, and AR{Z < (s,n)} c Y, = $.

Hence we have An{S < s} = An{S _ s}N{T < oo}

(because { S < oo } c { T < x } by hypothesis), UAn{Z < (s,n)}
nr-I

E .

So A E Gs. We conclude that YFz gs. Remark Let Z = (S,T) be a predictable stopping time. Since Fz = 9s (by Proposition 2.2),

- ,F (by assumption),

we are motivated to make the following definition for the predictable projection. Definition 2.3 Let X P x _ R be an TF� I,(R2)-measurable process. A predictable process Y: Q x 2+ R which satisfies E[ Xz 1 {> I q z] - Yz I Z





39

and is denoted PX. Note that because the process X is measurable and Z is a stopping time, the function Xz is T-measurable. Therefore, since ,Fz c JF, the conditional expectation E[ Xz I {z<> I ,Yz ] is defined when the process X is, for example, positive or bounded.

The Uniqueness of PX

We begin by providing ourselves with some tools that will enable us to prove that if a predictable projection PX for X exists, then it is unique. Definition 2.4 A subset B of n) x R2 is evanescent if there is a P-negligible set Nc Q such that B c NxR 2. Theorem 2.5 Let qO : Q x 2 -, R be a real-valued predictable process. If Tz 1 = 0 a.s. for every predictable stopping time Z then T = 0 outside an evanescent set.

Proof The proof is divided into four parts. First we show that if A c fd satisfies (1A)zl {z<,} = 0 a.s. for every predictable stopping time Z, then 1A - 0 outside an evanescent set. Let A E V be such that (1A)zI {z<.y = 0 a.s. for every predictable stopping time Z. Suppose P(7r[A]) * 0.

Then P(;r[A]) > 0 and so there is an E > 0 such that P(ir[A]) > E. By Theorem 1.21, for this E there is a predictable stopping time Z, such that

2.5a [Z,] c A, and

2.5bP([A]) _ E + P(,r[Z,]). We have the following chain of equalities and inequalities.









0 P({(1A)Zlz, 0)) by hypothesis,


- P({oIgGQ(,Z (L)) E A})

SP([ A A [ZJ])

= P(7r[ [ZWJ] ) by 2.5a, > P(,[A]) - E by 2.5b,

> C - E by assumption,

= 0, which is a contradiction. So we cannot have P(Yr[A]) * 0. Accordingly, IA = 0 outside an evanescent set. Second, we show that for any disjoint sets Ai e p and any numbers ai > 0, if


ailA, ) > 1 0 a.s. for every predictable stopping time Z then ailA, =0 outside an evanescent set. Let nEN, and for i = 1 ...n let Ai E and a >0. Assume that the sets A, are disjoint. We have

(n ailA) l{z<> = 0 a.s. for every predictable stopping ailA,)I q = as. or ver prditabe soppngtime Z


= for each index i, (ailA,)z lIzAo = 0 a.s. for every predictable stopping time Z

(since the sets Ai are disjoint),

= for each index i, (1Ai)z 1 Z = 0 a.s. for every predictable stopping time Z

(since each ai > 0),

= for each index i, Ai, = 0 outside an evanescent set (by the first part above),








n
=> ailA, = 0 outside an evanescent set. This completes the second part. Third, we show that if qo is positive and satisfies the hypothesis of the theorem, then T = 0 outside an evanescent set. Assume that the predictable process T is positive and satisfies oz 1 z<' = 0 a.s. for every predictable stopping time Z. Since (p is positive and predictable, there is a sequence ((,) of positive predictable step functions such that q,, n / T pointwise. So {p, > 0) n / {qT > 0) and ('On)ZlIZ<. n / PIozZ for every predictable stopping time Z.

Hence we have

2.5c P('r[ { Ton > 0 }]) / 7 P(,r[ { To > 01]) since P is sigma-additive (note 1r[{ (, > 0) ] and ;r[ {V > 0}] are in T by Corollary 1.8), and

2.5d for each index n, (pn)z I z<-c = 0 a.s. for every predictable stopping time Z.

kn
Now each step function To, can be expressed as ailA,, where each a, > 0,
i=1

and the sets Ai are disjoint.

In light of this, when we apply the result in the second part of the proof to 2.5d we obtain (p, = 0 outside an evanescent set; n e N. Then, by 2.5c we have (p = 0 outside an evanescent set. Fourth, we show that if (p is real-valued and satisfies the hypothesis of the theorem, then T = 0 outside an evanescent set. We have the following chain of implications.






42

(pz 1 0 a.s. for every predictable stopping time Z



- 0 a.s. for every predictable stopping time Z

- = 0 outside an evanescent set (note, Jqp is a positive

predictable process, so the third part of the proof applies); (p = 0 outside an evanescent set. We are now able to establish the uniqueness, up to an evanescent set, of PX. Corollary 2.6 Let (p, V K2 x R2 - R be two real-valued, predictable processes. If (pz 1 Vz a.s. for every predictable stopping time Z then (P = V outside an evanescent set.

Proof From the hypothesis we deduce that (T - y')z 1 4z
(Note that subtraction is valid since the processes T and V take values in a Banach space, R.)

The process V - V satisfies the hypothesis of Theorem 2.5, and therefore we conclude that - = 0 outside an evanescent set. Thus, / - outside an evanescent set. Proposition 2.7 Let X : Q x R2 --+ R be a real-valued, f� 1B(R2)-measurable process. If X has a predictable projection, then the projection is unique up to an evanescent set.

Proof Suppose (P Q2 x --+ R are two predictable projections for X. Then for every predictable stopping time Z we have






43

((p,)zl~z
((P2)z 1 {z<} a.s..

From Corollary 2.6 we conclude that T, = T2 outside an evanescent set, and the proposition is hence proved.

The Existence of PX

We will begin by presenting explicit forms of the predictable projection for two particular processes, X = H I [o,u]x[o,v] and X = I (z,I). Both forms will be used later in the paper, and one form will intimate that, desirably, a predictable process is its own projection. We will close by proving that every bounded, measurable, real-valued process possesses a predictable projection. Proposition 2.8 Let H r L(F), let u,v e R,, and let X: K x R2 --+ R be defined by XS,,U,) = H(m) 1 [o,.]x[o,v](s,t). Denote by (E[ H 13". ]), the function s '-* E[ H I JF, ], which is chosen (Dinculeanu 2000, p. 181) to be right continuous with left limits (cadlag). Then X has a predictable projection, and we have

(PX)S,(' y) = (E[ H Y-]) ,(t) 1 [o,,]j[o,v](s, t). Proof Let Y,,,(u):= (E[ H I T. ]), (t) I [o,t]4o,v](s.t) for every tu E Q and every s, t c R,. First we verify that the process (Y,,,) is left continuous. Since (E[ H 1F3. ]), is cadlag, (E[ H I Y. ]), is left continuous. Also, 1 [o,u]x[o,v] is left continuous. Therefore, (Y,,,) is left continuous. Next we check that for every s,t E R,, the map Y,,, is T,,I-measurable. Let st E R,+.

If s > u or t > v then Ys,t = 0, which is ,Ts,,-measurable.






44

If s< u and t< v then Yst = (E[ H -F.

= E[HI _]

(note, since E[ H . - ] is cadlag and s is a predictable stopping time, the Stopping Theorem (Metivier 1982, p. 87) applies),





Lastly we confirm that E[ Xz 1 {z< > I ,z ] = Yz I {q<.> a.s. for every predictable stopping time Z.

Let Z = (S,T) be a predictable stopping time. We have Yz 1 Zo = YST I
= (E[ H I F. ])s- I{z<(uv)>

= E[ H I Fis- ] 1 Z<(u,,)

(note, since E[ H I F. ] is cadlag and S is a predictable stopping

time, the Stopping Theorem (Metivier 1982, p. 87) applies). Also, E[ Xz I I Fz ] = E[ H 1 Z<(uv)> I Fz ] SE[ H I Fz ] I lz<(u,,)} since { Z < (u, v) } ,Fz,

- E[ H I Fs- ] 1 {z<(.uv) since ,z = Gs by Proposition 2.2, and since 9s = Fs- by assumption. Therefore we have E[ Xz 1 q





45

the predictable projection of 1 (z,.) is itself. Therefore we have P(O (z.)) - I (Z,.). Proof Let Z' be a stopping time. To prove the first assertion of the proposition, we must show that {Z < Z' < oo}n{Z' < (s, 0 } G, for every s, tR R+. Let s,t e R +.

Note for every r,q R R+ with r< s we have {Z < (r,q)} cFr,q


= GSr


Also, for every r,q E R with r < s and q < t we have A:={r < S' < s}fn{T' < t} = {Z' < (s,t)}fl{Z' < (r,t)}c



SGs, and

B:={S' < s}f{q < T' < = {Z' < (s,t)}n{Z' < (s,q)}c


- 'T .
= Gs.=

Hence we have {(r,q) < Z < (s, t)} An B



Therefore we have

{Z < Z' < oo}n{Z' < (s,t)} {Z < < (s,t)} U {Z < (r,q)}n{(r,q) < Z< (s,t)}
r,qcQ
r

e s.
We now prove the second assertion of the proposition. Then, we may be able to infer






46

that for any predictable process (X,,) that has a predictable projection PX we have PX= X.

Observe that the process 1 (z,) is left continuous. Also, for each s,t ( R + we have I (z,)(s,t) = l Z<(s,0. Since I z<(S,,)> is 37s,t-measurable, then I (z)(s,t) is also -F,t-measurable. Now let Z' be a predictable stopping time. We have E[ (1 (z,.))z, I Jz'<} I D I ] = E[ 1 {z '
- (1 (Zo))z, I z'
We have proved that P(1 (z,.)) = I(z). Theorem 2.10 Let X' � x+ --. R be a bounded, real-valued, 3� 11(R+)measurable process. Then X' has a predictable projection. Proof The proof is inspired by Dellacherie and Meyer's proof for the existence of the predictable projection of a one parameter process. The proof will unfold in five steps. In the first step we show that if X, and X2 are measurable, real-valued processes having predictable projections PX1 and PX2 respectively, then X, < X2 implies that PX 1 < PX2 outside an evanescent set. Thus, in particular, if IX I < K then IPXI < K outside an evanescent set. Let XI, X2 :1 x R2-+ R be two T� I(R2)-measurable processes having predictable projections PXI and PX2 respectively. Let Z be a predictable stopping time. We have






47



E E[ (XI)z I z
: (PX )Z I{Z
> (PX2- PXI)zlz<.> > 0 a.s.

((Px2 - PXI) 1PXPX<%)Z1{z
Since the predictable stopping time Z was arbitrary, we deduce that if X1 < X2 then (PX2 - PXI) lx,2--Px, 0 outside an evanescent set. We show secondly that if X is a measurable, real-valued process and (Xn) is a sequence of uniformly bounded, measurable processes such that each Xn has a predictable projection and Xn n / X, then X has a predictable projection and we have PX = liminf, PX .

Let (X") be a sequence of measurable processes Xn : Q x R2 -- R, n E N, and let X �Qx + - R be a measurable process. Assume that the sequence (Xn) is uniformly bounded, that each Xn has a predictable projection PXn, and that X" n, / X pointwise. Let Z be a predictable stopping time. Then (X.)z I Zo " ,/ Xz I z I ,z] n / E[ Xz z<> I ,z ] a.s. and in L' (P). Thus we have






48

E[ Xz I Z I ,'z] = lim E[ (Xn)z I z n

- liminf (PXn)z I iz<>, a.s.
n

- (liminf PX"") lz<1 .
n )z

In our third step we prove that if (X") is a sequence of bounded, measurable processes each having a predictable projection, and X is a measurable process such that Xn --- X uniformly as n - oo, then X has a predictable projection and we have PX = liminf, PX . Let (X") be a sequence of bounded, measurable processes Xn x --+ R, n e N, and let X �g2 x 2+ _ R be a measurable process. Assume that each X" has a predictable projection PX", and that X" _ X uniformly as n - 0.

Since each Xn is bounded and X" -- X uniformly, X is bounded. Hence, E[ Xz I {z I ,Fz] exists for every predictable stopping time Z. We have E[ (Xn)z I qo ,z ] - E[ Xz I jz<.o I ,z ] a.s. and uniformly as n oo, for every predictable stopping time Z. So for every predictable stopping time Z we have E[ Xz 1 {z<> I ] = lim Ef (X")zl z< } I , z] a.s.
n

= liminf (PX")z 1 a.s.
n


-(liminf npXn )iz I~}






49

In this fourth step we verify that the projection p is linear. Let X,Y" g2 x R- IR be two processes that have projections PX, PY respectively, and let a,b e R.

We will prove that aX + bY has a predictable projection, and P(aX + bY) = a PX + b PY. In fact, for every predictable stopping time Z we have E[ (aX + bY)z 1z
- a(PX)z 1 {z<- + b(PY)z 1 z< a.s.

- (a PX + b PY) z 1 Z<0o>.

Lastly, let B be the set of all real-valued, bounded, Yf� g(RE)-measurable processes, and let ?f be the set of all 'F� Ig(R2)-measurable, bounded, realvalued processes which admit a predictable projection. We will show that R = B. From the four steps above we deduce that 'H is a vector space that is closed under bounded monotone convergence and uniform convergence. Further, h contains the process 1, since P1 = 1. Let C be the class of bounded, measurable processes X : nx 2 _, R of the form XS,1(u) = H(w) I[o,u][o,,](s,t), where H E LD(,T) and u,v E R. Then C is closed under multiplication. Also, C c 7- by Proposition 2.8. Thus, by the Monotone Class Theorem (Dellacherie & Meyer 1975, p. 14-I), 7- contains all processes which are bounded and oa(C)-measurable. But ca(C) = y� 1J(R2). Hence H = B.














CHAPTER 3
PREDICTABLE DUAL PROJECTIONS In this chapter we present the predictable dual projection XP for a two parameter process X. It will be shown that if XP exists then it is unique, and that in the presence of a step filtration, XP exists when X is right continuous, measurable, and has integrable variation.

We begin with the definition of a step filtration.

Step Filtrations (YI)

Definition 3.1 Let E > 0. A filtration (Y') is a right continuous E-step filtration if

Y = ' for every a > 0, where LaJ denotes the largest integer that is less
ac [a] C

than or equal to a.

A filtration (Y') is a left continuous E-step filtration if YF' = 'T' for every a > 0, where Fal denotes the smallest integer that is greater than or equal to a. Remark A step filtration (FT) is of interest to us because

" it facilitates a property that is assumed of the filtration (), namely that
Gs = Fs- for every predictable stopping time S for (9,), and
" it causes (G,) to be a left continuous step filtration, a property that will help us
later to demonstrate the existence of a predictable dual projection.

Proposition 3.2 Let E > 0, and assume that (YI) is an E-step filtration. Then

" the filtration (G,) is a left continuous E-step filtration, and
" for every predictable stopping time S for (G,) we have Gs = FsProof We prove the first assertion of the proposition. Let so E R,.








We have Go =Y,,= ca (U 0 r)


Y ,so if so =E for every n eN
Fs0- if so = nE for some n e N

since (,F,) is a right continuous (see Notation 1.1 f)) E-step filtration. Hence, the filtration (g) satisfies G,, = GrlI for every a > 0, and Go = 9'. As such, (g.) is a left continuous E-step filtration. Next we prove the second assertion of the proposition. Let S be a predictable stopping time for (G,). We must show that 9s = YsFirst we show that Ys- c Gs. Let B be a generator of Ys-. Then B = A n{S > s} for some s- R+, and AEcFs. Hence, for every tc R, we have

Bn{S< t} = An{S> s}n{S< t}

= r0 if s>t
A n{s
C 91.

So B c gs. Since gs is a --algebra, it follows that Ys- c Gs. Next we show that gs c s-. Let A E Gs. We have A (An{S< 0)) U U[An{S< nE}n{S > (n-)E}1



Bou UBnn{S>(n-1)E},
nr-I








where B0 = An{S< o},

c Go since A c gs and S is a stopping time for (G),

= 'TO



andwhere B, = An{S< nE}, n cN,

S,, since A c Gs and S is a stopping time for (9,),

= CT(nj)r.

We deduce that A E Ys-.

Predictable Dual Projections XP We begin by introducing some terms (Dinculeanu 2000, p. 363-390) we will use during the definition of the predictable dual projection, as well as beyond. Definition 3.3 Let X : x R2 -- R be a two parameter process.

3.3a (X,,) is right continuous if for every s0,t0 e R and tu E Q, we have

lir X5,,(U) = Xo00(w).
(s,t)-.(so,to)
to
3.3b Let o e Q and let s < s' and t < t' be elements of R+. The increment of X(tur) on the rectangle R:= (s, s ] x (t, t'], denoted AR X(rU) or A (st),(S , X(rU), is defined by AR X(CU) = X',,,(u) + X,,, (tu) - - X ,(m). One might think of Xz(o) as measuring the "area" of the rectangle [0 , z], for every z G R2. Then, one would see that AR X(m) delivers the area of the rectangle R.

3.3c (X,,) is increasing if for every Qi e Q and every z < z' from R2 we have X. < X,.






53

(XSI) is incrementally increasing if for every o c K and every z < z' from R2 we have A ,X(c) > 0.

3.3d Let o c 92 and let I,J c R+ be intervals. The variation of the function

+ - on the rectangle I x J, denoted var( X(ay), I x J ), is defined by

var( X(m), I x J) = sup E I A(s,s,+,1]x(jj,,,] X() 1, where the supremum is taken over ij

all divisions so < s, < ... < s, of points from I, and all divisions to < tj < ... < tm of points from J.

3.3e The variation process IXI: X R* R+�is given by IXlz(r) = var(X(w) , (-o0, z]) for every m E n and z E R2, after extending X to Q x R2 by setting X, = 0 for every w e 2 \ +.

(Xl,,) has finite (bounded) variation if for every ru c 0, the function IlX.(u) is finite (bounded).

(XS) has integrable variation if

" (Xs,,) is F� I(R2)-measurable, and
" the total variation IXl := sup IXk is P-integrable.


3.3f Let U E Q. Denote by K the ring of subsets of R2 generated by the rectangles [0 , z]. The measure associated with the function X(u) is the additive set function mx(W) : 7Z -, R defined by mx(W)( [0 , z]) = Xz(o) for every z e R2. Note that it follows that mx()( (s, s'] x (t, t'] ) = A(t, ]X(,']X(tu) for every 0 < s < s' and 0 < t < t'. Remarks If (Xsj) has finite variation then the variation process IlX is increasing and incrementally increasing. If (Xs,/) is incrementally increasing and right continuous then






54

for each ay c n, the measure mx(,) is sigma-additive on 7. Furthermore, if the process (X,1) is right continuous and has bounded variation then for each w f, the measure mx(,) can be extended uniquely to a sigma-additive measure on R(R 2) that has finite variation Imx(,)l on g(R2), and we have Imx()l - mx()I on 1R. Therefore, if (Xst) is right continuous and has integrable variation, then for each u G 2 and each g(R)-measurable, Imx(,)-integrable, real-valued function f: R2 -+ R, the Stieltjes integral f f dmx(,) is defined and is often written J f dX(zu).

If (XS,,) is right continuous and 3� l(R2)-measurable, and has integrable variation, then for any bounded, Y� g(R2)-measurable, real-valued process T, the expectation E[ f p dXz ] is defined and is finite.

Definition 3.4 Let X : .2 x R2 - R be a right continuous, Y� fl(R2)-measurable process with integrable variation IXI. A right continuous, predictable process Y" Q x I+ R with integrable variation IYI is called a predictable dual projection for X if for every bounded, F� BJ(R2)-measurable, real-valued process T we have E[ J TdY] = E[ J Pqi dX ]. The predictable dual projection Y of X is denoted XP.

The Uniqueness of XP

We now show that if a predictable dual projection XP for X exists, then it is unique up to an evanescent set.

Proposition 3.5 Let X : x 2R __ -+ be a right continuous, measurable process with integrable variation. Assume that X has a predictable dual projection Y. Then Y is unique up to an evanescent set.






55

Proof Suppose that X has two predictable dual projections, Y, and Y2. Let Z be a stopping time.

Set A:= { (Yl)z lz< o> > (Y2)z lz<> } and T(tu,s,t) := l[oz](cu,s,t) IA(UJ) lz<}(U). Note that A c ,F and To is real-valued, measurable, and bounded. We have E[JTdY, ] = E[ JPT dX] = E[ f T dY2 ]. So J (Y,)z lz< q, dP = J(Y2)z l z<.> dP (see Definition 3.3f).
A A

Therefore J ((Y,)z lZ< - (Y2)1 i z<->) dP = 0.
A

Similarly, setting B as { (Y)z lz<.} < (Y2)z 1 Z<-. } and Tp'(tu,s,t) as 1[0,zj(,s,t) IB(G) l{z< y(w) leads to J ((Y2)z l B

Since P is a positive measure, we conclude that (Y])z I {z<.> = (Y2)z 1 zo almost surely. In particular, we have (Yi)z 1 jz<.y = (Y2)z l z
Since Y, and Y2 are predictable, by Corollary 2.6 we have Y, = Y2 outside an evanescent set.

Predictable Dual Projections of Measures

We have defined the predictable dual projection XP for a process X. In this section we will present the predictable dual projection mP for a measure m. We will call upon mP in order to establish the existence of XP. Definition 3.6 Let E be a Banach space and let m : y-� I,(R2) -, E be a sigmaadditive measure. We say that m is a stochastic measure if m vanishes on evane-






56

scent sets. As an abbreviation we say that the measure m is stochastic. Definition 3.7 Let m +� (R ) -R R be a sigma-additive measure. The measure mp: � F I (1 ) -, R defined by mP(A) J P(1,) dm for every A c .� f(R2) is called the predictable dual projection for m. Proposition 3.8 Let m �F� 1 (R2) - R be sigma-additive and stochastic. The following three assertions are true.

" The measure mP is sigma-additive.
" The measure mp has finite variation ImPI which satisfies ImPI < ImIP.
" The measure mP is stochastic.

Proof First we show that mP is additive. Let A, B be disjoint sets from YO 1(R2). mP(A U B) = JP(AUB)dm


= J P(1A + IB) dm since A and B are disjoint,


- J (P(lA) + P(lB) ) dm since the measure m is a stochastic measure,

- P(1A) dm + J P(1B) dm since both integrals exist,

= mP(A) + mP(B).

Next we show that mp is sigma-additive. Let (A,) be a sequence from Y� I+() that decreases to 0.

Then P(IA) decreases pointwise to 0 (apply the second step in the proof of Theorem 2.10 to the sequence (1 - IAJ,). So J P(IA) dm -- 0 as n - oc, since m is sigma-additive with finite variation. Thus, by equality, mP(A.) -- 0 as n - oo.






57

We next prove the second assertion of the proposition. First, note that a real-valued sigma-additive measure on a a-algebra has finite variation. Accordingly, the measure Iml is a real-valued sigma-additive measure on the a-algebra Y� IB(R2). Thus by the first assertion of the proposition, the measure ImIP is sigma-additive, and so is finite. Further, for every set A c Y� R(R2) we have ImP(A)l =I f P(1A) dm I

< J P(A) diml

-ImIP(A).

