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Appendix 1 Strong Supermartingales
APPENDIX 1 STRONG SUPERMARTINGALES
T h i s appendix c o n t a i n s m a r t i n g a l e t h e o r y ( o r a t l e a s t t h e p r e l i m i n a r y r e s u l t s ) w i t h no r i g h t c o n t i n u i t y o f paths and no "usual c o n d i t i o n s " on t h e f a m i l y o f 0 - f i e l d s . From t h i s we deduce i n p a r t i c u l a r t h e p r o j e c t i o n theorems w i t h no hypothesis on t h e f i l t r a t i o n . However, t h e t h e o r y which we develop a l s o has i m p o r t a n t consequences under t h e usual c o n d i t i o n s : we s h a l l f o r example deduce the t h e o r y o f o p t i o n a l s t o p p i n g i n continuous time, due m a i n l y t o Mertens. Throughout t h e appendix
(a, F", P )
denotes a n o t n e c e s s a r i l y
complete p r o b a b i l i t y space w i t h a f i l t r a t i o n (F:)t20
FE-
=
F;
finally
.
(Fit)
Then t h e f a m i l y
( w i t h the convention
i s r i g h t continuous b u t n o t complete, and
( F i ) , o b t a i n e d by c o m p l e t i n g Fit. R e c a l l ( I V . 6 1 ) t h a t t h e
Ft) denotes t h e usual augmentation o f
F" and a j o i n i n g a l l t h e n e g l i g i b l e s e t s t o o p t i o n a l o - f i e l d 0 on
IR,
x
n i s generated by t h e processes adapted t o (F;)
w i t h c a d l a g paths, and t h a t t h e o p t i o n a l c r o s s - s e c t i o n theorem g i v e s t h e same r e s u l t as under t h e usual c o n d i t i o n s (IV.84). F o r c l a r i t y , we keep t h e word s t o p p i n g t i m e ( w i t h o u t q u a l i f i c a t i o n ) f o r s t o p p i n g times o f (F;).
The
p r e d i c t a b l e c r o s s - s e c t i o n theorem ( I V . 8 5 ) r e q u i r e s no p r e c a u t i o n , as t h e p r e d i c t a b l e times a r e t h e same f o r t h e f i l t r a t i o n s (F;)
and (F;').
T h i s appendix i s e x t r a o r d i n a r i l y l o n g and must be t h o u g h t o f as b e l o n g i n g t o b o t h Chapters V I and V I I : t h a t i s why i t has i t s own numbering.
STRONG MARTINGALES AND SUPERMARTINGALES
1
DEFINITION. A r e a l o p t i o n a l process X i s an o p t i o n a l s t r o n g m a r t i n g a l e (resp. supermartingale) if ( 1 ) For e v e r y bounded s t o p p i n g time T, XT i s i n t e g r a b l e
39 3
APPENDIX I
394
-
STRONG SUPERMARTINGALES
( 2 ) For every p a i r of bounded stopping times S , T such t h a t S
5
T,
A cadlag supermartingale i s an optional strong supermartingale (no. VI.10: the usual conditions a r e not needed t h e r e ) and under the usual conditions every optional strong martingale i s , on every bounded i n t e r v a l , the optional projection o f a constant process and hence i s cadlag. B u t even under the usual conditions there e x i s t many optional strong supermartingales which a r e not cadlag. For example, the optional projection of a ( n o t necessarily r i g h t continuous) decreasing process i s always an optional strong supermartingale ( t h i s i s the case f o r the l e f t p o t e n t i a l s of no. VI.89). S i m i l a r l y , the l i m i t of a decreasing sequence of cadlag p o s i t i v e supermartingales i s an optional strong supermartingale b u t i s i n general no longer cadlag. Besides Definition 1 we have t h e following d e f i n i t i o n , which i s a l s o important i n p r i n c i p l e b u t which we s h a l l scarcely study. 2
DEFINITION. A p r e d i c t a b l e process X is a predictable strong martingale if (r e s p . supermartingale) -
( 1 ) For every bounded predictable time T , XT i s i n t e g r a b l e . ( 2 ) For every p a i r of predictable times S , T such t h a t S
5
PRELIMINARY PROPERTIES
Some i nequal i t i e s of martingale theory follow from the stopping theorem only: here i s an example. 3
For a l l
Let X be a n optional o r predictable s t r o n g supermartingale. R, we have
T E
T,
395
STRONG MARTINGALES AND SUPERMARTINGALES
We g i v e a q u i c k p r o o f , f o r example i n t h e o p t i o n a l case. R e c a l l t h a t the F-measurable f u n c t i o n supt5.,lXt(
i s i n general denoted by X * and
l e t h denote t h e sup on t h e r i g h t hand s i d e . We b e g i n by n o t i n g t h a t , for every stopping time S
lEIIXsI]
5 T,
prove ( 3 . 1 ) , g i v e n a number
E
>
5
3h ( c f . V,
( o p t i o n a l c r o s s - s e c t i o n theorem). We s e t S = T
We l e t
E
( 1 2 . 2 ) ) . Then t o
0 choose a s t o p p i n g T such t h a t
t e n d t o 0 and then t a k e t h e l i m i t i n
A
T
and then
t o o b t a i n (3.1) w i t h
t h e broad i n e q u a l i t y . The more p r e c i s e i n e q u a l i t i e s o f t h e t y p e V.(20.1),
V.(20.2)
and V.(24.1)
can be shown analogously.
Doob's i n e q u a l i t i e s on upcrossings and downcrossings a r e s t i l l v a l i d f o r s t r o n g supermartingales, b u t t h e y a r e c l e a r l y more d i f f i c u l t t o prove. Wehad communication o f works o f Th. E i s e l e [l]and R. Berkemeier 111 on t h i s s u b j e c t , b u t we do n o t know whether they have been p u b l i s h e d . Here i s a fundamental r e s u l t , due t o Mertens under t h e usual
-
conditions 4
THEOREM.
o u r p r o o f i s moreover e s s e n t i a l l y t h e same as h i s .
L e t X be an o p t i o n a l o r p r e d i c t a b l e s t r o n g s u p e r m a r t i n g a l e .
F o r almost a l l of
IR+
w E
n, t h e p a t h X(,)
i s bounded on e v e r y compact i n t e r v a l
and has r i g h t and l e f t l i m i t s .
As t h e o - f i e l d s a r e n o t complete, we emphasize t h a t t h e s e t of all w
E
n such t h a t X , ( W ) does n o t possess t h e above p r o p e r t y can
be enclosed i n a n e g l i g i b l e s e t , b u t i s n o t n e c e s s a r i l y F"-measurable.
proof.
We s h a l l t r e a t the o p t i o n a l case i n d e t a i l and l e a v e t h e p r e d i c t -
a b l e case t o the r e a d e r .
APPENDIX I
396
-
STRONG SUPERMARTINGALES
We s h a l l prove the existence of r i g h t and l e f t l i m i t s in 3: inequality ( 3 . 1 ) will on the o t h e r hand imply t h a t these l i m i t s a r e i n
R. We s e t Y
= X/( 1
+ I X I ) . For every sequence (S,) of uniformly
bounded stopping times, e i t h e r increasing o r decreasing, the convergence exists a . s . , so theorem for supermartingales t e l l s us t h a t limn X Sn t h e n a l s o does limn YSn,and we see t h a t limn E[YS ] e x i s t s by dominated n convergence. Hence the conditions hold f o r applying Theorem V I .48 (VI.49 f o r the p r e d i c t a b l e c a s e ) , except t h a t t h e usual conditions do not hold f o r the family of a - f i e l d s . We now take up the proof of VI.48,
taking care about t h e
a - f i e l d s . The processes U a n d V a r e progressive with respect t o the augmented family ( F t ) ; hence t h e debut S i s a s t o p p i n g time of this family, b u t we can use IV.59 t o modify i t i n t o an F-measurable s e t of measure zero (without change of n o t a t i o n ) t o make i t i n t o a stopping time of t h e family ( F t + ) . Then t h e s t o c h a s t i c i n t e r v a l IIS, mU i s and the corresponding stopping optional w i t h r e s p e c t t o the family times S o , S1, ... a r e constructed using the optional cross-section thus these a r e t r u e stopping times theorem applied t o the family and i t i s no longer d i f f i c u l t t o prove t h a t S = m a . s . The second p a r t of ( a ) ( c r i t e r i o n f o r r i g h t c o n t i n u i t y ) does n o t generalize t o a r b i t r a r y f i l t r a t i o n s . We now pass t o ( b ) . Here again t h e processes U a n d V a r e
(Fi)
(Fi):
predictable only w then we modify t h e notation) u s i n g I V (F;) and the proof 5
REMARKS.
t h respect t o the family ( F t ) , s i m i l a r l y f o r K , b u t predictable stopping time T (without changing t h e 78 t o make i t i n t o a stopping time of the family then proceeds without d i f f i c u l t y .
( a ) Let D be a countable dense subset of R,.
By Theorem 4
every r e s u l t on convergence a t i n f i n i t y proved f o r t h e process ( X t ) t c D extends t o the whole of the process ( X t ) t 6 R + . Thus i t i s unnecessary here t o reproduce the statements of the convergence theorems. ( b ) There e x i s t s a process X,, which i s optional with r e s p e c t t o t h e family and i n d i s t i n g u i s h a b l e from the process of r i g h t l i m i t s of X, and a predictable process X-, which i s indistinguishable
(Fi,)
from the process of l e f t l i m i t s of X. The construction of such a process X- i s very simple: i t i s s u f f i c i e n t t o take
397
STRONG MARTINGALES AND SUPERMARTINGALES
(5.1
1
X
't- = liminfsED,s++t
b u t t h e c o n s t r u c t i o n o f X,
s for t
= X,
X,-
> 0,
i s much more d e l i c a t e . We e x p l a i n e d i t i n
no. VI.5 ( a ) . The processes X,
and X - c o n s t r u c t e d i n t h i s no. have much
more p r e c i s e p r o p e r t i e s : X+ i s o p t i o n a l w i t h r e s p e c t t o
P
{p <
up,
a]
mu,
=
0 and X,
but a t instant
p,
notexist or iti s infinite. verified that
p
(Fit)
and r i g h t
o f t h e f a m i l y ( F " ) such t h a t t+ has f i n i t e l e f t l i m i t s on no, p E and i s z e r o on
continuous. There e x i s t s a s t o p p i n g t i m e on
{p
<
a},
p
e i t h e r the l e f t l i m i t o f X+ does
With t h i s d e s c r i p t i o n , i t i s moreover e a s i l y
i s p r e d i c t a b l e . As f o r X-,
i n VI.(5.2)
the following
d e f i n i t i o n was proposed X
=
t h e l e f t l i m i t o f X,
i f i t exists,
X- = 0 o t h e r w i s e .
( c ) L e t X be an o p t i o n a l o r p r e d i c t a b l e s t r o n g s u p e r m a r t i n g a l e . Then X- i s a p r e d i c t a b l e s t r o n g s u p e r m a r t i n g a l e and X+an o p t i o n a l s t r o n g s u p e r m a r t i n g a l e w i t h r e s p e c t t o t h e f a m i l y (Fit).
I f T i s a predictable
t i m e ( r e s p . s t o p p i n g t i m e ) then
(5.2)
XT-
2
IEIXTIF;-]
( r e s p . XT
E[XY+IFf]).
2
( d ) Theorem 4 i s i n t e r e s t i n g even under the usual c o n d i t i o n s . For example, l e t (Xn) be a d e c r e a s i n g sequence o f r i g h t continuous p o s i t i v e supermartingales and l e t X be i t s l i m i t . The process X i s a s t r o n g s u p e r m a r t i n g a l e and hence i t has a process o f r i g h t l i m i t s X, and X
2
X + b y " r i g h t upper s e m i - c o n t i n u i t y "
( t h i s a l s o f o l l o w s from t h e
second i n e q u a l i t y (5.2), which can be w r i t t e n as X T usual c o n d i t i o n s ) . F o r a l l
E
>
0, t h e s e t { X
2
X+
+
2 E}
XTt
under t h e
has no accumul-
a t i o n p o i n t a t f i n i t e d i s t a n c e . T h i s l i t t l e remark i s t h e a b s t r a c t form o f a convergence theorem o f p o t e n t i a l t h e o r y which can be s t a t e d as f o l l o w s : l e t (f,)
be a decreasing sequence o f e x c e s s i v e f u n c t i o n s
and l e t f be i t s l i m i t ; f i s a s t r o n g l y supermedian f u n c t i o n ( t h e analogue o f s t r o n g supermartingales i n p o t e n t i a l t h e o r y ) which has an excessive r e g u l a r i z a t i o n f
5
f . For a l l
E
>
0, t h e s e t { f
t
f t
F}
has
no r e g u l a r p o i n t and t h e s e t { f > f } i s s e m i - p o l a r . ( e ) L e t X be a p o s i t i v e o p t i o n a l s t r o n g s u p e r m a r t i n g a l e which i s completed w i t h Xa= 0. The s t r o n g s u p e r m a r t i n g a l e i n e q u a l i t y and the
APPENDIX I - STRONG SUPERMARTINGALES
398
o p t i o n a l c r o s s - s e c t i o n theorem g i v e e a s i l y : i f S i s a s t o p p i n g t i m e such t h a t X s = 0
i n d i s t i n g u i s h a b l e from 0
0" US,
s., X
mu.
