d IXI )
A u A; [ 0, U J W 1 1 W
In fact, on the left hand side of (2'), we get =
- = = <í X dP,z> =
J ^2 M w AJ [0, u] w A u J A u
/AdP - E(1A), which is the left hand side of (4.2.3). As
for the right hand side, JMdy|^| = E(JlHwd|X|y) (by Theorem
Ü.1.K, since || S ||h|-||z| < |z||) = E (1 aÍ[0iU]„d | X |w) , which
is the right hand side of (it.2.3), thus proving the equality.
Now, since (4.2.3) holds for all A e F, there is a P-negligible
set (depending on u and z) N(u,z)C n outside of which
= , d IXI . Since both sides of this equation are
u J [ 0, u ] v 1 1 v M
right continuous, there is a negligible set N9x) - N(u,z)
u rational
outside of which = , dIXI for all u e R2.
u ■'[O.u] v ' 'v +
Let S be a countable dense subset of Z and set N = ( N(z);
zeS
is negligible. Also, since |X|^ is integrable, there is a third
negligible set outside of which |x|^(w) < ®. Let, now, w t
NQU Nl U N2 be fixed. The function X_(w) = X(w) is an EQ-valued
2
function defined on R+, having bounded variation |X(w) | = |x|#(w).
Then (Theorem 3.1.1) it determines a Stieltjes measure y on
X(w)
2 i i
B(R ) with finite variation |y , . I. By Theorem 3.2.2, we have
+ a (w;
I» Then by Theorem 1.5.7, there exists a function
lwX(w) I
1 |X(w)
G : R ■* Z’ such that
w +
1") l__z> is y,x(w),-integrable for every z e Z and
~ ^MdtllX(w) l f°r M C B(R^* In faot’
taking f - 1 in 1.5.7(2), we obtain (as for 1.2')):
M
113
' ^dvJ |X(w) I
^1Mdy|X(w) | = •ÍMdy|x(w) |-
Taking M = [0,u], we obtain __

__ - , and
A { W ) U
-fMdy|X(w)] = i[0,U].dlXl.(w); henCe “
/[0 u-j^d |X | . (w). Putting this together with what we had
2
earlier for , we now have, for Z z S, u e R ,
u +’
lCOtu]-dlX].(w) = .z> - JC0,u].Z>.d|x|,(w).
1 2
There Is then a ^|x(w)|"ne®ligible set N (w,z)C R+ outside of which
we have = . In fact, the two integrals above forra
w
2
measures on B(R ). By taking differences, we have, for u____d|X|(w) = J ^, d|X|(w), and these rectangles,
2
along with those [0,u] generate B(R ); hence -
■+ w
y|X(w) fa¬
llow, the set N1(w) - l In'(w,z) Is y. .-negligible, and for
zeS lX(w)l
u i N (w) we have = for all z e S; hence for all
u w
z e Z since S is dense in Z. Since Z is norming, we have
H (w) = G (u) for u i n'(w).
u w
Let A = {(u,w): |H (w)I < l). A is then y^-measurable (in fact,
A - AqI^I N with AQ e M, N p|x|-negligible. Then N is px-negligible
so A is px-measurable). For each w, consider the section
A(w) = {u||H (w)| < 1 }; since for w i NQ\J N1 we have
K\ * ly|x(w) |'a-e- and HU(W) = GwCu) u|X(w) |'a-e-’ we deduce that
A(w) = |u: |Hu(w)| < 1} = {u: |Gw(u) ¡ < l) (a.e.) is U|x(w)|"
negligible. Then P |x|CA) - E(/ j1A(w)(u)dIxIu
I" I A z
I _1 (w)d|v| (w) - 0 P-a.s. Denote the exceptional set by N. For
R 2
+
w i N, the section 1^(w) is d|V|^(w)-negligible, so
x”(w) -» X^(w) d|v| (w)- almost everywhere, and |x|\w) | < Jx^Cw) |.
Now, since E(||Xzjd|v| ) < ”, there is a P-negligible set n'' e F
such that for w t N1, J|X (w) |d|V | (w) < ®, l.e., |X(w) ¡ is
d| V| (w)-integrable. Then for w t N N1 we have
Jxn(w)| i |X(w) | e L1(d|V|(w)) and |xn(w) ¡ ■* |x(w)Jd|V|_(w)-a.e,
so by Lebesgue / |Xn(w)[d|v| (w) -*■ / |x (w)|d]v| (w) for
R¿ z z R¿ z z
+ +
w t N UN1 and in particular f Xn(w)dV (w) -> f _X (w)dV (w)
'2 Z z J 2 z z
H n
+ +
for w i N (J N1 .
Repeating the procedure, since the map w - J |x (w)|d|v| (w)
r2 z z
+
is P-integrable, J |x^(w) |d|V| (w) £ J|x (w) |d|V| (w) P-a.s. and
J |Xz(w) |d|V| (v) ■* /|X (w) |d|V| (w) P-a.s., we can apply Lebesgue
again and deduce that we also have convergence in L^P); in
particular, E(J Jx"|d|v| J - E(J |x Jd|v| ). Moreover, for each n
R¿ z z RZ z z
we have ECl/x^dVj) < E(/|x"|d|vJ) < ECj |Xz |d | V | J < »; hence
Jx^(w)dlMw) e LR(P) for each n. (Note: We must 3how this map is
measurable; this will come out of a later computation.) We already
showed that Jxn(w)dV (w) -> Jx (w)dV (w) P*a.s. so by Lebesgue the
121
limit is P-integrable and we have E(fxndV ) •* E( fx dV ). (in
J z z J z z
particular E(/x dV ) is defined). Next, we show that x t L1_(p ):
for each n, U|v|(|xn|) =. E( j |x" |d | V | ¿ S E( J |Xz|d|V|2) < -, so by
Fatou we have p|v|(lim inf|xn|j) < lim inf P|v|(|xn() < ”, in
particular lim inf fxn| is u |,; |-integrable. But ¡x| = lim|Xn¡
a.e. so |X| is p, .-integrable. Also, x - lim Xn is p -measurable,
' ' n v
so X e L^,(uv). Moreover, pv(Xn) •* uv(X) by Lebesgue again, since
|xnI < |X| e L1 (U|V| ).