It follows that the measure mp has finite variation satisfying ImPI < ImIP. Lastly we prove the third assertion of the proposition. Since the measure m is a stochastic measure, for every evanescent set A e Y� R(R 2) we have m(A) = 0. It follows that for every evanescent set A we have Iml(A) = 0. Let A F� Yo( 1R) be an evanescent set. We must show that mP(A) = 0. Since the set A is evanescent, there is a P-negligible set N c Q such that A c N x R. We have ImP(A)I = I J P(IA) dm I

< J P(IA) dlml

< J P(1NAR)2) diml (refer to the first step in the proof of Theorem 2.10 after noting that Iml >_ 0 is stochastic); SJ INxR2 diml because P(1N.R2) = N2R+ (the proof of this is similar to that of Proposition 2.9);

- Iml(N x R2), which is 0.






58

Processes Associated With Stochastic, R-Valued Measures

We now prove a theorem that will underpin our construction of a predictable dual projection in the next section.

Theorem 3.9 Let m :� I(R) -, R be a sigma-additive, stochastic measure. There is a right continuous, T� 1g(R2)-measurable process X : Q x R2 --+ R with integrable variation IXi such that for every process V : Q x R-2+ R in LI(Iml) we have

" Jq dm : E[(JpdX] and

*" f dlml E[ f , dXlz ].

Proof Since the measure m is real-valued and sigma-additive on a c--algebra, the measure m has finite variation.

So Iml is a finite, positive, stochastic, sigma-additive measure on ,F� 1G(R2). There is (Dinculeanu 2000, p. 390) an increasing, incrementally increasing, right continuous, y� 1g(R2)-measurable process V: 0 x R2+ R with integrable variation IVI such that

3.9a Iml(M) = E[ J IM dV ] for every M + ,.� 1(12). For each z C R2 define the measure m Y -, R by mz(A) = m( [0, z] x A) for every set A e F.

Note that the measure mz is sigma-additive with finite variation lmzl. Since m is a stochastic measure, we have m: << P. (Note that mz << P means that if the set A c Y" satisfies P(A) = 0, then mz(A) = 0). So by the classical Radon-Nikodym theorem (Dinculeanu 2000, p. 36), there is an






59

-measurable, P-integrable function Yz .Q , R such that

3.9b mz(A) J Y_ dP for every A E Y.
A

Next, let z, = (s,t) and Z2= (s',t'), and consider the measure mz2 - mi, : F --. R. The measure mz2 - mz is sigma-additive and has finite variation. Further, we have i-n2 - mi << P. So by the classical Radon-Nikodym theorem (Dinculeanu 2000, p. 36), there is an Y-measurable, P-integrable function Y, -such that

3.9c (mZ2 - mz')(A) = J Y-,2 dP and
A

3.9d Im2 - m2 I(A) = J Yz,2I dP for every A e ,F.
A

From 3.9b and 3.9c we deduce that for every zI < z2 we have Y2z - Y-2, =Y,2 a.s.; that is, outside a P-negligible set NZ,2. Therefore, from 3.9d we obtain

3.9e Imzz - mz' I(A) f J YZ - Y2, dP for every A c Y.
A

Also, the measure m2 + mi - m(s, ) - m(S,t) :F -, R is sigma-additive, has finite variation, and is absolutely continuous with respect to P. So by the classical RadonNikodym theorem (Dinculeanu 2000, p. 36), there is an Y-measurable, P-integrable function YZQ2 R such that

3.9f (mz2 + m' - m(s) - m(S))(A) J YIZ,2 dP and
A

3.9g 1m-2 + m, - m(st) - m(S,)(A) f JIY-,I2 dP for every A c F.
A


From 3.9b and 3.9f we deduce that for every zi < Z2 we have






60

Y + Y'IY' outside a P-negligible set N' Therefore from 3.9g we obtain

3.9h Im-2 + mZI - m(S' .)- m (St)I(A) = IYZ2 + YZ, - Y,', - YSt' dP for every set
A

A e .

For every zI = (s,t) < z2= (s',t') and A E- we have I(m-'2 - mz')(A)l = IM( [0 , Z21 x A )- m( [0 , zj] x A )1

= Im( ([0 z21- [0, z) x A)1 < Im( ([0, Z2]- [0, z]) x A). Hence for every A E Y we have ImZ2 - mz,lI(A) < Iml( ([0 , Z21 - [0, Z1) xA )

SE[ J 1 ([oz2]-[o,z])xA dV] by 39a,


f J (VZ2 - V=,) dP (see Definition 3.3f).
A

Soby 3.9e we have J IYZ2- Y1IdP < J (VZ2- V,)dP for every AcY.
A A

Since all functions involved are Y-measurable, we have

3.9i JYz - YzI < Vz2- VZ, outside a P-negligible set MZ,=2 Further, for every zI = (s,t) < z2= (s',t') and A e ,T we have I(mZ2 + mI, - M(S ,I - m(S'I ))(A)I = Im( (z, , Z21 x A )1

- Iml( (Z] , Z21 x A).

Hence for every A c ,T we have ImZ2 + mI, - m (SA) - m(St)I(A) < ImI( (z1 , z2] x A )









f E[J1(AZz]X dV ] from 3.9a,

- J~IVdP.
A

So by 3.9h we have f [A I2 YJ dP < f Az,2 V dP for every A e F.
A A

Since all functions involved are J-measurable, we conclude that

3.9j IA:,_2 YI 5 Az,:2 V outside a P-negligible set M' Set M U (M1, u M',2). Then the set M is P-negligible.
Z"1, '2CE Q2

Fix M E Mc. We now show that for any z e R and any sequence (z.) from Q2 that decreases to z, the limit lim Yz,,(tu) exists. In fact, let z + li, and let zn \ z with z + Q2 for every n e N. Then we have Vo) -, Vz(o) since V is right continuous. So the sequence (Vz(z))" is Cauchy. Hence by result 3.9i, the sequence (Y.(tu)), is Cauchy. Since R is complete, the function Y.,,(w) has a limit as z, \ z. Next we show that this limit is independent of the choice of (z,). In fact, let z,, z' \ z, with z,, z' Q2 for every m,n ( N. Then there are real numbers L, and L2 such that Y. () --+ L1 and Y_,(m) -- L2. We are able to find subsequences (z,,) and (z') of (zn) and (z',) respectively, such that the sequence z,,,, zm', Zn,, ZM2, Zn,, ZM... , denoted (u,), decreases to z from Q2. We have Y,,(u) -- L as n -- oo, for some L E R. We deduce that LI = L = L2. Define the process X : n x R 2 -_ R by






62

li{ Y.(n) if m c Mc and z - R2
XJUI n'-, z.
X(u()) C zQ

0 otherwise
The process X is right continuous, and for each z (- R2, the random variable X: is Y-measurable. Hence the process X is T� I?(R2)-measurable. In addition, for every z, < Z2 e + we have

3.9k IAz1-2Xl < Az,=2V on fQ (apply the definition of X to 3.9j). We now show that the process X has integrable variation. Let n e N and let Ri = (zi, z'], i =1 ... n, be disjoint rectangles in R2.

n n
We have IARXl < AR, V by 3.9k,
i=1 iil

< IVoA.

Since the process X is right continuous, we may take the supremum over all finitely many disjoint rectangles with endpoints in Q2 to obtain IXl. _ IVl. and IXko G Y. Integration with respect to P is permitted, and we have JiXlI dP < JIVI dP

< co.

This establishes that the process X has integrable variation. We note from the third remark on page 66 that the measures mx(,) and mm(,) can be extended uniquely to a sigma-additive measure on 1R(R2) for almost all tu e . Next we show that for every z E R2+ we have X, = Y- almost surely. Fix z G R2. Let (u,) be a sequence from Q2 decreasing to z. For every index n, we have from 3.9i IY.. - YzI < V. - V. outside a P-negligible set MU.=.








0o
Set M' :=U M,,=.
n=]

Then the set M' is P-negligible. Further, the inequality IY,, - Y-i < V,,- Vz is valid on the set M'c for every index n. Let n -- o. We obtain IXz - Yzl < 0 on M'c. In other words, we have X_ = Yz almost everywhere. Finally we show that for every P E L'(Iml) we have J q dm = E[p "dX,] and f p dlml = E[ fJ pz dlXl-]. Note from the last remark on page 67 that the expectation E[ f IA dX ] is defined and finite for every set A e ,T� 1(RI). Let the measure px: ,,� Iy(R2) -_ R be defined by yx(A) = E[ J 1A dX ] for every set A c F� (FQR+2). Then the measure px is sigma-additive (Dinculeanu 2000, p. 394). Now let z + and A e ,F.

We have m( [0, z] x A) = mz(A) by definition, f Yz dP by 3.9b,
A

f Xz dP since X = Yz a.s.,
A

- E[ J 1 [o]�A dX ] (see Definition 3.3f,

- px( [0, z] x A).

We may similarly show that m( (z1 , z21 x A) = J A,_,X dP
A


=px( (z] , z21 x A ) for every Z| < z2 from R+.






64

We deduce that the sigma-additive measures m and yx agree on a ring of generators of 2-� 1g(R2), and this ring includes the whole space K2 x R2 Hence, we have m - /x on y� 1(R2). We apply a theorem (Dinculeanu 2000, p. 394) to deduce that for every process (p c LI(Iml) we have J ( dm = E[ J qp dX]

an "( dlml = E[ f "p dlXlz ].
and J~~I E~zI~l


The Existence of XP

In this section we establish conditions under which we are guaranteed the existence of the predictable dual projection XP for a two parameter process X. Many of the tools that were furnished in the preceding sections will be employed. Theorem 3.10 Let X : x - R be a right continuous, 0� I3(RE)-measurable process with integrable variation. Assume that for every o g1(R2)-measurable, bounded process T:Lx R2 -- R we have E[ J dX] = E[ PT dX]. Then X is adapted to (,,f,,).

Proof Fix u,v E=- R2. We must show that Xu,, is F. = gu-measurable. Since the process X is measurable and has integrable variation, we have Xu, e LI(T). Further, LI(Gu) is a closed subspace of L'(,). Nowsuppose that Xu, c L(31) \L'(G). Then there is a continuous linear functional T : LI (J) -- R that is 0 on LI (G) and is nonzero at XU'..

Since the dual of L'(F) is L�(,f), there is an essentially-bounded, ,-measurable function H -: K2 - R such that T(g) = f Hg dP for every g c









We have E[ H I] = 0. In fact, let A c .. Then we have IA c L'(g.). So we have 0 - T(IA) (since T is 0 on L'(g.)),

f HIA dP


f HdP.
A

Since A E -g was arbitrary, we conclude that E[ H .] = 0. Set p := H I [o,]x[o,v]. Then (Pq,),,(zu) = (E[ H I Y. ]),(t) [ou]x[ov](st) by Proposition 2.8, where the function x ,-, E[ H I ox ] is chosen to be cadlag. We have pq = 0 outside an evanescent set. In fact, let s,t c R,. First assume that s > u. Then we have I [0,1(s) = 0. So we have I [o,][o,](s, t) = 0. Next, assume that s < u. Then we have Y', c g. So for every A e Y, we have J E[ HI Y, ]dP JHdP
A A

- J E[ H I g. ] dP (since A e 9u),
A

0.

Hence we have E[ H j F-] - 0 outside a P-negligible set N,. Set N:= U Nr.
r
Then we have E[ H I,-] = 0 outside the P-negligible set N, for every r e Q with r < u.






66

Since the function x F-+ E[ H F, ] is left continuous, we deduce that E[ H I Fs ] = 0 outside the set N.

We conclude that Pp = 0 outside the evanescent set N x R2. Therefore we have

0 = E[J PpdX]


= E[ J (p dX ] by hypothesis,


= E[ J H 1 [o,u]x[o,v] dX]

E[ H X,,

=T(X,,,,).

But T(X,,) # 0. From this contradiction we deduce that X,, c L( Corollary 3.11 Let E > 0, and assume that the filtration (Yt) is a right continuous E-step filtration. Let X � x+ --+ R be a right continuous, 3� 1 (R+)-measurable process with integrable variation. Assume that for every F0 1I(R2)-measurable, bounded process (p :0x R2 , IR we have E[ ( dX] = E[ JP qdX]. Then X is a predictable process for the double filtration (FY,t). Proof We will show that the process (X,,,) is the limit of left continuous, adapted processes. Note that since the process (X,,1) is right continuous, it is the pointwise limit as n - oo of the sigma step processes


:= X0,0 1 + xX I 1(,LL ] , i - ic ' n E N.
iF 1 i - 2 i

In addition, the process (Xs,,) is adapted to (,,t) by Theorem 3.10. Observe that the process X0,0 110o4o] is left continuous and adapted to (F,,,).






67

Now fix n, i, and j c N, and consider the process X _,i := X,, j, I , Is O )
2" 2 ] 2 ' 2

Since the process (X,.,) is adapted to ( F.), the function X ,c j. is ,T ,c j measurable; that is, G,, -measurable.


Since (,) is a right continuous E-step filtration, then (G,) is a left continuous E-step filtration (see Proposition 3.2).

Hence, the function X,, ,, is G,-measurable for every s e ((2> , Consequently, the process X",J is adapted to (G). Further, the process Xn",J is left continuous. We conclude that the process (X,,) is predictable. Remark If a cross section theorem for optional subsets of f2 x R2+ can be established, then using such a theorem, we would be able to obtain the result in Corollary

3.11 without the requirement that (F) is a step filtration. In such an event, we would be able to demonstrate the existence of a predictable dual projection without requiring that (,T) is a step filtration.

Theorem 3.12 Let E > 0, and assume that the filtration (f) is a right continuous E-step filtration. Let X x+ --+ R be a right continuous, Y� 1(l0)-measurable process with integrable variation. Then X has a predictable dual projection for the double filtration ( 'F,).

Proof The proof will be very similar to the proof in Dinculeanu 2000, p. 278, which supplies the existence of a predictable dual projection for a one parameter process. Since the process (X,,) is right continuous, measurable, and has integrable variation,






68

then there exists (Dinculeanu 2000, p. 394) a sigma-additive, stochastic measure pjx: ,7� T 3(R ) --, R with finite variation Ipxl such that for any process T in L'(px) we have Jpdx= E[ f (dX ]. Consider the measure (+x)p :� 13(R2) -By Proposition 3.8, the measure (px)P is stochastic, sigma-additive, and has finite variation.

Thus, by Theorem 3.9, there is a right continuous, ,� 1I(R2)-measurable process

x - _ with integrable variation IYI-, such that for every bounded,

-� 1(R+)-measurable process +, � x -- R we have

3.12aJ p d(x)P = E[ J (p dY ]. On the other hand, for every bounded, measurable process q we have

3.121b d(/x)P = E[ f Pq0 dX ]. In fact, for every measurable step process qp we have 312c J (p d(px)P = . PT d/x. Now apply Lebesgue's theorem (Rudin 1976, p. 321) in L'((px)P) and L'(/ux) to obtain result 3.12c for every bounded, measurable process T. So we have J T d(px)P = f PT dux = E[ J P(p dX ] for every bounded, measurable process T'.

Combining results 3.12a and 3.12b gives

3.12d E[ f 1'q' dX ] - E[ J ( dY ] for every bounded, measurable process (p. Also, note that the process Y is predictable.






69

In fact, for every bounded, measurable process ( we have E[f PpdY] f PT d(px)P by 3.12a,
f PT dux by 3.12c,



= J d(/ux)P by 3.12c, - E[JdY] by 3.12a.

Now apply Corollary 3.11 to the process Y. Since the process Y is predictable, right continuous, has integrable variation, and satisfies 3.12d, then by Definition 3.4 we have Y = XP. Remark If the process X is right continuous and measurable, and has integrable variation, then the limit lim X., denoted X-0 exists for every zo E R2, as z/zO
Z
well as as when zo is one of the infinities. Note that the limit X,_ is denoted X.v, while the limits X,,1-- and XS,-_ are respectively denoted X.,, and Xs,.. Proposition 3.13 Let E > 0, and assume that the filtration (,fF) is a right continuous

-step filtration. Let X : Q X R2 -+ R be a right continuous, measurable process with integrable variation. Assume that X is adapted to (,cs,,), and hence is predictable. Then XP = X.

Proof By Theorem 3.12, X has a predictable dual projection Y. We will show in five steps that X - Y = 0 outside an evanescent set. First we show that for every A e T0 and t E R, we have J (X",t - Y,,) dP = 0.
A

Let A E ,Yo and t E R,. Then we have P(1Ax[(o), (-,t)]) = 1Ax[(O,O),








So we have f X.,, dP -E[ f I Ax[(O,O) , (-,t)] dX]
A

= E[ J P(1A�[(oo) (cA0,)]) dX ]


E[ J 1 Ax[(OO), (00,)] dY ]


J Y,,, dP.
A

Hence we have J (Xo, - Y.,t) dP = 0.
A

Second, we show that if Xo,,, = Y.,, for every t E R, then X = Y outside an evanescent set. Assume that X,,, = Y.,t for every t E R+. Let s,t e R, and let A E G,. Set S :=S A + 0 01A and Z:= (S,t). The functions Z, (S,0), and (0,t) are stopping times, and so by Proposition 2.9 we may write

= P(1 R2 + I (ZGO ) - ((SO) 00) - ((O,t), )



1 1 0 ) -*


So we have J Xz dP = E[ fJ (oo),z] dX] E[ f P1 [(oo0), Z] dX E[J I [(o0o)Z I] dY]


f Yz dP.






71

Therefore we have 0 - J (Xz - Yz) dP f (X,, - Y,,,) dP + J (X , - Y ,)
A A'

(note, the integral [ (X, - Yz') dP is equal to J (xs, - Y,,) dP + J (X, - Y,)dV
A Ac

when Z'= (S IA + s' IA' , t) for any s' > s; now let s' /oo );

= J (X,t - Y,,) dP (since X. = Y.,, by assumption).
A

Since the set A e gs was arbitrary, and since the random variable X,,t - Y,, is g,measurable, we conclude that X,, - Y,,t = 0 outside a P-negligible set Nsj. Since X and Y are right continuous, we deduce that X - Y = 0 outside an evanescent set.

Third, we show that ((X,, - Y-t)lR2(s,t))P = 0. Let AEY0 and re R+. Bywriting Ax{0}x [0,r] =Ax [ \ ( we deduce with the help of Proposition 1.3 that the sets A x R2 where A E ,Y0, and the sets (Z , oo) where Z is a stopping time, together generate p. Next, note that the process (X.,,- Y.,,)1,R((st) is right continuous. In fact, let G E Q and let t > to. Then we have [(0,0), (ao,t)] \ [(0,0), (octo)] as t \ to.

So we have mx(,)( [(0,0) , (oo,t)]) - mx(.)( [(0,0) , (co,to)]) as t \ to; that is, we have X,,,(tu) -- X.,to(r) as t\ to.






72

Moreover, the process (X,,, - Y,,I,))1 R(s,t) is measurable and has integrable variation. Hence by Theorem 3.12, the process (X ,, - Y.,) IR2(s,t) has a predictable dual projection.

We claim that it is the zero process. Observe, the zero process 0 Q x R2 -- R is right continuous, predictable, and has integrable variation. Also, for every bounded, measurable, real-valued process (p we have E[ J PT d((X.,t - Y.,,) 12(s,t))] - 0 = E[ J p dO]. In fact, observe that for every stopping time Z = (S,T) we have

E[ J l(Z,.) d((X.,, - Yo,1) IR2(s,t)) ]

- E[ (X , - Yco) + (X.,T - Y.,T) - (X. - Y.) - (X.,T - Y.,T) ]

0,

while for every set A c 0Fo we have E[ J 1AXR2 d((Xoot - Y ),t) 1I )] = J (Xo. - Y.) dP
A

= 0 by the first step, after letting t /oo. So since the sets A x R2 and (Z, oo) generate p (where A E Yo and Z is a stopping time), we deduce that E[ J Vi d((Xo,, - Y-,) lR2(s,t))] = 0 for every bounded, predictable, real-valued process V/. In particular, for every bounded, measurable, real-valued process T we have E[ J PT d((X.,, - Y.,J) IR2(s,t))] = 0, SE[ fJ dO].






73

From Definition 3.4 we obtain ((X1,1 - Y ,,) lR2(s,t))P = 0. Penultimately we show that X - Y = (X.,. - Y,,.) 1R , up to an evanescent set. Let the process X' : f x R 2 -- be defined by X',, = X,, - (X- Yo,,,t) IR 2(s,t) for every s,t E P+.

The process (X',,) is right continuous, measurable, and has integrable variation. So by Theorem 3.12, the process (X',,) has a predictable dual projection. We have (X')P = XP - ((X . - Y I.) 1R)p since the predictable dual projection operator is linear;

= Y - 0 by the third step.

Also, we have X'M.t = Y , for every t R +. Therefore, by the second step we have X' = Y outside an evanescent set.

This means that for every s,t c R, we have X,,, - Y,, = (X.,, - Y.,') lR2(s,t) outside a P-negligible set that does not depend on s and t. Lastly we show that X = Y outside an evanescent set. Let to E R+. From the penultimate step we have in particular the result Xoo- Yoo = X :o- Y.,10 almost surely. Since X and Y are adapted to the filtration (F,,'), we conclude that the function Xo,,- Yo,1, is Fo-measurable. It follows that the function X.,10 - Y.,to is ,omeasurable. So from the first step we deduce that X.oo - Y*,1o = 0 almost surely. Since X and Y are right continuous, we conclude that the process (XIo1 - Y-) 1 R2 (S,t) is 0 outside an evanescent set. Then, we conclude from the penultimate step that X = Y outside an evanescent set. We may take XP = X.






74

Corollary 3.14 Let E > 0, and assume that the filtration (,,) is a right continuous E-step filtration. Let X : E X R2+ --+ R be a right continuous, measurable process with integrable variation. Then X is predictable if and only if we have E[ f PT dX ] = E[ f (p dX ] for every bounded, measurable, real-valued process (P. Proof This result follows from Corollary 3.11 and Proposition 3.13. Proposition 3.15 Let E > 0, and assume that the filtration (,T,) is a right continuous E-step filtration. Let X,Y : f X R2 - R be two right continuous, predictable processes with integrable variation. If X0,0 = Yoo, Xo,a = Y0 , and X.,o = Y,o, and if E[Az, X] = E[Az,. Y] for every stopping time Z, then X - Y outside an evanescent set.

Proof Set A X - Y. Assume that Xo,o = Yo, X0,. = Yo,w, and X,o = Y.,o, and that E[Az,,X] = E[Az,,Y] for every stopping time Z. Then we have

3.15a Ao,o = 0, Ao,m = 0, and A.,0- 0, and

3.15b E[Az,.A] - 0 for every stopping time Z. Let Be Y O, and set Z := (0,0) 1B + 00 1 B. The function Z is a stopping time, and so applying 3.15a and 3.15b we obtain

3.15cfA. dP = 0 for every BE Yo.
B

Since A is right continuous and predictable, and has integrable variation, then there is (Dinculeanu 2000, p. 394) a sigma-additive, stochastic measure PA :F� I(R2) -+ R with finite variation such that for every bounded, measurable, real-valued process p we have









3.1d f (p dA - E[ f (dA ].

For every stopping time Z we have p( (Z, 0o) f= J1 (z,) dPA


SE[ J 1(z,.,,) dA ] by 3.15d,

= E[Az2,A]

- 0 by 3.15b. Also, for every set B E Y0 we have y(B x R ) JIBxR2 dA


= E[ J1 BxR dA] by 3.15d,


= JA dP
B

= 0 by 3.15c.