Then l e t T be t h e e s s e n t i a l l o w e r e n v e l o p e o f t h e s e t o f s t o p p i n g times S such t h a t X s = 0 a.s.;
T i s equal a.s. t o a s t o p p i n g t i m e o f t h e f a m i l y
and t h e o p t i o n a l c r o s s ( F & ) , X i s i n d i s t i n g u i s h a b l e from 0 on JT, % > 0% KO, TU except on an evanescent
s e c t i o n theorem shows t h a t X ~
s e t . L e t S be a p r e d i c t a b l e s t o p p i n g t i m e such t h a t Xs-
IS <
m};
t h e i n e q u a l i t y Xs-
t
IE[Xs/F5-] a.s.
= 0 a.s.
on
( 5 . 2 ) i m p l i e s Xs=O on
{S < -1 and'hence everywhere, and hence S 2 T a . s . The s e t I X - = 03 n UO, TI] i s p r e d i c t a b l e w i t h c o u n t a b l e c r o s s - s e c t i o n s , hence i t i s i n d i s t i n g u i s h a b l e from a sequence o f p r e d i c t a b l e graphs Sn and,
-
a p p l y i n g t h e above t o each one, i t f o l l o w s t h a t X - i s > 0 on - UO, T U e x c e p t on an evanescent s e t . S i m i l a r l y l e t U be a s t o p p i n g time o f t h e f a m i l y ( F t + ) such t h a t Xu+ = 0 a.s. on { U
<
(and hence everywhere),
= 0 a.s., hence the i n e q u a l i t y 1 Xu+ t IEIXU+l/nIF,J+l a s . i m p l i e s U + l / n 2 T and f i n a l l y U 2 T. Arguing as above on t h e s e t { X t = 01 n
UO, TI],
we see t h a t X+
3
> 0 on UO, TU, e x c e p t on an evanescent s e t .
Thus we o b t a i n a v e r s i o n o f Theorem VI.17,
valid f o r positive optional
s t r o n g supermartingales, due t o Th. E i s e l e .
THE PROJECTION THEOREMS Our aim i n t h i s s e c t i o n i s t o prove t h e f o l l o w i n g theorem. 6
THEOREM.
Let X -
be a p o s i t i v e o r bounded measurable process.
( a ) There e x i s t s a p r e d i c t a b l e process Y such t h a t
(6.1)
*
Y T = IEIXTIFy-l a . s . f o r e v e r y bounded p r e d i c t a b l e t i m e T
2.
We l e a v e these very simple i n e q u a l i t i e s t o t h e reader.
The r e s t r i c t i o n t o bounded s t o p p i n g times i s i n t e n d e d o n l y t o a v o i d unnecessary d e t a i l s . The e x t e n s i o n t o f i n i t e s t o p p i n g t i m e s o r a r b i t r a r y s t o p p i n g times when t h e processes a r e indexed by C O Y m l , i s immediate.
THE PROJECTION THEOREMS
399
( b ) There e x i s t s an o p t i o n a l process Z such t h a t
(6.2)
ZT = IEIXTIF;]
a.s. f o r e v e r y bounded s t o p p i n g t i m e T .
Y and Z a r e unique t o w i t h i n evanescent processes. They a r e c a l l e d t h e ( p r e d i c t a b l e , o p t i o n a l ) p r o j e c t i o n s o f X and a r e denoted by Y = pX,
z
= O x .
P r o o f . The uniqueness f o l l o w s from t h e c r o s s - s e c t i o n theorems
(IV.84-85). To prove t h e e x i s t e n c e , we s h a l l reduce i t t o t h e case of t h e usual c o n d i t i o n s u s i n g t h e f o l l o w i n g lemma, which i s o f i n t e r e s t i n itself.
7
LEMMA.
( a ) Every process which i s p r e d i c t a b l e r e l a t i v e t o t h e
f a m i l y ( F t ) i s i n d i s t i n g u i s h a b l e f r o m a process which i s p r e d i c t a b l e r e l a t i v e t o (FJ ( b ) Every process which i s o p t i o n a l r e l a t i v e t o t h e f a m i l y (Ft) i s i n d i s t i n g u i s h a b l e f r o m a process which i s o p t i o n a l re1 a t i ve t o ( F i + )
.
P r o o f . Using monotone classes, we a r e reduced t o p r o v i n g t h e lemma f o r s t o c h a s t i c i n t e r v a l s and we then a p p l y IV.78 i n t h e p r e d i c t a b l e case and V1.59 i n t h e o p t i o n a l case. We now come t o t h e e x i s t e n c e p r o o f f o r 6 . We choose v e r s i o n s o f p r o j e c t i o n s o f X r e l a t i v e t o ( F t ) : by Lemma 7 we can choose
a
v e r s i o n o f t h e p r e d i c t a b l e p r o j e c t i o n which i s p r e d i c t a b l e r e l a t i v e t o ( F i ) and denote i t by Y. Sirlce e v e r y ,:edictable predictable time of (Ft),
time T o f
(Fi)
is a
Y s a t i s f i e s (6.1) and t h a t proves ( a ) . I n t h e
o p t i o n a l case we choose a v e r s i o n which i s o p t i o n a l r e l a t i v e t o
(Fi+),
which we denote by 5 ; i t s a t i s f i e s (6.2) b u t i n general i t i s n o t
( F i ) and we must m o d i f y i t . VI.46, t h e s e t H = { ( t , 0) : Yt(u) # c t ( u ) 1 i s c o n t a i n e d
optional relative to By
i n a c o u n t a b l e union o f graphs o f random v a r i a b l e s . We a p p l y Theorem
IV.88 t o i t i n o r d e r t o w r i t e i t i n t h e form H = K u L
APPENDIX I - STRONG SUPERMARTINGALES
400
(Fi),
where K i s such t h a t f o r every stopping time T of P{w : ( T ( w ) , W ) E K} = 0 and L i s contained i n a countable union of graphs o f stopping times T n . These graphs can be assumed t o be d i s j o i n t in IR, x R. We use xn t o denote a version of E[X T n I { T n < m } IF;nl and f o r a l l ( t , U ) we s e t Z t ( u ) = Y,(w)
if
(t, W) 4 U UTnll n
if
( t , W)
(6.3) Zt(W)
=
x,(w)
E
UTq.
Clearly the process Z i s o p t i o n a l , Let T be a bounded stopping time of We s e t A n = A n IT = T n } , which belongs the family (F:) and l e t A E t o F" and F,: and B = A \ U A n , which belongs t o F;. n Tn On An we have ZT = xn = E[X IF" 1 = E[X I IF" I = Tn Tn T n An Tn E[X I IF; 1 a n d hence T An
Fi.
On B, T # Tn f o r a l l n and hence ( T ( u ) , W ) 4 L . As I P I ( T ( w ) , W ) E K ? = 0, we have a . s . ( T ( w ) , W ) 4 H, i n o t h e r words, Y,(u) = 5 ( w ) a . s . On the T other hand, ZT = Y T on B . Thus
To obtain ( 6 . 2 ) i t only remains t o add the r e l a t i o n s ( 6 . 4 ) and ( 6 . 5 ) . 8
( a ) I f X,(W) = X ( W ) , X E L 1 ( F ) , we obtain the existence of versions of the conditional expectations E[XIFy] o r lE[XIF;-l which are optional o r predictable strong martingales. Denoting these versions by Z a n d Y i t i s immediately v e r i f i e d t h a t Y = Z- as under t h e usual = "(Y,). conditions,and t h a t Z = "(Z,) REMARKS.
( b ) As under the usual conditions, the s e t { O X # p X } i s a countable union of graphs of stopping times. No proof i s necessary as i t follows by the very construction of "X above. 9
We i l l u s t r a t e Theorem 6 by two examples. F i r s t , a consequence due t o Kunita [ l ] , taken by Yoeurp-Yor [ l ] under s l i g h t l y d i f f e r e n t
401
THE PROJECTION THEOREMS
hypotheses and f i n a l l y f r e e d o f a1 1 unnecessary hypotheses by Horowi t z . Let ( a , Fa,
IP) be a p r o b a b i l i t y space and l e t Q be another law on (n, F " ) . A p o s i t i v e r . v . U i s c a l l e d a p r e c i s e d e n s i t y o f Cp r e l a t i v e
to P i f - t h e a b s o l u t e l y continuous p a r t o f Q r e l a t i v e t o P i s U . I p
( t h u s U i s I P - i n t e g r a b l e and hence P-a.s.
-
t h e s i n g u l a r p a r t o f Cp i s c a r r i e d by {U =
I t i s very easy t o see t h a t
(IP
t
tm).
U i s then determined t o w i t h i n a
Q ) - n e g l i g i b l e f u n c t i o n . On t h e o t h e r hand, a p r e c i s e d e n s i t y can
be computed law
finite),
(IP
t
as f o l l o w s : choose a d e n s i t y V o f Ip w i t h r e s p e c t t o the
2
C p ) / Z , such t h a t 0 s V s 2, and s e t U = V - 1. We now g i v e a f i l t r a t i o n ( F ; ) . We have t h e f o l l o w i n g
theorem'. THEOREM.
I n t h e above n o t a t i o n , t h e r e e x i s t s a unique o p t i o n a l s t r o n g
s u p e r m a r t i n g a l e (U,)
( i n d e x e d by L O ,
ml)
such t h a t , f o r e v e r y s t o p p i n g
~
__ t i m e T,
UT i s a p r e c i s e d e n s i t y o f Q r e l a t i v e t o Ip ~ on t h e _0 - f i e l _ d
(F;)._
Proof. By Theorem 6 a p p l i e d t o t h e law ( P t Q ) / Z t h e r e e x i s t s an o p t i o n a l s t r o n g m a r t i n g a l e (V,)
under t h i s law such t h a t Vm i s a
d e n s i t y o f P w i t h r e s p e c t t o ( P t Q)/2 on F:.
P w i t h r e s p e c t t o ( P +Cp)/Z Ut = 2/Vt
-
on F; f o r e v e r y s t o p p i n g t i m e T. I f we s e t
1, t h e process U i s o p t i o n a l and
Cp w i t h r e s p e c t t o Ip on
Then Vt i s a d e n s i t y o f
UT i s a p r e c i s e d e n s i t y o f
F i f o r e v e r y s t o p p i n g t i m e T. I t f o l l o w s
immediately t h a t U i s a s t r o n g s u p e r m a r t i n g a l e and the uniqueness f o l l o w s from t h e c r o s s - s e c t i o n theorem. REMARK.
I t can e a s i l y be shown t h a t s i m i l a r l y t h e process 1/U
provides precise densities o f
IP
w i t h respect t o
Here i s t h e second example
~
4.
Dellacherie [ l o ] )
~~
There i s an analogous statement i n t h e p r e d i c t a b l e case. See Horowitz
[]I.
APPENDIX I - STRONG SUPERMARTINGALES
402
Every supermartingale X has a modification Z which i s an
10 THEOREM.
optional strong supermartingale. &f. We do not give our primitive proof, b u t an improved version communicated by E . Lenglart. Let Dn denote the s e t of dyadic numbers of the form k/2" ( k an i n t e g e r ) , D the s e t of a l l dyadic numbers and H the countable s e t of points t where the decreasing function E[Xt] i s not r i g h t continuous. Let X, = U be the process of r i g h t l i m i t s of X . We know (nos. 4 and 5 ) t h a t U i s a strong supermartingale of the family and t h a t XT t IEIUTIFf] a . s . f o r every bounded stopping time T. Let V = U'
(Fit)
be the optional projection of U ( i f reassurance i s required t h a t t h i s projection e x i s t s , note t h a t V i s bounded below on every f i n i t e i n t e r v a l LO, n l by the strong martingale (EIUnI . By the above X t V . Finally we s e t
Fi]))
and show t h a t Z i s a modification of X and an optional strong supermarti ngal e .
( 1 ) I t i s s u f f i c i e n t t o check t h a t Z t = V t = X t a . s . a t a point t H . B u t a t such a p o i n t I E I X t l = (IE[X.]), = IEIXttl = EIVtl, which w i t h the i n e q u a l i t y X t 2 V t a . s . implies X t = Vt a . s . ( i n t h e
+
preceding equal i t i e s we have used the uniform integrabi 1 i t y property V.30).
( 2 ) Z i s an optional process. To show t h a t i t i s a strong supermartingale, we consider two bounded stopping times S a n d T such t h a t S 2 T a n d l e t A E FP. We s e t Tn = T
if
T
E
H,
Tn
=
inf{t
E
Dn: t
>
TI
if
T 4 H.