Finally, we show that, for each n, we have u,,(Xn) - E(ixndV ),
V J z z
so we get the desired equality by passing to limits. Being a step
k
process, we can write Xn = 1 1 X,, M. e M, X, e E. Then we have
1 = 1 Mi 1 1 1
uv(Xn) = Jf) = (Mj) = EXj(E(|l^ dV^)) (by Theorem ¿1.2.1)
- EE(x. fl dV ) - EE( Í1 x dV ) (by Theorem H.l .3) = E([(I1 , x,)dV )
i* z J Mj 1 z J Mj i z
* EfJx^dV^) (and in particular the map Jx^(w)dVz(w) is P-
measurable). Letting n + », we obtain p,,(X) = E(fx dV ), and the
V 1 z z
theorem is completely proved. I
Remarks.
1) By taking E = R in the statement, we have the following: If
X is any scalar-valued measurable process, then X e l'Cpy) iff
122
E(f |X |zd|V | ) < (X is automatically separably valued.) Then
E(|xzdVz) is defined, and Uy(X) = Etjx^dV^). Finally, equality
(H.1.2) is proved the same way as (t.1.1), by taking step processes
and passing to limits.
2) The correspondence is not one-to-one: as we shall see In the
next section, a stochastic measure with values in L(E,F) is generated
by a stochastic function{not necessarily measurable) with values in a
subspace of L(E,F").
We can also generate stochastic measures from stochastic
functions (not necessarily measurable) with raw integrable variation,
as the next theorem shows.
Theorem t.2.3. Let E,F be two Banach spaces and Z C F' a subspace
2
norming for F. Let B:R+x£l + L(E,F) be a right-continuous stochastic
function satisfying the following conditions:
i) B has raw integrable variation |b|.
ii) For every x e E and z e Z, is a real-valued process
(measurable!) with raw integrable variation ||.
Then there exists a stochastic measure m: M ■* L(E,Z') with finite
variation |m| satisfying the following conditions:
1) If X is an E-valued process and If X is y, .-integrable, then
lBl
X e L„(m), the integral E(<í X (w)dB (w),z>) is defined for every
& U U
Rt
Z E Z,
= E(< fx dB ,z>),
1 u u
123
and
M;|x||) < E(/|Xu|d|B|u).
2) If, in addition, Bx is separably valued for every x e E and
if X is y i i-integrable, then the integral E( f „X dB ) is defined, and
Í31 jr2 u u
+
m(X) - E(f X dB ).
V u u
+
3) The measure m has values in L(E,F) in each of the following
cases:
a) F - Z\
b) For every x t E and v e R^, the convex equilibrated
(balanced) cover of the set (3 (w)x: w e ill is
v
relatively o(F,Z)-compact in F.
o
c) For every x e E and v e R , the function 3 x is F-
+ v
measurable and almost separably valued; in particular,
this is the case if F is separable.
Proof. Let p|B| be the measure generated by 13] via Theorem i).1 .¿l.
For every x e E and z e Z the variation of the process <3x,z>
satisfies ||v < |b| ¡x||z) (cf. proof of Prop. 4.1 .3).
Let z be the stochastic measure generated by :
mv ,(M) - E(J 1 d) for M E M. The mapping (x,z) •* m (M)
X,Z ^ X'Z
+
is linear in each argument: in fact, m (M) =
*i X2»z
E(i 21Md__**) “ E(JlMd) = E(/lMd(+<3x2,z>))
R
- E(/i Md + /l Md) - E(/lMd)+E(/lMd) =
m (M) + m (M). The computation for z Is completely analogous.
x-j » 2 *2* ^
Also, we have |mx z(M)| = |E(/ 1^d****)¡ £ E(\j 1 d****|) S
1 FT R^ V
+ +
E***|) 2 E = m (M) = E(/ .1 d***).
X t z n v
(More precisely, any continuous, bilinear function f(x,z): ExZ ■+ R
is continuous and linear in each component. Then the map x •+ f(x,*)
is a continuous linear map from E into Z'. In our situation, we
have m(M)x = mx ^(M), so = mx Z(H), i.e., for M t M, we
have ra(M) z L(E,Z').) We also have |m(M) I £ y, ,(M), since
lBl
|m(M) | = sup |m(M)x| - sup K .(M>l - sup ( sup [m (M) J) £
:|<1
Jx|£1
x,
IS1
x,z
sup ( sup Sx|Iz|ui(M)) = u, ,(M). This gives us a map
[x|<1 |z|£1 'Bl
m: M -* L(E,Z'). We now verify that m is o-additive and has finite
variation |m| (in particular |m| £ U|B|):
First of all, m is additive. Let M,N z M be disjoint; we show
m(Ml^/N) ■= m(M) + m(N) , i.e., that m(MVJN)x = m(M)x + m(N)x for all
125
x e E. This amounts to showing that =
for all x e E, z e Z. Now, = E(f 1 d****)
1 HUN v
+
' E(J 2(lM+1N)d) ’ E(l 21Md + / 21Nd) ’
R + Rt R+.
E(/ 1 d****) + EC/ ?1 d****) = + =
R R
+ +
, which shows that m is additivie. If, now,
A + 0, |m(A )| + 0 since Im(A )1 £ u,„,(A ) and the latter is
n * n ’ n " | B | n
o-additlve. Then m is o-additive as well.
As for the variation, y, , is a bounded, positive measure
!BI
satisfying |m| £ u|B|; hence |m| £ p|g| since the variation is the
smallest positive measure dominating the norm. In particular, m has
finite variation.
Now we prove assertions (1)-(3).
Ad (1): Let X be an E-valued process. From the inequality
hi - P|B| it follows that if X e Lg(u|B|), then X e Lg(|m|)
(since any sequence of step functions Cauchy in l].ui , is then also
E | B |
Cauchy in Lg(|m|)¡ hence X e Lg(m) since X is H-measurable, and
M(|*l> - U|B|(|XJ) = e(J 2!x|v rd |B | v) , which is the second part of
(1).
The first part of (1) is satisfied for any M-measurable step
n
process X * I 1 x., M e M disjoint, x. e E. In fact, for
i-1 i 1 1
n n
z e Z, we have = = < I m(M.)x.,z>
i-i Mi 1 i=i 1 1
126
n n r n ,
- I <■(«.)* ,z> = l E( 1 d****) = I E() (by
1-1 1 1 i-1 R2 Mi V 1 i>1 R2 M1 V 1
+ +
Prop. 11.1.3) - E( I ) - E() -
1-1 v 1 J v i
E() (again by lJ.1.3) - E() =
E(). Now, let X e Lg(y|g|) and let Xn be a sequence of
measurable step functions such that Xn ■* X u|B|-a.e. and |Xn || < |x||
everywhere. Let A be a u|B|-negllglble set outside of which X is
separably valued, J|x |d|B|v < » (more precisely, for (w,u) i A,
^[0,u]lXvCw)!dlBlv(w) < slnce E(/ 2lxvIdlBlv> <
R
+
J 2 lxv<«) ld [B1 v(w) < - P-a.s.; hence ir0>u] lx./w) |d|B |y(w) < »
U|g|-a.e. since it is a P-measure), and X •> X. There then exists a
P-negligible set N e F such that for w i N, the section A(w) is
d 131 _(w)-negligible (in fact, E(J1A (w)d(3|y(w)) = P|b|(a5 ” 0 ->
/ 21 A(w)d I9 I • ^ ” 0 p'a's*)> so for w t N we have d|B| (w)-a.e.:
R
i) X (w) is separably valued
11) |X%)| < |x,(w)|
HI) X%) •» X.Cw).