We have shown that the measure PA is 0 on the generators of p. It follows that the measure PA is 0 on p. So we have

3.15e J T dPA = 0 for every bounded, predictable, real-valued process T. Now let M E 3�0 f(R2). We have PA(M) f " MdPA = E[JIMdA] by 3.15d,

- E[P(1M) dA] by Corollary3.14, JP(IM) dpA


by 3.15e.


= 0






76

This means that the measure PA is 0 on F� R(R2). Let z R2 and Cc. We have 0 = /A(CX[0,ZI)

-f 1 c.[oz] djUA


= E[ J lc[o] dA ] by 3.15d,


J Az dP.
C

Since the set C c F was arbitrary, we conclude that A, = 0 outside a P-negligible set N.Z.

Since the process A is right continuous, we deduce that A = 0 outside an evanescent set. Consequently we have X = Y outside an evanescent set.














CHAPTER 4
VECTOR-VALUED PREDICTABLE DUAL PROJECTIONS

The results in the preceding chapters were derived for a real-valued, two parameter process (Xi,,). Now, in this chapter, we regard a two parameter process (W,,) with values in a Banach space E. Under this expanded gaze, the properties "finite variation" and "integrable variation" slip from focus: a sigma-additive measure on a aalgebra does not necessarily have finite variation when it is vector-valued. Hence, the conditions under which W possesses a predictable dual projection become thrown into question.

Our goal is to present a necessary and sufficient condition under which the process (WSt) with integrable semivariation has a predictable dual projection WP with integrable semivariation, in the presence of a step filtration ((,).

Predictable Dual Projections WP Notation and Terminology 4.1 The following is the framework of the sequel.

4.1a E, F, and G are Banach spaces over R. See 4.1 c for another assumption about the space E.

4.1b Let M be a Banach space.

" The dual of M is denoted M*.
* The unit ball {x E M I Ix1 < 1} is denoted Mi.
" Let x c M and x* e M*. Then x*(x) is denoted .
" We write co 9: M to indicate that M does not contain a subspace
which is isomorphic to the Banach space co.






78

4.1c We assume that c0 r E. Note that this automatically occurs if E = E**.

4.1d The space of continuous linear operators from F into G is denoted L(F,G). We write E c L(F,G) to mean that E is continuously embedded into L(F,G); that is, IXYIG < IXIELYIF forevery x cF and yeG.

4.1e W : x 12 -, E c L(F,G) is a function.

4.1f Let M be a Banach space. A subspace U c M* is a norming space for M if for every x e M we have IxI = sup I< x,u >1.
uEU1

4.1g m: B(012) -, E c L(F,G) is a sigma-additive measure, and U c G* is a norming space for G.

4.lh For every set A c 1 (R12), the semivariation of m on A relative to F,G, denoted mFG(A), is defined by mFG(A) = sup E m(A) x, , where the supreG

mum is taken over all finite families (Ai),i of disjoint sets from (S(R2) contained in A and all families (xi)i-i of elements from F 1. Note that if F = R and G = E, then m has bounded semivariation mFG = mRE.

4.1i Let u r U. The set function m + (R) -- F* is given by = forevery xe F and AE-(R+). Note that mu is sigma-additive, with sigma-additive variation Imul. Further, we have myG = sup ImuI, and so if m has finite semivariation relative to F,G then for
UCUl

every u e U, the measure mu has finite variation.

4.1j Let u c U. The process W, : x R22 -, F* is defined by






79

= for every xe F and (uj,z) gx2R.

4.1k For every function f: R2_ , F we define mFG(f) suplis dmj, where the supremum is taken over all F-valued, I,(R2) step functions s satisfying Isl _ IflIf f is m-measurable then we have mFG(f) = sup f Jfj diml.
ucU1

We denote by ,TF(mFG) the set of all m-measurable functions f: R2 -- F with mFG(f) < oo. Note that if f C ,FF(mFG) then f e LI(mu) for every u e U.

4.11 Let f C ,FF(mFG). The integral J f dm is the continuous linear functional u1-,J*fdm, on Z.

4.1m Let w - K2 and let ,J c R+ be intervals. The semivariation of the function W(tu, ) on the rectangle I x J relative to F,G, denoted svarFG(W(, ), I x J), is defined by svarFG(W(CU, I x J) = sup E A(,,s,,+,x(,j+] W(ni, ) xj where
ij IG

the supremum is taken over all divisions so < s, < ... < s. of points from I and all divisions to < t1 < ... < t. of points from J and all families (xij),.m of elements of F1.

4.1 n The semivariation process WFG x -- R� relative to F,G is defined by WFG(tu,z) = svarFG(W(1,), (-00 , z]) for every u E Q and z R2, after extending W to ( x 2 by setting W(, z') - 0 for every z' e2\Eti. We have WFG(tu,z) = sup IW, I(t,z) forevery w cQ and ze ll2.
uUI






80

4.1o W has finite (bounded) semivariation relative to F,G if for every u E L2 the function WFG(o, ) is finite (bounded).

W has integrable semivariation relative to F,G if the total semivariation

FG 0:= sup WF,G(,z) is in L1(P).


Note that if W has integrable semivariation relative to F,G then W,, has integrable variation for every u c U (Dinculeanu 2000, p. 403). Remarks Since by assumption co T E, if W is right continuous and has bounded semivariation relative to F,G then for every u (, the measure mW(,) can be extended uniquely to a sigma-additive measure on Ig(R2) that has finite semivariation relative to F,G and R,E (Dinculeanu 2000, p. 405). Therefore, if W is right continuous and has integrable semivariation relative to F,G and R,E, then for every tu E K and every function f in the space ,F((MW(W)F)' the integral f f dmW(,") is defined and is often written J f dWw,).

If W is right continuous and measurable and has integrable semivariation relative to F,G and R,E, then for every bounded, y� g(R2)-measurable, F-valued (R-valued) process (p, the expectation E[ J ( dW] is defined and is in G (in E) (Dinculeanu 2000, p. 409).

Definition 4.2 Let W g2 X R2 -- E be right continuous and measurable with integrable semivariation relative to R,E. A right continuous predictable process V" x -+ E with integrable variation (integrable semivariation relative to R,E)






81

is called a predictable dual projection with integrable variation (integrable semivariation relative to R,E) for X if for every bounded, T� 9 (R')-measurable, real-valued process (p we have E[ J (p dV ] = E[ J PT dW ]. The process V is denoted WP.


The Uniqueness of WP

We now show that if a predictable dual projection WP for W exists, then it is unique up to an evanescent set once E is separable. Proposition 4.3 Let W: x R2 --+ E be a right continuous, measurable process with integrable semivariation relative to R,E. Assume that W has a predictable dual projection V with integrable semivariation relative to R,E. If E is separable, then V is unique up to an evanescent set.

Proof Assume that E is separable. Then there is a countable set ZO c E* that is a norming for E.

Assume that W has two predictable dual projections V, V with integrable semivariation relative to R,E.

Fix x* c Zo. Note that the real-valued process < W, x* > is right continuous since W is right continuous, measurable since W is measurable, and has integrable variation since W has integrable semivariation; see Definition 4.1o. Similarly, the real-valued process < V, x* > is right continuous, predictable, and has integrable variation.

Further, for every bounded, measurable, real-valued process (P we have the following chain of equalities.






82

E[ JT d] :


-


= E[ f P(p d

We conclude from Definition 3.4 that < V, x*> = < W, x* >P. Similarly, we have < V', x* > - < W, x* >P. From Proposition 3.5 we deduce that < V, x* > = < V , x* > outside an evanescent set N,.

Since x* e Z0 was arbitrary, and since Z0 is a countable norming for E, we conclude that V = V outside the evanescent set N U Nx,.
x.CZo

Processes Associated With Stochastic, E-Valued Measures

The following theorem will be used in the next section to help establish a necessary and sufficient condition under which a predictable dual projection WP for W exists, in the presence of a step filtration (,T,). Theorem 4.4 Let m "F� 17(R2) -, E c L(R,E) be a sigma-additive, stochastic measure. Assume the following:

" E is reflexive.
" E is separable, and therefore the dual E* contains a countable, dense,
Q-linear subspace U that is a norming space for E.
* There is a number M c R+ such that sup E Imx.I(Ai X R2l) < M, where the

supremum is taken over all disjoint finite families (Ai), from Y and all finite
families (x*)i from U 1.

Then there is a right continuous, y� 1I(R2)-measurable process V: Q x 2 -, E such that the following three assertions hold.









* V has measurable, integrable semivariation VRt.
" For every bounded, f� g(R2)-measurable, real-valued process T we have
J T dm = E[fT dV].
� When (,T) is an -step filtration for some c > 0, then V is predictable if and
only if fq dm = J P dm for every bounded, T� g(R2)-measurable, realvalued process (.

Proof Let x* ( U. The measure x " � 11(R) - R is sigma-additive and stochastic. By Theorem 3.9 there is a process Vx* : g x *2 -+ R that possesses the following properties.

4.4a Vx* is right continuous.

4.4b Vx* is ,Y-� I,(R2)-measurable and has ,f-measurable, P integrable variation.

4.4c For every bounded, measurable process (p : Q x -2 + R we have J V dm , = E[ f ( dVx* ] and J Il djmx*I = E[ J Ipl dlVf*l]. Let x*, x* eU, Z ECR2, and AE3 . Then x*+x e U. We have f Vx;+x(mz) dP(u) - E[ J 1A[o,] dVx;+x ]
A

f 1AJ�[o0] dm*, +; by 4.4c, = .AI," ] dm, x*> + < j" d2,



J IA.oz] dmx* + J IAx[oz] dmx; JE[f 1A[oz] dV] + E[ J lAxz] dVx1o ] J (Vx (, z) + Vx2(m,z)) dP(t).
A






84

The processes Vxj+*2, Vx*, and Vx; are ,F� R(R+)-measurable, so the functions Vx;+x*(, z), Vx*(, z), and Vx;(, z) are T-measurable. Since A c T was arbitrary, we conclude that Vx*+x*( , z) = VX;( , z) + Vx(, z) outside a P-negligible set Nz, ,,.. In view of 4.4a, and since Q2 is dense in R, we have

4.4d Vx*+ix = Vx; + Vx2 outside a P-negligible set Nx,;,2. Similarly, for q c Q and x* E U we have

4.4e Vqx* = qVx* outside a P-negligible set Nq,.. Set N: U Nx * U (q U N'*
,x U qCQ,r~cU

Then N is P-negligible.

Set V'x(u,z):= f Vx*(tu,z) if tu E Nc, z R2, and x* e U
L 0 if zu c N, z +, and x* E U Then for v e K2, Z E12, x*,x* c U, and q c Q we have

4.4f V'x+x (w,z) - Vx(w,z) + V'*(w,z) by 4.4d, and

4.4g V111(w,z) - qV'x1(tu,z) by 4.4e. Further, for every x* E U, the process V'x possesses properties 4.4a, 4.4b, and

4.4c. Note that 4.4f and 4.4g together say that the map x* - V'x*(U,z) from U into R is Q-linear for every o E Q and z C R2. Next we show that this map is uniformly continuous for every (w,z) outside an evanescent set.

Since the map is Q-linear, it is enough to show that f(w) := sup IV'x*[k(w) is finite x*GUI






85

for every tu outside a P-negligible set. By 4.4c, for every x* G U and A G T we have

4.4h Im,*l(A x R+ J IV'xl dP.
A

Let U1= ( xi}N, let A,,. { IV 'l > IV'xjlI for j=1 ... n}, let BI,.= A1,., and let Bi, = Ai,, U Bjn, where n e N and i= 1 ... n.
1
Note that for each n e N, the sets Bi,, are in F- and are pairwise disjoint. We have the following result:

4.4i fdP Jsup IV'x dp x~eWi

f lim max JV' ,J dP
n-.-o i= n

- lim J max IV'4, dP
n--.*ooi= l ...n

by the Monotone Convergence Theorem (Rudin 1976, p. 318),
n

lim J IV'IxI. dP
n-co i- B

n
lim , Im~.I(B,,, x R ) by 4.4h,
n-* i

< lim M by hypothesis,
f--aoc

=M

< 00.

We conclude that f < oo outside a P-negligible set N". It follows that for u c N "c and z E R+, the map x* ,-, V'x*(w,z) from U into R is uniformly continuous.






86
{2

For each x* c U set V"x*(,z):f Vx(,z) ifwN" and zeR 0 if u eN and zGR2 Note that for every x* r U, the process V'x* possesses properties 4.4a, 4.4b, and

4.4c, and also V" satisfies 4.4f, 4.4g, 4.4h, and 4.4i. Further, for every 0 E Q and z E:- R2, the map x* -. V"x*(uy,z) from U into R is uniformly continuous. Thus, for every f e Q and z G R2, we may extend uniquely the map x* V"x*(w,z) to a continuous, R-linear functional V(1U,z) on E* given by

4.4j < V(t,,z), x* > = V"x*(,z) for x* c U, and 4.4k < V(tu,z), x* > lim V"x*(tuz) for x* c E* \ U,
n

where (xn,) is any sequence from U such that x, -- x as n - oo. Denote by V the E**-valued process (rw,z) -, V(tx,z) on n x RI2+. Note that the process V is E-valued since by hypothesis E is reflexive. We now show that V is right continuous. Let (U E Q, ZO, Z E R2, and c > 0. Assume that z>zo. We have I V(CU,z) - V(,zO) I E I < V(Uz) - V(u,zO), x* > I +
3

for some x* E U

since U is a norming for E, < V'(CU,Z) - V X-(W,zo) I + -+
3 3,

for some choice x* =x, c U; see result 4.4k,

< C +
3 3 3

once I z - zo I < J, for some 0, > 0; see 4.4a.






87

Next we show that V is measurable. Let x* c E*. There is a sequence (x,) from U such that < V, x* > = lim VX;
n

pointwise; see result 4.4k.

For each index n the process V"x* is f� ((Ri2)-measurable by 4.4b. Therefore < V, x* > is ,,� 1(R)-measurable. Since E is separable, it follows (Dinculeanu 2000, p. 9) that the process V is T� fl(R2)-measurable. We now show that V has integrable semivariation VRE. Recall that the process V is measurable, and the space U I is countable. Hence, from the equality VRE = sup Vx*I we deduce that the semivariation VRE is
x*GUI

,F� Ij(R2+)-measurable. Moreover, the semivariation VRE is increasing, and so the expectation E[ (VRE) ] is defined. It remains to show that E[ (VRE). is finite. We have E[(VRE) Jsup JIldP = J sup IV'lx* 1 dP; see 4.4j, < M; see 4.4i. Penultimately, we show that for every bounded, measurable, real-valued process To we have J T dm = E[ J T dV ]. Let +p Qxt -- R be a bounded, ,F� ( +)measurable process. Then both f T dm and E[ J q dV] are defined, and are elements of E. For every x* - U we have the following chain of equalities.









= ( dm,*

= E[ f T dV x*] by 4.4c, = E[J'q d] by 4.4j,


= .


Since U is a norming for E, we conclude that J q dm = E[ J T dV ]. Lastly, we show that when (3,) is an E-step filtration for some E > 0, V is predictable if and only if J q' dm = J PT dm for every bounded, measurable, real-valued process T#.

Let E > 0, and let (Y,) be an E-step filtration. We have the following implications.
J" q din = J" Pp dm for every bounded, 7� ?,f(R2)-measurable, real-valued process (p

< f J q' dm, x*> - < J PT, dm, x* > for every bounded, measurable, real-valued process T' and every x*E U f J dm,. = f PT dm,. for every bounded, measurable, real-valued process q and every x* c U

= E[ J T dV x It = E[ f PTp dV'x* ] for every bounded, measurable, real-valued process T and every x* E U; see 4.4c, <=* E[ J d < V, x*>] - E[ J PT d < V, x* > ] for every bounded, measurable, real-valued process T, and every x* E U; see 4.4j,






89

< K V, x* > is predictable for every x* c U by Corollary 3.14, S< V, x* > is predictable for every x* c E*, since U is dense in E*, = V is predictable (Dinculeanu 2000, p. 9).

The Existence of WP

We now present a necessary and sufficient criterion for WP to exist in the presence of a step filtration (,,). We will assume that E is both separable and reflexive. Theorem 4.5 Let E > 0 and assume that the filtration (,") is an E-step filtration. Let W x I + --+ E be a right continuous, ,� 13(R0)-measurable process with integrable semivariation WRE. Assume that

" E is reflexive; and
" E is separable, and therefore the dual E* contains a countable,
dense Q-linear subspace U that is a norming space for E.

Then W has a predictable dual projection with integrable semivariation relative to R,E if and only if there is a number M E R+ such that sup E E[ f P(AR2)(tu,z) dlW,.I(tu,z) ] < M, where the supremum is taken over all
ieJ

disjoint, finite families (Ai)i 1 from F, and all finite families (x*)ij from U1. Proof Assume that W has a predictable dual projection V with integrable semivariation VR,E. Set M := E[ VE ]. Then M < oo. Let (A,)i be a disjoint, finite family from YF, and let (x;)i~l be a finite family from U . Note that for every index i, the process V,- is right continuous and predictable, and has integrable variation IVp 1. Further, for every index i we have

4.5a IW IIP = IVx*I = I(W, )Pl.






90

In fact, let V be a bounded, measurable, real-valued process, and let x* c U. Since E[fPV dW] = E[J dV], then . Hence we have E[ J P q dW,* ] = E[ J V, dV,* ], and so (W,.)P It follows (Dinculeanu 2000, p. 394) that E[ J PV dIW,. j] E[ J V dlVx* I We conclude that IW,.IP = IVx*l = I(Wx*)PI. Hence we have E E[ P(1Ax.R) dIW ,;] E[ J 1 ,xA., dIV,,I ] by 4.5a,
i~l iEl

=E f IV, dP
ic" A,


icl A,


f J RI) dP
UAt

< E[ (V,)]


=-M.

Conversely, assume that there is a number M E R, such that we have sup EjE[ J (lAi.R ) dIW,*I] < M, where the supremumn is taken over all disjoint,
uic

finite families (Ai), j from F, and all finite families (x*)icI from U1. We will show that W has a predictable dual projection V. There is a sigma-additive, stochastic measure p �F� I (R) -- E (Dinculeanu 2000, p. 409) such that for every bounded, measurable, real-valued process V, we have

4.5b fqdy = E[ fJ vdW].






91

Note that for every x* c E*, the real-valued measure u,* is sigma-additive and stochastic. Consider the measure pP �F� pj(R2) -, E given by pP(M) J P(l1m) d/u for every M E T� I(R2). Observe that for every x* c E* we have

4.5c (pP).* = (pu.*)P (consult Definition 3.7). Let x* e E* and M c ,F& Ia(R2). We have (yP),.(M) = E[ J P(lM) dW:. ] = E[ f 1 M d(Wx,)P].

Therefore we have

4.5d I(yP)-.I(M) = E[ j Im dI(W,.)Pl ] (Dinculeanu 2000, p. 394),

- E[ JIM dIW.IP], consult the proof of 4.5a,

= E[ f P(IM) dlW,.I .

Next we show that for every bounded, measurable, real-valued process ' we have

4.5e J V d P = J P dp. In fact, let x* e U and let V be a bounded, measurable, real-valued process. We have < J duP, x*> JV d(pP)X* f -Py d/,*, see result 3.12b in the proof of Theorem

3.12,

-< f Py dp , x* >.

Since U is a norming for E, we conclude that J V' dyp = J PV dp. We now show that the measure p is sigma-additive and stochastic. From 4.5c and Proposition 3.8 we deduce that the measure pP is weakly sigma-






92

additive. Then the measure pP is sigma-additive (Dinculeanu 2000, p. 57). Next, let M T o� R(R2) be evanescent. For every x* e U we have < pP(M), x* > - (pP),.(M)

(P.)P(M)

0 by Proposition 3.8. It follows that yp(M) = 0. We conclude that pP is stochastic. The measure pP also has the following property:
sup I(4,*I(A, x R )

I finite ic/
(A4i)i~l c 3F disjoint
(xi)m~l C U]

- sup E E[ f p(1A,�R2) dlW,*I] by 4.5d,
Ifinite ic'
(Ai)iE! c Y disjoint
(xi)i.l C U1

< M.

By Theorem 4.4 there is a right continuous, measurable process V ) x 2- E with integrable semivariation VRE such that for every bounded, measurable, realvalued process (p we have 4.5fJ'Td/P = E[J(pdV]. Let (p be a bounded, measurable, real-valued process. We have

4.5g E[ J p dW f p(p dp by 4.5b, f (p dtP by 4.5e,

- E[ J p dV by 4.5f.









We also have f Jpq dyPi f" P(p dy f f (p dy P

From Theorem 4.4 we conclude that V is

= V.


by 4.5e, taking V/ to be Pvp, by 4.5e, taking V to be (p. predictable. Then by 4.5g we have WP














CHAPTER 5
AN EXTENSION OF THE RADON-NIKODYM THEOREM
TO MEASURES WITH FINITE SEMIVARIATION

We have noted in Chapter 4 that a sigma-additive measure m on a sigma-algebra does not necessarily have finite variation, once the measure is vector-valued. Many results in Measure Theory do not hold in the absence of finite variation. We see an example of this when we compare Theorems 3.12 and 4.5. Therefore, many results in classical Measure Theory may not be extended to embrace Banach spaces. Of particular interest to us is whether the classical Radon-Nikodym theorem (which was evoked in Theorem 3.9) may be extended to a vector-valued measure m with finite semivariation. In this chapter we demonstrate that an extension of the Radon-Nikodym theorem, with all its classical features intact, is possible if and only if m has a-finite variation. We will close by presenting one occasion on which a Banach space-valued measure is guaranteed to have a-finite variation.

The framework for this chapter consists of Banach spaces F and G, a measurable space (X,X), and sigma-additive measures v : -- R+ and m :X -, L(F,G), and a norming space Z c G* for G. We will assume that the spaces F and Z are separable, and that both F0 c F and Z0 c Z are countable, dense subspaces.

We begin by presenting a useful proposition that will be called upon several times later on. The notations and terminology that we will use have been introduced at the start of Chapter 4.