Tn i s a stopping time ,and i t i s n o t d i f f i c u l t t o check t h a t Tn decreases t o T: through s t r i c t l y g r e a t e r values i f T 4 H and s t a t i o n a r i l y i f T E H . We define Sn s i m i l a r l y from S ; Sn 5 Tn does not necessarily hold on t h e s e t IS f H , T E H } , b u t t o ensure t h a t t h i s i n e q u a l i t y holds (without a l t e r i n g t h e manner in which Sn converges t o S , described above), i t i s s u f f i c i e n t t o replace Sn by Sn A T n , which we do without changing the n o t a t i o n .
DUAL PROJECTIONS
403
We then have ( t a k i n g t h e l i m i t under t h e
1 sign
being
j u s t i f i e d by t h e u n i f o r m i n t e g r a b i l i t y p r o p e r t y V.30)
's
limn
n
= "S'{S~HI
+
X
~
l
{
~
E
and ~ ~s i m i l a r l y f o r T
which i s t h e s t r o n g s u p e r m a r t i n g a l e i n e q u a l i t y . The theorem i s proved. L e t W be a n o t h e r o p t i o n a l s t r o n g s u p e r m a r t i n g a l e which i s a
REMARK.
m o d i f i c a t i o n o f X . We have t o w i t h i n an evanescent process: W+ = U, W
2
0
(W+), hence W
= V and f i n a l l y W 2 Z on HC x
2 OU
n.
On t h e o t h e r hand W and Z a r e b o t h m o d i f i c a t i o n s o f X and hence W and Z a r e i n d i s t i n g u i s h a b l e on t h e s e t H x a , because H i s c o u n t a b l e . Thus W
t
Z t o w i t h i n an evanescent process. Thus t h e m o d i f i c a t i o n which we
have c o n s t r u c t e d i s t h e s m a l l e s t p o s s i b l e .
DUAL PROJECTIONS As i n Chapter V I we d e f i n e p r o j e c t i o n s o f 8-measures and random measures, dual p r o j e c t i o n s o f i n t e g r a b l e i n c r e a s i n g processes, e t c . However t h e r e i s one p o i n t which must be examined i n d e t a i l , namely t h e r e l a t i o n between ( o p t i o n a l ) p r e d i c t a b l e 8-measures and ( o p t i o n a l ) p r e d i c t a b l e i n c r e a s i n g processes: f o r o u r d e f i n i t i o n o f i n c r e a s i n g processes r e q u i r e s t h a t t h e i r p a t h s & lbe i n c r e a s i n g , r i g h t continuous and f i n i t e , which i s i n c o m p a t i b l e w i t h a d a p t a t i o n t o the f a m i l y
(F;) when t h e usual c o n d i t i o n s do n o t h o l d ; t h u s something
must be l o s t on one s i d e o r the o t h e r . We b e g i n w i t h t h e analogue o f VI.57. 11
THEOREM.
L e t A be an i n t e g r a b l e i n c r e a s i n g process which i s
i n d i s t i n g u i s h a b l e from a process B which i s p r e d i c t a b l e ( o p t i o n a l 1 relative to foDtional1.
(F;).
Then t h e measure
)-I
associated w i t h A i s predictable
APPENDIX I
404
-
STRONG SUPERMARTINGALES
P r o o f . We s h a l l use t h e r e s u l t s o f no. 6, where we worked on 19, and n o t on
w,
x
R
x
a . Hence f o r s i m p l i c i t y we assume t h a t A does n o t
jump a t i n f i n i t y ; t h i s i s no r e a l r e s t r i c t i o n i n g e n e r a l i t y
((VI.53) ( e l ) . The p r e d i c t a b l e case i s immediate: f o r A i s p r e d i c t a b l e r e l a t i v e t o ( F t ) , hence ,(X)
= V ( ~ X )f o r e v e r y bounded measurable
process X, where p X denotes t h e p r e d i c t a b l e p r o j e c t i o n r e l a t i v e t o ( Ft). B u t by Lemma 7 pX i s i n d i s t i n g u i s h a b l e from the p r e d i c t a b l e p r o j e c t i o n r e l a t i v e t o (F;)
A
d
and on t h e o t h e r hand
p
has no mass on evanescent sets.
We now t r e a t t h e o p t i o n a l case. We decompose A i n t o AC and
,
i t s continuous and d i s c o n t i n u o u s p a r t s ; AC i s p r e d i c t a b l e r e l a t i v e
t o (Ft) and hence i n d i s t i n g u i s h a b l e from a p r e d i c t a b l e process r e l a t i v e
(F;) by Lemma 7 . We denote t h i s process b y Bc and s e t Bd = B - BC, which i s o p t i o n a l r e l a t i v e t o ( F i ) . Since t h e measure a s s o c i a t e d w i t h to
AC i s p r e d i c t a b l e by t h e p r e c e d i n g paragraph, i t i s s u f f i c i e n t t o d I n o t h e r words, we can reduce i t s t u d y t h e measure a s s o c i a t e d w i t h A d t o t h e case where A i s p u r e l y d i s c o n t i n u o u s and suppress a l l the
.
we s e t Ct =
SUPreq,r
Bt i f t
.
>
0, C,
= 0. T h i s process i s
p r e d i c t a b l e r e l a t i v e t o ( F i ) and i n d i s t i n g u i s h a b l e from t h e process (At-).
Hence t h e s e t { C
+ B}
i s indistinguishablefromthe set { A
+ A-}.
B u t t h e f i r s t i s o p t i o n a l , whereas t h e second i s a u n i o n o f graphs o f r . v . Then by t h e end o f IV.88 ( c ) we see t h a t {B f C), and hence
{ A # A - }, i s i n d i s t i n g u i s h a b l e from t h e union o f a sequence o f graphs UUnD
o f s t o p p i n g times o f
(Fi).
On t h e o t h e r hand t h e r . v . 6n = A A ~ ,
and hence i t i s a.s. F" 'u,)' IU~<~I Un Un measurable. We conclude by n o t i n g t h a t , f o r every bounded measure
i s a.s. equal t o (B
-
process X w i t h o p t i o n a l p r o j e c t i o n Z r e l a t i v e t o
(F;),
where a l l t h e terms a r e meaningful s i n c e 6n = 0 on {Un = - 1 . REMARK.
I n p r a c t i c e , t h i s r e s u l t i s used i n a s l i g h t l y d i f f e r e n t way
from t h a t suggested by t h e statement: i t s B t h a t i s given: an o p t i o n a l process such t h a t , f o r almost a l l
w E
n, t h e p a t h B . ( w ) i s
f i n i t e , i n c r e a s i n g and r i g h t continuous. Then we d e f i n e t h e measure
405
DUAL PROJECTIONS m
u(X) = lE[j
XsdBs]
0
IE
where t h e f u n c t i o n under t h e
( X p o s i t i v e and measurable)
s i g n i s o n l y d e f i n e d a.s. and LI i s
assumed t o be bounded. Then t h e measure LI i s o p t i o n a l . F o r l e t N be a n e g l i g i b l e s e t such t h a t , f o r
w
4 N,
B . ( w ) i s i n c r e a s i n g , f i n i t e and
r i g h t c o n t i n u o u s . We s e t
As
LI
i s a l s o t h e measure a s s o c i a t e d w i t h t h e i n t e g r a b l e i n c r e a s i n g
process A, Theorem 11 t e l l s us t h a t u i s indeed o p t i o n a l . Conversely, l e t u be a p r e d i c t a b l e o r o p t i o n a l bounded
12
IP-measure on
IR,
x
n; we wish t o r e p r e s e n t i t u s i n g a process w i t h
p r o p e r t i e s as p r e c i s e as p o s s i b l e . THEOREM.
IP-measure LI has a r e p r e s e n t a t i o n
The p r e d i c t a b l e ( o p t i o n a l )
u(X) =
(12.1)
IE[/
XSdBSl LO,-[
where B i s a p r e d i c t a b l e ( o p t i o n a l ) process a l l o f whose paths a r e i n c r e a s i n g and r i g h t continuous, w i t h t h e p o s s i b l e e x c e p t i o n o f a s i n g l e value o f t f o r which Bt <
m
and Bt+ =
-.
Moreover, B can be r e p r e s e n t e d as a sum zn Bn o f t r u e predictable
( o p t i o n a l ) i n t e g r a b l e i n c r e a s i n g processes.
P r o o f . We b e g i n w i t h t h e p r e d i c t a b l e case. The measure
u i s predictable
r e l a t i v e t o ( F t ) and hence has a r e p r e s e n t a t i o n (VI.52)
where cx i s an i n t e g r a b l e i n c r e a s i n g process which i s z e r o a t 0, adapted t o t h e f a m i l y (Ft) and continuous, t h e A,, a r e p o s i t i v e constants and t h e Tn a r e p r e d i c t a b l e times o f t h e f a m i l y ( F t ) . We a l s o i n t r o d u c e t h e s t o p p i n g t i m e s o f t h e f a m i l y (Ft)
APPENDIX I
406
sn
=
STRONG SUPERMARTINGALES
infit: a
>
t -
nl
which a r e predictable ( S n i s t h e debut of a r i g h t closed p r e d i c t a b l e s e t ) , and the increasing processes bounded by 1
We now regularize these processes. For B~ we choose a predictable time
-
Tn of the family
(Fi)
such t h a t
Tn
=
T n a . s . (IV.78) and we s e t
For a n we choose f o r r r a t i o n a l a r . v .
s:
equal a . s . t o a: and then f o r t real
0 we s e t
and then f o r t real
2
>
which i s F;+-measurable
and
0
an increasing process which i s adapted t o ( F i + ) , bounded by 1 and indistinguishable from an. We then denote t h e continuous p a r t o f r;" by -n~1 ( a t t h e same time removing the j u m p a t 0); t h i s increasing n process i s s t i l l indistinguishable from ~1 , s t i l l adapted to ( F { + ) and, as i t i s continuous, i t i s predictable r e l a t i v e t o ( F ; ) . remains t o s e t
I t only
(12.2) This process i s indistinguishable from A , i t i s p r e d i c t a b l e with , r e s p e c t t o ( F : ) and i t s paths a r e a l l increasing. I f B t ( w ) < m y t h e s e r i e s ( 1 2 . 2 ) i s uniformly convergent on CO, t l and hence the path B { w ) i s r i g h t continuous on COY t C , hence B can have only the type of r i g h t discontinuity indicated in the statement. As lELB-1 c m y note a l s o t h a t t h i s lack o f r i g h t continuity can occur only on a s e t o f measure zero.
We now come t o the optional case, which i s as usual a l i t t l e
DUAL P ROJ E CTI ONS
407
more d e l i c a t e . L e t p be a bounded IP-measure on u(X) = p('X) as a sum
IR+
x
n such t h a t
f o r e v e r y bounded measurable process X . We decompose u
p = v
+
IT
o f two p o s i t i v e measures w i t h t h e f o l l o w i n g
properties i s c a r r i e d by a c o u n t a b l e u n i o n o f graphs o f s t o p p i n g times o f (F:Iy v has no mass on any graph o f a s t o p p i n g t i m e
( i t i s s u f f i c i e n t t o define
as t h e l e a s t upper bound o f t h e measures
IT
bounded above by V. c a r r i e d by a c o u n t a b l e u n i o n o f graphs o f s t o p p i n g times and t o s e t u = p
-
I t i s immediately v e r i f i e d t h a t t h e r e
IT).
e x i s t s a s e t H which i s a c o u n t a b l e u n i o n o f graphs o f s t o p p i n g times such t h a t
IT
=
IH*p;
hence
i s o p t i o n a l and the same i s t r u e o f v by
IT
taking the difference.
As t h e s e t where
and 'X
OX
d i f f e r i s a countable u n i o n o f
graphs o f s t o p p i n g times, we have v ( X ) = .(OX)
= v ( ~ X ) ; hence u i s
p r e d i c t a b l e and we can a p p l y t h e f i r s t p a r t o f t h e p r o o f t o i t . Hence i t remains t o c o n s i d e r IT.
We r e p r e s e n t H as a u n i o n o f d i s j o i n t graphs o f s t o p p i n g t i m e s Un. Then
IT
= I H * u i s t h e sum o f t h e measures
i t i s s u f f i c i e n t t o r e p r e s e n t each measure
IT
n
.
As
I UUnll - u
IT^
IT^
=
and
i s c a r r i e d by
t h e graph o f Un, we can w r i t e ."X)
IE[X
=
'n
6 ]
where 6n i s an i n t e g r a b l e p o s i t i v e r . v . On t h e o t h e r hand
rrn
is
o p t i o n a l ; t a k i n g X = 0 o u t s i d e t h e graph o f Un, we see t h a t 6 n i s a.s. equal t o a p o s i t i v e F" -measurable r . v . Un 6
n
=
zm
X
I
I t o n l y remains t o w r i t e
nm Hnm
a.s
and t h e n t o where t h e Xnm a r e c o n s t a n t s and t h e tinm elements o f f " Un s e t Tnm = U n on Hnm and = +m on H i m y t o o b t a i n t h e r e p r e s e n t a t i o n
and hence t h e " i n c r e a s i n g process" a s s o c i a t e d w i t h n, which i s
APPENDIX I - STRONG SUPERMARTINGALES
408
'nm A nm 1 I t r T n m l ' REMARK.