Since | |x (w) |d |B| (w) < ® for w ¿ N, X (w) is d131 (w)
R V
+
integrable, and x"(w) ♦ X.(w) in L¿(d|B|^(w)), so by Lebesgue
127
|x”(w)dBy(w) * Jx^(w)dB (w). Then, by continuity, for w t N, z e Z,
we have -<• j moreover,
R2 V v R2 v
|| < IzI-|/x^(w)dBv(w)| < |z|.f|x"(w)|d|3|v(w) <
fiz |/|Xy(w) |d |3 | ^(w), Now, the function w -> J |x (w) |d|B| (w) is P-
R
+■
integrable, so by Lebesgue is P-integrable and
Ft v v
+
E( -*• E() for all z e Z. For each
n, » E() as we saw above. Finally, X is
M-measurable by assumption, and [x| e L1(p|B|) C L1( |m|)¡ hence
X e Lg(m), and |m(Xn) - m(X)| £ ]m|(|xn - X|) * 0 by Lebesgue (since
|Xn| < | X |, Xn •* X |m|-a.e.), i.e., m(Xn) -*• m(X). Then »
for all z £ Z. Passing to limits, we obtain =
E() for all z e Z, which completes the proof of (1).
Ad (2): Suppose, now, that Bx is separably valued for every
x e E. Let X e l!(ui ,), let Xn be step processes convering to
e, J 31
X U|B|-a.e. with |xn | < |x|| for ail n. We shall show fir3t of all
that the map w -» J 2Xy(w)dBy(w) is integrable for all n; write
n
T 1^ x^, £ M disjoint, x^ £ E. For each i, Bx^ is separably
valued. Also, by (ii), is measurable for z e Z. Since Z is
norming, Bx^ is weakly measurable, so Bx^, being separably valued, is
128
strongly measurable, with integrable variation (in fact, 13x.] S
x.| by Prop. 11.1.3). By Theorem it.2.1, E(J 1 d(Bx.)) exists.
1 RZ Mi 1
+
E(J ?1M d(Bx )) = E E(/ (1 x ) d3 ) (by Prop. it.1.3)
R I i-1 Rz i 1 u J
+ +
E( E jl x.dB ) = E(/( E 1 x,)d3 ) = E(ixndB ) exists. We proved
, ,1 M l u J , , M, i u - u u
n
Then E
1 = 1 Rf "i
n
i=1 i
1 = 1 i
in (1) that JXn(w)dB (w) ■+ ix (w)dB (w) P-a.s. Moreover, for each n,
|/X%)d3u(w)| é j |x”(w) |d|B |u(w) £ J|X (w) |d|B| (w). By assumption,
the latter is P-integrable, so by Lebesgue Jxu(w)dBu(w) is P-
integrable, and we have E(Jx^(w)dBu(w)) -*■ E(Jx^CwJdB^tw)). We have
from before that m(Xn) -> m(X). It remains to prove that
m(Xn) = E(JxndB ) for all n. For all z t Z, we have =
= *** = E = Em (M.) =
j_1 Hj i 1 i i i x^z l
EE(/lM d) = EE() = 5ZECz>)
E(Z) = EC) = E()
. (Note: We can now do this last step since
Jx^(w)dBu(w) is P-integrable; it was not in part (1)!) Both m(Xn)
and E( |XndB ) are Z'-valued, so this means that m(Xn) =
1 u u
E(J 2X«d3u) for all n. Passing to the limit, we obtain
m(X) = E(/ 2XudBu) ; in particular, the double integral on the right is
defined.
129
Ad (3):
(a) is trivial,
(b): Let x e E, v e R+. Since the set C - co|Bv(w)x: weíj}
(balanced closed convex hull) is oCF,Z)-compact, the natural embedding
of C in Z*, the algebraic dual of Z, is o(Z*,Z)-compact (see Dunford and
Schwartz [8]). There is then a family (z.) of elements of Z such that
i íeI
C - PllytZ*: | | £ if (any closed convex set is an intersection
isl
of half-planes; we can use balls since C is equilibrated). Then we
have || £ 1 for all i e I, w e Q. Let M = [0,u] x A,
A e F; we have | | = |E( fl r _ , ,d***)| -
1 i " 1 1 [0,uJxA v * i "
|E(1A)I £ E(1 |J) £ 1; hence m([0,u]xA)x e r,
i.e., m([D,u]xA) e L(E,F). By taking differences, we have
m((u,u']xA) e L(E,F), and also finite disjoint unions of such sets.
We shall use the monotone class theorem to prove that m(M) e L(E,F)
for all M e M.
Let M * (m e M: m(M) e L(E,F)]. We show first that is a
monotone class: Let M e M , M + M. Then m(M )x e FC Z'. We have
n n n
Jm(Mn)x - m(M)x| £ - m(H)|[x[ 0 by o-additivity of m. Hence
m(M^)x + m(M)x in the metric topology of Z'. Since F is closed in
Z' for the metric topology, m(M)x e F as well, i.e., m(M) e L(E,F).
The proof for + M is exactly the same.
Now, let C be the algebra generated by sets of the form
(u,u']xA, A £ F. C consists of finite unions and complements of such
sets. We have shown that if M is a finite union of such sets, then
130
c p
m(M) e L(E,F). As for complements, m(M ) = m(R+Xii) - m(M), and
2
m(R+xQ)x = lim m([(0,0), (n,n)]xii)x e F by closure of F in Z' as
before. Thus, if m(M) e L(E,F), then m(MC) e L(E,F). Then CC M , C
is an algebra, so M = a(C)Ch>/ by the monotone class theorem, l.e., m
takes values in L(E,F).
p
c) Suppose that for x e E, v e R , the function 3 x is F-
+ v
measurable and almost separably valued (in particular, if F is
separable, then B^x is separably valued and weakly measurable by (ii);
hence B^x is F-measurable). Then for every A e F, x e E, the function
I^B^x is integrable; in fact, B^x is F-measurable, by assumption
and we have |Bvx| £ |3VIIX| E l'cp). We also have, as before,
= E(/lr n .d****) = E() =
; [0,v]xA • ‘ [0,v]xA
= (again, we can move the
1A [0,v] • A v
expectation inside since l^B^x is integrable). Since this holds for
all z e Z, we conclude m([0,v]xA)x ■ EO^B^x) e F. Then
m([0,v]xA) e L(E,F), and we conclude by the same monotone class
argument as in (b). I
Remarks.