Full Text

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PREDICTABLE PROJECTIONS AND PREDICTABLE DUAL PROJECTIONS OF A TWO PARAMETER STOCHASTIC PROCESS By PETER GRAY A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006

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ACKNOWLEDGMENTS I am indebted to my supervisor, Dr. Nicolae Dinculeanu, for his infinite patience with me while I slowly learned the material. Also, I am grateful to the many excellent teachers that I have had during my journey at the University of Florida. Finally, for the friendly banter and hearty laughs that I hold in fond memory, I thank Julia, Connie, and Gretchen. II

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TABLE OF CONTENTS ACKNOWLEDGMENTS ii ABSTRACT CHAPTER 1 THE CROSS SECTION THEOREM 1 Introduction 1 Predictable ^-Algebras 2 Stopping Times 6 Projections 7 t[A] g Sets fZs 1 1 The Cross Section Theorem 21 2 PREDICTABLE PROJECTIONS 37 Projections 37 The Uniqueness of 39 The Existence of /Â’X 43 3 PREDICTABLE DUAL PROJECTIONS 50 Step Filtrations (^,) 50 Predictable Dual Projections X/' 52 The Uniqueness of XF 54 Predictable Dual Projections of Measures 55 Processes Associated With Stochastic, K-Valued Measures 58 The Existence of X^ 64 4 VECTOR-VALUED PREDICTABLE DUAL PROJECTIONS 77 Predictable Dual Projections W^ 77 The Uniqueness of W^ 81 Processes Associated With Stochastic, E-Valued Measures 82 The Existence of W^ 89 iii

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5 AN EXTENSION OF THE RADON-NIKODYM THEOREM TO MEASURES WITH FINITE SEMIVARIATION 94 6 SUMMARY AND CONCLUSIONS 106 REFERENCE LIST 107 BIOGRAPHICAL SKETCH 108 IV

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PREDICTABLE PROJECTIONS AND PREDICTABLE DUAL PROJECTIONS OF A TWO PARAMETER STOCHASTIC PROCESS By PETER GRAY August 2006 Chair; Nicolae Dinculeanu Major Department: Mathematics The framework of this dissertation consists of a probability space (Q, a filtration such that if is a filtration satisfying = 3^ sfor every s e then we have Qs = 3^sfor every predictable stopping time S for a double filtration {3^s,i)s,ie^i^ such that 3^s,t = 3^sfor every > 0; and a Banach space E. We study initially a real-valued, two parameter stochastic process X : Q x K, and then we extend some of our results to a vector-valued process Y : Q x ^ In Chapter 1 we start by defining the predictable o-algebra p of subsets of Q X to be the a-algebra generated by the left continuous processes X that are adapted to the double filtration {3^s,t)s,K'SL^. Then we prove the main result of the chapter, the cross section theorem for sets in p . In Chapter 2 we define the predictable projection of a measurable process X : Q X IR2 to be a predictable process ^X ; x oj such that for every V

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stopping time Z we have E [ Xz 1 \3^z] = (^X)z 1 almost surely. Then, using the cross section theorem, we show that the predictable projection is unique up to an evanescent set. In addition, we demonstrate that every bounded, measurable process X has a predictable projection. in Chapter 3 we define the predictable dual projection of a right continuous, measurable process X : Q x integrable variation to be a right continuous, predictable process X^ : Q x k with integrable variation such that for each bounded, measurable process (p :ClxKl R we have E[j Pep dX ] = E[ | (p dX^ ]. Then we show that the predictable dual projection is unique up to an evanescent set. We also establish that the predictable dual projection of the process X exists if the filtration is a step filtration. In Chapters 4 and 5 we turn our attention from real-valued processes X to vectorvalued processes Y. In this setting, our formulations are based not on finite variation and integrable variation, but on finite semivariation and integrable semiva nation. VI

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CHAPTER 1 THE CROSS SECTION THEOREM Introduction The theory surrounding one parameter stochastic processes has applications in many fields. In finance for instance, a filtration contains information that is known up to time t about a market; a martingale (X,),er^ for reflects the price of stock options; a predictable process (H/),er^ houses the number of shares to be held at time t\ and a stopping time S for (^/),^r^ indicates when stocks should be sold for optimal profit. Note a discrete (or step) filtration suffices for good results in many markets. The predictable projection (^X,),eR^ and the predictable dual projection (XO/eR+ for the process (X,),eR^ both play a role in one parameter stochastic theory. For an example of this we retread the finance stage that was set above. The random variable PXs, which is the predictable projection (^X,),er^ evaluated at a (predictable) stopping time S, may be regarded as an updated version of the expected selling price E[Xs] of stock options, given the market information 3^ sThe goal of this dissertation is to extend the definition and existence of the predictable projection and the predictable dual projection to a two parameter process (Xj,/)j,,eR^. This extension is difficult because, while the set IR+ of positive real numbers is totally ordered, the set of ordered pairs of positive real numbers is not. In order to reduce slightly the complexity that we face, we will retain a one parameter 1

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2 flavor: our framework will be built around the double filtration = ^sn where is a right continuous, complete filtration. Predictable a-Aigebras The main result of this chapter is a cross section theorem for predictable subsets of QxIR 2 relative to the double filtration satisfying 3^s,t= 3^sfor s,t> 0. The cross section theorem will be derived with invaluable help from the Monotone Class theorem, which may be found in the text Probabilities and Potential (Dellacherie & Meyer 1975, p. 13-1). In this section we introduce the predictable a-algebra p of subsets of the space Q X K+. Notation and Terminology 1.1 The following will be used in the sequel. 1.1a(Q,^,P) is a probability space. 1 .1 b K+ is the set of non-negative real numbers. N is the set of natural numbers. Q+ is the set of positive rational numbers. 1R+ is the set IR+ x K+, and Q+ is the set Q+ x Q+. 1.1cR(lR+) istheBorel cr-algebra generated by the intervals {s , t] of 0^+. is the Borel a-algebra generated by the rectangles (s , r] x (« , v] of Rl. I.ld A function X ; Q x ^ is called a two parameter process, and is denoted (X.,). I.le ^(g) B(K+) is the
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3 1 ,1f (^/),€r, is a filtration and therefore satisfies • for each r > 0, is a 0 (that is, {3,) is right continuous). S>t See 1.1j below for another assumption about {3,). 1.1gA function S : Q [0 , c»] is a stopping time for the filtration (3,) if { S < r } g 3, for every t > 0. Let S,T be two stopping times. The stochastic interval (S , T] is the set { (cT,r) G n X K+ I S(cj) < 1 < T(c 7) }, while [S , T) is the set { (c7,r) e D X IR+ I S(cj) < / < T(cj) }. The stochastic intervals (S , T) and [S , T] are defined in a similar fashion. 1.1hThe predictable cr-algebra V of subsets of Q x K+ is the a-algebra generated by the sets A x (^ , r] and B x {0}, where A g 3^and B g ^o. A stopping time S is predictable if the stochastic interval [S , oo) is a predictable set. 1.1i Let S be a stopping time. 3 s denotes the cr-algebra {AeJ^j An{S 0 } while 3sdenotes the cr-algebra generated by the sets in 3o as well as sets of the form A n { S > r }, where t > 0 and A g 3,. is the filtration defined by the following rules.

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4 • ^0 = ^0 and • Qs = for every 5’>0. We will write simply {Q,), and we will assume that the filtration (^,) is such that Gs = J^sfor every predictable stopping time S for the filtration (Gs)For example, this is the case when for every 5 e 1 R+, where [sj is the largest integer less than or equal to 5. 1.1k i 3 ^s,t)s,eR+ is a double filtration and therefore satisfies • for each s,t> 0 , 3 ^s,t is a cr-algebra contained in 3 ^, and • 3 ^sj <= 3 ^u,v if s0. 1.1I Let B cOx B(c 7) is the set | (t(7,j:,r) e B }. ;r[B], called the projection of B, is the set { t(7 e n I {fD,s,i) e B for some s,t e K+ }. 1.1m The point (00, oo) will be denoted by 00. Therefore, the inequality {s,t) < 00 means that 5 < qo and t < 00. I.ln Let g : n ^ [0 , 00] X [0 , 00] be a function. [g] denotes the set { (c;,5,r) g x j g(gj) = (5^f) called the graph of g. We now commence our study of the real-valued two parameter process (Xsj). We begin by defining the predictable
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5 lim X^^(cj) = X (uj). Note that this limit is a pointwise limit. (i,/)— (io,«o) 0x(-s',0 + Xp x 1 /ovxfik=0 k,m=0 m=0 where 1 is the indicator function, and « g N. Since (Xs,,) is left continuous, Y" ^ X pointwise as n^oo. Fix m,«, andA:, and consider the process := X™. *. If™. .atLixf*. itiiUO’ J^\ n > n -I Since X^ ± is ^^-measurable, R is the pointwise limit of processes of the form p ;=1 where for every index i we have a, g 1R and A, g But for every index i we have A, x (-^ , -^] = (S, , T,], r on A. I 00 on aJ where S, = , which is a stopping time for (Qs)', and

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6 f on A. ' / = ^ f , which is also a stopping time for (Qs). I 00 on A. p Therefore, R is the pointwise limit of processes of the form ^ a, 1 /=! where for every index i we have a, e K, and S/, T, are stopping times for (Os)By considering in a similar manner the processes Xq i 1 {o>x(i, *±i-] and ^~,o ^ obtain p e aa-(^(Sj , T,] x (n , 5 |], A x {0} x (r2 S2]. (S2 , T2] x {0} where ri,A-2,,si,52 e IR+, Ae Oo, and Si,S 2 ,Ti, and T 2 are stopping times for (Os): = aa{(S , 00 ) X [0 , n], A X {0} X [0 , ^ 2 ] }, where r,,P 2 e K+, A g Oo, and S is a stopping time for (Os). In fact we have p = crfl{(S , 00 ) x [o , ri], A x {O} x [o , a2 ]} because the sets (S , 00 ) x [0 , ri] and A x {0} x [0 , ^ 2 ] are elements of p. Stopping Times Definition 1.4 Let Z : — > [0 , qo] x [0 , 00 ] be a function. 1.4a The function Z is a stopping time for the filtration (J^sj) if { Z < z} e for every z e K?. 1 .4b The stopping time Z = (S,T) is predictable if • S is a predictable stopping time for (Os), and • {S
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7 Proof Let We must show that {Z< (s,/)}e ^s,iFor each index n and each (r,M)€lR2 we have {Z„< (r,«)} = { S„ < r}n{T„< «} e 3^r,u by hypothesis. GO Set t: := V T„ i=n Then limsup T„ = lim J’„. n n We have { Z < (5,r) } = { (sup S„ , lim Tl) < {s,t ) } n n = {sup S„< s}n{lim Tl p=N k=\ Ln=1 m=k 00 00 00 nu nn{T„, < , + e,>n .... p=N k=l _n=l m=k 00 00 CO = > n {2„, < >n ....] p=N m=k e ^i./+CATN e N being arbitrary. Since is right continuous in t, we conclude that {Z<{s,t)} g

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8 Proposition 1.6 Let n e N, and let Z, = (S„T,) i = 1 ... « be a finite set of stopping times for {3^ sj) such that each S, is a stopping time for {Qs). Set A, := {inf S* = S,} n {S, S,} n (S, ^ S 2 } n ... n (S, ^ S;-i}, i = 1 ... n, k S,(cj) = oo for each index / then T(cj) = oo => S(c 7 ) = oo. Proof We first prove that S is a stopping time for {Gs). Note that the sets A, / = 1 ... n are painwise disjoint. Further, we have A, e Gsi for each / (Metivier 1982, 00 p. 20), and A, = Q. It follows that S is a stopping time for (Gs). i=l Next we prove that Z is a stopping time for {3’sj). Let (5 :,/)gK 2. We must show that {Z<(s,/)}e GsBy hypothesis, (Z,<( 5 ,/)}e Gs for each /. n So we have { Z < {s,t) } = (J { Z, < (5,/) } n A, »=i = U({S, i=I G Gs. Lastly we prove the third assertion of the proposition. We have T(cr) = co => T,(ct) = qo for some index / such that gj e A,, => S/(ct) = 00 by hypothesis. S(cj) = 00 since m e A/.

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9 Projections ;r[A] In this section we establish a key result concerning the projection k[A] of an ^0 B(IR?)-measurable set A. It will emerge that 7t[A] is ^-measurable. Proposition 1 .7 Let A e J ^0 and let g be a function such that k[ [g] ] g and [g] c A on ;r[A] n ;r[ [g] ]. Then n-[ [g] n A ] g Proof Let A be the collection of sets B g ^0 such that for any function h : Q — [0 , oo] X [0 , oo] satisfying n[ [h] ] g and [h] c B on ;r[B] n n[ [g] ] we have ;r[ [h] n B ] g Let n be the ring generated by the sets ( C x ( 5 ,/] x («,v] ) n QxK? and fixK?, where C g ^ and s,t,u,v g K. First we show that A contains TZ. Let B G 7^ and let a function h be such that k[ [h] ] g ^ and [h] c B on K[B]nn[ [h]]. Without loss in generality we consider B = C x ( 5 ,/] x (m,v], where C g ^ and s < t and u < V. We have 7 t[ [h] n B ] = ( ;r[ [h] ] ) n C g H ence, B g A Next we show that ^ is a monotone class. Let (B„) be a monotone sequence from A. Assume first that (B„) is increasing. Set B:=[JB„. Let a function hi satisfy ;r[[hi]]G ^ and [hijcB on n ;r[B] n ;r[ [hi] ]. Let h' : f 2 — >• [0 , 00] x [0 , 00] be a function that satisfies the con-

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10 ditrons set out below. There are four conditions, and they involve the function hi. We require that ;r[ [h ] ] = ;r[[hi]], h' = h, on (;r[B]r, [h']eBi on 4Bi]n;r[hi], and [h']cB, on (;r[B,] \ 4B,_i] ) n 4h,] for / = 2,3,4... . Such a function h' exists. Note that we have k[ [h ] ] e and [h'] c Bj on ;r[Bj] n n[ [h ] ] for i e N. Therefore for each index / we have n[ [h'] n B, ] e ^ (since each B, is in A). Hence we have ;r[ [hi] n B ] = ;r[ [h'j n B ] from the definition of h', = ^[[h'jnUB,] i = u >'[ Ih'l n B, 1 / e .r. Next, assume that (B„) is decreasing. Set B' := p B„. Let a function ha satisfy n[ [ha] ] e ^ and [ha] c B' on n ;r[B'] n n[ [ha] ]. Let h" ^ [0 , co] x [0 , oo] be a function that satisfies the conditions set out below. We require that k{ [h"] ] = n[ [ha] ], h" = ha on (;r[Bi]ru4B'], and [h"]cB, on (;r[B,] \ ;r[B,+i]) n 7r[ha] for / e N. Such a function h" exists. Note that we have ;r[ [h"] ] e and [h"] cz B, on ;r[B,] n n[ [h"] ] for i e N.

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11 Therefore for each index i we have ;r[ [h"] n B, ] e ^ (since each B, is in A). Hence we have n[ [h 2 ] n B ] = tc[ [h"] n B ] (from the definition of h '), = 4[h"]nr|B,] i = n I J (since [h"] is a graph), i G We have shown that is a monotone class which contains the ring of generators of B(IR?). Because this ring contains the whole space n x we conclude from the Monotone Class theorem that A is equal to 3^® ^([K^). The statement of the proposition is now seen to be true. Corollary 1.8 Let A e IX(K2). Then ;r[A] e Proof We are able to find a function h ; n — > [0 , oo] x [0 , oo] such that 7c[ [h] ] = Q and [h] c A on ;r[A] n n[ [h] ]. From Proposition 1.7 we obtain ;r[ [h] n A ] g 3^. For such a function h we have ;r[ [h] n A ] = «-[A], Hence we have n^[A] g 3^. Sets K.S We continue to prepare for the Cross Section theorem by introducing a special collection JCs of predictable sets with compact cross sections. Definition 1.9 Let K c Q x 1.9a We say that K has compact cross sections if for each cj g ;r[K], the cross section K(cr) a is compact. 1.9b The cross sectional closure of K, denoted K*, is given by K*(c 7) = ( K(cj) )* for every m e Q, where ( K(c7) )* denotes the closure of the

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12 cross section K(cj) e (The closure includes all adherent points.) Proposition 1.10 Let N e N, and let (K„) be a sequence of subsets of x with compact cross sections. Assume that each set K„ has the following properties: n[B* n K„] G ^ for every ^0 B(K+)-measurable set B; and there is a stopping time Z„ = (S„,T„) such that • S„ is a predictable stopping time for (Qs), • [Z„]cK„, • {Z„ < oo} = ;r[K„], • T„(tJ 7 ) = 00 => S„(cj) = 00 for every cr g Q, and • (5,/) G K„(cj) s > S„(cj) for every tu e Q. Set K := (J K„ and K' := Q K„. n— 1 n— 1 Then K and K' both have compact cross sections, and K has all the above properties of the sets K„. Further, if the sequence (K„) is decreasing, then the set K' also has all the above properties. Proof The finite union and the countable intersection of compact subsets of K? is compact. Therefore, K and K' have compact cross sections. Next, let B g ^0 We have ;r[B* n K] = ;r[B* n |J K„] n=l = U ;r[B* n K„] n=l G , 4 B* n K'] = ;r[B* n n K„] ff=l and

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13 = p ;r[B* n K„] if (K„) is decreasing, n=l since each set B* n K„ has compact cross sections; G We now show that K has the second property set out in the proposition. Set Z := mint ( Z„ ). (See Proposition 1 .6 for the definition of "minf.) n=l...JV From Proposition 1 .6 we know that Z is a stopping time, and that if we write Z = (S,T) then S is a stopping time for {Qs). Note that the stopping time S is a predictable stopping time because S is the minimum of finitely many predictable stopping times S„, n-=\...N. Next, let vj e [Z], Then Z{uj) < 00. There is an index n such that Z(cj) = Z„(c7). From this equality we deduce that Z„(CJ) < 00 . By hypothesis we have (n7,Z„(cr)) e K„. Hence we have (cj,Z(c7)) = (cj,Z„(oj)) eK„ cK. We now show that { Z < oo } = ^[K]. N We have ;r[ K ] = ;r[ (J K„ ] n=l = n=l

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14 N = [J{Z„ n < «.}) rt=\ N = [J{S„ < c»} by a property of the function Z„, n=] N = U [{S« S{m) = oo for every m e Q.. Lastly we show that (5,r)eK(rz7) implies that s> S(cr) for every cr g Q. We have (s,t) g K(cj) (cr,-s,f) e K => {uj,s,t) G K„ for some index n, => s > S„(cj) by hypothesis, > S(ct) since S = inf S„. n=I..JV To complete the proof, assume that the sequence (K„) is decreasing. We verify that the set K' has all five properties that were listed. Let Z := (sup S„ , limsup T„). Note that Z is a stopping time. n n To see this, it is enough to show (by Proposition 1 .5) that the sequence (S„) is increasing. Let men and «o e N, and assume first that Z„„+i(nj) < oo. Then (cj, S„„+i(cr), T„„+i(cj)) g K„„+i by a property of the set K„„+i.

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15 But we have the containment K„„+i c K„„. So we have {m, S„„+,(cj), T„„+,(cj)) € K„„(cj). By a property of the set K„„ we have S„„+,(n 7 ) > S„„(cr). Next, assume that Z„„+i(c7) = oo. Then S„„+i(ct) = oo or T„„+i(t; 7 ) = oo. We have S„„+i(cj) = oo by a property of the set K„„+i . Hence S„„+i(c7) > S„„(nj), and we have shown that Z is a stopping time. Now note, the stopping time sup S„ is a predictable stopping time for (Qs)n This follows since each S„ is a predictable stopping time for {Q^), and since (S„) is increasing. We now prove that [Z] c K. Let Z{m) < oo. We must show that (sj, Z(uj)) e K'. Since Z(ct) < oo, we have sup S„(nr) < oo. n Therefore S„(cj) < oo for every index n. It follows from a property of K„ that Z„(t
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16 By the compactness of the cross sections involved, it follows that (c 7 , Z(ct)) = lim (cj, Z„*(ct)) belongs to each of the sets K„^, A:€N. k So we have (nr, Z(gt)) g pj K„^ k = K'. We now show that { Z < oo } = n[ K']. First we establish that { Z < oo } = n< Z„ < oo }. n In fact, let C 7 g { Z < oo }. Suppose there is a number no e N such that Z„„(jzj) < oo. Then either S„„(ct) = oo or T„„(cj) = oo. Since T„„(cj) = oo => = oo, we obtain S„„(cj) = oo. Since (S„) is increasing, we have S„(ct) = oo for each m > «oIt follows that sup S„(gj) = 00 . n Therefore Z{tu) = (sup Sn(tu ) , limsup T„(?zt)) < oo, and we have reached a n n contradiction. Because of this contradiction, we conclude that Z„{m) < oo for every A7 G N. Hence, we have { Z < 00} c n< Z„ < CO }. n Next let or G PI { Z„ < 00 }. Then for each n we have Z„(ti 7 ) < 00. n Note for each n we have [Z„] cz K„. Therefore for each n we have (c;,Z„(c 7 )) g K„ c Ki.

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17 Since Ki has compact cross sections, there is a number M in K? depends on cf) such that (s,t) < M V (s,r) e Ki(cj). Thus, for each n we have Z„{uj) < M. Consequently we have Z(c 7 ) = (sup S„(cj) , limsup T„(cj)) < M. n n Accordingly, we have cj e { Z < oo }. Hence, we have p|{Z„ sup S„(cj) = oo for every m e Q. n n In fact for each uj g Q vje have limsup T„(ct) = 00 => Z{uj) < 00 n Z„o(c 7 ) < 00 for some index «o, since {Z either S„„(cj) = oo or T„o(cr) = oo => s„„(cj) = 00 (as T„„(cj) = 00 => S„o(c 7 ) = oo), => sup S„{m) = 00.

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18 Lastly we prove that (s,r) G K'(c7) => S> sup S„(c7) for every n In fact, for each cj e Q we have (Sj) G K'(bj) => 00 (u,s,/) e K' = f^K„ n=l => (g7,s,/) g K„ for every n, => (s,t) G K„(c;) for every n, => s > S„{uj) for every n by a property of K„, s > sup S„{m). n Definition 1.11 We define the set K to be the collection of finite unions N U [S, + e, , T, A rii ] x [r, , r\ ], where rii, N & H, e, > 0, r,, r\ g and S;, T, are j=i stopping times for {Qs), for every index /. We define the set fCs to be the collection of countable intersections of sets from K. Proposition 1.12 The set K& is closed under finite union and countable intersection. Further, the elements of Ks have all the properties that were presented in Proposition 1 . 10 . Proof It is evident that Ks is closed under countable intersection. Next we show that Ks is closed under finite union. Let A/g N and let (K,)/=i a/ be a finite family from Ks. 00 For each index i we may write K, = f| where each set is an element of K. 7= I M A/ 00 Then we have (J K, = (J H i=l i=l 7= I

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19 n u ^2,2 u ... u jiji, -Jm =1 which is a countable intersection of elements of fC, since each set Ki_„ u Kij^ U ... U Kmjm is an element of K,. M Accordingly, the set K, is an element of ICsi=\ We now prove that the elements of ICs possess all the properties that were presented in Proposition 1.10. Let S,T be stopping times for (Gs), let e > 0, let n e N, and let r,r' e K+. In view of Proposition 1.10, it is enough to show that the set Ko := [S -Ie , Ta« ] x [r , r ] possesses ail the aforementioned properties. Without loss in generality, we assume that S + e < Tao and r < r'. Let B G B(IR^). We will show that ;r[B* n Ko] e 3^. Let (S„) be a sequence of positive numbers decreasing to 0. For each index m, denote by K„ the set [S + e , Tao + 5„] x[r-S„ ,/ + 5„], and by K; the set (S + e-S„ ,TAn + 5„) x{r-5„,r + 5„). 00 We have ;r[B* D Ko] = p| ;r[B n K„]. m=\ CX5 CO In fact, we have p| n[B n K„] c p| n[B* n K„] since B c B*, m=\ CO = ;r[B* n Pi K„] since (K„) is decreasing, and since each set B* n K„ has compact cross sections.