R e t u r n i n g t o t h e r e s u l t s o f nos. 11 and 12, we o b t a i n a
r e s u l t which i s o f i n t e r e s t i n i t s e l f : i f A i s an i n t e g r a b l e i n c r e a s i n g process which i s i n d i s t i n g u i s h a b l e from a p r e d i c t a b l e process, t h e n
B which i s i n d i s t i n g u i s h a b l e from A and i s a
t h e r e e x i s t s a process
sum o f p r e d i c t a b l e i n t e g r a b l e i n c r e a s i n g processes ( a l l t h e p r o p e r t i e s h o l d s i m u l t a n e o u s l y f o r each term i n t h e sum). S i m i l a r l y f o r t h e o p t i o n a l case.
RETURN TO STRONG SUPERMARTINGALES
13
L e t X be a s t r o n g s u p e r m a r t i n g a l e which f o r s i m p l i c i t y we assume t o be indexed by L O , m l ( r i g h t c l o s e d ) . We a s s o c i a t e w i t h i t a measurable process ,X,
which i s i n d i s t i n g u i s h a b l e from t h e process o f
r i g h t l i m i t s o f X ( f o r example t h e v e r s i o n c o n s t r u c t e d i n no. 5 ( b ) , b u t t h e r e i s no need t o be so p r e c i s e ) , and a p r e d i c t a b l e process X-, which i s i n d i s t i n g u i s h a b l e f r o m t h e process o f l e f t l i m i t s o f X . The r e l a t i o n s (5.2)
can be w r i t t e n
x
(13.1)
2
px,
x
r "(X,).
As we d i d under t h e usual c o n d i t i o n s ( V I . , p X = X- ( i n c l u d i n g a t
m
V I I . ) we c a l l X r e g u l a r i f
i n t h e s i t u a t i o n i n which we a r e i n t e r e s t e d ) .
I f X belongs t o c l a s s ( 0 ) t h i s means, as i n no. VII.10, t h a t X i s continuous i n e x p e c t a t i o n : i f (T,) i s an i n c r e a s i n g sequence o f
s t o p p i n g times and T = l i m n Tn, then IEIXTl = l i m n I E I X T second i n e q u a l i t y ( 5 . 2 )
i n t r o d u c e s a new n o t i o n :
n
left
1. B u t t h e
LEMMA. With t h e above n o t a t i o n , t h e f o l l o w i n g p r o p e r t i e s a r e equivalent: (a)
x
= "(x,).
( b ) F o r e v e r y s t o p p i n g t i m e T, IEIXT] = l i m IEIXT+l,,nl. n ( c ) F o r e v e r y s t o p p i n g t i m e T and e v e r y d e c r e a s i n g sequence o f s t o p p i n g times Tn
J.
T,
IEIXTl = l i m n lE[X
I.
Tn I f these p r o p e r t i e s h o l d , X i s c a l l e d r i g h t continuous i n e x p e c t a t i o n .
P r o o f . C l e a r l y ( c ) a ( b ) . Since the r . v . XT+l,n
are u n i f o r m l y i n t e g r a b l e
RETURN TO STRONG SUPERMARTINGALES ( b ) can be w r i t t e n as IEIXTl = IE [ X T + l ;
(V.30),
XT
IE [XT+
2
I FT-I , whence
on t h e o t h e r hand
I FTl .
t h e equal it y XT = IE [XTt
f o r e v e r y s t o p p i n g t i m e T, we see t h a t ( b )
3
409
(a).
Since t h i s h o l d s
F i n a l l y , i n the
iC~T+~,
n o t a t i o n o f ( c ) , l i m n IE[X I = I E [ l i m n x I = ~ X ~ I + P Tn Tn where A i s t h e s e t 13: n : Tn = T I , which belongs t o F;. Given ( a ) , t h i s sum i s equal t o IEIXTl and the l a s t i m p l i c a t i o n ( a )
( c ) i s proved.
Under the usual c o n d i t i o n s , p r o p e r t i e s ( a ) , ( b ) , ( c ) mean s i m p l y t h a t X i s r i g h t continuous b u t t h i s i s n o t t r u e i n general, as t h e example o f m a r t i n g a l e s immediately shows. The f o l l o w i n g n o t i o n i s due t o Bismut 131 (who c a l l s i t a
14
quasi-stopping time). It i s e s p e c i a l l y useful f o r studying strong
s upermart ngales w i t h o u t t h e usual c o n d i t i o n s , b u t i s a l s o o f i n t e r e s t i n " c l a s s c a l " s i t u a t i o n s . A g e n e r a l i z a t i o n can be found i n Maingueneau [ll. A s p l i t stopping time (abbreviated s.s.t.)
i s an ordered
H , T), where T i s a s t o p p i n g t i m e and H an element o f F; such t h a t TH i s p r e d i c t a b l e (hence H i n f a c t belongs t o F;-). T i s pair
T
=
c a l l e d bounded, f i n i t e ,
...
i f T i s bounded, f i n i t e ,
...
The i n t u i t i v e
i d e a o f such an o r d e r e d p a i r i s t h e f o l l o w i n g : d i v i d e each p o i n t x i n
Rt
i n t o two p o i n t s , one again denoted x and t h e o t h e r x - and extend
t h e o r d e r r e l a t i o n by t h e conventions xy
5
x and y s x- i f and o n l y i f y
f u n c t i o n w i t h values i n
<
Et doubled
<
x . Then
x, y- s x- i f and o n l y i f T
can be considered as a
i n t h i s way, which takes t h e value
T- on H and t h e v a l u e T on HC. I f X i s a s t r o n g s u p e r m a r t i n g a l e or more g e n e r a l l y a process which a . s . admits l e f t l i m i t s , we s e t (14.1)
X
T
=X
w i t h t h e c o n v e n t i o n t h a t X,wise ( i f 10,
T
=
X,
X
i f X,-
C
=XTonH
has n o t been s p e c i f i e d o t h e r -
i s n o t a . s . f i n i t e , i t i s necessary f o r X t o be indexed by
ml).
I f X i s optional,
o - f i e l d F:, < T
XT
i s measurable w i t h r e s p e c t t o t h e
consisting o f the A L e t u = (G, S ) and
u
onH,
T-
E
T =
i f S s T and H n { S = T I
c
F y such t h a t A n H belongs t o F;-. (H, T) be two s . s . t . ;
we w r i t e
G n IS = T I , conforming w i t h t h e
o r d e r r e l a t i o n d e f i n e d above on t h e s p l i t h a l f - l i n e . I t i s very
APPENDIX I - STRONG SUPERMARTINGALES
41 0
easy t o check t h a t F; c F;. With every stopping time T can be associated the s . s . t .
(a,
T ) which we i d e n t i f y with T . I f T i s p r e d i c t a b l e , we can a s s o c i a t e w i t h i t the s . s . t . ( n , T ) , which we denote by T - . Then t h e fol 1 owing theorem general i zes t h e two stopping theorems VI.10 and VI.14. Let X be an optional strong supermartingale indexed by
15 THEOREM. C O Y -1.
We adopt the convention X,-
be two s . s . t . such t h a t u s
T.
=
X,.
KU = (G,
S ) pnd
T
= (H, T )
Then -
with e q u a l i t y i f X i s an optional strong martingale on LO, m l Proof. Suppose f i r s t t h a t X i s an optional strong martingale. Then i t i s immediate t h a t X T = I E I X m I F ~ l ,Xu = IEIXmIFil and (15.1) with e q u a l i t y follows immediately. martingale positive.
We now consider the general case. Subtracting the strong IEIXmIFil from X , we a r e reduced t o the case where X i s We wish t o e s t a b l i s h the i n e q u a l i t y
We begin by t r e a t i n g the case where A = n . We a r e f r e e t o truncate X a t n and then take t h e limit,and hence we may assume t h a t X i s bounded. Let (R,) be a sequence which a . s . f o r e t e l l s SG (IV.77) and l e t Sn = Rn A S ; then Xu = lirn S a . s . and hence i t i s s u f f i c i e n t t o n Sn check t h a t
j xs
n
IP
2
jXTIP.
Let (U,) be a sequence which a . s . f o r e t e l l s TH; the r e l a t i o n u implies t h a t t h e sequence T, = S, v (Urn A T ) i s then such t h a t
I T
XT a . s . ; on t h e o t h e r h a n d f o r a l l n and a l l m we have J X , IP n IP by the d e f i n i t i o n of s t r o n g supermartingales, whence the Tm required i n e q u a l i t y . X
Tm I X
-+
2
41 1
APPLICATION TO INEQUALITIES
To deal w i t h t h e general case, we proceed as f o r o r d i n a r y s t o p p i n g t i m e s : we a p p l y t h e above t o t h e s . s . t . -cA = ( H n A,
uA = ( G n A,
SA) and
TA).
APPLICATION TO INEQUALITIES We now s e t o u t t o extend t h e i n e q u a l i t i e s o f Chapter V I . § 3
16
t o v e r y general s i t u a t i o n s and a t t h e same t i m e t o u n i f y them.
two
We c o n s i d e r denote by A- and A+;
i n t e g r a b l e i n c r e a s i n g processes which we
t h e f i r s t i s p r e d i c t a b l e r e l a t i v e t o ( F i ) and
zero a t 0 b u t may jump a t i n f i n i t y , t h e second i s o p t i o n a l r e l a t i v e t o
(Fi),
may jump a t 0 b u t n o t a t i n f i n i t y and i s p u r e l y d i s c o n t i n u o u s .
+
+ A_,
We denote by A t h e p r e d i c t a b l e i n t e g r a b l e " i n c r e a s i n g " process A-
which i s n e i t h e r r i g h t n o r l e f t c o n t i n u o u s . The p o t e n t i a l Z generated by A i s by d e f i n i t i o n t h e o p t i o n a l p r o j e c t i o n o f t h e process ( A m - A t ) t y t h a t i s t h e process (16.1)
- At
Zt = E[A,IFi]
where t h e process lEIAmIFi] must be i n t e r p r e t e d as an o p t i o n a l s t r o n g m a r t i n g a l e . Our aim i s t o e s t i m a t e A,
as a f u n c t i o n o f Z. Under t h e
usual c o n d i t i o n s , t h e s i t u a t i o n i n c l u d e s t h e two cases o f no. VI.89: t h e p r e d i c t a b l e case corresponds t o a r i g h t continuous process A and t h e o p t i o n a l case t o a l e f t continuous process A. Under t h e "unusual" c o n d i t i o n s , o u r hypotheses a r e r a t h e r t o o r e s t r i c t i v e , b u t see no. 19. We saw i n no. V I I . 3 t h a t A determines A+ and A - . 17
We a p p l y formula (91.4) t o t h e r i g h t continuous process A+ = A-
+ A'.
The l a t t e r i n t e g r a l i s equal t o ~ ( ( A A ; ) ~- (AA:)~). We thus o b t a i n A:
=
\(Am
-
At)dAt+
+
((Am
-
A t - ) d A i + j(A,
-
At+)dA;
.
41 2
I -
APPENDIX
STRONG SUPERMARTINGALES
-
We now i n t e g r a t e ; we can r e p l a c e (A, (Zt),
(A,
-
At-)
by i t s p r e d i c t a b l e p r o j e c t i o n (Zt-)
by i t s o p t i o n a l p r o j e c t i o n (Ztt) (17.1)
At) b y i t s o p t i o n a l p r o j e c t i o n
IE[A:l
relative to (Fit).
lE[j ZsdAst
=
t
j ZS-dA;
- At.)
and (A, We o b t a i n
j ZStdA:l.
t
We thus g e n e r a l i z e t h e two "energy formulae" o f no. VI.94. 18
We s h a l l now g e n e r a l i z e t h e fundamental theorem V I .99. As noted i n no. 100, i t a l l amounts t o g e n e r a l i z i n g t h e "maximal lemma" (100.1) and t o t h i s end we argue d i r e c t l y , u s i n g t h e s p l i t s t o p p i n g times i n t r o d u c e d i n no. 14. I t w i l l be convenient t o assume t h a t A does n o t jump a t i n f i n i t y , which i s no r e s t r i c t i o n i n g e n e r a l i t y ( V I . 5 3 ( c ) ,
V I I . 15).
T h i s t i m e we s e t
at
t
A t t A-t; a i s a r i g h t continuous
=
o p t i o n a l i n c r e a s i n g process and the o p t i o n a l s e t J = { ( t , A}
i s a stochastic interval
at(.) CS-
z A}
[TT,
i s a stopping time o f
mu,
z
: at(")
whose debut T(w) = i n f { t : t
(Fi+).
s A 5 a, t h e s e t s { A 2 I } and { a
M)
Let H =
{O
: AT(w)
z
A};
? E
Q,
as
A } d i f f e r o n l y by a subset o f
t h e graph o f T and hence
i s a p r e d i c t a b l e s e t . On t h e o t h e r hand, TH i s a s t o p p i n g t i m e o f (Fit),
and hence I T H , m U
is
p r e d i c t a b l e and f i n a l l y
i s p r e d i c t a b l e , which means t h a t TH i s a p r e d i c t a b l e time (IV.69) and hence t h a t
T =
(H, T) i s a s . s . t .