1) This theorem shows that if B has values in L(E,F), then m
B
has values in a subspace of L(E,F"). Moreover, we do not have in
general Im | = p, ,. Later we will establish some conditions
B | B |
sufficient for equality.
2) The correspondence B ■* m is not injective. For an example
involving measures associated with functions, see Dinculeanu [6,
p. 273].
131
iJ.3 Vector-Valued Stochastic Functions Associated
With Measures
In this section we consider the converse; starting with a
stochastic measure m with finite variation, we will find a stochastic
function B with integrable variation such that m Is associated with B
in the sense of Theorem H.2.3. The precise result is the following:
Theorem 4.3.1. Let E,F be two Banach spaces and Z Cl F' a subspace
normlng fo" F. Let m: M -> L(E,F) be a stochastic measure with finite
variation |m|. Then there exists a right continuous stochastic
function B: R^xi! ->■ L(E,Z') satisfying:
i) B has raw integrable variation |b|.
ii) For every x c E and z e Z, is a real-valued raw
process with integrable variation ||.
Moreover, we having the following:
1) If X is an E-valued measurable process we have X e lVih)
E
if and only if X e L^Cp. .), In this case the integral
E 13 ]
E() Is defined for every z e Z,
R
+
= E(), and
;02 u u
n
+
IH<|X|) - E(/ 2|Xu|d|B|u), i.e., |m[ = p|B|.
2) If F is separable (or more generally if Bx is separably
valued for every x e E), then Bx is measurable for every x e E.
If B Is separably valued, then B is measurable.
132
3) We can choose B with values in L(E,F) in each of the
following cases;
a) F is the dual of a Banach space H and we choose Z = H;
hence F = Z'.
b) For every x e E, the convex equilibrated cover of the set
(Jxdm: $ simple process, J|$|d|m| < 1 ) is relatively o(F,Z)-
compact in F.
c) E is separable and F has the Radon-Nikodym property (we
say F e RNP); in this case B can be chosen such that Bx is measurable
and separably valued for every x e E, hence
m(X) = E(JxudBu) for X e L^m).
d) The range of m is contained in a subspace G L(E,F)
having the RNP: in this case B can be chosen measurable, with
separable range contained in G; hence
m() = E(Jif^dB^) for <|> e L1(m).
4) If p is a lifting of P, we can choose B uniquely up to an
evanescent set, such that p[B ] = B for every v e R^ (see Definition
V V +
1.5.5(b)).
Proof. Let V be the integrable increasing raw process associated with
!m| via Theorem H.1.4:
Im!(M) = E(f 1 dV ) for H £ M.
11 JH u
133
2 2
Denote the rectangle [0,z] by R^; for z e R~ set ra (A) = m(RzxA)
for A e F. We verify that mZ: F ♦ L(E,F) Is a o-addltlve measure
with finite variation |mZ|, and that mZ is absolutely P-continuous:
i) m_ is q-additlve: First of all, mz is additive. Let
A,B e F, disjoint. Then R^xA and R^xS are disjoint, so we have
mZ(A^JB) = m (R x(A{J 3)) = m((R xA)^,'(R x3)) = m(R xA) + m(R x3)
Z 2 Z Z Z
= mZ(A) + mZ(B), Now, let (A ) t F, A + 0. Then (R xA )+(R x 0)
n n z n z
= 0, so lim m (A ) = lim m(R xA ) - ra(0) = 0. so m is indeed
n z n
n n
o-additive.
2
ii) m has finite variation: we show in particular that
|m2| < [m |Z, where |m|Z(A) = |m|(RzxA). Let A e F, and let (A ^),
n
1 - 1,...,n be disjoint sets from F with Í^/A C A. We have, since
i-1 1
n n n
»x(U*,) - U(S *A.)CR xA, I |mZ(A ) I
2 1=1 1 iVi 2 1 2 1=1 1
n
n
£ |m(R xA.) | <
i = 1 2
E |m|(R^xA.) £ |m|(R^xA) = |m|“(A). Taking supremum, we obtain
|mZ|(A) £ |m|Z(A).
ill) rr,Z << P: In fact, we have |mZ(A) | = |m(RzxA) |
£ |ra|(R^xA) = E(j 21R xAdV ) = E(1 V ); hence ]m|z « P, so
R+ 2
2
m << P as well.
Applying the Extended Radon Nlkodym Theorem (1.5.8), we get, for
2 0
each z e R+, a function : n ♦ L(E,Z') satisfying:
o 1
1) |Bz| is P-integrable, and for iji e L (|m |), we have
id m
J|Bj*dP.
134
1 7
2)**** is P-lntegrable for all f e L^(|m |) and zQ £ Z,
and = i****dP.
‘ 0 ' z 0
so "
3) If p is a lifting of L (P), we can choose (Bz) uniquely
0 0
P-a.s. such that pCJ - Bz> i.e., for all A £ F, x e E, zQ e Z,
0 «> o o
we have 1ft e L (P), and p(1ft) = 1p(A)‘ If’
addition, there exists a>0 such that |mz| £ aP, then we can choose
o 0 0 0 m
B uniquely everywhere such that p(B ) = B , i.e., **** £ L (P)
z z z z 0
0 0
for all x e E, z e Z, and p(****) = **** for all x,z .
v Z U Z U (J
4) If one of the conditions in 3(a) or 3(b) is satisfied, then
0
B takes values in L(E,F).
z
Now, in particular, taking i|> = 1 , A e F in (1 ) we obtain
1 ') |mZ J (A) - EOjBj) for A e F.
Also, taking first f ■ x, x e E, and then f = xlfl, A e F, we get
0
2') is integrable for x e E, zQ e Z, and
= = /****dP = E(1A****), for
A e F, x e E, z0 e Z.
0
(Notice also that from (1'), if B is bounded, the condition
z
|mZ| £ aP in (3) is satisfied.)
From (1') and the inequality |mz| £ |m|z we obtain
fl|B°|) = |mZ|(A) £ |m|Z(A) - |m| (IMcA) - E(/lR dVu> = E(1 V ),
z
D 0
i.e., E(1 |B 1) £ E(1 V ) for all A £ F; hence |B | £ V P-a.s.