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20 This last set is equal to the projection ;r[B* n Ko]. On the other hand, for each index m we have ;r[B* n Ko] <= 7t[B* n K^j since Ko c K;,, = ;r[B n K;,] since every cross section of the set K; is an open ball when S + € < oo, c ;r[B n K,„] since K^ c K„. 00 Therefore, 4 B* n Ko] c p| ^B n K„]. m=] oo So we have ^B* n Ko] = p| ;r[B n K„] e iF hy Corollary 1 .8, since K„ g ^0 for every index m. We now reveal the remaining properties of the set Ko. Set Z := (S + e , r). Note that Z is a stopping time. In fact, let ( 5 ,r)GK 2 . Then {Z< (5,/)} {S < 5:} if r < r 0 otherwise In either case we have {Z < (5,/)}g QsWe complete the proof by making the following five observations. The stopping time S + e is a predictable stopping time since e > 0. The graph [Z] is a subset of Ko, since by assumption S + e < Taw and r < r' . We have {Z < 00} = {S + g < 00} = ;r[ [S -f6 , TAw] ] since S + g < Ta«, = ;r[Ko].

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21 For each or e Q we have r(cj) = oo ==> S(rar) + e = oo, since r < ao. For each oj e Q. we have {s,t)eKo{nj) => 5 e [S(cr) + e , T(ct)A« ] and r e [r , r ] => s > (S + e)(oj). The Cross Section Theorem We begin with some important precursors to the cross section theorem for predictable subsets of n X [R2. Definition 1.13 Let be the ring of subsets of K? generated by the sets (z , z ] such that z,z' € with z < z . The measure is the set function from into 1R+ that is given by niiz.z']) = the area of the rectangle (z , z] for every z < z', and which is additively extended to 9^. Remark The measure fi is the Lebesgue measure on and can be extended to a sigma-additive measure on with values in [0 , oo]. We still denote by n this sigma-additive extension. Lemma 1.14 Let B and tn e Q. Then B(c7) e B(K?). Proof Let Fen, z,z' g K?, and (B„)eQxlR2. Then (F x (z , z'])(cr) g while (B, -B 2 )(cj) = B,(o7)-B2(c 7) and (|JB„)(nj) = (JB„(c7). n n Notation 1.15 Let A,B g ® B(IR2), x e K+, and z g We denote by ABa^ the set { G n I /i( (A n n X [0 , z\){m ) ) > A/i( (B n n X [0 , z]){m) ) }. When AeB and 0
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22 projection of the portion of A n x [0 , z] that fills some Borel cross section of B n Q X [0 , z] by a factor of 1 0OA percent or more. Proposition 1.16 Let A, B e ^ 0 gnd let z e Then for every A g IR+ we have ABa^ e n Proof Let R be the ring generated by the sets (J F/ x (z, , zj], where « g 1^, /=i F, G and z,, z) g Rl for every index /. rto Fix Ao G R. Write Ao n x [0 , z] = jj Fq,, x Bq,,, where the sets Fo,/ g ^ are mu1=1 tually disjoint, and where each set Bo,, is an element of B(K+). Let A be the collection of sets C in ^0 R(1R2) such that for every A g the set AoCa,z is an element of We first show that A contains the ring R. Let C\ e R and let A g K+. Write C|n^lx[ 0 ,z] = Fi,, x Bi,,, where the sets Fi,,g^ are mutually disjoint, /=i and where each set Bi,, is an element of Then AoC, = UFo,, nF|_,, where the union is taken over all pairs (/,y) such that (v) /i(Bo.,) > A/i(Bi^). Therefore we have AoC, e 3^. Accordingly we have Ci g yl, since A g K+ was arbitrary. Next, let (C„) be an increasing sequence from A.

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23 Set C := U C„. n=\ We will show that C e Note for each e n we have /z( (C n X [0 , z])(cj) ) = lim,^-«o /i( (C„ n Q X [0 , z])(cr) ), since ju is sigma-additive. Let A G K+, and let (S„) be a sequence of positive numbers decreasing to 0, with X Si > 0 . 00 00 We have AqCa^^ U D (since n is finite on [0 , z]), m=l n=\ G Since A g K+ was arbitrary, we conclude that C g Now let iC'„) be a decreasing sequence from A. Set C' := p c;. n=\ We will show that C' g .4. Note for each c; g we have (C n n X [0 , z])(cj) ) = lim„^ /i( (C'„ n Q X [0 , z])(c7) ), since is sigma-additive and since ^u( (C, n O x [O , z])(nj) ) < oo for every tu e Q. Let A G K+. GO We have AqCa^ = [J AoC^ Since A g [R+ was arbitrary, we conclude that n=l C G A. By applying the Monotone Class theorem we deduce that for every Ao g R, C G J^<8» and A g K+, we have AoCa^ g Now let A be the collection of sets A' in S(IK?) such that for every A g K+, the set A'Ba^ is an element of

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24 By the above we have R cz A’ . Next, let (A'„) be an increasing sequence from A'. Set A';= n=l We will show that A' e Let A e E+. 00 Since n is sigma-additive we have A B^^ = |Ja|,Ba^ n=l G r. Since A g K+ was arbitrary, we conclude that A' g A. Lastly, let (A") be a decreasing sequence from A'. Set A" := f]Al n=\ We will show that A" g A' . Let A g and let (5„) be a sequence of positive numbers decreasing to 0. 00 00 Since jx is sigma-additive we have A"Ba^. m=l n=l G Since A g 1+ was arbitrary, we conclude that A” g X . We have shown that X is a monotone class containing R. By applying the Monotone Class theorem we deduce that for every A g K+, the set ABa^ is an element of 3^. Proposition 1.17 Let z g 1^, let (A„) be an increasing sequence from 00 !f® S(K?), and let A = A„. Let B be a measurable subset of A, let g > 0, n=\

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25 and let 0 < A < I . There is an index Ne N such that P{n[ B n H x [0 , z] ]) P{ (B n An)B,. ) < e. Proof Since n is sigma-additive, for each cj € we have /|( (B n A„ n D X [0 , Z])(C7) ) (B n A n n X [0 , z]){m ) ) (B n Q X [0 , z])(tu ) ). Therefore, since A < 1 and since z e IR? is finite, for each m e Q there is an index Niuj) such that /i( (B n Ajv(o,) D Q x [0 , z])(cj) ) > A/x( (B n Q x [0 , z]){v }) ). Hence we have (B n A„)Ba^/'(Bn A)B^;:. Since each set (BnA„)BA^ is an element of ^ (see Proposition 1.16), we have P{ (B n ) / P( (B n A)Ba^. ) = Pi BBa^ ) = /’(;r[ B n X [0 , z] ]). The statement of the proposition now follows. Proposition 1.18 Let z e [R+, Ao e p,andeo > 0. Let Bo be a measurable subset of Ao. Then there is an element Ko of Ks such that Ko c Ao n x [0 , z] and P(7C[ Bo n Q X [0 . z] ]) P(k[ Bo n Ko n Q X [0 , z] ]) < eo. Proof Let A be the collection of sets A e p such that for every e > 0, 0 < A < 1 , and B (= A, there is an element K of ICs such that K c A n Q x [0 , z] and PiK[ B n n X [0 , z] ]) P( (B n K)Bv) < e. Let S denote the ring generated by the sets (S , T] x (5 , r] and D x {0} x (« , v], such that S,T are stopping times for (g,), D e ^0, and e [R+. Note that we have p = aa{S).

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26 It is our goal to show that ^ is a monotone class containing S. It will follow from the Monotone Class theorem that Let A G 5. I t fi m We may write A = U (Sj , T,] X (r, , 5 ,] u U [0^, , UJ X (y, , wj] where S,, T,, and /=! 7=1 Uy are Stopping times for {Qs), Ay g ^o, and ri,Si,Vj,Wj are real numbers, for every index / and every index j. n Without loss in generality, we will consider A = (J (S, , T,] x (r, , 5 ,], and we will asr=l sume that the sets A' := (S, , T,] x (r, , 5 ,] are pairwise disjoint. Let € > 0, 0 < A < 1, and let B be a measurable subset of A. Let (e„) be a sequence of positive numbers decreasing to 0. For each L e N and each index / denote by Klj the set [S, + , T, A I] x [r, + , 5 ,] g JCs. For each index i we have Klj y A' as L 00 . By Proposition 1.17, for each index i there is a number I, g N such that p(k[ b n A' n n X [0 , z] ]) />( (b n an KlJ(b n A'^ ) < fn Set K ;= (J Kl,,. 1=1 We have K n n x [0 , z] g K n fi x [0 , z] c A n fi x [0 , z], and p{k[ b n fi X [0 , z] ]) n (B n K n fi X [0 , z])B;i^, ) = P( ;r[ B n fi X [0 , z] ] \ (Bn KjB^^: ) since (B n K n fi x [0 , z])Ba^= (B n K)By^. c B n fi x [0 , z].

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27 = p( ;r[ B n y A' n n X [0,z] ] \ (B n y K,, ,)(B n U A'),^ = />^y ;r[ B n A' n n X [0,z] ] \ y (B n ,)(B n A');i^,^ since the sets A' are disjoint, < P\^i ;r[ B n A' n Q X [0,z] ] \ (B n ,)(B n A'),^, ) rt < p(n[ B n A' n n X [o,z] ] \ (B n ,)(B n a');^^,) /=! ;=1 < €. Thus, A G Since A g 5 was arbitrary, we have 5 c 00 Next, let (A^) be an increasing sequence from A. Set A' := Ua: We will show n=l that a' g a. Let B' be a measurable subset of A', and let g > 0 and 0 < A < 1. There are numbers Ai,A 2 such that 0WCnQx[0,z]]) />((CdK)Ca,^) < f. But ;r[ C n Li X [0 , z] ] = (B' n A^jBl,^,.

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28 Hence we have P{ (B' n ) P( (C n K)Ca,^) < f . Note that we have (B' n K)B^^ 3 (C n K)Ca 2 ^. Hence we have P{n[ B' n Q x [0 , z] ]) P( (B' n K)B^^, ) < P{n[ B' n n X [0 , z] ]) P( (C n K)Cx,^ ) = P{n[ B' n n X [0 , z] ]) n (B’ n a;)b1.^, ) + P{ (B' n a;,)b;,^ ) n (c n K)C,,. ) < + 2^2 = e. We conclude that A' e A. 00 Lastly, let (A”) be a decreasing sequence from A. Set A" := P| A" . We will n=l show that a" € Let B" be a measurable subset of A", and let e>0 and 0 < A < 1. Let a' e K+ satisfy A < A' < 1. There is a sequence (A„) from [0, 1) 00 such that A„ = A'. n=\ Note that the set B" is a measurable subset of A','. So there is a set Ki e ICs such that K, c A',' n x [0 , z] and P{n[ B" n Tl x [0 , z] ]) P{ (B" n Ki)BI,^, ) < 1 • The set Ci ;= [ ( (B" n Ki)B^,^ ) x j p| g" ^ n Q x [0 , z] is a measurable subset of A 2 . So there is a set K 2 g Ks such that K 2 c A 2 n Q x [0 , z] and P(;r[C,nfix[0,z]]) /»((C,n K 2 )C, 2 ,^.) < Note that 4 C, n a x [0 , z] ] = (B" n

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29 The set C2 := [ ( (Ci n K2)C, ) x K? ] n Ci n K2 n x [0 , z] is a measurable subset of A3. So there is a set K3 e /Cg such that K3 c A3 n fi x [0 , z] and P(^[ C2 n o X [ 0 , z] ]) />( (C2 n K3)C2 23.) < Note that ;r[ C2 n x [0 , z] ] = (Ci n K2)C, Continuing inductively we obtain a sequence (K„) from /Cg such that for each index n we have K„+i c A'^^, n n x [0 , z] and P(;r[ C„ n Q X [0 , z] ]) P( (C„ n K^,)C„ 2^,^. ) < where C„ = [ ( (C^^i n K„)C„_, 2„^, ) x K? ] n C„_, n K„ n fi X [0 , z] and ;r[ C„ n Q X [0 , z] ] = (C^i n K„)C„_, 2„^„ n = 2 , 3 , 4 .... So for each index n we have p( 7 t[ b" n n X [ 0 , z] ]) P( (C„ n k^.)c„ 2^.., ) = P(;r[ B" n fi X [0 , z] ]) P( (b" n ) + p{k\^ C] n X [0 , z] ]) p( (Cl n K 2 )C| 2 j^ ) + ...+ p{k{ c„ n X [ 0 , z] ]) n (c„ n k„,,)c„ 2„,.) 2 ^ 2^ 2'“' < e. The sequence ( (C„ n K^i)C„ 2^,^ )« 's decreasing, and so we may write P{n[ B" n X [0 , z] ]) P{ lim„^„o(C„ n K„+i )C« ) = P{n[ B" n D X [0 , z] ]) lim^ P{ (C„ n K,^i)C„ 2„,,^ ) < e. Now for each index n we have

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30 (C„ n K^ m-i )C„ , c (B" n Kin Kan... nK„+i)B';n.i . Hence, we have ;=1 lim„_ (C„ n K„,i)C„ e B’" := f] (B" n K, n ... n K„„)B" 1+1 n=l But we have B" c (B ^n In fact, let me B". ( „ n+l \ n+1 (B nQK.nQx[0,z])(cT)J > ]^A,/i((B"nOx[0,z])(c7)) for every index n. So we have ( CO \ 00 (b" n Q K„ n Q X [0,z])(n7) J > nA„/i((B"nnx[0,z])(tiT)) = A. V^(B n n X [0,z])(t<7)^ > A^^(B n n X [o,z])(c7)^. 00 This means that m e (B" n P| K„)B^^.. n=l We remark here that if we had considered only A = A' = 0, then all we would have 00 been guaranteed at this point is oj g ;t[B"* n P| K„], which would have been unhelpn=l ful. We now have 7’(;r[B" n x [0 , z]]) /*((B nniBl.) < €. Set K := PI K„ n=\

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31 We have K g JCs, K c A ' n x [0 , z] (since K„ e A" n fi x [0 , z] for every n), and P{k[ B" n x [0 , z] ]) P{ (B” n K)B^^ ) < e. Accordingly, we have A" e A. We have shown that ^ is a monotone class that contains the ring S of generators of Since S generates the whole space OxK^ we have A = p. Then for the sets Ao e p and Bo <= Ao, the value eo > 0, and any number A e [0 , 1), we may find an element Ko e ICs so that we have Ko c Ao n x [0 , z] and P{k[ Bo n Q x [0 , z] ]) P(k[ Bo D Ko n x [0 , z] ]) < P(k[ Bo n n X [0 , z] ]) P{ (Bo n Ko)Bq ) since (Bo n Ko)Bq e ;r[ Bo n Ko n iA x [0 , z] ], < eoThe statement of the theorem is now seen to be true. Corollary 1.19 Let Ae p, and let e>0. There is an ^-measurable function / : — *• [0 , oo] x [0 , oo] such that • [/] e A and • P(;r[A]) P{k{ [/] ]) < e. Proof Let (z„) be a sequence from such that z„ oo. For each index n there is a set K„ e K& such that K„ c A n x [0 , z„] and P(;r[ A n X [0 , z„] ]) P(;r[K„]) < (take Bo = Ao = A in Theorem 1 .18). 00 Set D := U K„. We have D cz A (since z„ Z' ), and n=l OO 00 n^A]) P(k[D]) = P(4|jAnfix[0,z„]]) n^[|J K„]) n=\ rt=l 00 00 = /’( U ^ ^ ^ ^ J \ U n[K„'\ ) n=l n=l

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32 1 \ ^[KJ) Vn=l 00 A n Q X [0 , z„] ] \ ;r[K„]) n=] GO s E + n=\ = €. For each index n there is an ^-measurable function Z„ : fit — [0 , oo] x [0 , oo] such that [Z„] c K„ and ;r[[Z„]] = ;r[K„] (see Proposition 1.12). 00 /•= ^l”*>r[[Z,]] + n-l n=2 n\iZ„]] X |J n^Z,\] i=\ Then / is an ^-measurable function such that [/] e D and n[\f\] = ;r[D]. Accordingly, we have [/] c A and P(;r[A]) P{k[ [/]])< c. Theorem 1.20 Let Ao g eo > 0, and let / : Q -> [0 , oo] x [0 , oo] be an measurable function such that [/] c Ao. There is an element Ko g Ks such that • Ko c Ao and • W[/]]) WMnKo]) < eo. Proof Let A be the collection of predictable sets A such that for any e > 0 and any ^-measurable function ^ [0 , oo] x [0 , oo] with [g] e A there is an element K G /Ca such that K c A and P{n[ [g] ]) P(^n[ K n [g] ]) < e. Let S,T be stopping times for {Gs), and let s,t e R+. First we show that the predictable set A := (S , T] x (5 , /] is in A. Let e > 0 and let g be an J^-measurable function such that [g] e A. Let (e„) be

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33 a sequence of positive numbers decreasing to 0. For each index n, let K„ = [S + e„ , Ta«] y-ls + en ,t] e /C^. We have K„ A. Hence we have P{n[ [g] ]) = P{n[ [g] n A ]) 00 = Pin[ [g] n U K„ ]) rt=I ou «=l = lim„^ 7>(4 [g] n K„ ]) since P is sigma-additive and since each set [^] n K„ ] is an element of see Corollary 1.8, < P{^[ [g] n Kjv ]) + e for an existing index N. Note that we have e Ks, Kn a A, and P{k[ [g] ]) P{n{ [g] n K;v ]) < e. Hence, we have A e We may similarly show that the set B x {0} x (« , v] is an element of A, where B e Qo and u,v e K+. Next we show that A is closed under countable unions. It will follow that a monotone increasing sequence from A has its limit in A, and also that A contains the ring S generated by the sets (S , T] x (.« , r] and B x {0} X (m , v] such that S,T are stopping times for (Qs), B g ^o, and g K+. 00 Let (A„) be a sequence from A. Set A' := [J A„. n=l Let e>0 and let g bean ^-measurable function such that [g] cA'.

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34 For each index n, the set [^]nA„ c A„ is the graph of the .^-measurable function Hence for each index n there is an element K„ e ICs such that K„ c A„ and W[^]nA„]) P(4[^]nK„]) < We have P(j:[ [g] ]) = P(n[ [g] n A' ]) 00 = A4[g]n|jA«]) n=\ 00 = A|j4[^]nA„]) n=\ N, «=i for an existing index N] e N. Ni Set K ;= U K„ g ICsThen K c A , and we have n=l W [g] ]) [g] n K ]) < P(|J n[ [g] n A„ ]) P(\J n[ [?] n K„ ]) + f n=l n=l A'l < n A„ ] \ ;r[ [^] n K„ ])) + f n=l Ni < Z^Wt^JnA„]\4[^]nK„]) + f n=l N, n=l < €. We conclude that A' e A.

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35 Lastly, let (A„) be a decreasing sequence from A. Let A" := P|A„. We will show n=I that A e Since A contains S, and since aa(S) = p, we will then conclude that A = p. The statement of the theorem will follow as a consequence. Let e > 0, and let g bean J^-measurable function such that [g] c A". Then [^] c Ai, and so there is an element Ki g ICs such that Ki c Ai and Pi^Ug]]) W[^]nK, ]) < f. The measurable graph [g] n Ki is contained in Ai, and so there is an element K2 e fCs such that K2 cz A2 and P(k[ [g] n Ki ]) P(k[ [^] n Ki n K2 ]) < The measurable graph [g] n Ki n K2 is contained in A3, and so there is an element K3 G ICs such that K3 c A3 and P(;r[ [g] fi K| n K2 ]) P(k[ [^] D Ki n K2 D K3 ]) < -J2^ Continuing inductively we obtain a sequence (K„) from ICs such that for each index n-l n n we have K„ c A„ and P(k[ [^] n riK.]) />WbmnK,]) < ir. l=\ 00 Set K := Pi K„. We have Kg/C^, K e A", and n=\ WL?]]) W[^]nK]) 00 = P(^[ Ig] ]) P{n[ [g] n PI K„ ]) n=\ CO n ~ n
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36 = P{k[ [g] ]) P{n[ [g] n K, ]) + '^(p{k[ [g] n n K, ]) P{n[ [g] n K, ]) n =2 V '=1 '=• 0 2 " n=l = £. We conclude that A" e We close Chapter 1 with the main result of the chapter, the cross section theorem. Theorem 1.21 Let Ag p, and let e > 0. There is a stopping time Z such that • [Z] c: A and • 7^(;r[A]) P{n[ [Z] ]) < e. Proof By Corollary 1.19 there is an ^-measurable function / satisfying 1.21a [/] (z A and 1.21b P(4A]) WW]) < f. By Theorem 1 .20 there is an element K g so that 1.21c K e A, ^.2^6P{n[\f\^) WWnK]) < f, and 1.21e there is a stopping time Z such that [Z] ci K and ;r[K] = 7t[ [Z] ]. We have [Z] cz K by 1.21e, (z A by 1.21c; and /’(;r[A]) < /^(;r[ [/] ]) + y by 1.21b, < /'(rr[ [/] n K ]) +6 by 1.21d, < P(;r[K]) + e = P(k[ [Z] ]) + G by 1.21e.

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CHAPTER 2 PREDICTABLE PROJECTIONS In this chapter we define the predictable projection for a measurable two parameter process (X^,,). In addition, we demonstrate that the projection ^X exists, and is unique. Projections pX In the paragraphs that immediately follow, we motivate the definition of the predictable projection. Definition 2.1 Let Z be a stopping time for (^v). The cr-algebra -T'z is defined by = { A G ^ I A n { Z < (5,/) } e V {s,t) e }. Proposition 2.2 Let Z = (S,T) be a stopping time such that {S
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38 Let A e QsThen A e ^ and A n { S < 5 } e for every s e K+. Let s,t G K+. We have A n { Z < (5,0 } = A n { S < 5 } n { Z <{s,t)} G since {Z < (s,t)} g 3^s,i and Gs = So A G 3^ z. Lastly we show that 3^z <= GsLet A G 3^z, and let 5 g 1R+ and n g N. Then A g and A n { Z < (^,«) } g = Gs. Hence we have An{S < 5 } = An{S < 5 }n{T < 00} (because {S [R be an ^(g) (J([R5)-measurable process. A predictable process Y : O x [Rj ^ 0^ which satisfies E[ Xz 1 {z | T'z ] = Yz 1 {z<®> a.s. for every predictable stopping time Z is called a predictable projection for X,

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39 and is denoted pX. Note that because the process X is measurable and Z is a stopping time, the function Xz is J'-measurable. Therefore, since 3^z c: the conditional expectation E[ Xz 1 {z = 0 a.s. for every predictable stopping time Z then

= 0 a.s. for every predictable stopping time Z, then ^A = 0 outside an evanescent set. Let A e p be such that (1z)z1{z = 0 a.s. for every predictable stopping time Z. Suppose /Â’(;r[A]) * 0. Then /Â’(;r[A]) > 0 and so there is an e > 0 such that P(;r[A]) > e. By Theorem 1 .21 , for this e there is a predictable stopping time Z^ such that 2.5a [Zf] cz A, and 2.5bP(;r[A]) < e + P{k[Z,]). We have the following chain of equalities and inequalities.