The s e t {A, 2 A} = ( T < - } u { T = -, A, 2 A } belongs t o F Y and i t s i n t e r s e c t i o n w i t h H i s equal t o H and hence i s Fy--measurable; hence i t belongs t o FY. F i n a l l y , AT 5 A , whence t h e i n e q u a l i t y
by 15 a p p l i e d t o the s t r o n g m a r t i n g a l e (Zt by a s t r o n g m a r t i n g a l e
(IE [Mm I Fg] ) ,
+
At).
we deduce t h a t
I f Z i s bounded above
APPLICATION TO INEQUALITIES
41 3
(18.1) t h a t i s (100.1) and a l l t h e consequences o f t h i s i n e q u a l i t y proved i n Chapter V I remain t r u e i n t h e general case. 19
We r e t u r n t o t h e hypotheses 16: i n f a c t , i f we a r e w o r k i n g under t h e "unusual" c o n d i t i o n s , i t was p o i n t e d o u t e a r l i e r t h a t A+ and A- a r e o n l y v e r y r a r e l y t r u e o p t i o n a l o r p r e d i c t a b l e i n c r e a s i n g processes: they a r e c o n s t r u c t e d from IP-measures by t h e procedure o f no. 12, i n o t h e r words A- and A+ a r e b o t h processes w i t h i n c r e a s i n g paths, t h e f i r s t i s p r e d i c t a b l e and t h e second o p t i o n a l , b u t r i g h t c o n t i n u i t y does n o t h o l d i d e n t i c a l l y : t h e r e may e x i s t w s e t o f measure zero
-
f o r which A+(.)
o r A:(w)
-
forming a
i s n o t r i g h t continuous.
How do we overcome t h i s d i f f i c u l t y ? By no. 12, A
+
and A- a r e
o f analogous processes, b u t
which a r e i d e n t i c a l l y r i g h t continuous and hence A i t s e l f i s a sum o f the corresponding processes An and Z t h e sum o f t h e corresponding n . We s e t
processes Z
c i k = z k, A n,
5 k = z k, Z n.
I f Z i s bounded above by t h e s t r o n g m a r t i n g a l e
(iEM[ , I F ; ] ) ,
a fortiori
so i s gk and, s l i g h t l y m o d i f y i n g t h e i n t e g r a t i o n s e t s , we o b t a i n
then, l e t t i n g k t e n d t o i n f i n i t y ,
and f i n a l l y , t a k i n g l i m i t s gives i n e q u a l i t y (100.1) i n complete generality (19.1) Moreover t h e same r e s u l t s a r e v a l i d i f t h e hypotheses on A+ and Aa r e s l i g h t l y m o d i f i e d : i t can be assumed t h a t these two processes a r e t r u e i n t e g r a b l e i n c r e a s i n g processes which a r e i n d i s t i n g u i s h a b l e , the
APPENDIX I - STRONG SUPERMARTINGALES
414
f i r s t from an optional process a n d the second from a predictable process. To see t h i s , we must use t h e Remark of n o . 1 2 , which reduces these hypotheses t o the preceding ones.
MERTENS DECOMPOSITION
We s h a l l now generalize the r e s u l t s a t t h e beginning of Chapter VII. As in no, 19, the theorems a r e valid under the usual conditions and under the most general conditions, b u t t h e r e i s a degree of additional complication i n t h e general case, r e s u l t i n g from the more d e l i c a t e manipulation of the notion of an increasing process ( n o . 1 2 ) . To avoid a ponderous statement, we a d o p t t h e usual conditions and then in n o . 21 i n d i c a t e t h e modifications t o be introduced in the general case. 20
Here then i s the r e s u l t under the usual conditions, which i s due t o Mertens. THEOREM. Let Z be a p o s i t i v e optional strong supermartingale of c l a s s ( D ) (with t h e convention Z o - = Z,, Zm = 0 ) . Then t h e r e e x i s t s a
predictable process A with increasing ( b u t not necessarily r i g h t o r l e f t continuous) paths such t h a t (20.1) IE[A,l
<
-,
ZT
=
IEIAmIFTl - AT f o r every stopping time T .
A i s unique and the following inequality between processes holds:
In p a r t i c u l a r , A i s r i g h t continuous i f f Z i s r i g h t continuous and l e f t continuous i f f z = PZ ( i . e . i f z is r e g u l a r ) . Proof. The reader i s requested t o r e f e r t o no. VII.6 a n d t o follow the modifications. In n o . 6 i t i s s u f f i c i e n t t o suppress "cadlag"
1
Under the usual conditions "(Z+)
=
Z+.
41 5
MERTENS DECOMPOSITION
t h r o u g h o u t and s i m i l a r l y i n no. 7 . I n no. 8 t h e p r o o f shows t h a t
A
i
= 0 i f f Z i s r i g h t continuous; i n t h e general case we have s i m p l y
which, when X i s t h e i n d i c a t o r o f UT, WE, g i v e s
t h a t i s ( 2 0 . 1 ) . No. V I I . 1 0 g i v e s w i t h o u t m o d i f i c a t i o n t h e f i r s t e q u a l i t y o f ( 2 0 . 2 ) and the second f o l l o w s s i m i l a r l y . F i n a l l y t h e uniqueness o f A f o l l o w s f r o m t h e f a c t t h a t b e i n g g i v e n A i s e q u i v a l e n t t o b e i n g g i v e n t h e o r d e r e d p a i r (A-, A')
b y V I I . 3 , which i s i t s e l f
e q u i v a l e n t t o b e i n g g i v e n t h e l i n e a r form J ( V I I . 2 ) and f i n a l l y t o b e i n g g i v e n Z (V11.6). 21
We now c o n s i d e r t h e general ("unusual " ) c o n d i t i o n s . F i r s t we must c a r e f u l l y r e - r e a d t h e p r o o f o f Theorem V I I . 2 : a l l goes w e l l where c o n s t r u c t i n g measures i s concerned, b u t when i t comes t o i n c r e a s i n g processes, nos. 11-12 o f t h i s appendix l e a v e us t h e c h o i c e between two possibilities:
-
t o t a k e A+ and A- t o be two w e i n t e g r a b l e i n c r e a s i n g
processes, b u t which a r e o n l y i n d i s t i n g u i s h a b l e f r o m o p t i o n a l o r p r e d i c t a b l e processes;
-
=
t o take A+ and A- t o be two processes which are r e s p e c t -
i v e l y o p t i o n a l and p r e d i c t a b l e , w i t h i n c r e a s i n g paths, b u t i d e n t i c a l l y r i g h t c o n t i n u o u s and f i n i t e .
The f i r s t c h o i c e leads t o the simp e r r e s u l t . F o r i t enables us t o s e t A = A- t Af ( a process w i t h f i n i t e values which i s i n d i s t i n g u i s h a b l e from a p r e d i c t a b l e process
and then A+ = A- + A+,
which i s r i g h t c o n t i n u o u s . Then t h e measurab l i t y c o n d i t i o n s on A can merely be expressed as f o l l o w s : A i s i n d i s t i n g u i s h a b l e f r o m a p r e d i c t a b l e process and A+ from an o p t i o n a l process. That a p a r t , t h e r e i s no i m p o r t a n t m o d i f i c a t i o n . Note however t h a t r i g h t c o n t i n u i t y o f A i s no l o n g e r e q u i v a l e n t t o r i g h t c o n t i n u i t y
APPENDIX I - STRONG SUPERMARTINGALES
41 6
of Z in expectation. We s h a l l give no more d e t a i l s , f o r everything would have t o be redone under the general conditions: local martingale theory, Doob decomposition of a r b i t r a r y supermartingales (VII . l Z ) , e t c . Frankly, such a systematic study does n o t seem worthwhile a t t h e moment.
THE SNELL ENVELOPE We s h a l l now i l l u s t r a t e the i n t e r e s t of strong supermartingales (even under the usual conditions!) by presenting the elements o f the theory o f t h e Snell envelope. A h i s t o r i c a l word i s not out of place here. The raison d'i2tre o f t h i s theory i s t h e problem of optimal stopping: given an a r b i t r a r y process X , f i n d a stopping time T f o r which IEIXT] i s as large as p o s s i b l e . I t was i n order t o study optimal stopping in t h e d i s c r e t e case t h a t Snell in [11 introduced the notion of a submartingale and the corresponding stopping theorem, along with t h e envelope which now bears h i s name. Snell ' s works were extended t o the continuous case by J . F . Mertens, whose a r t i c l e s 131, [41 a r e fundamental t o the notion o f a strong supermartingale (and Mertens decomposition) and the notion of a strongly supermedian function i n potential theory. Again q u i t e r e c e n t l y , t h e theory of the Snell envelope and optimal stopping was taken up i n a r t i c l e s by Bismut [31, Bismut-Skalli [11 and Maingueneau [ l ] (using s p l i t stopping times, which were introduced on t h i s occasion). We say nothing here on the theory of optimal stopping, the l i t e r a t u r e on t h i s s u b j e c t i s enormous and largely orientated towards a p p l i c a t i o n s ; the reader can consult S h i r i a i e v ' s book [ l l . We hope t o return t o i t l a t e r w h e n we study the notion of a r e d u i t e i n potential theory. 22
Let Y denote an optional process, indexed by [O, m[ a n d belonging t o c l a s s [Dl: the r.v. Y T , where T runs through the s e t o f a l l bounded stopping times, form a uniformly i n t e g r a b l e s e t . We adopt the convention Ym = 0. Then YT i s defined f o r every stopping time T , f i n i t e o r otherwise, and the s e t o f Y T (where T runs through the s e t I of a l l stopping times) i s a l s o uniformly i n t e g r a b l e . We now give the fundamental existence theorem f o r the Snell
41 7
THE SNELL ENVELOPE envelope. As we a r e w i t h o u t t h e usual c o n d i t i o n s , we do n o t h e s i t a t e t o use a " s l e d g e hammer" theorem. F o r an elementary p r o o f (under t h e usual c o n d i t i o n s ) see Maingueneau [ l ] , p. 458. THEOREM.
( a ) There e x i s t s a p o s i t i v e o p t i o n a l s t r o n g supermartingale
Z w i t h the f o l l o w i n g property: Z
2
Y and f o r e v e r y p o s i t i v e o p t i o n a l s t r o n g s u p e r m a r t i n g a l e
Z ' which bounds Y above, Z '
2
Z ( t h e i n e q u a l i t i e s h o l d as usual except
on evanescent s e t s ) . Z i s unique ( t o w i t h i n evanescent s e t s ) . I t i s c a l l e d t h e S n e l l envelope o f Y. Moreover, Z belongs t o c l a s s ( 0 ) .
( b ) For e v e r y s t o p p i n g t i m e T we have a.s. (22.1) P r o o f . We f i x T and c o n s i d e r t h e s e t o f r . v . through t h e s e t o f s t o p p i n g times S
t
E I Y S I F i l , where S runs
T. We make t h e fundamental
remark t h a t t h i s s e t i s d i r e c t e d and i n c r e a s i n g . For i f S , and S, s t o p p i n g times t T, l e t Ui U, v U,
=
=
IEIYsi I F T I ( i = 1, 2 ) ; we have
where R = S , i f U,
EIYRIFYl,
are
t
U,,
R = S, i f U,
<
U2.
Then l e t Z ( T ) denote t h e supess on t h e r i g h t hand s i d e o f ( 2 2 . 1 ) - we c a r e f u l l y w r i t e Z(T) and n o t ZT because a c l a s s o f r . v .
i s i n v o l v e d and
we do n o t y e t know whether we can "paste t o g e t h e r " t h e Z(T) i n t o a process Z. Note some p r o p e r t i e s o f these r . v . : ( a ) Z(T)
2
0 (take S =
( b ) For a l l A
E
jA
(22.2)
+m
i n (22.1));
Z(T)
j-s
Fy-measurable.
FY,
IP
= SUPStTj
A
ys IP
(an immediate consequence o f t h e f a c t t h a t t h e r . v . E I Y S I F y ] form an increasing directed set).
( c ) - I f -T and U a r e two s t o p p i n g times w i t h T
5
U, W Z ( T ) t
EIZ(U)IFoTl a.s. and Z(T) = Z(U) a . s . on I T = U3 ( a n easy consequence o f (22.2)).
APPENDIX I - STRONG SUPERMARTINGALES
418
( d ) The r . v .
M = sups I E I Y s l < P{Z(T)
2
cl
5
m;
Z(T) a r e u n i f o r m l y i n t e g r a b l e . For l e t
by (22.2) w i t h A = n, we have IEIZTl
M/c. Then by (22.2) again
!I Z ( T ) r c I Z(T) P As P { Z ( T ) 2 c l
M and hence
5
sup
5
S2T
IZ(T)tc)
Ys
P.