A Z A Z Z Z
2
Let z = (s,t), z' = (s',t') be points in R , z < z'. Denote by
D , the set R , \ R , and by R
zz z z zz
in
the rectangle t (s,t), (s', t') ]
135
We have m(D xA) - m((R \ R )xA) = ra((R ,xA) \ (R xA)) =
¿¿ z z z z
m(R ,xA) - m(R xA) = mZ (A) - raZ(A). Likewise, ra(R ,xA) =
, ZZ
m (A) - m ''(A) - mSt (A) + mSt(A). Then for x e E, zQ e Z,
zy z 7 '
we have <(ra -m )(A)x,zQ> = - =
0 0 0 0
E(1a) - E(1a). The same
computation gives <(mS - mS t - mst + mSt)(A)x,zQ> =
o o
E(1 <(A (B°))x,z >). Now, since p[B = B etc., we have
zz' u s c 3 t
0 0 0 0 0 o
PÍX' - B ] = B , - B , and p[¿ (B )] = AD (B ). In fact (we
Z Z Z Z n , n ,
ZZ ZZ
give the proof for the first; the second is the same), for A e F,
x e E, zQ e Z, we have
<(B,' - Vx-V\ - E L“(p)
since each term is. Also, p(<(B , - B )x,z„>1„) =■ p(****1 -
Z z U A z 0 A
o o oo
1A) " P(****1 ) - p(****1A = <3 ,x,z.>1
W’pU) = <(V ~ Bz)x-V 1 p(A) ’ S° p[Bz' ‘ Bz] = Bz' ' Bz’
0
and the same for ar (B ). Then, by Proposition 1.5.6, both
zz'
I 0 0 0 0 0
|3Z* “ Bz| and |ar (B )|. are P-measurable. Also, |B , - B |
zz' z z
oo o o
• + |B I S V „ + V £ 2V hence |B , - B I is P-integrable;
z z z z z z z
o o
similarly, |¿ (B ) | < ¿4V hence Iad (B )| is P-integrable as
* Z * H ,
zz zz
o o
well. Also, by properties of liftings, <(B , - B )x,z> and
z z
o
<(Ar (B ))x,zq> are measurable for x E g, z e Z (see property 2
136
after Defn. 1.5.5). By the "converse" of the generalized Radon-
Nikodym Theorem (Theorem 1.5.9), there exist measures
m^: F * L(E.Z') and mR: F * L(E,Z') (the measures have the same
o
values as the function B since Z' is a dual (cf. part 3(a) of the
statement of this theorem) with finite variation |mD| and |mR|
such that:
0 0
1) = E(1 <(B . - B )x,z_>) for A e F, x e E,
D U A z z 0
zQ e Z (by taking f = x1fl in 1.5.9), and =
o
E(1A<(AR (B ))x,zQ>) likewise. Also,
zz'
ii) |mD|(A) = E(1a|B°, - Bj), and |mR|(A) = E(1A|AR (B°)¡)
ZZ '
for A c F (we take ty = in 1.5.9).
0 0
From (i) we have =■ E(1, <(B , - B )x,z„>) -
u u A z z 0
<(m2 - mZ)(A)x,z > from earlier. Likewise, =
U K U
S't ' s't
<(m - m
s t ^ s t
m + m )(A)x,Zq>. Both these hold for all A,
x, Zn so we have mn(A)x = (mz - mZ)(A)x, and mR(A)x
(m‘
0
s't'
s't
D
st'
+ mst)(A)x for all A, x; hence m^ = mZ - m2
and m * m
n
S't'
s't St'
m - m
st
(and in particular m^, mR
have values in L(E,F)),
By (ii), we have |mZ - m21(A) = |mD|(A) = E(1|Bz, - B^J),
q ' q *t" q t- * q j- 0
and similarly |m - m - m + m |(A) = E(1a|ar (B )||) for
zz'
A e F. On the other hand, we have |(m - m )(A)| = |m(Dzz,xA)j <
|m|(Dzz,xA); hence |mZ - mZ|(A) < |m|(D ,xA) = |m|Z (A) - |m|Z(A)
- E(1AVz'} - e(1aV - E(VVZ'-V2^ hence e(1a¡b¡' - 3J> á
>37
E( 1 (V í )) for all A e F, so |3 , - B I < V , - V P-a.s. for
A z z 1 z z‘zz
each z < z'. The same computations for the rectangle yield
o
|ar (3 )[ í (V) P-a.s. If we take z,z' with rational
zz' zz'
0
coordinates, we can find a common negligible set and modify B
on it to get the Inequalities everywhere for all z,z' rational.
Next, let z be fixed. We show that for any sequence r + z, r
n n
0
rational, the sequence (B (w)) is Cauchy for all w. In fact, the
n
sequence (V (w)) is Cauchy for all w since V is right continuous,
n
i.e., for any e>0, there exists n^ such that n,m > implies
It 0 0
|V (w) - V (w)| < e. Then, for n,m £ n , we have |B (w) - B (w)|
J n m n r m
0
£ |V (w) - V (w)| < e. Thus, for any w, the sequence (B (w)) is
n m n
o
Cauchy in L(E,Z') complete, so lim B (w) exists.
r
n n
Now, let (rnK(sn) be two sequences of rationale decreasing to
z. We can construct a sequence (v^) decreasing to z, containing
0
subsequences of both (r ) and (s ), Then lim B (w) exists; moreover,
n n v.
J J
0 0
since lim B (w) and lim B (w) exist, and subsequences of both are
n rn n 3n
O
contained in (3 (w)), all three limits are equal. In particular,
VJ
0 0
lim B (w) = lim B (w), so we get the same limit for any sequence of
n rn n Sn
2
rationals decreasing to z. Then, for every w e fi, z € R ,
B (w) = lim B (w) exists. The stochastic function B thus
z , r
r + z
r rational
defined is right continuous.
138
In fact, let e>0: there exists a neighborhood to the right of z
so that if r is rational and lies inside the neighborhood, then
|b^(w) - B^(w) | < | . For any z' in this neighborhood, there exists a
similar neighborhood for it, and the intersection of these has
nonempty interior. Let r be in their intersection, r rational. We
have |Bz(w) - Bz«(w)| = |Bz(w) - B°(w) . B°(w) - B7.(w)|
|B (w) - B (w)| + |B (w) - B ,(w) | < |
r r'
e £
z r
continuous.
Thus 3 is right
Some more properties of B are the following:
а) For wz, r>w. Then |B - 3 | ■
o o
|B - B + B -B +B - B I < V - V . Letting q+z, we get
■q z z w w r1 q r o
o
|Bz - Bw + Bw - B^[ £ Vz - by definition of B and right continuity
of V. Letting r+w likewise, we get |3 - B I £ V - V . We
1 z w‘ z w
similarly have ¡A (B)| £ A (V).