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40 0 = ^({(1^)z,10}) by hypothesis, = d e O I (d, Z^(d)) e A }) = n4An[z,]]) = W[Z.]]) > /^(«-[A]) € > € e = 0 , by 2.5a, by 2.5b, by assumption, which is a contradiction. So we cannot have P(;r[A]) ^ 0. Accordingly, 1^ = 0 outside an evanescent set. Second, we show that for any disjoint sets A/ g ^ and any numbers a, > 0, if ^ a/1^_ I 1{z = 0 a.s. for every predictable stopping time Z then V<=i J Z '=l = 0 outside an evanescent set. Let A7 G N, and for i = 1 ... « let A, g p and a, >0. Assume that the sets A, are disjoint. We have 1 {z = 0 a.s. for every predictable stopping time Z z => for each index /, (a,1/(,)z 1{z = 0 a.s. for every predictable stopping time Z (since the sets A, are disjoint), ^ for each index /, (1z/)z 1{z 0), => for each index /, = 0 outside an evanescent set (by the first part above).

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41 n => ^ Oily). = 0 outside an evanescent set. This completes the second part. ;=l Third, we show that if (p is positive and satisfies the hypothesis of the theorem, then (p = 0 outside an evanescent set. Assume that the predictable process (p is positive and satisfies = 0 a.s. for every predictable stopping time Z. Since q> is positive and predictable, there is a sequence {(pn) of positive predictable step functions such that (p„ " /

0} " / { (jo > 0 } and {(p„)z 1 {z " for every predictable stopping time Z. Hence we have 2.5c P(7t[ { > 0 } ]) " y P{n[ {(p > 0 } ]) since P is sigma-additive (note k[{(p„ > 0 } ] and n[{(p > 0 } ] are in 3^ by Corollary 1 .8 ), and 2.5d for each index n, ((p„)z'\{z 0, (=1 and the sets A, are disjoint. In light of this, when we apply the result in the second part of the proof to 2.5d we obtain
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42 (pz 1 {z = 0 a.s. for every predictable stopping time Z => | = |Z 1 {Z| = 0 a.s. for every predictable stopping time Z => l^l = 0 outside an evanescent set (note, \(p\ is a positive predictable process, so the third part of the proof applies); => (^ = 0 outside an evanescent set. We are now able to establish the uniqueness, up to an evanescent set, of ^X. Corollary 2.6 Let (p,\i/ : D. x Rl —> R be two real-valued, predictable processes. If = wz 1 {z a.s. for every predictable stopping time Z then (p = y/ outside an evanescent set. Proof From the hypothesis we deduce that {cp y/)z‘^{z
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43 (9Ji)z 1 {z E[ Xz 1 ^ 2 I ^z ] a.s. = {(pi)z 1 {z a.s. . From Corollary 2.6 we conclude that (p\ = (pi outside an evanescent set, and the proposition is hence proved. The Existence of pX We will begin by presenting explicit forms of the predictable projection for two particular processes, X = H 1[o,„]x[o,v] and X = 1(2,®). Both forms will be used later in the paper, and one form will intimate that, desirably, a predictable process is its own projection. We will close by proving that every bounded, measurable, real-valued process possesses a predictable projection. Proposition 2.8 Let H e L”(^), let m,v g 1R+, and let X ; L 2 x ^ bg defined by Xsj{of) = H(cj) 1[o,„]x[o,v](^,0Denote by (E[ H | ])j the function 5 E[ H I ], which is chosen (Dinculeanu 2000, p. 181) to be right continuous with left limits (cadlag). Then X has a predictable projection, and we have (OX),,, (07) = (E[ H I J^.]),-(ot) 1 [O,«]x[O,v]('S7 0Proof Let '= (E[ H | ])i-(cr) 1[o,«]x[o,v](5',0 for every cr e Q and every s,t G K+. First we verify that the process (Y^,,) is left continuous. Since (E[ H | ])^ is cadlag, (E[ H | ])i is left continuous. Also, 1[o,„]x[o,v] is left continuous. Therefore, (Yi,,) is left continuous. Next we check that for every g 1K+, the map Y^,, is -measurable. Let s,t e K+. If 5 > M or r > V then Ys,t = 0, which is ^^/-measurable.

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44 If s < M and t < V then = (E[ H | ])^_ = E[ H I ] (note, since E[ H | ] is cadlag and 5 is a predictable stopping time, the Stopping Theorem (Metivier 1982, p. 87) applies), e Gs = ^ s,tLastly we confirm that E[ Xz 1 {z<®> | ] = Yz1{z a.s. for every predictable stopping time Z. Let Z = (S,T) be a predictable stopping time. We have Yz 1 {Z = Y^J 1 ^z = E[ H I ] 1 {z<(«,v)> (note, since E[ H | ] is cadlag and S is a predictable stopping time, the Stopping Theorem (Metivier 1982, p. 87) applies). Also, E[Xz1 {Z I ] = E[ H I ^z ] 1 {Z<(«.v)> since { Z < (m, v) } e 3^z, = E[ H I 3^s] 1{z<(«.v)> since J^z = Gs by Proposition 2.2, and since Gs = ^sby assumption. Therefore we have E[ Xz 1 | ^z ] = Yz 1 {z. Proposition 2.9 Let Z be a stopping time. For every stopping time Z we have {Z
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45 the predictable projection of 1(z,<») is itself. Therefore we have ^(1(z.oo)) = 1(z,c»). Proof Let Z' be a stopping time. To prove the first assertion of the proposition, we must show that {Z < Z' < oo}n{Z' < {s,t)}G Q, for every s,te^+. Let s,t G H+. Note for every r,q g 1R+ with r< s we have {Z < J^r,q = Qr c= Qs. Also, for every r, 9 e IR+ with r< s and q< t we have A:={r < S' < ^}n{T' < /} = {Z' < (5,/)}n{Z' < {r,t)Y ^ ^ s,t = and B:={S' < 5}n{9
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46 that for any predictable process (Xj,,) that has a predictable projection we have ^X = X. Observe that the process 1 (z,®) is left continuous. Also, for each we have 1(z,oo)(5,r) = 1{z<(s,o}Since 1{z<(^,/)} is .?^i,rmeasurable, then 1(z,oo)(v,r) is also ^j^-measurable. Now let Z' be a predictable stopping time. We have = 1 j^ 2 ‘ We have proved that ^’(1 (z,oo)) = 1(z.oo). Theorem 2.10 Let X' : O x jjg g bounded, real-valued, measurable process. Then X' has a predictable projection. Proof The proof is inspired by Dellacherie and Meyer’s proof for the existence of the predictable projection of a one parameter process. The proof will unfold in five steps. In the first step we show that if Xi and X 2 are measurable, real-valued processes having predictable projections /’Xi and pXi respectively, then Xi < X 2 implies that ^Xi < py .2 outside an evanescent set. Thus, in particular, if |X]| < K then I^Xil < K outside an evanescent set. Let X], X 2 ;Q X (R be two B(IR^)-measurable processes having predictable projections ^Xi and PX 2 respectively. Let Z be a predictable stopping time. We have

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47 X| < X2 => (Xi)z1{Z < (X2)z1{Z<00> ^ E[ (X,)z 1 {Z«»} I ] < E[ (X 2 )z 1 {Z I ] a.s. => (^Xi)z 1 {Z< 00 > < (^X 2 )z 1 {Z > 0 a.s. {(^Xj X,) 1 1 .(z = 0 a.s.. Since the predictable stopping time Z was arbitrary, we deduce that if Xi < X 2 then (PX 2 ^Xi) 1 .(px2-^’a', = 0 outside an evanescent set (apply Corollary 2.6 after noting that the process (PX 2 ^Xi) 1 is predictable and real-valued). Hence we have X| < X 2 => ^X 2 ^X| > 0 outside an evanescent set. We show secondly that if X is a measurable, real-valued process and (X") is a sequence of uniformly bounded, measurable processes such that each X" has a predictable projection and X" " y X, then X has a predictable projection and we have ^X = liminf„ /’X". Let (X") be a sequence of measurable processes X" ; Q x IR, ne N, and let X : Q X []j be a measurable process. Assume that the sequence (X") is uniformly bounded, that each X" has a predictable projection ^X", and that X" " y X pointwise. Let Z be a predictable stopping time. Then (X")z 1 " y Xz 1 {z pointwise. So by Lebesgue’s theorem (Rudin 1976, p. 321) in V{P) we have E[ (X")z 1 I ^z ] a.s. and in L' (/>). Thus we have

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48 E[ Xz1 {Z a.s. = ^liminf 1^z X uniformly as n — > oo, then X has a predictable projection and we have ^X = liminf„ ^X". Let (X") be a sequence of bounded, measurable processes X" : Q x ^ rt e N, and let X : O x Kj be a measurable process. Assume that each X" has a predictable projection ^X", and that X" ^ X uniformly as n —> CO. Since each X" is bounded and X" — X uniformly, X is bounded. Hence, E[ Xz 1 | ] exists for every predictable stopping time Z. We have E[ (X")z 1 {z | ] -»• E[ Xz 1 {z | ^z ] a.s. and uniformly as « — > oo, for every predictable stopping time Z. So for every predictable stopping time Z we have E[Xz1 {Z I ^z ] a.s. n = liminf (^X")z 1 a.s. Tliminf ^X"^ I {Z
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49 In this fourth step we verify that the projection ^ is linear. Let X,Y : Q X 1 R 2 jjg processes that have projections /Â’X, respectively, and let o,Z> e IE. We will prove that aX + ^>Y has a predictable projection, and P{aX + bY) = a pX + bPY. In fact, for every predictable stopping time Z we have E[ (flX + bY)z 1 {z I ] = aE[ Xz 1 \3^z] + bE{ Yz 1 | ^z ] = a{PX)z 1 {z + b{PY)z 1 {z a.S. = (a^X + ^>'Â’Y)^1^z
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CHAPTER 3 PREDICTABLE DUAL PROJECTIONS In this chapter we present the predictable dual projection X^’ for a two parameter process X. It will be shown that if X^ exists then it is unique, and that in the presence of a step filtration, X^ exists when X is right continuous, measurable, and has integrable variation. We begin with the definition of a step filtration. Step Filtrations (^,) Definition 3.1 Let e > 0. A filtration is a right continuous e-step filtration if ^'ae " ^Lajf for evory a > 0, where [aj denotes the largest integer that is less than or equal to a. A filtration is a left continuous e-step filtration if for every a>0, where f a“| denotes the smallest integer that is greater than or equal to a. Remark A step filtration (^,) is of interest to us because • it facilitates a property that is assumed of the filtration {Qs), namely that Qs = 3^sfor every predictable stopping time S for (Qs), and • it causes (Qs) to be a left continuous step filtration, a property that will help us later to demonstrate the existence of a predictable dual projection. Proposition 3.2 Let e > 0, and assume that (^,) is an e-step filtration. Then • the filtration (Qs) is a left continuous e-step filtration, and • for every predictable stopping time S for (Qs) we have Qs = 3^ s-Proof We prove the first assertion of the proposition. Let sq g 50

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51 We have J if So ^ ne for every n g N 1 3^sa-( if So = n€ for some n g N since (^,) is a right continuous (see Notation 1.1 f)) e-step filtration. Hence, the filtration {Gs) satisfies for every «>0, and Go = G^. As such, (G,) is a left continuous e-step filtration. Next we prove the second assertion of the proposition. Let S be a predictable stopping time for (Gs). We must show that Gs = ^s-First we show that ^sGsLet B be a generator of 3^sThen B = An{S > 5 } for some s e K+, and A g 3^^Hence, for every r g K+ we have Bn{S< /} = An{s> 5}n{s< t} f 0 if 5 > r lAn{5 (A7-l)e}] CO := Bo U U B„ n { S > (« 1 )e },

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52 where Bo = An{S< 0}, e ^0 since Ae Qs and S is a stopping time for (^^), = ^0 c= ^s: and where B„ = A n { S < }, n g N, € since A e and S is a stopping time for (^^), = ^ (n-l)€We deduce that A e 3^s-Predictable Dual Projections Xp We begin by introducing some terms (Dinculeanu 2000, p. 363-390) we will use during the definition of the predictable dual projection, as well as beyond. Definition 3.3 Let X : 12 x []j l^g g parameter process. 3.3a (Xsj) is right continuous if for every so,to g K and m e Q, we have lim X^,(tJ7) ioSs /o
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53 (X^,,) is incrementally increasing if for every m e Q and every z < z' from we have X(nr) > 0. 3.3d Let or € Q and let l,J c 1+ be intervals. The variation of the function X(cj) : IR2 ^ K on the rectangle I x J, denoted var( X(cr) , I x J ), is defined by var( X(c7) , I X J ) = sup ^ |where the supremum is taken over ij all divisions 5o < < ... < of points from I, and all divisions ro
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54 for each cr e fl, the measure mx(n,) is sigma-additive on TZ. Furthermore, if the process (X^,,) is right continuous and has bounded variation then for each cj e Q, the measure mx(ny) can be extended uniquely to a sigma-additive measure on ( 1 ( 1 ^) that has finite variation |mx(^)| on and we have |mx(^)| = m|X(^)| on TZ. Therefore, if (X 5 ,,) is right continuous and has integrable variation, then for each uj & Q. and each B(Kj)-measurable, |mx(o,) [-integrable, real-valued function f : K? j-^e Stieitjes integral | f dmx(n,) is defined and is often written | f dX(cj). If (Xj,,) is right continuous and ^0 B(K+)-measurable, and has integrable variation, then for any bounded, ^0 B(lR+)-measurable, real-valued process cp, the expectation E[ I (p: dX] is defined and is finite. Definition 3.4 Let X : n x 3 continuous, 5^0 B(K?)-measurable process with integrable variation jXj. A right continuous, predictable process Y : Q X K with integrable variation |Y| is called a predictable dual projection for X if for every bounded, ^0 B(K+)-measurable, real-valued process q> we have E[ J
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55 Proof Suppose that X has two predictable dual projections, Yi and Y 2 . Let Z be a stopping time. Set A;={(Yi)z1{z«x.} > (Y 2 )z 1{z } and (p(o7,5,r) := 1[o.z](cr,.s,r) 1^(cj) 1^z(nj). Note that A e ^ and cp is real-valued, measurable, and bounded. We have E[ | dY, ] = E[ | > dX ] = E[ J (p dY 2 ]. So I (Yi)z 1{z = (Y2)z1 almost surely. In particular, we have (Y|)z1{z = (Y2)z1{z«x>} a.s. for every predictable stopping time Z. Since Yi and Y 2 are predictable, by Corollary 2.6 we have Yi = Y 2 outside an evanescent set. Predictable Dual Projections of Measures We have defined the predictable dual projection Xp for a process X. In this section we will present the predictable dual projection m^ for a measure m. We will call upon m^ in order to establish the existence of XF. Definition 3.6 Let E be a Banach space and let m : ^(gi E be a sigmaadditive measure. We say that m is a stochastic measure if m vanishes on evane-

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56 scent sets. As an abbreviation we say that the measure m is stochastic. Definition 3.7 Let m : ^0 (l(IR+) — >• K be a sigma-additive measure. The measure mP : ^0 ^ K defined by mP(A) = | /’(1 a) dm for every A e J ^0 is called the predictable dual projection for m. Proposition 3.8 Let m : ^0 IR be sigma-additive and stochastic. The following three assertions are true. • The measure m^ is sigma-additive. • The measure m^ has finite variation |m^| which satisfies |m^| < |m|^. • The measure m^ is stochastic. Proof First we show that m^ is additive. Let A, B be disjoint sets from ^0 m/’CAu B) = j P{^AuB) dm = + ^B) dm = I (^(1^) + ^(1b) ) dm = |^’(1^)dm + |p(lB)dm since A and B are disjoint, since the measure m is a stochastic measure, since both integrals exist. = m^(A) + m^(B). Next we show that m^ is sigma-additive. Let (A„) be a sequence from ^0 R(IRJ) that decreases to 0. Then P{^A„) decreases pointwise to 0 (apply the second step in the proof of Theorem 2.10 to the sequence (1 So |^(1^„)dm -^0 as « — > oo, since m is sigma-additive with finite variation. Thus, by equality, mP(A„) — 0 as « — 00.

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57 We next prove the second assertion of the proposition. First, note that a real-valued sigma-additive measure on a a-algebra has finite variation. Accordingly, the measure |m| is a real-valued sigma-additive measure on the cr-algebra B(fR5). Thus by the first assertion of the proposition, the measure |m|^ is sigma-additive, and so is finite. Further, for every set A e 3^® we have |mP(A)| = I J^’(1^)dm | < |p(1^)d|m| = |mp(A). It follows that the measure m^ has finite variation satisfying |m^| < |m|^. Lastly we prove the third assertion of the proposition. Since the measure m is a stochastic measure, for every evanescent set A e we have m(A) = 0. It follows that for every evanescent set A we have |m|(A) = 0. Let A e ^0 be an evanescent set. We must show that m^(A) = 0. Since the set A is evanescent, there is a P-negligible set N e Q such that A e N x We have |mP(A)| = | j ^(1^) dm | < j^(1^)d|m| (refer to the first step in the proof of Theorem 2.10 after noting that |m| > 0 is stochastic); because (the proof of this is similar to that of Proposition 2.9); |m|(N X Rl), which is 0.

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58 Processes Associated With Stochastic, IR-Valued Measures We now prove a theorem that will underpin our construction of a predictable dual projection in the next section. Theorem 3.9 Let m : 3^® — » IR be a sigma-additive, stochastic measure. There is a right continuous, 3 ^® R(K?)-measurable process X ; Q x Kj ^ [r vvith integrable variation |X| such that for every process ^:Qx 1 R 2 _„I]j Ip, L*(|m|) we have • I (p dm = E[ I dX:: ] and • | For each z e define the measure m^ : ^ ^ 1 by m-(A) = m( [0 , z] x A ) for every set A e Note that the measure m"" is sigma-additive with finite variation |m‘|. Since m is a stochastic measure, we have m' « P. (Note that m= « P means that if the set A e ^ satisfies /’(A) = 0, then m"(A) = 0). So by the classical Radon-Nikodym theorem (Dinculeanu 2000, p. 36), there is an

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59 ^-measurable, /*-integrable function Yj : Q — > K such that 3.9b m^(A) = I Y; dP for every A g A Next, let Z] = {s,t) and Z 2 = {s' ,t), and consider the measure m-^ m^' : ^ ^ DS. The measure m*^ m*' is sigma-additive and has finite variation. Further, we have m *2 m^> « P. So by the classical Radon-Nikodym theorem (Dinculeanu 2000, p. 36), there is an ^-measurable, P-integrable function Ys,,^ ; Q — > IR, such that 3.9c (m '2 m'")(A) = j Y.,.-^ dP and A 3.9d Im''^ m-‘|(A) = J |Yz,;J dP for every A g 5^. A From 3.9b and 3.9c we deduce that for every zi < zi we have Y.^ Y-, = Y-.^^ a.s.; that is, outside a P-negligible set Therefore, from 3.9d we obtain 3.9e |m "2 m‘>|(A) = | lY::^ Y^JdP for every A g A Also, the measure m-"^ + m-'> m("' '> ) ^ ^ K is sigma-additive, has finite variation, and is absolutely continuous with respect to P. So by the classical RadonNikodym theorem (Dinculeanu 2000, p. 36), there is an ^-measurable, P-integrable function Yl,,^ : Q — K such that 3.9f (m -2 + m-' m^^’P m(*''))(A) = j Y',.j dP and A 3.9g |m-^ + m^' m^^ m^*’' ^|(A) = | |Yl,.J dP for every A g A From 3.9b and 3.9f we deduce that for every z\ < zi we have

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60 Y, + Y. Y' Y ' = Y' outside a P-negligible set N' 22 -1 s ,t S,l Z]Z2 ^ ^ ” 1-2 Therefore from 3.9g we obtain 3.9h |m -2 + m"‘ m^^')|(A) = | |Y,^ -iY^^ Y^^^Y^^'|dP for every set A Ae^. For every z\ = (s,t) < Z 2 = {s' ,i) and A g we have m-')(A)| = lm( [0 , Z 2 ] X A ) m( [0 , zi] X A )| = |m(([0,z2]-[0,z,])xA)| < |m|(([0.z2]-[0,z,])xA). Hence for every A e ^ we have |m-'2 m“''|(A) < |m|( ([0 , Z2] [0 , zi]) X A ) = E[ J 1([o^,,Ho.'.])x^ dV] by 3.9a, = I (Vz 2 V,,) dP (see Definition 3.3f). A So by 3.9e we have | jY^^ Y,,| dP < | V^,) dP for every A g A A Since all functions involved are ^-measurable, we have 3.9i |Y,, Y;,| < Vz, outside a P-negligible set Uz^ziFurther, for every zi = {s,t) < zi = {s' ,i) and A g we have |(m-^ + m=" m^^ P m^"'-' ^)(A)| = |m( (zi , Z 2 ] x A )| < |m|( (zi , Z2] X A ). Hence for every A g we have lm -2 + m-' m^^Pm^''’'^|(A) < |m|( (zi , Z2] x A )

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61 = E[| dV] from 3.9a, = fA.,,VdP. A So by 3.9h we have | Y| dP < | V dP for every A g A A Since all functions involved are ^-measurable, we conclude that 3.9j |A.|.^ Y| < V outside a P-negligible set ML. Set M := U u Then the set M is P-negligible. Zl L 2 . We are able to find subsequences (z„J and {z^^) of (z„) and {z„,) respectively, such that the sequence z„^,z„^,Znj,z'm^,z„,,z,„^,...,6eno\e6 (m„), decreases to z from Q^. We have Y„„(£• L as n —> 00 , for some L g K. We deduce that Li = L = L 2 . Define the process X : Q x IRj ^ by

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62 lim Y (m) if cr e and z e Mfi ' ' ^ X.-(cj) = ^ u„\z («„) c Q| 0 otherwise The process X is right continuous, and for each z g the random variable X; is ^-measurable. Hence the process X is B(K^)-measurable. In addition, for every zi < Z 2 g 0^+ we have 3.9k |A,,^ 2 X| < A-izjV on Q (apply the definition of X to 3.9j). We now show that the process X has integrable variation. Let « G N and let R, = (z, , z|], / = 1 ... n, be disjoint rectangles in K+. n n We have ^ |A«,X| < ^ Ar,V by 3.9k, /=i (=1 < IVIoo. Since the process X is right continuous, we may take the supremum over all finitely many disjoint rectangles with endpoints in Qj to obtain |X|«> < |V|„o and |X|oo e Integration with respect to P is permitted, and we have I |XU dP < J |VU dP < 00 . This establishes that the process X has integrable variation. We note from the third remark on page 66 that the measures mx(c;) and mp^Ka,) can be extended uniquely to a sigma-additive measure on for almost all cj g L2. Next we show that for every z g j^gve X^ = Y; almost surely. Fix zgKJ. Let (u„) be a sequence from Qj decreasing to z. For every index n, we have from 3.9i |Y„„ Y^l < V„„ V, outside a P-negligible set Mu„z-

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63 Set M' := U M„„,. «=i Then the set M' is /’-negligible. Further, the inequality |Y„„ Y;| < V„„ is valid on the set M'^ for every index n. Let «— >oo. We obtain |X, Y;| < 0 on M In other words, we have X, = Y. almost everywhere. Finally we show that for every e L'(|m|) we have | (p dm = E[ | 9 , dX, ] and I .p d|m| E[ I d|X|, 1. Note from the last remark on page 67 that the expectation E[ 1 dX ] is defined and finite for every set A e 3^® (S(K^). Let the measure pLx'. 3^® B(H+) — IR be defined by /ix(A) = E[ J 1 ^ dX ] for every set A e 3^® B(IR?). Then the measure ^ix is sigma-additive (Dinculeanu 2000, p. 394). Now let z g and A G m( [0 , z] X A ) = m*(A) by definition. = I Y.dP by 3.9b, A = I X, dP since X^ = Y, a.s. , A = E[ jl[o,.]x^dX] (see Definition 3.3f), = nxi, [0 , z] X A ). We may similarly show that m( (zi , Z 2 ] x A ) = | Aj,;.,X 6P A = iAx{ (zi , 22 ] X A ) for every zi < Z 2 from IR+.