M/c i n d e p e n d e n t l y o f T and t h e r . v . Ys a r e u n i f o r m l y
i n egrable, the r i g h t hand s i d e tends t o 0 as c *
m,
u n i f o r m l y i n T and
S. Hence t h e s t a t e d r e s u l t .
And now we r e t u r n t o no. V I I . 6 . I f X i s a cadlag process wh ch i s bounded on 10, -31 and o f t h e form (6.1)
x
= XoIno
Y 1
t
...
+
xnlDTn,mn
we s e t
and we t a k e t h e whole o f t h e p r o o f o f t h e Mertens decomposition
( n o . 21 above). We o b t a i n a r e p r e s e n t a t i o n o f J i n t h e form
as i n no. 21, then, i f we denote by Z t h e process which i s t h e o p t i o n a l
p r o j e c t i o n o f t h e process (A,
t
+ A, -
At
-
t
At-),
which i s a p o s i t i v e
o p t i o n a l s t r o n g supermartingale, we f i n d t h a t t h e measure a s s o c i a t e d f o r e v e r y s t o p p i n g t i m e T. Thus
w i t h Z i s J , and hence Z(T) = ZT a.s.
we have e s t a b l i s h e d t h e e x i s t e n c e o f a p o s i t i v e o p t i o n a l s t r o n g supermartingale o f class
(D), which s a t i s f i e s (22.1)
The r e s t i s obvious. Taking S = T i n (22.1), we see t h a t
ZT
and 2
2 Y b y t h e o p t i o n a l c r o s s - s e c t i o n theorem. I f Z ' i s T another p o s i t i v e o p t i o n a l s t r o n g s u p e r m a r t i n g a l e bounding Y above, we
2
Y
have f o r every s t o p p i n g t i m e T and e v e r y s t o p p i n g t i m e S
and t a k i n g supess
i n S we o b t a i n
Zt
2
2
T
ZT a . s . and hence Z ' 2 Z by t h e
THE SNELL ENVELOPE
41 9
o p t i o n a l c r o s s - s e c t i o n theorem. F i n a l l y , t h e S n e l l envelope Z i s c h a r a c t e r i z e d u n i q u e l y as t h e s m a l l e s t p o s i t i v e o p t i o n a l s t r o n g s u p e r m a r t i n g a l e bounding Y above ( e x c e p t on evanescent s e t s ) . 23
REMARK.
( a ) I n s t e a d o f d e s c r i b i n g Z i n t h e statement as a p o s i t i v e
o p t i o n a l s t r o n g s u p e r m a r t i n g a l e , Z c o u l d be d e s c r i b e d as an o p t i o n a l s t r o n g s u p e r m a r t i n g a l e on LO, m l , bounding Y above on LO, m l . I n t h i s form t h e r e s u l t extends t o processes Y indexed by COY m l . I n s t e a d o f going through t h e whole argument again f o r t h i s case, t h e s i m p l e s t t h i n g t o do i s t o i n t r o d u c e t h e o p t i o n a l s t r o n g m a r t i n g a l e Mt =
$1,
iE[Ym/
t h e o p t i o n a l process Y t = Yt
i t s S n e l l envelope ( Z t ) ; bounding Y above on C O Y
-
Mt ( z e r o a t i n f i n i t y ) and
then the smallest strong supermartingale ml
is Z = Z
+ M.
( b ) The f o l l o w i n g i s a consequence o f t h e p r o o f o f 22. For e v e r y s t o p p i n g t i m e T 5 +m l e t Z(T) be a c l a s s under a.s. e q u a l i t y , b e l o n g i n g t o L1(F;),
such t h a t , f o r S
5
T, Z(S)
2
I E [ Z ( T ) ( F i I a.s. and
Z(S) = Z(T) a.s. on I S = TI. Then t h e r e e x i s t s a unique o p t i o n a l s t r o n g s u p e r m a r t i n g a l e Z such t h a t ZT = Z(T) a.s. f o r e v e r y s t o p p i n g time
T (reduce i t t o t h e p o s i t i v e case by s u b t r a c t i n g IE [ Z ( m ) IF;],
t r u n c a t e a t n t o reduce i t t o the bounded case, c o n s t r u c t t h e f u n c t i o n a l J and proceed as a t t h e end o f 22, then l e t n t e n d t o +-). There i s an analogous r e s u l t when l i m i t i n g t o bounded T. ( c ) We a d o p t t h e usual c o n d i t i o n s . I t i s n o t d i f f i c u l t t o check t h a t , f o r e v e r y f i n i t e s t o p p i n g t i m e T,
and t o deduce t h a t Z = Y v 2,.
I n p a r t i c u l a r i f Y i s r i g h t continuous,
Z i s cadlag. Here i s an i n t e r e s t i n g consequence o f t h e e x i s t e n c e o f S n e l l envelopes. I t i s due t o Mertens. 24
THEOREM.
Let Y -
be an o p t i o n a l process. F o r Y t o b e l o n g t o c l a s s ( D ) ,
i t i s necessary and s u f f i c i e n t t h a t t h e r e e x i s t an o p t i o n a l s t r o n g
m a r t i n g a l e Mt = lEIMmIFi] such t h a t sets).
IYI
5
M ( e x c e p t on evanescent
APPENDIX I - STRONG SUPERMARTINGALES
420
Proof. The condition i s obviously s u f f i c i e n t . To show t h a t i t i s necessary, we note t h a t the p o s i t i v e process I Y I belongs t o c l a s s ( D ) and hence has a Snell envelope which i t s e l f belongs to c l a s s ( 0 ) . Thus Z has a Mertens decomposition ZT = lEIAm/FT] - AT where A i s a predictable increasing process (with no r i g h t o r l e f t c o n t i n u i t y ) . I t i s then s u f f i c i e n t t o take M, = Am. 25
REMARK. Note t h a t IE[A,l = IE[Z,l = supT I E I I Y T I l and t h a t the martingale M i s the optional projection of the non-adapted process U t = Am such t h a t IE[U*l = IE[A,I. In [21 Emery gave another proof of Theorem 24 (under the usual conditions) and made i t more precise as fol 1 ows : Let Y be an optional process of c l a s s ( D ) . Then there e x i s t s a measurable process U such t h a t Y i s the optional projection of U & IE[U*l = s u p T I E [ / Y T I l .I f Y i s cadlag, U can be taken t o be cadlag. This r e s u l t may becompared t o a more d i f f i c u l t theorem of Bismut [ Z I : under the usual conditions,
E
>
Let Y be a regular cadlag process of c l a s s ( D ) . Then f o r a l l 0 there e x i s t s a measurable process U with continuous paths such
t h a t Y i s the optional projection o f U and IE[U*l
5
supTIEIIYTIl +
E.
T h i s appendix c o n t a i n s m a r t i n g a l e t h e o r y ( o r a t l e a s t t h e p r e l i m i n a r y r e s u l t s ) w i t h no r i g h t c o n t i n u i t y o f paths and no "usual c o n d i t i o n s " on t h e f a m i l y o f 0 - f i e l d s . From t h i s we deduce i n p a r t i c u l a r t h e p r o j e c t i o n theorems w i t h no hypothesis on t h e f i l t r a t i o n . However, t h e t h e o r y which we develop a l s o has i m p o r t a n t consequences under t h e usual c o n d i t i o n s : we s h a l l f o r example deduce the t h e o r y o f o p t i o n a l s t o p p i n g i n continuous time, due m a i n l y t o Mertens. Throughout t h e appendix
(a, F", P )
denotes a n o t n e c e s s a r i l y
complete p r o b a b i l i t y space w i t h a f i l t r a t i o n (F:)t20
FE-
=
F;
finally
.
(Fit)
Then t h e f a m i l y
( w i t h the convention
i s r i g h t continuous b u t n o t complete, and
( F i ) , o b t a i n e d by c o m p l e t i n g Fit. R e c a l l ( I V . 6 1 ) t h a t t h e
Ft) denotes t h e usual augmentation o f
F" and a j o i n i n g a l l t h e n e g l i g i b l e s e t s t o o p t i o n a l o - f i e l d 0 on
IR,
x
n i s generated by t h e processes adapted t o (F;)
w i t h c a d l a g paths, and t h a t t h e o p t i o n a l c r o s s - s e c t i o n theorem g i v e s t h e same r e s u l t as under t h e usual c o n d i t i o n s (IV.84). F o r c l a r i t y , we keep t h e word s t o p p i n g t i m e ( w i t h o u t q u a l i f i c a t i o n ) f o r s t o p p i n g times o f (F;).
The
p r e d i c t a b l e c r o s s - s e c t i o n theorem ( I V . 8 5 ) r e q u i r e s no p r e c a u t i o n , as t h e p r e d i c t a b l e times a r e t h e same f o r t h e f i l t r a t i o n s (F;)
and (F;').
T h i s appendix i s e x t r a o r d i n a r i l y l o n g and must be t h o u g h t o f as b e l o n g i n g t o b o t h Chapters V I and V I I : t h a t i s why i t has i t s own numbering.
STRONG MARTINGALES AND SUPERMARTINGALES
1
DEFINITION. A r e a l o p t i o n a l process X i s an o p t i o n a l s t r o n g m a r t i n g a l e (resp. supermartingale) if ( 1 ) For e v e r y bounded s t o p p i n g time T, XT i s i n t e g r a b l e
39 3
APPENDIX I
394
-
STRONG SUPERMARTINGALES
( 2 ) For every p a i r of bounded stopping times S , T such t h a t S
5
T,
A cadlag supermartingale i s an optional strong supermartingale (no. VI.10: the usual conditions a r e not needed t h e r e ) and under the usual conditions every optional strong martingale i s , on every bounded i n t e r v a l , the optional projection o f a constant process and hence i s cadlag. B u t even under the usual conditions there e x i s t many optional strong supermartingales which a r e not cadlag. For example, the optional projection of a ( n o t necessarily r i g h t continuous) decreasing process i s always an optional strong supermartingale ( t h i s i s the case f o r the l e f t p o t e n t i a l s of no. VI.89). S i m i l a r l y , the l i m i t of a decreasing sequence of cadlag p o s i t i v e supermartingales i s an optional strong supermartingale b u t i s i n general no longer cadlag. Besides Definition 1 we have t h e following d e f i n i t i o n , which i s a l s o important i n p r i n c i p l e b u t which we s h a l l scarcely study. 2
DEFINITION. A p r e d i c t a b l e process X is a predictable strong martingale if (r e s p . supermartingale) -
( 1 ) For every bounded predictable time T , XT i s i n t e g r a b l e . ( 2 ) For every p a i r of predictable times S , T such t h a t S
5
PRELIMINARY PROPERTIES
Some i nequal i t i e s of martingale theory follow from the stopping theorem only: here i s an example. 3
For a l l
Let X be a n optional o r predictable s t r o n g supermartingale. R, we have
T E
T,
395
STRONG MARTINGALES AND SUPERMARTINGALES
We g i v e a q u i c k p r o o f , f o r example i n t h e o p t i o n a l case. R e c a l l t h a t the F-measurable f u n c t i o n supt5.,lXt(
i s i n general denoted by X * and
l e t h denote t h e sup on t h e r i g h t hand s i d e . We b e g i n by n o t i n g t h a t , for every stopping time S
lEIIXsI]
5 T,
prove ( 3 . 1 ) , g i v e n a number
E
>
5
3h ( c f . V,
( o p t i o n a l c r o s s - s e c t i o n theorem). We s e t S = T
We l e t
E
( 1 2 . 2 ) ) . Then t o
0 choose a s t o p p i n g T such t h a t
t e n d t o 0 and then t a k e t h e l i m i t i n
A
T
and then
t o o b t a i n (3.1) w i t h
t h e broad i n e q u a l i t y . The more p r e c i s e i n e q u a l i t i e s o f t h e t y p e V.(20.1),
V.(20.2)
and V.(24.1)
can be shown analogously.
Doob's i n e q u a l i t i e s on upcrossings and downcrossings a r e s t i l l v a l i d f o r s t r o n g supermartingales, b u t t h e y a r e c l e a r l y more d i f f i c u l t t o prove. Wehad communication o f works o f Th. E i s e l e [l]and R. Berkemeier 111 on t h i s s u b j e c t , b u t we do n o t know whether they have been p u b l i s h e d . Here i s a fundamental r e s u l t , due t o Mertens under t h e usual
-
conditions 4
THEOREM.
o u r p r o o f i s moreover e s s e n t i a l l y t h e same as h i s .
L e t X be an o p t i o n a l o r p r e d i c t a b l e s t r o n g s u p e r m a r t i n g a l e .
F o r almost a l l of
IR+
w E
n, t h e p a t h X(,)
i s bounded on e v e r y compact i n t e r v a l
and has r i g h t and l e f t l i m i t s .
As t h e o - f i e l d s a r e n o t complete, we emphasize t h a t t h e s e t of all w
E
n such t h a t X , ( W ) does n o t possess t h e above p r o p e r t y can
be enclosed i n a n e g l i g i b l e s e t , b u t i s n o t n e c e s s a r i l y F"-measurable.
proof.