R , R t
zz zz
o 2
б) For each zt B = B a.s.: In fact, for z e R , r rational,
o o
r>z we have Ib - B I $ V - V a.s. (in fact there Is a common
1 r z1 r z
negligible set outside of which this holds for all r>z) letting r + z,
D 0 0
we have |b - B | = lim|B - B | < lim(V - V ) - 0 a.s., i.e.,
r + z r + z
0
B = B P-a.s.
z z
Next, we show that B has raw integrable variation |3|. Since
0 DO
B - B P-a.s., B,-B = B , - B a,s.f and A_ (3) =
zz z zz z R ,
zz
o
A (B ) a.s.; hence p[B , - B ] = 3' - B , and n[A (B)l
z z z z’ R /
139
- ¿ (B) (property (5) following Definition 1.5.5), so by 1.5.7,
zz'
|Bz, - Bz¡ and |ar (B)J are measurable; hence the finite sums we
zz'
use to compute Va^ fl] (B(. _ Q)) , Var [Q> t] (B(()>.,) , and
iar.,„ , ,.,(B) are also measurable.
L(0,0),(s,t)J
Moreover, since B is right continuous, we can compute the
variation using partitions consisting of rational points; the first
two terms from the one-dimensional result, the third by Proposition
2.2.5. Each of these limits can then be taken along a sequence, so
Var[o,s](B(.fo))‘ Var[0,t](B(0,-)) * Var[(0,0),(s,t)]****]) - v(>it) - Y(0it)
- V
(s,0) (0,0)
of grills as before, we have Var
[(0,0),(s,t)]!B) £ V(s,t) ' V(0,t)
" V(s + V^0 ^ a.s. Adding up (1)-(H), we obtain |B|^g ^ -
^(O.O)1 + Var[0,s]iB(-,0)) + Var[0,t](B(0,-)) + Var[(0,0),(s,t)]CB)
■ V(0,0) + (V(s,D) " V(0,0)) * (v(0,t) “ v(0,0)) + a-S- Thu9' for £ R'- l3l(s,t) -
V(a a.s. Since | B [ is increasing, and 131 ^ ^ ^ a.s., |b|
is finite outside an evanescent set*; hence |b| Is right continuous
outside an evanescent set. Then, since both |b| and V are right
continuous, |b| £ V outside an evanescent set. In particular,
|B|b £ V_, so B has integrable variation, and this completes the proof
of (1).
* More precisely, for example is a negligible set N such that
n
w i N => 13], (w) < V .(w); hence IBI ¿ V,
n 1 (n,n) (n,n) 1 1 (n,n)
outside N^. Then NQ = is negligible and |B[ < ■> outside
this set. n
mi
Now for (ii). For x e E, z. e Z we have **** = ****
0 z 0 z 0
a.s., so by (2') is integrable; moreover, is right
continuous since 3 is, and has integrable variation (we showed in the
proof of 4.1.3 that |****(w)] < |B|(w) |x | ]z |). Also, for each
2
x e R , **** is a
+ z 0 z z 0
real-valued raw process with integrable variation, which is (ii). We
now turn to assertions (1)-(4).
Proof of (1); For M = [0,z]xA, A e F, we have, for x,zQ:
z ®
= = E(1.**** (by (2')) = E(1,****)
U U A Z U AZU
E(1Ai 21[0,z]d)- ^en ll =
R
|E(|lMd)| < E(/lMd|3|w||x|[z0P - E(J 21Md|3|w)lx|||z0||,
R
+
so |m(M)| £ E(/1Md|B|^) ; hence |m|(M) £ E(/1 d|B| ). On the other
hand, from |B| £ V we have E(Jl d|B| ) £ E(Jl^dV ) - |m|(M); hence
|m|(M) = E(J1Md|3| ) for M = [0,z]xA. 3y additivity, then, as usual,
we have |m|(M) = E(/1Md|B|w) - u j g j(M) for M = RxA, A e F, R a
2
rectangle of the semiring generating B(R+) (there are four kinds; cf.
Theorems 3.2.2 and 4.2.1 ). As |m|, y|^| are both o-additive, and
sets of the form M = RxA form a semiring generating M, we have
H - 1,1131 *
Now, let X be E-valued, measurable. If X e L^(m), then
1*1 c L1(|m|) * L1(y|B|) »> X e L^(u|b|). Conversely, if
X E Lgji^|Bp, then |x| e L1 (u | B |) -> |x| c L1 ( | m ] ) ; hence
X e Lj,(m) since X is measurable.
E
We already showed that |m| = m|b|’ whloh establishes the second
equality in (1). As for the first, we note that by Theorem H.2.3,
E() is defined for zQ e Z. To prove the equality, we
R
+
note first, that by Theorem H.2.2 there exists a stochastic measure
m: M ■» L(E,Z') corresponding to B satisfying -
E(). (Note that since Z' is the dual of a Banach space,
w w u
+
m has values in the same space of operators as Bj cf. statement (3a)
of this theorem.) We shall show that m - m; as both are o-additive it
will suffice, as above, to prove for sets of the form M - [0,z]xA,
A s F.
Let M = [0,z]xA, A e F; let x e E, z^ e Z (which is a norming
subspace of (Z')'). We have = -
E«/(1Mx)dBv,z0>) = E(/lMd) (by 0.3) - E(1 A)
0
-E(1a) = . Then, for x E E, zQ e Z,
= ; hence m(M) * m(M) for M * [0,z]xA. As
2
before, we conclude that m = i on all of B(R+)xF. Then for
X e Lp(m) we have = = E(), which
completes the proof of (1).
Proof of (2): If Bx Is separably valued for x e E, then Bx is
measurable for x e E since it is weakly measurable by (ii). If B is
separably valued, then B is measurable since is measurable for
all x e E, Zq e Z, by (ii) [6, Proposition 2U, p. 106],
0
Proof of (!|) : If p is a lifting of P, we can choose
uniquely a.s. for all z; in particular, outside an evanescent set for
all z rational. Then 3 is determined uniquely outside this evanescent
2
set; we already showed that p[B ] = B^ for every v e R^.