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64 We deduce that the sigma-additive measures m and nx agree on a ring of generators of and this ring includes the whole space x Hence, we have m = on We apply a theorem (Dinculeanu 2000, p. In this section we establish conditions under which we are guaranteed the existence of the predictable dual projection for a two parameter process X. Many of the tools that were furnished in the preceding sections will be employed. Theorem 3.10 Let X ; Q x ik pe a right continuous, S(IR2)-measurable process with integrable variation. Assume that for every ^0 B(lR5)-measurable, bounded process (jp : Q x []j pgyg E[ | (^ dX ] = E[ J dX ]. Then X is adapted to Proof Fix u,vsRl. We must show that X„,v is 3^u,v = ^«-measurable. Since the process X is measurable and has integrable variation, we have X„_v e L'(^). Further, L'(^„) is a closed subspace of L'(^). Now suppose that X„,v e L'(^) \ L’(^„). Then there is a continuous linear functional T : L’(^) ^ IR that is 0 on L'(^„) and is nonzero at X„,v. Since the dual of L*(^) is L”(^), there is an essentially-bounded, ^-measurable 394) to deduce that for every process The Existence of function H : Q ^ K such that T(g) = | Hg dP for every g e L'(^).

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65 We have E[ H | ^„ ] = 0. In fact, let A e Then we have e L’(^«)So we have 0 = T(1^) (since T is 0 on V{Qu)), = j 6P = J H dP. Since A e was arbitrary, we conclude that E[ H | ^„ ] = 0. Set (p:= H 1[o.„]x[o,v]Then (»^,,(cj) = (E[ H | ^. ])^_(t<7) 1 [o,„]x[o,v](.s',0 by Proposition 2.8, where the function x E[ H | ^ ^ ] is chosen to be cadlag. We have p

u. Then we have 1[o,«](5) = 0. So we have 1[o.«]x[o,v](i',0 = 0. Next, assume that s
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66 Since the function x E[ H | ] is left continuous, we deduce that E[ H | 3^s] = 0 outside the set N. We conclude that P(p = 0 outside the evanescent set N x IRj Therefore we have 0 = E[ J V dX ] = E[ I ^ dX ] by hypothesis, = E[ I H 1 [o,u]x[o,v] dX ] = E[ H X„.v ] = T(X„,v). But T(X„,v) * 0. From this contradiction we deduce that X„,v e Corollary 3.11 Let e > 0, and assume that the filtration (J^,) is a right continuous e-step filtration. Let X : Q x ^ be a right continuous, ^0 S(K+)-measurable process with integrable variation. Assume that for every ^0 B([R2)_measurabie, bounded process


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67 Now fix n, i, and j e N, and consider the process := X is, is. 2 " Â’ 2 " Since the process (X^,,) is adapted to the function X^ is. is 2 " Â’ 2 " measurable; that is, -measurable. Since (J^,) is a right continuous e-step filtration, then {Qs) is a left continuous e-step filtration (see Proposition 3.2). Hence, the function X^ ^ is ^^-measurable for every 5 g Consequently, the process X"'-' is adapted to (Gs)Further, the process X"" is left continuous. We conclude that the process (X^,,) is predictable. Remark If a cross section theorem for optional subsets of Q x can be established, then using such a theorem, we would be able to obtain the result in Corollary 3.1 1 without the requirement that (IF,) is a step filtration. In such an event, we would be able to demonstrate the existence of a predictable dual projection without requiring that (^,) is a step filtration. Theorem 3.12 Let e > 0, and assume that the filtration (J^,) is a right continuous e-step filtration. Let X : Q x IR^ k be a right continuous, ^0 R(lR^)-measurable process with integrable variation. Then X has a predictable dual projection for the double filtration (^s,t)Proof The proof will be very similar to the proof in Dinculeanu 2000, p. 278, which supplies the existence of a predictable dual projection for a one parameter process. Since the process (X^_,) is right continuous, measurable, and has integrable variation

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68 then there exists (Dinculeanu 2000, p. 394) a sigma-additive, stochastic measure Hx ' B(IR+) — K with finite variation |/x,v| such that for any process (p in Consider the measure ^<8» — > K. By Proposition 3.8, the measure (nxY is stochastic, sigma-additive, and has finite variation. Thus, by Theorem 3.9, there is a right continuous, B(K+)-measurable process Y : Q X 1R2 with integrable variation |Y|,, such that for every bounded, ^(g) B(lK+)-measurable process ^ : Q x ^ we have ZA2a \ (p 6{^xr = E[J we have 3.12c I (p d{fixY = I ^(P d/iA-Now apply Lebesgue’s theorem (Rudin 1976, p. 321) in U{{^xY) and L'C/ia-) to obtain result 3.12c for every bounded, measurable process cp. So we have | cp di^xY = J = E[ J dX ] for every bounded, measurable process (p. Combining results 3.12a and 3.12b gives 3.1 2d E[ I dX ] = E[ I ^ dY ] for every bounded, measurable process (p. Also, note that the process Y is predictable.

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69 In fact, for every bounded, measurable process cp we have E[ j dY ] = j P(p d(/ixK by 3.12a, = j P(p d^x by 3.12c, = J (pdifixV by 3.12c, = E[J .pdY] by 3.12a. Now apply Corollary 3.1 1 to the process Y. Since the process Y is predictable, right continuous, has integrable variation, and satisfies 3.12d, then by Definition 3.4 we have Y = Xp. Remark If the process X is right continuous and measurable, and has integrable variation, then the limit lim X,, denoted X.^__, exists for every zo e Kj, as z/zo Z0, and assume that the filtration (J^,) is a right continuous 6-step filtration. Let X : Q x 3 continuous, measurable process with integrable variation. Assume that X is adapted to and hence is predictable. Then Xp = X. Proof By Theorem 3.12, X has a predictable dual projection Y. We will show in five steps that X Y = 0 outside an evanescent set. First we show that for every A e and r g K+ we have j (Xoo,< Y„o,,) dP = 0. A Let Ag^ and /gK^. Then we have ^(l^.t(o.o).K/)]) = 1 m(o.o).(oo./)j-

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70 So we have I Xoo,, dP = E[ j 1 A x4x[(0,0) , (OO,/)] dY ] = J dP. A Hence we have J (Xoo,/ Yoo,/) dP = 0. A Second, we show that if Xoo,/ = Yoo,/ for every t e D^+ then X = Y outside an evanescent set. Assume that Xoo,/ = Yoo,/ for every reK+. Let .s,/g[ 1^+ and let Set S;= 5 + 00 1,4c and Z := (S,r). The functions Z, (S,0), and (0,r) are stopping times, and so by Proposition 2.9 we may write ^('^[(0,0),Z]) = ((S.O) , ®) "*((0,/),®)) I ^”*[(0,0),Z] ] I ”^[(0,0),Z] ] jYzdP.

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71 Therefore we have 0 = | (Xz Yz) dP = I (X., Y,,) dP + I (Xoo, Y„,) 6P A (note, the integral | (X^^ Y^>) 6P is equal to J(x,., -Y„)d/> + |(X-,-Y.,)dP A A‘ when Z = (s1^ + s' , t) for any s' > s\ now let s' y ao)] = j {Xsj Ys,t) dP (since X*,, = Y„o,/ by assumption). A Since the set A e was arbitrary, and since the random variable Xst Ys,i is Qsmeasurable, we conclude that X^,, Ys,, = 0 outside a P-negligible set N*,,. Since X and Y are right continuous, we deduce that X Y = 0 outside an evanescent set. Third, we show that ((X*,/ Yoo,/)1 852(5,/))^ = 0. Let A e ^0 and r e K+. By writing A x {0} x [0 , r] = A x (Rj p j []j2 ^ ^ we deduce with the help of Proposition 1 .3 that the sets A x vvhere A e ^0, and the sets (Z , 00) where Z is a stopping time, together generate p. Next, note that the process (Xao,r Yoo./)1 152(5,/) is right continuous. In fact, let m e Q and let t > toThen we have [(0,0) , ('*,0] \ [(0,0) , (o),/o)] as t \ toSo we have mx(tiT)( [(0,0) , (00,/)] ) ^ mji'(o,)( [(0,0) , (oo,/o)] ) as / \ to\ that is, we have Xoo,,(cr) Xoo./„(n7) as t \ to-

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72 Moreover, the process (Xoo,/ Ya,,,)lR2(5,r) is measurable and has integrable variation. Hence by Theorem 3.12, the process (X*,, has a predictable dual projection. We claim that it is the zero process. Observe, the zero process 0 : O x ^ K is right continuous, predictable, and has integrable variation. Also, for every bounded, measurable, real-valued process

d((Xa,., Y„,,) lR2(i?,0) ] = 0, = E[J^dO).

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73 From Definition 3.4 we obtain ((Xoo./ Y*,,) = 0Penultimately we show that X-Y = (X*,. Yao,.)1n2 up to an evanescent set. Let the process X' : Q x l^e defined by X' , = X^,, (X»,, Y*./) 1r|U0 for every 5,r e K+. The process (X' is right continuous, measurable, and has integrable variation. So by Theorem 3.12, the process (X' ,) has a predictable dual projection. We have (X')^’ = Xp ((X„. since the predictable dual projection operator is linear; = Y 0 by the third step. Also, we have Xi, = Yoo,t for every r e K +. Therefore, by the second step we have X' = Y outside an evanescent set. This means that for every e (R+ we have X^,, -Y^,, = (Xoo./ Yoo,/) 1 r 2(5,0 outside a /’-negligible set that does not depend on s and t. Lastly we show that X = Y outside an evanescent set. Let to G K+. From the penultimate step we have in particular the result Xo .,0 Yo,,o = Xaojo Y®,,o almost surely. Since X and Y are adapted to the filtration (5^^), we conclude that the function Xo,/o Yo,/o is ^o-measurable. It follows that the function Xoo,,„ Yoo,
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74 Corollary 3.14 Let e > 0, and assume that the filtration (^,) is a right continuous e-step filtration. Let X : O x — > DS be a right continuous, measurable process with integrable variation. Then X is predictable if and only if we have E[ J dX ] = E[ I dX ] for every bounded, measurable, real-valued process (p. Proof This result follows from Corollary 3.1 1 and Proposition 3.13. Proposition 3.15 Let e > 0, and assume that the filtration (^,) is a right continuous e-step filtration. Let X,Y : O x Pjg|.^l continuous, predictable processes with integrable variation. If Xo.o= Yo.o, Xo,® = Yo,®, and X®.o = Y®,o, and if E[Az,® X] = E[Az,® Y] for every stopping time Z, then X = Y outside an evanescent set. Proof Set A ;= X Y. Assume that Xo,o = Yo.o, Xo.® = Yo.®, and X®.o = Y®.o, and that E[Az,®X] = E[Az,®Y] for every stopping time Z. Then we have 3.15a Ao,o= 0, Ao,® = 0, and A®.o = 0, and 3.15b E[Az,® A] = 0 for every stopping time Z. Let B e ^ 0 , and set Z := (0,0) 1g + oolgc. The function Z is a stopping time, and so applying 3.15a and 3.15b we obtain 3.15c J A® dP = 0 for every B g J^oB Since A is right continuous and predictable, and has integrable variation, then there is (Dinculeanu 2000, p. 394) a sigma-additive, stochastic measure Ha ' 3^® B(Kj) ^ IR with finite variation such that for every bounded, measurable. real-valued process q> we have

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75 3.15d|(pdjU^ = E[ J ^ dA ]. For every stopping time Z we have /^( (Z , oo) ) = I 1 (Z,oo) d^lA = E[| 1(z„o) dA] by 3.15d, = E[Az,ooA] = 0 by 3.15b. Also, for every set B e we have /i(B X 0^2) ^ I = E[| IfixRi dA] by 3.15d, = j Aoo dP B = 0 by 3.15c. We have shown that the measure is 0 on the generators of p. It follows that the measure ha is 0 on p. So we have 3.1 5e J (p dfiA = 0 for every bounded, predictable, real-valued process (p. Now let M e ^0 |^gve /iz(M) = J 1 m d/i^ = E[jlMdA] by 3.15d, = E[r'(lA/) dA] by Corollary 3.14, = \P{^M) d^A = 0 by 3.15e.

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76 This means that the measure ha is 0 on ^0 Let z e K? and C e We have 0 = ha{C x [0 , z]) = 1 1 cx[o^] = E[jlcx[o.-]dA] = I A, ap. by 3.1 5d, c Since the set C e was arbitrary, we conclude that A, = 0 outside a P-negligible set N.„ Since the process A is right continuous, we deduce that A = 0 outside an evanescent set. Consequently we have X = Y outside an evanescent set.

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CHAPTER 4 VECTOR-VALUED PREDICTABLE DUAL PROJECTIONS The results in the preceding chapters were derived for a real-valued, two parameter process (X^_,). Now, in this chapter, we regard a two parameter process (W,,) with values in a Banach space E. Under this expanded gaze, the properties "finite variation" and "integrable variation" slip from focus: a sigma-additive measure on a oalgebra does not necessarily have finite variation when it is vector-valued. Hence, the conditions under which W possesses a predictable dual projection become thrown into question. Our goal is to present a necessary and sufficient condition under which the process (W^ ,) with integrable semivariation has a predictable dual projection \Np with integrable semivariation, in the presence of a step filtration (^,). Predictable Dual Projections Notation and Terminology 4.1 The following is the framework of the sequel. 4.1a E, F, and G are Banach spaces over K. See 4.1c for another assumption about the space E. 4.1b Let M be a Banach space. • The dual of M is denoted M*. • The unit ball {xeM||x|<1} is denoted M|. • Let X G M and x* g M*. Then x* (x) is denoted . • We write Co £ M to indicate that M does not contain a subspace which is isomorphic to the Banach space Co. 77

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78 4.1c We assume that Co E. Note that this automatically occurs if E = E**. 4.1 d The space of continuous linear operators from F into G is denoted L(F,G). We write E c L(F,G) to mean that E is continuously embedded into L(F,G); that is, \xy\c Wflylf for every xeF and >' e G. 4.1e \N : D.xRl E c: L(F,G) is a function. 4. If Let M be a Banach space. A subspace U c M* is a norming space for M if for every x e M we have \x\ = sup |< jc , m >|. u^U\ 4.1g m : e e L(F,G) is a sigma-additive measure, and U cz G* is a norming space for G. 4.1 h For every set AeS(IK+), the semivariation of m on A relative to F,G, denoted m^(j(A), is defined by m/,g(A) = sup X m(A.)x, /e/ where the supremum is taken over all finite families (A,),e/ of disjoint sets from contained in A and all families (jc,),e/ of elements from Fi. Note that if F = and G = E, then m has bounded semivariation m^^^ =rn;^£.. 4.1i Let M e U. The set function m„ : F* is given by < X , m„(A) > = < m(A) x ,u> for every jc g F and A g B(IRJ). Note that m„ is sigma-additive, with sigma-additive variation |m„|. Further, we have m^(j = sup |m„|, and so if m has finite semivariation relative to F,G then for uer/i every m g U, the measure m„ has finite variation. 4.1j Let M G U. The process W„ : Q x p* jg defined by

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79 < X , \Nu{nj,z) > = < \N{uf,z) X ,u> for every jc e F and (n},z) g Q x 4.1k For every function f : K? ^ F we define m/,^(f) ;= supjjsdm where the supremum is taken over all F-valued, R(IR2) step functions s satisfying |s| < ^ f If f is m-measurable then we have m/^^(f) = sup J |f| d|m„|. We denote by the set of all m-measurable functions f : IR? — > F with m/,^(f) < c». Note that if f g then feL'(m„) for every mgU. 4. II Let f e The integral | f dm is the continuous linear functional M >--)• I f dm„ on Z. 4.1m Let uj e D. and let l,J cz 1R+ be intervals. The semivariation of the function W(c7, ) on the rectangle I x J relative to F,G, denoted svarF.G(W(c;, ) , I x J), is defined by svarF.G(W(cT, ) , I x J) = sup '^(.?i..Si+i1x(ti, [r^ relative to F.G is defined by = svarF,G(W(c7, ) , (-go , z]) for every tu g Q. and ze IRj, after extending W to X (R2 by setting W( ,z') = 0 for every z' e IR^ \ IR^. We have \Nf,(.{(n,z) = sup |W^|(tiT,z) for every cj g D and zgIR^. mU\

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80 4.1oW has finite (bounded) semivariation relative to F,G if for every cj e Q the function W^g(c7, ) is finite (bounded). W has integrable semivariation relative to F,G if the total semivariation Note that if W has integrable semivariation relative to F,G then W„ has integrable variation for every m g U (Dinculeanu 2000, p. 403). Remarks Since by assumption Co £ E, if W is right continuous and has bounded semivariation relative to F,G then for every cj e O, the measure m^^^^ ^ can be extended uniquely to a sigma-additive measure on B(IR?) that has finite semivariation relative to F,G and K,E (Dinculeanu 2000, p. 405). Therefore, if W is right continuous and has integrable semivariation relative to F,G and IR,E, then for every men and every function f in the space the integral j f dm^(^ ) is defined and is often written | f dW(cj, ). If W is right continuous and measurable and has integrable semivariation relative to F,G and K,E, then for every bounded, B(H5)-measurable, F-valued (K-valued) process (p, the expectation E[ | (/> dW ] is defined and is in G (in E) (Dinculeanu 2000, p. 409). Definition 4.2 Let W : O x ^ jjg continuous and measurable with integrable semivariation relative to K,E. A right continuous predictable process V ; Q X integrable variation (integrable semivariation relative to IR,E )

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81 is called a predictable dual projection with integrable variation (integrable semivariation relative to 1R,E) for X if for every bounded, 3^® B(IR+)-measurable, real-valued process q> we have E[ J ^ dV ] = E[ J dW ]. The process V is denoted W^. The Uniqueness of We now show that if a predictable dual projection for W exists, then it is unique up to an evanescent set once E is separable. Proposition 4.3 Let W : Q x e be a right continuous, measurable process with integrable semivariation relative to K,E. Assume that W has a predictable dual projection V with integrable semivariation relative to K,E. If E is separable, then V is unique up to an evanescent set. Proof Assume that E is separable. Then there is a countable set Zo c E* that is a norming for E. Assume that W has two predictable dual projections V, V with integrable semivariation relative to IR,E. Fix X* G Zo. Note that the real-valued process is right continuous since W is right continuous, measurable since W is measurable, and has integrable variation since W has integrable semivariation: see Definition 4.1 o. Similarly, the real-valued process < V , x* > is right continuous, predictable, and has integrable variation. Further, for every bounded, measurable, real-valued process


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82 E[ I ^ d < V , X* > ] = < E[ J (/) dV ] , X* > = < E[ j dW ] , X* > = E[\P(p d<\N ,x*>]. We conclude from Definition 3.4 that <\/ , x* > = <\N , x* >p. Similarly, we have < V' , x* > = < W , x* >p. From Proposition 3.5 we deduce that < V , x* > = < V , x* > outside an evanescent set Nx* . Since x* e Zo was arbitrary, and since Zo is a countable norming for E, we conclude that V = V' outside the evanescent set N := N^*. Jt’eZo Processes Associated With Stochastic, E-Valued Measures The following theorem will be used in the next section to help establish a necessary and sufficient condition under which a predictable dual projection \Np for W exists, in the presence of a step filtration (^,). Theorem 4.4 Let m : ^0 B(K2) -*Eci L(1R,E) be a sigma-additive, stochastic measure. Assume the following: • E is reflexive. • E is separable, and therefore the dual E* contains a countable, dense, Q-linear subspace U that is a norming space for E. • There is a number M e 1+ such that sup ^ |mx*|(A, x < M, where the ieJ supremum is taken over all disjoint finite families (A/),e/ from ^ and all finite families (x*)(W from Ui. Then there is a right continuous, ^0 BflRjj-measurable process V : Q x IRj e such that the following three assertions hold.

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83 • V has measurable, integrable semivariation • For every bounded, B(IR+)-measurable, real-valued process (p we have • When (3^,) is an e-step filtration for some e > 0, then V is predictable if and valued process (p. Proof Let jc* e U. The measure m;^* ; ^0 K is sigma-additive and stochastic. By Theorem 3.9 there is a process V"'* : O x possesses the following properties. 4.4a V^* is right continuous. 4.4b V^* is J ^0 S(K+)-measurable and has ^-measurable, P integrable variation. 4.4c For every bounded, measurable process (p :QxRl —> R we have for every bounded, ^0 B(K 5 )-measurable, real|^odm;c* = E[J(pdV^*] and J|^|d|m^‘| = E[ | |(p| d|V^*| ] Letx*,JC 2 ^U, zeK^, and A e Then jc*+X 2 g U. We have J V^?+^^(ct,z) dP{uj) = E[ J dV^^^j ] A by 4.4c, + I (V^i^(ti7,z) + V^2(t(7,z)) dP{tu). A

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84 The processes and are B(IR^)-measurable, so the functions V^r+j;!( V^t( , z), and V^ 2 ( , z) are ^-measurable. Since A e ^ was arbitrary, we conclude that , z) = V-'?( , z) + V^ 2 ( , z) outside a P-negligible set In view of 4.4a, and since Q+ is dense in we have 4.4d V^i +^2 = outside a P-negligible set Similarly, for ^ e Q and jc* e U we have 4.4eV^’= q\/^* outside a P-negligible set N^^.. Set N :{ U N,.,. ) u ( U N„. y \qEQjc*eU Then N is P-negligible. Set V^*(c7,z) := V"" {m,z) if cj e z e K+, and x* g U 0 if CT G N, z 6 and x* g U Then for or g Q, z g KJ, x, , x| g U, and ^ g Q we have 4.4fV^>^^2(u,z) = V'‘i(tjT,z) + V'^2(o7,z) by 4.4d, and 4.4g V ^r(cT,z) qV'^^(tu,z) by 4.4e. Further, for every x* g U, the process V'^* possesses properties 4.4a, 4.4b, and 4.4c. Note that 4.4f and 4.4g together say that the map x* i-^V'^"(aj,z) from U into K is Q-linear for every oj g n and z g Next we show that this map is uniformly continuous for every (uj,z) outside an evanescent set. Since the map is Q-linear, it is enough to show that f(c7) sup |V ''*|a,(c7) is finite ;c*gC/|

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85 for every tu outside a ^-negligible set. By 4.4c, for every ;c* g U and A g ^ we have 4.4h |m;,.|(Ax 12) = I |V'^‘|^d/>. A Let Ui = {x*}m, let A,,„ := {|V'*-*|a. > |V'^/‘loo for j=^ ...«}, let Bi,„ = Ai.„, and let B,;„ = A,„ \ [J By,„, where n eN and i=^ ... n. j are in ^ and are painwise disjoint. We have the following result: 4.4i f fd/* = [ sup |V'-‘ loodP x'eU\ = f lim max iV’^'loodP J n—^oo i=i l...n = lim j max |V'"‘UdP n—»co 1=1. ..n by the Monotone Convergence Theorem (Rudin 1976, p. 318), n = lim |°o dP n-.cc n = lim 2] |m;,*|(B,,„ X 12 ) by 4.4h, n->co ,= i < lim M by hypothesis. = M < 00 . We conclude that f
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86 (n7,z) if C7 G N ^ and z g 0 if CT G N and z g IR^ Note that for every x* e U, the process V""'* possesses properties 4.4a, 4.4b, and 4.4c, and also V" satisfies 4.4f, 4.4g, 4.4h, and 4.4i. Further, for every ar e Q and zgIR^, the map x* V"^’(cr,z) from U into R is uniformly continuous. Thus, for every cr g Q and z g we may extend uniquely the map x* V^'{u},z) to a continuous, IR-linear functional y{xn,z) on E* given by 4.4j < V(?5T,z) , X* > = V '^*(?zT,z) for x* g U, and 4.4k < V(rx7,z) , X* > lim V"*"(n7,z) for x* e E* \ U, n where {x*„) is any sequence from U such that x*„~^x* as n —> oo. Denote by V the E**-valued process (tu,z) V(cj,z) on Q x [Rj Note that the process V is E-valued since by hypothesis E is reflexive. We now show that V is right continuous. Let C7 G Q, zo, z e IRj, and e > 0. Assume that z > zq. We have | V(ct,z) \/{uj,zo) Ie < | < y{oj,z) V(cr,zo) , x* > | + y for some jc* g Ui since U is a norming for E, < I V"^"(c7,z) V"^"*(n7,zo) I + f + f for some choice x% =x'^g U; see result 4.4k, For each x* g U set V"^*(gt,z) once \ z zo \ 0; see 4.4a.