We s h a l l t r e a t the o p t i o n a l case i n d e t a i l and l e a v e t h e p r e d i c t -
a b l e case t o the r e a d e r .
APPENDIX I
396
-
STRONG SUPERMARTINGALES
We s h a l l prove the existence of r i g h t and l e f t l i m i t s in 3: inequality ( 3 . 1 ) will on the o t h e r hand imply t h a t these l i m i t s a r e i n
R. We s e t Y
= X/( 1
+ I X I ) . For every sequence (S,) of uniformly
bounded stopping times, e i t h e r increasing o r decreasing, the convergence exists a . s . , so theorem for supermartingales t e l l s us t h a t limn X Sn t h e n a l s o does limn YSn,and we see t h a t limn E[YS ] e x i s t s by dominated n convergence. Hence the conditions hold f o r applying Theorem V I .48 (VI.49 f o r the p r e d i c t a b l e c a s e ) , except t h a t t h e usual conditions do not hold f o r the family of a - f i e l d s . We now take up the proof of VI.48,
taking care about t h e
a - f i e l d s . The processes U a n d V a r e progressive with respect t o the augmented family ( F t ) ; hence t h e debut S i s a s t o p p i n g time of this family, b u t we can use IV.59 t o modify i t i n t o an F-measurable s e t of measure zero (without change of n o t a t i o n ) t o make i t i n t o a stopping time of t h e family ( F t + ) . Then t h e s t o c h a s t i c i n t e r v a l IIS, mU i s and the corresponding stopping optional w i t h r e s p e c t t o the family times S o , S1, ... a r e constructed using the optional cross-section thus these a r e t r u e stopping times theorem applied t o the family and i t i s no longer d i f f i c u l t t o prove t h a t S = m a . s . The second p a r t of ( a ) ( c r i t e r i o n f o r r i g h t c o n t i n u i t y ) does n o t generalize t o a r b i t r a r y f i l t r a t i o n s . We now pass t o ( b ) . Here again t h e processes U a n d V a r e
(Fi)
(Fi):
predictable only w then we modify t h e notation) u s i n g I V (F;) and the proof 5
REMARKS.
t h respect t o the family ( F t ) , s i m i l a r l y f o r K , b u t predictable stopping time T (without changing t h e 78 t o make i t i n t o a stopping time of the family then proceeds without d i f f i c u l t y .
( a ) Let D be a countable dense subset of R,.
By Theorem 4
every r e s u l t on convergence a t i n f i n i t y proved f o r t h e process ( X t ) t c D extends t o the whole of the process ( X t ) t 6 R + . Thus i t i s unnecessary here t o reproduce the statements of the convergence theorems. ( b ) There e x i s t s a process X,, which i s optional with r e s p e c t t o t h e family and i n d i s t i n g u i s h a b l e from the process of r i g h t l i m i t s of X, and a predictable process X-, which i s indistinguishable
(Fi,)
from the process of l e f t l i m i t s of X. The construction of such a process X- i s very simple: i t i s s u f f i c i e n t t o take
397
STRONG MARTINGALES AND SUPERMARTINGALES
(5.1
1
X
't- = liminfsED,s++t
b u t t h e c o n s t r u c t i o n o f X,
s for t
= X,
X,-
> 0,
i s much more d e l i c a t e . We e x p l a i n e d i t i n
no. VI.5 ( a ) . The processes X,
and X - c o n s t r u c t e d i n t h i s no. have much
more p r e c i s e p r o p e r t i e s : X+ i s o p t i o n a l w i t h r e s p e c t t o
P
{p <
up,
a]
mu,
=
0 and X,
but a t instant
p,
notexist or iti s infinite. verified that
p
(Fit)
and r i g h t
o f t h e f a m i l y ( F " ) such t h a t t+ has f i n i t e l e f t l i m i t s on no, p E and i s z e r o on
continuous. There e x i s t s a s t o p p i n g t i m e on
{p
<
a},
p
e i t h e r the l e f t l i m i t o f X+ does
With t h i s d e s c r i p t i o n , i t i s moreover e a s i l y
i s p r e d i c t a b l e . As f o r X-,
i n VI.(5.2)
the following
d e f i n i t i o n was proposed X
=
t h e l e f t l i m i t o f X,
i f i t exists,
X- = 0 o t h e r w i s e .
( c ) L e t X be an o p t i o n a l o r p r e d i c t a b l e s t r o n g s u p e r m a r t i n g a l e . Then X- i s a p r e d i c t a b l e s t r o n g s u p e r m a r t i n g a l e and X+an o p t i o n a l s t r o n g s u p e r m a r t i n g a l e w i t h r e s p e c t t o t h e f a m i l y (Fit).
I f T i s a predictable
t i m e ( r e s p . s t o p p i n g t i m e ) then
(5.2)
XT-
2
IEIXTIF;-]
( r e s p . XT
E[XY+IFf]).
2
( d ) Theorem 4 i s i n t e r e s t i n g even under the usual c o n d i t i o n s . For example, l e t (Xn) be a d e c r e a s i n g sequence o f r i g h t continuous p o s i t i v e supermartingales and l e t X be i t s l i m i t . The process X i s a s t r o n g s u p e r m a r t i n g a l e and hence i t has a process o f r i g h t l i m i t s X, and X
2
X + b y " r i g h t upper s e m i - c o n t i n u i t y "
( t h i s a l s o f o l l o w s from t h e
second i n e q u a l i t y (5.2), which can be w r i t t e n as X T usual c o n d i t i o n s ) . F o r a l l
E
>
0, t h e s e t { X
2
X+
+
2 E}
XTt
under t h e
has no accumul-
a t i o n p o i n t a t f i n i t e d i s t a n c e . T h i s l i t t l e remark i s t h e a b s t r a c t form o f a convergence theorem o f p o t e n t i a l t h e o r y which can be s t a t e d as f o l l o w s : l e t (f,)
be a decreasing sequence o f e x c e s s i v e f u n c t i o n s
and l e t f be i t s l i m i t ; f i s a s t r o n g l y supermedian f u n c t i o n ( t h e analogue o f s t r o n g supermartingales i n p o t e n t i a l t h e o r y ) which has an excessive r e g u l a r i z a t i o n f
5
f . For a l l
E
>
0, t h e s e t { f
t
f t
F}
has
no r e g u l a r p o i n t and t h e s e t { f > f } i s s e m i - p o l a r . ( e ) L e t X be a p o s i t i v e o p t i o n a l s t r o n g s u p e r m a r t i n g a l e which i s completed w i t h Xa= 0. The s t r o n g s u p e r m a r t i n g a l e i n e q u a l i t y and the
APPENDIX I - STRONG SUPERMARTINGALES
398
o p t i o n a l c r o s s - s e c t i o n theorem g i v e e a s i l y : i f S i s a s t o p p i n g t i m e such t h a t X s = 0
i n d i s t i n g u i s h a b l e from 0
0" US,
s., X
mu.
Then l e t T be t h e e s s e n t i a l l o w e r e n v e l o p e o f t h e s e t o f s t o p p i n g times S such t h a t X s = 0 a.s.;
T i s equal a.s. t o a s t o p p i n g t i m e o f t h e f a m i l y
and t h e o p t i o n a l c r o s s ( F & ) , X i s i n d i s t i n g u i s h a b l e from 0 on JT, % > 0% KO, TU except on an evanescent
s e c t i o n theorem shows t h a t X ~
s e t . L e t S be a p r e d i c t a b l e s t o p p i n g t i m e such t h a t Xs-
IS <
m};
t h e i n e q u a l i t y Xs-
t
IE[Xs/F5-] a.s.
= 0 a.s.
on
( 5 . 2 ) i m p l i e s Xs=O on
{S < -1 and'hence everywhere, and hence S 2 T a . s . The s e t I X - = 03 n UO, TI] i s p r e d i c t a b l e w i t h c o u n t a b l e c r o s s - s e c t i o n s , hence i t i s i n d i s t i n g u i s h a b l e from a sequence o f p r e d i c t a b l e graphs Sn and,
-
a p p l y i n g t h e above t o each one, i t f o l l o w s t h a t X - i s > 0 on - UO, T U e x c e p t on an evanescent s e t . S i m i l a r l y l e t U be a s t o p p i n g time o f t h e f a m i l y ( F t + ) such t h a t Xu+ = 0 a.s. on { U
<
(and hence everywhere),
= 0 a.s., hence the i n e q u a l i t y 1 Xu+ t IEIXU+l/nIF,J+l a s . i m p l i e s U + l / n 2 T and f i n a l l y U 2 T. Arguing as above on t h e s e t { X t = 01 n
UO, TI],
we see t h a t X+
3
> 0 on UO, TU, e x c e p t on an evanescent s e t .
Thus we o b t a i n a v e r s i o n o f Theorem VI.17,
valid f o r positive optional
s t r o n g supermartingales, due t o Th. E i s e l e .
THE PROJECTION THEOREMS Our aim i n t h i s s e c t i o n i s t o prove t h e f o l l o w i n g theorem. 6
THEOREM.
Let X -
be a p o s i t i v e o r bounded measurable process.
( a ) There e x i s t s a p r e d i c t a b l e process Y such t h a t
(6.1)
*
Y T = IEIXTIFy-l a . s . f o r e v e r y bounded p r e d i c t a b l e t i m e T
2.
We l e a v e these very simple i n e q u a l i t i e s t o t h e reader.
The r e s t r i c t i o n t o bounded s t o p p i n g times i s i n t e n d e d o n l y t o a v o i d unnecessary d e t a i l s . The e x t e n s i o n t o f i n i t e s t o p p i n g t i m e s o r a r b i t r a r y s t o p p i n g times when t h e processes a r e indexed by C O Y m l , i s immediate.
THE PROJECTION THEOREMS
399
( b ) There e x i s t s an o p t i o n a l process Z such t h a t
(6.2)
ZT = IEIXTIF;]
a.s. f o r e v e r y bounded s t o p p i n g t i m e T .
Y and Z a r e unique t o w i t h i n evanescent processes. They a r e c a l l e d t h e ( p r e d i c t a b l e , o p t i o n a l ) p r o j e c t i o n s o f X and a r e denoted by Y = pX,
z
= O x .
P r o o f . The uniqueness f o l l o w s from t h e c r o s s - s e c t i o n theorems
(IV.84-85). To prove t h e e x i s t e n c e , we s h a l l reduce i t t o t h e case of t h e usual c o n d i t i o n s u s i n g t h e f o l l o w i n g lemma, which i s o f i n t e r e s t i n itself.
7
LEMMA.
( a ) Every process which i s p r e d i c t a b l e r e l a t i v e t o t h e
f a m i l y ( F t ) i s i n d i s t i n g u i s h a b l e f r o m a process which i s p r e d i c t a b l e r e l a t i v e t o (FJ ( b ) Every process which i s o p t i o n a l r e l a t i v e t o t h e f a m i l y (Ft) i s i n d i s t i n g u i s h a b l e f r o m a process which i s o p t i o n a l re1 a t i ve t o ( F i + )
.
P r o o f . Using monotone classes, we a r e reduced t o p r o v i n g t h e lemma f o r s t o c h a s t i c i n t e r v a l s and we then a p p l y IV.78 i n t h e p r e d i c t a b l e case and V1.59 i n t h e o p t i o n a l case. We now come t o t h e e x i s t e n c e p r o o f f o r 6 . We choose v e r s i o n s o f p r o j e c t i o n s o f X r e l a t i v e t o ( F t ) : by Lemma 7 we can choose
a
v e r s i o n o f t h e p r e d i c t a b l e p r o j e c t i o n which i s p r e d i c t a b l e r e l a t i v e t o ( F i ) and denote i t by Y. Sirlce e v e r y ,:edictable predictable time of (Ft),
time T o f
(Fi)
is a
Y s a t i s f i e s (6.1) and t h a t proves ( a ) . I n t h e
o p t i o n a l case we choose a v e r s i o n which i s o p t i o n a l r e l a t i v e t o
(Fi+),
which we denote by 5 ; i t s a t i s f i e s (6.2) b u t i n general i t i s n o t
( F i ) and we must m o d i f y i t . VI.46, t h e s e t H = { ( t , 0) : Yt(u) # c t ( u ) 1 i s c o n t a i n e d
optional relative to By
i n a c o u n t a b l e union o f graphs o f random v a r i a b l e s . We a p p l y Theorem
IV.88 t o i t i n o r d e r t o w r i t e i t i n t h e form H = K u L
APPENDIX I - STRONG SUPERMARTINGALES
400
(Fi),
where K i s such t h a t f o r every stopping time T of P{w : ( T ( w ) , W ) E K} = 0 and L i s contained i n a countable union of graphs o f stopping times T n . These graphs can be assumed t o be d i s j o i n t in IR, x R. We use xn t o denote a version of E[X T n I { T n < m } IF;nl and f o r a l l ( t , U ) we s e t Z t ( u ) = Y,(w)
if
(t, W) 4 U UTnll n
if
( t , W)
(6.3) Zt(W)
=
x,(w)
E
UTq.