Proof of (3):
0
a), h): If one of these is satisfied, then 3 takes values in
L(E,F); hence B takes values in L(E,F) since L(E,F) is closed in
o
L(E,Z') for the metric topology (recall that B = 11m B ).
z r ’
r + z
r rational
c): Assume E is separable, F t RNP. Let x e E, z e R^. The
measure y: F-* F defined by y(A) = mz(A)x for A e F is a-additive
(since |mZ(A)x| < |mZ(A)|-jx| so An+0 => |mZ(An)| * Q =>
||J(An) | * 0; y is evidently additive) and has finite variation
|u| £ |mz|■ |x| £ |m|z|x|. Then y << P since |m|Z << P, as we showed
already. Since F t RNP, there is a Bochner-lntegrable function
B' e Li such that y(A) = E(1,B' ) for A e F. We can choose
x, z r A x,z
B' separably valued; therefore, we can consider F separable. More
precisely, let S be a countable dense set in E. For x e S, we have
lim B' = B' a.s. In fact, for A eF; E(1„B' ) = y(A) =
u + 2 X.U X,Z A *.u
u rational
mU(A)x = m([0,u]xA)x + m([0,z]xA)x - mZ(A)x = E(1AB' t) > henoe
B' ■» B' a.s. Then ¡B' (w); x e S, z rational} is separable;
X i U X | Z X | z
hence [b^ z(w): x e E, z rational} is separable. For any z,
BJ - lim B' a.s., and we modify B' on the exceptional set;
x,z w*z x,u x,z
u rational
we can do this and still get a R-N derivative of y. Hence we can take
F separable. We can then choose Z separable in F", normlng for F.
2
Let B: R+xí¡ -* L(E,Z') be the stochastic function associated with m
for this choice of Z. We have, then, for in a countable dense
subset ZAC Z, and for A ef E(1„**** = E(<1„B' , z„>)
0 A x,z 0 A x,z 0
' " **__ - * E(1a) =
E(1 __**); hence **** = **** a.s. for each z e Z„.
A z 0 x,z 0 z 0 0
Now, since is countable, there is a common negligible set;
since Z is dense in Z we have B' = B x outside this negligible
0 x,z z o c
set. There is then a common negligible set such that 3' = B x
x,z z
for all z rational, x in a countable dense set of E. Then, by right
continuity of B and closure of L(E,F) in L(E,Z'), we have
B - B' e L(E,F) outside this negligible set, i.e., up to
evanescence. By modifying B on this evanescent set, we obtain B with
values in L(E,F). Moreover, since B^x - B' , BzX is integrable (in
particular Bx is measurable and separably valued by right continuity)
and so, by H.2.3(2), for X e lI(di) , we have m(X) = E(i dB )
E J 2 v v
n
4-
(m is the measure associated with B via ^.2.3), and in
particular, m(M)x = md^x) = ECjl^xdB^), which completes (c).
d) Assume the range of m is contained In GCL(E,F) with
G e RNP. We write G - L(R,G) and apply (c): R is separable,
G £ RNP. Then B has values in L(R,G) = GC L(E,F), and Ba Is
measurable for a e R; hence B is measurable. Also, for ¡p e l/(m)
we have by (c) m($) = E (J (j vdB ^), which is (d) , and completes the proof
of this theorem. B
Remark. In the last part, once we have 3 measurable, we can also get
the equality by applying Theorem H.2.1.
.4 On the Equality |m| = p
M
In Theorem 4.?.3, we began with a stochastic function B with
Integrable variation, and associated a measure m with finite
variation, and we proved that |mI £ p, ,. We now consider some cases
lBl
where this is in fact an equality.
Theorem 4,4,1 Let E,F be two Banach spaces and Z C F' a subspace
2
norming for F. Let B: R + xfl -* L(E,F) be a right continuous stochastic
function satisfying conditions (1) and (Ii) of Theorem 4.2.3, and let
m: M * L(E.Z') be the corresponding measure with finite variation |m|
satisfying
= E(<[ X dB ,z>)
J ? v v
R
+
for any E-valued measurable process X e l!,(p, ,). We have the
E I B [
equality |m| = jj |B¡ , l.e.,
|m|(|xl) = E(J 2|Xv|d|3|y) for X e L^(|m]>
+
in each of the following cases:
1) There is a lifting p of P such that p[B ] - B for
z z
2
z e R .
+
2) E is separable and there is a countable subset S CZ z norming
for F.
3) E Is separable and B^x is integrable for every x e E and
4) 3 is measurable and B^ is integrable for z e R2.
1 ¡46
Proof. Case (¡4) has been dealt with In Theorem ¡4.2.1; we include it
here for completeness.
We have a measure m: M -» L(E,Z') with finite variation. By
Theorem 4.3.1, there exists a stochastic function B' with values also
in L,(E,Z') (by 3(a)) satisfying (i) and (ii) of Theorem 4.3.1, such
p i
that p[B'] = B' for z e R and such that for every X t L (m) we have
z z + E
= E() for e 1
R +
(note: here we consider Z embedded in Z" as a norming subspace in
order to apply the theorem), and
| m | ( | X |) = E (j p|X |d|B'| ).
R"
+
Now, for X = l^x with M e H, xe E, and for zQ e Z, we have
E(/lHd****) - E(/lMd). In fact, E(JlMd**~~)
■ Cby = E() = =
E(^ * E(:) ‘ E(/1Md)‘ Taking
PI = [0,z]xA with A eFwe obtain
E(i1[0,z]xAd) = E(1A)- and
E(^1[0,z]xAd) ' E(1A) - E(1A)
for A ef so (1.1) ~~** = **** a.s. for x e E,
z e Z. We shall prove that In cases (1)-(3), B and B' are
147
indistinguishable; hence 13') « jB| up to evanescence, so
y|3| = U|B'I = M-
p
1) Assume p[B ] = B for all z e R . He have, from above,
z z +
2
p[B^] » p[B ] for z e R+ as well. Then from (1.1) we conclude that
Bz = B' a.s. for each z (property 4 following Defn. 1.5.5). Since
both are right continuous, they are indistinguishable.
2) Assume E is separable, let Cl E be a countable dense set,
SC2 a countable subset norming for F. We have =
a.s. for all x e Eq, zq e S. There is then a common
p
negligible set N such that the equality is valid for all z e R+
rational, x e E^, zQ e S. By right continuity, then, this holds
2
outside N for all z e R . Since S is norming, we have B'x = B x
+ z z
2
outside N for all z e R+, x e Eg. Since EQ is dense in E, we have
p
B'x - B x for all z £ R , x e E (still outside N) ; hence B' = B
z z + z z
2
outside the evanescent set R+xN, i.e., B and B' are indistinguishable.