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87 Next we show that V is measurable. Let X* e E*. There is a sequence (x*) from U such that = lim V""'" n pointwise; see result 4.4k. For each index n the process is 5^® B(lR+)-measurable by 4.4b. Therefore < V , X* > is S(lR|)-measurable. Since E is separable, it follows (Dinculeanu 2000, p. 9) that the process V is B(K+)-measurable. We now show that V has integrable semivariation Recall that the process V is measurable, and the space Ui is countable. Hence, from the equality = sup |V;t*| we deduce that the semivariation is 3^® B(K5)-measurable. Moreover, the semivariation is increasing, and so the expectation E[ ] is defined. It remains to show that E[ ] is finite. We have E[ TVug'j ] = f sup | < V , x* > |oo dP = f sup |V"^*loodP; see 4.4j, < M; see 4.4i. Penultimately, we show that for every bounded, measurable, real-valued process q> we have | ^ dm = E[ | (^ dV ]. Let q> \QxRl ^ R be a bounded, measurable process. Then both | (p dm and E[ j ^ dV ] are defined, and are elements of E. For every x* e U we have the following chain of equalities.

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88 < J ^ dm , X* > = J ^ dm^* = E[ I ] by 4.4j, = < E[ j (p dV ] , X* >. Since U is a norming for E, we conclude that J ^ dm = E[ | ^ dV ]. Lastly, we show that when (J^,) is an e-step filtration for some e > 0, V is predictable if and only if | ^ dm = | dm for every bounded, measurable, real-valued process (p. Let e > 0, and let {3^,) be an e-step filtration. We have the following implications. I (jp dm = I /’(p dm for every bounded, 3^® B(0^|)-measurable, real-valued process

< J ^ dm , jc* > = < J ^'(p dm , X* > for every bounded, measurable, real-valued process (p and every x* g U <=> J ^ dm^‘ = j P(p dm;(* for every bounded, measurable, real-valued process

E[ I (p dV ] = E[jp(p dV"^‘ ] for every bounded, measurable, real-valued process

E[ I ^ d < V , X* > ] = E[ I d < V , X* > ] for every bounded, measurable. real-valued process cp, and every x* e L); see 4.4j,

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89 <=> is predictable for every x* € U by Corollary 3.14, » is predictable for every x* € E*, since U is dense in E*, « V is predictable (Dinculeanu 2000, p. 9). The Existence of We now present a necessary and sufficient criterion for to exist in the presence of a step filtration {^,). We will assume that E is both separable and reflexive. Theorem 4.5 Let € > 0 and assume that the filtration (3^,) is an e-step filtration. Let \N :D.xRl -> E be a right continuous, IS(lR+)-measurable process with integrable semivariation Assume that • E is reflexive; and • E is separable, and therefore the dual E* contains a countable, dense Q-linear subspace U that is a norming space for E. Then W has a predictable dual projection with integrable semivariation relative to K,E if and only if there is a number M e K+ such that sup ^ E[ I d|W,_*|(n7,2) ] < M, where the supremum is taken over all disjoint, finite families (A,),e/ from 3^, and all finite families (x*),e/ from Ui. Proof Assume that W has a predictable dual projection V with integrable semivariation V|| 5 g. Set M := E[ ]. Then M < oo. Let (A,),e/ be a disjoint, finite family from 3^, and let (x*),e/ be a finite family from Ui. Note that for every index i, the process Vx* is right continuous and predictable, and has integrable variation |V;c_*|. Further, for every index i we have 4.5a |W,.|^ = |V..| = |(W..K|.

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90 In fact, let 9 be a bounded, measurable, real-valued process, and let x* g U. Since E[^P(pd\N] = E[ | (p dV ], then < E[ | dW ] , x* > = < E[ j (p dV ] , x* >. Hence we have E[ | P(p d\Nx* ] = E[ | (p dV;^* ], and so = V^*. It follows (Dinculeanu 2000, p. 394) that E[ | P(p d|W;t*| ] = E[ j 9 d|V:t*| ]. We conclude that = l(W;c*)^lHence we have J^E[j d|W,.|] = d|V,.|] by 4.5a, Conversely, assume that there is a number M g IK+ such that we have sup XI E[ j ^( 1 ^,,
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91 Note that for every jc* e E*, the real-valued measure is sigma-additive and stochastic. Consider the measure : ^(g) ^ E given by ^p{M) = |^(1a/) d/x for every M g Observe that for every x* g E* we have 4.5c (consult Definition 3.7). Let X* G E* and M g We have (hp)x*{M) = E[}/Â’(1a/) dW;,* ] = E[|1m d(W,.)MTherefore we have 4.5d |(/i^):,.|(M) = E[|1 a/ d|(W;c*)^l ] (Dinculeanu 2000, p. 394), = E[ 1 1 A/ d|W;t*|^ ], consult the proof of 4.5a, . d|W,.|l. Next we show that for every bounded, measurable, real-valued process v' we have 4.5e j I// d/i^ = j 6fi. In fact, let j:* g U and let v' be a bounded, measurable, real-valued process. We have < jv^ 6fiP , x* > = jy/ d{iiP)x* = \pyf d/ixs see result 3.12b in the proof of Theorem 3.12, < ^ P\j/ , X* >. Since U is a norming for E, we conclude that j i// d//^ = j We now show that the measure is sigma-additive and stochastic. From 4.5c and Proposition 3.8 we deduce that the measure ^p is weakly sigma-

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92 additive. Then the measure is sigma-additive (Dinculeanu 2000, p. 57). Next, let M e ^(g) be evanescent. For every x* g U we have < , x* > = {fxP)x*{M) = (/^.*K(M) = 0 by Proposition 3.8. It follows that = 0. We conclude that is stochastic. The measure piP also has the following property: sup Z X K?) / finite ^ disjoint = sup EEtpClA.!!;) d|W,;|l by 4.5d, I finite c ^ disjoint (jf|)/G/ CT U\ < M. By Theorem 4.4 there is a right continuous, measurable process V : Q x — >• E with integrable semivariation Vi^^. such that for every bounded, measurable, realvalued process (p we have 4.5f j (p dfxP = E[ I (jo dV ]. Let ^ be a bounded, measurable, real-valued process. We have 4.5g E[ J P(p dW ] = \P(p dpi by 4.5b, = I (p dpiP by 4.5e, = E[ I dV ] by 4.5f.

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We also have j P(p = j d/i = fcpdfiP 93 by 4.5e, taking y/ to be Pq>, by 4.5e, taking y/ to be (p. From Theorem 4.4 we conclude that V is predictable. Then by 4.5g we have W'' = V.

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CHAPTER 5 AN EXTENSION OF THE RADON-NIKODYM THEOREM TO MEASURES WITH FINITE SEMIVARIATION We have noted in Chapter 4 that a sigma-additive measure m on a sigma-algebra does not necessarily have finite variation, once the measure is vector-valued. Many results in Measure Theory do not hold in the absence of finite variation. We see an example of this when we compare Theorems 3.12 and 4.5. Therefore, many results in classical Measure Theory may not be extended to embrace Banach spaces. Of particular interest to us is whether the classical Radon-Nikodym theorem (which was evoked in Theorem 3.9) may be extended to a vector-valued measure m with finite semivariation. In this chapter we demonstrate that an extension of the Radon-Nikodym theorem, with all its classical features intact, is possible if and only if m has cT-finite variation. We will close by presenting one occasion on which a Banach space-valued measure is guaranteed to have a-finite variation. The framework for this chapter consists of Banach spaces F and G, a measurable space (X,E), and sigma-additive measures v : E — and m : E — L(F,G), and a norming space Z c G* for G. We will assume that the spaces F and Z are separable, and that both Fo c F and Zo c Z are countable, dense subspaces. We begin by presenting a useful proposition that will be called upon several times later on. The notations and terminology that we will use have been introduced at the start of Chapter 4. 94

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95 Proposition 5.1 Assume that there is a family (^z)z€z of positive Z-measurable functions such that for each z e Z we have |m_-|(-) = j dv on Z. Then we (0 have |m|(*) = j sup V, dv < +oo on Z. (•) zeZo MSI Proof We begin by remarking that the measure m is not required to be sigmaadditive, just additive. Set H ;= sup V^. Note that H is Z-measurable. reZo In fact, for each z g Zo the function V; is Z-measurable. Hence, since Zo is countable, the function H is Z-measurable. Next we show that for each AeZ we have j H dv > |m|(A). A Let B G Z, and let z g Zo with |z| < 1 . We have J H dv > j V.dv B B = |m.-|(B). Therefore we have j H dv > sup |mz|(B) B -eZq W<1 > |m(B)lz,(F,G)Since the positive measure ..4 J H dv on Z is additive, we conclude that A I H dv > lm|(A) for every A g Z. A We now prove the converse inequality.

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96 Write {z e Zo I |z| < 1 } = (z,)/eN. For each index n set k„y.= {V= max(V.-, V.-„)} / = and set (-1 B„,i := A„,i, and B„,, := A„,, \ (J A„^ i = 2...n. 7=1 Let n G N. Then 5.1a B„,, e Z for / = S.lbthesets B„j, i=\...n are disjoint; 5.1c [JB„/ = X; and 1=1 5.1dmax(V.-, = Von the set B„,,. /= l...w. Let A e Z. We have j H dv = | sup dv A A /eN = j lim max(V.-, V.-„) dv A n->a> = lim J max(VV.-„) dv by the n-»co A Monotone Convergence theorem (Rudin 1976, p. 318), = lim S 1 V,; dv by 5.1c and 5. Id, «-<» '= AnB„j = lim Z |m.-,|(An B„,,) by 5.1a, n-*oo i= 1 n < lim Z |m|(A n B„,,) since |z,| < 1 for every index /, n-*^oo = lim |m|(A) by 5.1b and 5.1c, since |m| is additive; = |m|(A).

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97 Next we set out to express the measure m in terms of v by making use of a density function from X into L(F,Z*). We will see that for success, the measure m must necessarily have <7-finite variation. Theorem 5.2 Assume that the sigma-additive measure m has finite semivariation m^^, and that m « v. Then m has sigma-finite variation if and only if there is a function W ; X L(F,Z*) such that • for each xe f and z eZ, the function [\N,]x is in L*(v), and • for each x e F, z eZ, and A e Z we have < [m(A)]jc , z > = I [W,]x dv. A In this case the following two assertions hold: • For each zeZ the function [WzIf* is in L'(v), and for any cp e we have J (p d|m;| = j q> |W;|/.* dv. rs.i*' • For each zeZ and f e the function [W^]/* is in L'(v), and we have = | [W^]/’ dv. Proof We begin by remarking that the proof of the "only if implication will not require that the space Z be separable. Assume first that there is a function W ; X ^ L(F,Z*) such that for each x g F and zgZ the function [W..]x is in L'(v), and we have <[m(-)]x,z> = J [W^Jx dv on (•) S. We will show that the measure m has sigma-finite variation on E. Fix z e Z. For each xg F we have |[m.]x|(-) = j |[W,]x| dv on E (Dinculeanu (•) 2000, p.29). Note that the measure m.. : E ^ L(K,F*) is additive, the subspace F c F** is a separable norming space for F*, and the family (|[W,]x|)^^^ is a collection of positive, E-measurable functions.

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98 By Proposition 5.1 we have |m..|(*) = j |W;1 f* dv on I. () Next, note that the measure m : Z L(F,G) is additive, the subspace Z c G* is a separable norming space for G, and each member of the family is positive, and also is Z-measurable (because F is separable). Applying Proposition 5.1 a second time yields 5.2a |m|(*) = j sup |W.|f* dv < +00 on Z. (.) igZo N<1 We remark here that we have sup = |W|/.(/r_z*) on X, since the function z^Zq m |W|t(F.z*) is finite on X. We have the following properties; 5.2b The function \\N\l{f,z*) is Z-measurable (because Zo is countable). 5.2c For each or e X we have \\N{m)\nF,z*) < °oFor each « g N set B„ := {n-1 < \\N\l(f^*) < «}. Observe the following: 5.2d The sets B„ are elements of Z (see 5.2b), and are pairwise disjoint. 00 5.2e The sets B„ satisfy (J B„ = X (see 5.2c). n=l Note that the measure m has finite variation on each set B„ (see 5.2a and 5.2d). Accordingly, the measure m has sigma-finite variation (see 5.2e). Conversely, assume that the measure m has cr-finite variation. 00 Let (B„) be a sequence of disjoint sets from Z so that we have [J B„ = X, and n=l such that the measure m has finite variation on each set B„. Fix z G Z, and let « g N. The measure m is sigma-additive on the sigma-algebra

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99 EnB„. Further, by hypothesis, m has finite variation on There is (Dinculeanu 2000, p. 37) a function W" : B„ — * L(F,Z*) such that for each jc e F the function < [W"]x ,z> is in L'(v), and we have 5.2f < [m(-)]x , z > = I <[W"]jc,z> dv on EnB„. (•) Define the map W : X L(F,Z*) by W = X)W"1 b„. rt=l We now show that for each xe F, the function <[W]jc,z> is in L'(v), and we have [m(*)]x , z> = J [WjJx dv on Z. (•) We have [W,]x = < ^ [W"]x 1 , z > n=l = L<[W"]x,z>1b„. Since each function < [W"Jx , z > is v-measurable, we conclude that \\N:]x is vmeasurable. From 5.2f we deduce that for each index n, the value 1 1 < [W"]x 1 b„ , z > I dv is equal to the variation on B„ of the measure A n B„ j X dm^: on Z n B„ (Dinculeanu 2000, p. 29), which in turn is less than or equal AC\B„ to the value \\x\^B„ d|m;|. Hence we may write J|[W.]x| dv = |lE<[W"];c1s„,z>| dv n~\ jE I < 1fi„ , z > I dv (the sets B„ are disjoint). «=i = Efl<[W"]x1«„,z>| dv n=\ (by the Monotone Convergence theorem (Rudin 1976, p. 318)),

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100 00 < E J M 1s. d|m,| n=\ = 1 W d|m.-| < 00 (since m has finite semivariation m^^). Therefore the function [W.Jjc is in L'(v). Let A G Z and x g F. 00 We have j [W.]x dv = j ^ [W?]x 1 b„ dv A A Efiw:ixi,. dv n=l yl by Lebesgue’s theorem (Rudin 1976, p. 321), which applies since for each index k we have k Eiw^jxi. < I [W>| E L'(v); n=\ 00 = dm., n=l A by 5.2f, = j X dm; A 00 since |J B„ = X. «=i The result above allows us to deduce that for each F-valued Z-step function f we have [W..]f' g L‘(v), and 5.2g f [W;]f dv = J f’ dm.,. Now let f e There is a sequence of F-valued, Z-step functions f„ such

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101 thatf„ — <• f as « — c», and |f„| < |f| for every index n. We have [W.-]f = lim„ (because acts continuously on F), and so the function [W;]f„ is v-measurable for every index n. Thus we may write ||[WJf| dv = | liminf„ | [W.]f„| dv < liminf„ j |[W.]fJ dv (by Fatou’s Lemma (Rudin 1976, p. 320)), < liminf„ | |f„| d|m.-| [note, from b) we infer that for each index n, the value j |[W.-]f„| dv is the variation on X of the measure A j f„ dm.on E, which is domiA nated by the value j |f„| d|m.-|], = j |f| d|m,| (Rudin 1976, p. 321), < 00 (since fe^/r(m^^)). This establishes that the function [W^Jf is in L'(v). Now apply Lebesgue’s theorem (Rudin 1976, p. 321) to both sides of equality 5.2g. We get | [W-]f dv = j f dm.-. This proves the second assertion of the theorem. Lastly, we prove the first assertion of the theorem. For each x € F we have < [m(-)]x , z > = J [W.-]x dv on E. (•) Therefore for each jc g F we have |[m.-]jc|(*) = | |[W-]x| dv on E (Dinculeanu (•) 2000, p. 29). Note that the measure m^ : E ^ L(IR,F*) is additive, the subspace

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102 F e F** is a separable norming space for F*, and the family (|[W.Jx:|);teF is a collection of positive, measurable functions. By Proposition 5.1 we have 5.2h |m,|(.) = j |W,|f. dv on E. (•) So we have j |W..|/r. dv = |m.|(X) < 00 (since m has finite semivariation m^p^j). This means that |W,|f* e L'(v). From result 5.2h we infer that for any real-valued Z-step function q>' we have 5.2i J
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103 any set from E may contain a non-empty proper subset from Z. We now commence the proof of the theorem. Write {x* e Zo | |x*|< 1 } = Observe that the measure m has finite semivariation mj^^^ (Dinculeanu 2000, p. 70). Therefore, for each index i, the measure m^c^ ; E ^ K has finite variation |m;c,*| that is bounded by the value m 5 j^(X). 00 Define the set function A:E-^’K+by |mx;|Note that the measure A is a positive, finite, a-additive measure on E. Further, for each index i we have |m;(;|«A. Therefore, by the Radon-Nikodym theorem (Dinculeanu 2000, p. 36), for each index z there is a (positive) E-measurable function V, : X-> IR in L'(A) such that 5.4a |m;^.|(-) = j V, dA on E. (•) Suppose there is an element cj e X for which sup V,(z3j) = oo. i For each number N gM there is an index in such that > N. Set A := Pi { V/„ > }. Note that we have ,VeN • A e E (since each function is E-measurable), and • A ^ 0 (since zu e A). By hypothesis there is a subset B c; A from E such that m(B) ^ 0. It follows that A(A) > 0. In fact, suppose that A(A) = 0. Then since the measure A is a positive, additive measure on E, we have A(B) = 0. Therefore for each index / we have |m;(;|(B) = 0.

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104 Hence we have |m(B)| = < sup |< m(B) , X* >| i (since (Zo), is a norming for G), sup |m^.(B)l i sup |m;,;|(B) = 0 . This contradicts the fact that m(B) ^ 0, and so we conclude that A(A) > 0. Hence, there is a number No such that No > . " A(A) We have No)^{/\) = j iVo dA A < I dA from the definition of the set A, A = 1(A) by 5.4a, < (A)Since No > we have reached a contradiction. We must conclude that for A (A) each 07 G X we have sup V,(oj) < oo. Next, apply Proposition 5.1 to result 5.4a. We obtain 5.4b |m|(-) = j sup V, dA < qo on Z. (•) ' (We remark here that whether we regard the measure m as a function m : Z — » L(1R,G) or as a function m : Z G, the variation of m is the same, since the embedding G cz L(K,G) is an isometry.)

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105 For each number « g N set B„ := -|sup V. < n Then each set B„ is an element of Z. Further, since the function sup V, is finite i 00 on X, we have U B„ = X. n=l Observe that on each set B„, the measure m has variation |m| which is bounded by the value nX{B„). We conclude that the measure m has a-finite variation (apply the definition of the set B„ to result 5.4b).

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CHAPTER 6 SUMMARY AND CONCLUSIONS We have successfully extended the theory of one parameter stochastic processes (X,)/eiR+ to two parameter stochastic processes so far as the theory concerns the predictable projection and the predictable dual projection. However, the extension is based on a double filtration that satisfies J^s.i = for every s,t>Q, where is a step filtration. While such a filtration has useful applications (for example, in finance, the a-algebra might contain information about a market on day s, and that knowledge about the market may remain constant at ail times t within day s, and may be updated at the start of the next day, day 5 + 1), we still desire an extension theory based on a more general double filtration. Such a general filtration {3^ will depend on both 5 and t (like 3^s,t = S^svi), and will not necessarily have a step filtration underlying it. In my opinion, the best way to achieve the latter-mentioned is to first develop an extension theory for an optional projection and an optional dual projection. Then, using this theory, we may seek to modify Theorem 3.1 1 by relaxing the requirement that (^j)jer+ is a step filtration. Note that Theorem 3.1 1 is the main theorem in the dissertation that uses the property that (^i)jeR+ is a step filtration. 106

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REFERENCE LIST Dellacherie C. & Meyer P. Probabilities and Potential (1975), North-Holland Pub. Co., New York. Dinculeanu N. Vector Integration and Stochastic Integration in Banach Spaces (2000), John Wiley & Sons, Inc., New York. Metivier M. Semimartingales (1982), W. de Gruyter, New York. Rudin W. Principles of Mathematical Analysis (1976), McGraw-Hill, Inc., New York. 107

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BIOGRAPHICAL SKETCH I am a native of Jamaica, a small island in the Caribbean. My elementary education (Our Lady of the Angels Preparatory School, OLA), high school education (St. GeorgeÂ’s College, STGC), and my BSc degree (University of the West Indies, UWI), were all attained in my home country. Both OLA and STGC are run by American Jesuits living in Jamaica. At these schools, my appreciation for the sciences in general and mathematics in particular was nurtured. My mathematics teachers were excellent, and I can now see how key their contribution was to my love of math. After graduating from STGC as valedictorian of my graduating class, tertiary education beckoned. I earned a BSc degree (First Class Honors) from UWI after majoring in mathematics and computer science. All my math professors at UWI were thorough and caring, and in my mind they served as standard bearers during the six years (after graduating from UWI) that I taught mathematics at a technical college in Jamaica. When the college to which I was employed began its transition to a university, it was time to further my education. I arrived at the University of Florida with a PhD degree in mathematics as my target. However, I was unsure about which area to specialize in. My moment of clarity came one day during an analysis class led by Professor Nicolae Dinculeanu. There before me was Dr. Dinculeanu distilling from the Riemann integral all its essence, thereby creating a general theory of integration. I was hooked. 108

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109 A few months later, Dr. Dinculeanu graciously consented to be my research supervisor. He patiently guided me through the reams of background material, and then posed the task of extending the material to a second time parameter. Thanks to this challenging thesis problem, I have grown as a mathematician. Thanks to Dr. Dinculeanu, I have grown as a person.