3) Assume now that E is separable and B x is integrable for
z
2
every x e E, z e R+. Then B^x is almost separably valued; by right
continuity Bx is separably valued outside an evanescent set for
x e E. Since E is separable there is (as before) a common evanescent
set A outside of which Bx is separably valued for all x e E. We
modify B on A by setting it equal to zero on A and get a process B"
indistinguishable from B, with taking values in a separable space
Fp CT F for all x e E. Tnen
E({lMd****) = E(}lMd<3vX,z0>) = ;
1 m3
hence n is the measure associated with the stochastic function
2
B": R+xfJ •+ L(E,Fq). Since FQ is separable, there is a countable
subset SC Z norming for F . By (2), we have |m|(M) = E(fl d IB" I ).
Since B = B" outside an evanescent set, |b| = |3"| outside an
evanescent set, so ( „1„d|3l - f a.s. for any M e M;
J 02 M 1 1 v B 1 1 v
R+ R+
hence |m|(M) = E(Jl d|B")v) - E(/1 d|B| ) for M e M, i.e.,
|mJ = u |Bj, and this completes the proof. I
Remark. If we start with a stochastic measure and associate a
function, we always have |m| = M|g|> but if we start with a stochastic
function, we do not get equality—not even if the measure has values
in L(E,F). Equality (1.1) seems to be as close as we can come in
general; in order to get everywhere from there, it seems we need for E
and Z not to be "too large.
CHAPTER V
CONCLUSION
We have seen that the usual definition of the variation on a
rectangle of a function of two variables is insufficient to yield all
the properties necessary to extend the theory of Stieltjes measures to
functions of finite variation on the plane. We have given some
additional conditions sufficient to establish a proper definition of
the variation of a function, and although these were not shown to be
minimal, it would seem to be difficult to weaken them further.
We have shown that, starting with a two-parameter stochastic
function X with values in L(E,F), we can associate a measure with
values In L(E,Z') and that under certain conditions p has values in
x
L(E,F) as well. We have also established a similar correspondence,
starting with a measure and obtaining a stochastic function. We have
also shown that, if the spaces E and F are not "too large," we have
the equality
We hope that this lays the groundwork for exploring the question of
existence of optional and predictable projections of vector-valued
multiparameter processes.
BIBLIOGRAPHY
1) Bakry, D., Limites "Quadrantales" des Martingales, in Processus
Aléatolres á Deux Indices. Colloque ENST-CNET, Lecture Notes in
Mathematics no. 863, Springei—Verlag, New York, 1980, pp. 0-Ll9.
2) Cairoli, R., and Walsh, J.3., Stochastic Integrals in the Plane,
Acta Math., 13*1 (1975) , pp. 111-183.
3) Chevalier, L., Martingales Continue á Deux Parametres, Bull. Sc.
Math., 106(1982), pp. 19*62.
*0 Dellacherie, C., and Meyer, P.A., Probabilities and Potential B,
chaps. I-IV, North-Holland, New York, 1978.
5) Dellacherie, C., and Meyer, P.A., Probabilities and Potential,
chaps. V-VIII, North-Holland, New York, 1982.
6) Dinculeanu, N., Vector Measures, Pergamon Press, New York, 1967.
7) Dinculeanu, N., Vector-Valued Stochastic Processes I; Vector
Measures and Vector-Valued Stochastic Processes with Finite
Variation, Journal of Theoretical Probability (to appear).
8) Dunford, N., and Schwartz, J.T., Linear Operators, Part I;
General Theory, Pure and Applied Math., no. 7, Wiley-
Intersclence, New York, 1958.
9) Fouque, J.P., The Past of a Stopping Point and Stopping for Two-
Parameter Processes, Journal of Multivariate Analysis, 13(1983),
pp. 561-577.
10) Kussmaul, A.V., Stochastic Integration and Generalized
Martingales, Pitman, London, 1977.
11) Métivier, M., Semlmartingales, Walter de Gruyter, Berlin, 1982.
12) Meyer, P.A., Théorle Elémentalre de Processus á Deux Indices in
Processus Aléatoires á Deux Indices, Colloque ENST-CNET, Lecture
Notes in Mathematics no. 863, Springer-Verlag, New York, 1980,
pp. 1-39.
13) Millet, A., and Sucheston, L., On Regularity of Multiparameter
Amarts and Martingales, Z. Wahr., 56(1981), pp. 21-115.
1 50
151
19) Neveu, J., Discrete Parameter Martingales, North-Holland,
New York, 1975.
15) Nualart, D., and Sanz, M., The Conditional Independence Property
In Filtratlons Associated to Stopping Lines In Processus
Aléatolres á Deux Indices, Colloque ENST-CNET, Lecture Notes In
Mathematics no. 863, Sprlnger-Verlag, New York, 1980, pp.
202-210.
16) Radu, E., Mesures Stieltjes Vectorielles sur Rn, Bull. Math, de
la Soc. Sci. Math, de la R.S. de Roumanie, 9(1965), pp. 129-136.
17) Rao, K.M., On Decomposition Theorems of Meyer, Math. Scand.,
29(1969) , pp. 66-78.
18) Walsh, J.B., Optional Increasing Paths in Processus Aléatolres a
Deux Indices, Colloque ENST-CNET, Lecture Notes In Mathematics
no. 863, Springer-Verlag, New York, 1980, pp. 172-201.
BIOGRAPHICAL SKETCH
Charlea Lindsey was born in Lexington, Kentucky, on April 7,
1962, and lived there until 1969 when his family moved to Merritt
Island, Florida (where his parents still live). He graduated from
Merritt Island High School in June, 1979. He began his undergraduate
career in fall 1979 at the California Institute of Technology and
stayed there until December, 1980, when he transferred to Auburn
University. He attended Auburn the first six months of 1981, then
transferred to the University of Florida and has been there until the
present. He received his B.S. degree from the University of Florida
in April, 1983; his M.S. in December, 1989; and expects to receive his
Ph.D. in April, 1988.
152
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and 13
fully adequate, in scope and quality, as a dissertation for the degree
of Doctor of Philosophy.
NLcolae Dinculeanu, Chair
Professor of Mathematics
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the degree
of Doctor of Philosophy.
1
James Brooks
Professor of Mathematics
I certify that I have read this study and that in ny opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for t'ne degree
of Doctor of Philosophy.
CV -L
r\ 1, ,\v\
Louis Block
Professor of Mathematics
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the degree
of Doctor of Philosophy.
Jose
Professor of Mathematics
Glover
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the degree
of Doctor of Philosophy.
I certify that I h
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the degree
of Doctor of Philosophy.
William Dolbier
Professor of Chemistry
This dissertation was submitted to the Graduate Faculty of the
Department of Mathematics in the College of Liberal Arts and Sciences
and to the Graduate School and was accepted as partial fulfillment of
the requirements for the degree of Doctor of Philosophy.
April, 1988
**