Clearly the process Z i s o p t i o n a l , Let T be a bounded stopping time of We s e t A n = A n IT = T n } , which belongs the family (F:) and l e t A E t o F" and F,: and B = A \ U A n , which belongs t o F;. n Tn On An we have ZT = xn = E[X IF" 1 = E[X I IF" I = Tn Tn T n An Tn E[X I IF; 1 a n d hence T An
Fi.
On B, T # Tn f o r a l l n and hence ( T ( u ) , W ) 4 L . As I P I ( T ( w ) , W ) E K ? = 0, we have a . s . ( T ( w ) , W ) 4 H, i n o t h e r words, Y,(u) = 5 ( w ) a . s . On the T other hand, ZT = Y T on B . Thus
To obtain ( 6 . 2 ) i t only remains t o add the r e l a t i o n s ( 6 . 4 ) and ( 6 . 5 ) . 8
( a ) I f X,(W) = X ( W ) , X E L 1 ( F ) , we obtain the existence of versions of the conditional expectations E[XIFy] o r lE[XIF;-l which are optional o r predictable strong martingales. Denoting these versions by Z a n d Y i t i s immediately v e r i f i e d t h a t Y = Z- as under t h e usual = "(Y,). conditions,and t h a t Z = "(Z,) REMARKS.
( b ) As under the usual conditions, the s e t { O X # p X } i s a countable union of graphs of stopping times. No proof i s necessary as i t follows by the very construction of "X above. 9
We i l l u s t r a t e Theorem 6 by two examples. F i r s t , a consequence due t o Kunita [ l ] , taken by Yoeurp-Yor [ l ] under s l i g h t l y d i f f e r e n t
401
THE PROJECTION THEOREMS
hypotheses and f i n a l l y f r e e d o f a1 1 unnecessary hypotheses by Horowi t z . Let ( a , Fa,
IP) be a p r o b a b i l i t y space and l e t Q be another law on (n, F " ) . A p o s i t i v e r . v . U i s c a l l e d a p r e c i s e d e n s i t y o f Cp r e l a t i v e
to P i f - t h e a b s o l u t e l y continuous p a r t o f Q r e l a t i v e t o P i s U . I p
( t h u s U i s I P - i n t e g r a b l e and hence P-a.s.
-
t h e s i n g u l a r p a r t o f Cp i s c a r r i e d by {U =
I t i s very easy t o see t h a t
(IP
t
tm).
U i s then determined t o w i t h i n a
Q ) - n e g l i g i b l e f u n c t i o n . On t h e o t h e r hand, a p r e c i s e d e n s i t y can
be computed law
finite),
(IP
t
as f o l l o w s : choose a d e n s i t y V o f Ip w i t h r e s p e c t t o the
2
C p ) / Z , such t h a t 0 s V s 2, and s e t U = V - 1. We now g i v e a f i l t r a t i o n ( F ; ) . We have t h e f o l l o w i n g
theorem'. THEOREM.
I n t h e above n o t a t i o n , t h e r e e x i s t s a unique o p t i o n a l s t r o n g
s u p e r m a r t i n g a l e (U,)
( i n d e x e d by L O ,
ml)
such t h a t , f o r e v e r y s t o p p i n g
~
__ t i m e T,
UT i s a p r e c i s e d e n s i t y o f Q r e l a t i v e t o Ip ~ on t h e _0 - f i e l _ d
(F;)._
Proof. By Theorem 6 a p p l i e d t o t h e law ( P t Q ) / Z t h e r e e x i s t s an o p t i o n a l s t r o n g m a r t i n g a l e (V,)
under t h i s law such t h a t Vm i s a
d e n s i t y o f P w i t h r e s p e c t t o ( P t Q)/2 on F:.
P w i t h r e s p e c t t o ( P +Cp)/Z Ut = 2/Vt
-
on F; f o r e v e r y s t o p p i n g t i m e T. I f we s e t
1, t h e process U i s o p t i o n a l and
Cp w i t h r e s p e c t t o Ip on
Then Vt i s a d e n s i t y o f
UT i s a p r e c i s e d e n s i t y o f
F i f o r e v e r y s t o p p i n g t i m e T. I t f o l l o w s
immediately t h a t U i s a s t r o n g s u p e r m a r t i n g a l e and the uniqueness f o l l o w s from t h e c r o s s - s e c t i o n theorem. REMARK.
I t can e a s i l y be shown t h a t s i m i l a r l y t h e process 1/U
provides precise densities o f
IP
w i t h respect t o
Here i s t h e second example
~
4.
Dellacherie [ l o ] )
~~
There i s an analogous statement i n t h e p r e d i c t a b l e case. See Horowitz
[]I.
APPENDIX I - STRONG SUPERMARTINGALES
402
Every supermartingale X has a modification Z which i s an
10 THEOREM.
optional strong supermartingale. &f. We do not give our primitive proof, b u t an improved version communicated by E . Lenglart. Let Dn denote the s e t of dyadic numbers of the form k/2" ( k an i n t e g e r ) , D the s e t of a l l dyadic numbers and H the countable s e t of points t where the decreasing function E[Xt] i s not r i g h t continuous. Let X, = U be the process of r i g h t l i m i t s of X . We know (nos. 4 and 5 ) t h a t U i s a strong supermartingale of the family and t h a t XT t IEIUTIFf] a . s . f o r every bounded stopping time T. Let V = U'
(Fit)
be the optional projection of U ( i f reassurance i s required t h a t t h i s projection e x i s t s , note t h a t V i s bounded below on every f i n i t e i n t e r v a l LO, n l by the strong martingale (EIUnI . By the above X t V . Finally we s e t
Fi]))
and show t h a t Z i s a modification of X and an optional strong supermarti ngal e .
( 1 ) I t i s s u f f i c i e n t t o check t h a t Z t = V t = X t a . s . a t a point t H . B u t a t such a p o i n t I E I X t l = (IE[X.]), = IEIXttl = EIVtl, which w i t h the i n e q u a l i t y X t 2 V t a . s . implies X t = Vt a . s . ( i n t h e
+
preceding equal i t i e s we have used the uniform integrabi 1 i t y property V.30).
( 2 ) Z i s an optional process. To show t h a t i t i s a strong supermartingale, we consider two bounded stopping times S a n d T such t h a t S 2 T a n d l e t A E FP. We s e t Tn = T
if
T
E
H,
Tn
=
inf{t
E
Dn: t
>
TI
if
T 4 H.
Tn i s a stopping time ,and i t i s n o t d i f f i c u l t t o check t h a t Tn decreases t o T: through s t r i c t l y g r e a t e r values i f T 4 H and s t a t i o n a r i l y i f T E H . We define Sn s i m i l a r l y from S ; Sn 5 Tn does not necessarily hold on t h e s e t IS f H , T E H } , b u t t o ensure t h a t t h i s i n e q u a l i t y holds (without a l t e r i n g t h e manner in which Sn converges t o S , described above), i t i s s u f f i c i e n t t o replace Sn by Sn A T n , which we do without changing the n o t a t i o n .
DUAL PROJECTIONS
403
We then have ( t a k i n g t h e l i m i t under t h e
1 sign
being
j u s t i f i e d by t h e u n i f o r m i n t e g r a b i l i t y p r o p e r t y V.30)
's
limn
n
= "S'{S~HI
+
X
~
l
{
~
E
and ~ ~s i m i l a r l y f o r T
which i s t h e s t r o n g s u p e r m a r t i n g a l e i n e q u a l i t y . The theorem i s proved. L e t W be a n o t h e r o p t i o n a l s t r o n g s u p e r m a r t i n g a l e which i s a
REMARK.
m o d i f i c a t i o n o f X . We have t o w i t h i n an evanescent process: W+ = U, W
2
0
(W+), hence W
= V and f i n a l l y W 2 Z on HC x
2 OU
n.
On t h e o t h e r hand W and Z a r e b o t h m o d i f i c a t i o n s o f X and hence W and Z a r e i n d i s t i n g u i s h a b l e on t h e s e t H x a , because H i s c o u n t a b l e . Thus W
t
Z t o w i t h i n an evanescent process. Thus t h e m o d i f i c a t i o n which we
have c o n s t r u c t e d i s t h e s m a l l e s t p o s s i b l e .
DUAL PROJECTIONS As i n Chapter V I we d e f i n e p r o j e c t i o n s o f 8-measures and random measures, dual p r o j e c t i o n s o f i n t e g r a b l e i n c r e a s i n g processes, e t c . However t h e r e i s one p o i n t which must be examined i n d e t a i l , namely t h e r e l a t i o n between ( o p t i o n a l ) p r e d i c t a b l e 8-measures and ( o p t i o n a l ) p r e d i c t a b l e i n c r e a s i n g processes: f o r o u r d e f i n i t i o n o f i n c r e a s i n g processes r e q u i r e s t h a t t h e i r p a t h s & lbe i n c r e a s i n g , r i g h t continuous and f i n i t e , which i s i n c o m p a t i b l e w i t h a d a p t a t i o n t o the f a m i l y
(F;) when t h e usual c o n d i t i o n s do n o t h o l d ; t h u s something
must be l o s t on one s i d e o r the o t h e r . We b e g i n w i t h t h e analogue o f VI.57. 11
THEOREM.
L e t A be an i n t e g r a b l e i n c r e a s i n g process which i s
i n d i s t i n g u i s h a b l e from a process B which i s p r e d i c t a b l e ( o p t i o n a l 1 relative to foDtional1.
(F;).
Then t h e measure
)-I
associated w i t h A i s predictable
APPENDIX I
404
-
STRONG SUPERMARTINGALES
P r o o f . We s h a l l use t h e r e s u l t s o f no. 6, where we worked on 19, and n o t on
w,
x
R
x
a . Hence f o r s i m p l i c i t y we assume t h a t A does n o t
jump a t i n f i n i t y ; t h i s i s no r e a l r e s t r i c t i o n i n g e n e r a l i t y
((VI.53) ( e l ) . The p r e d i c t a b l e case i s immediate: f o r A i s p r e d i c t a b l e r e l a t i v e t o ( F t ) , hence ,(X)
= V ( ~ X )f o r e v e r y bounded measurable
process X, where p X denotes t h e p r e d i c t a b l e p r o j e c t i o n r e l a t i v e t o ( Ft). B u t by Lemma 7 pX i s i n d i s t i n g u i s h a b l e from the p r e d i c t a b l e p r o j e c t i o n r e l a t i v e t o (F;)
A
d
and on t h e o t h e r hand
p
has no mass on evanescent sets.
We now t r e a t t h e o p t i o n a l case. We decompose A i n t o AC and
,
i t s continuous and d i s c o n t i n u o u s p a r t s ; AC i s p r e d i c t a b l e r e l a t i v e
t o (Ft) and hence i n d i s t i n g u i s h a b l e from a p r e d i c t a b l e process r e l a t i v e
(F;) by Lemma 7 . We denote t h i s process b y Bc and s e t Bd = B - BC, which i s o p t i o n a l r e l a t i v e t o ( F i ) . Since t h e measure a s s o c i a t e d w i t h to
AC i s p r e d i c t a b l e by t h e p r e c e d i n g paragraph, i t i s s u f f i c i e n t t o d I n o t h e r words, we can reduce i t s t u d y t h e measure a s s o c i a t e d w i t h A d t o t h e case where A i s p u r e l y d i s c o n t i n u o u s and suppress a l l the
.
we s e t Ct =
SUPreq,r
Bt i f t
.
>
0, C,
= 0. T h i s process i s
p r e d i c t a b l e r e l a t i v e t o ( F i ) and i n d i s t i n g u i s h a b l e from t h e process (At-).
Hence t h e s e t { C
+ B}
i s indistinguishablefromthe set { A
+ A-}.
B u t t h e f i r s t i s o p t i o n a l , whereas t h e second i s a u n i o n o f graphs o f r . v . Then by t h e end o f IV.88 ( c ) we see t h a t {B f C), and hence
{ A # A - }, i s i n d i s t i n g u i s h a b l e from t h e union o f a sequence o f graphs UUnD
o f s t o p p i n g times o f
(Fi).
On t h e o t h e r hand t h e r . v . 6n = A A ~ ,
and hence i t i s a.s. F" 'u,)' IU~<~I Un Un measurable. We conclude by n o t i n g t h a t , f o r every bounded measure
i s a.s. equal t o (B
-
process X w i t h o p t i o n a l p r o j e c t i o n Z r e l a t i v e t o
(F;),
where a l l t h e terms a r e meaningful s i n c e 6n = 0 on {Un = - 1 . REMARK.
I n p r a c t i c e , t h i s r e s u l t i s used i n a s l i g h t l y d i f f e r e n t way
from t h a t suggested by t h e statement: i t s B t h a t i s given: an o p t i o n a l process such t h a t , f o r almost a l l
w E
n, t h e p a t h B . ( w ) i s
f i n i t e , i n c r e a s i n g and r i g h t continuous. Then we d e f i n e t h e measure
405
DUAL PROJECTIONS m
u(X) = lE[j
XsdBs]
0
IE
where t h e f u n c t i o n under t h e
( X p o s i t i v e and measurable)
s i g n i s o n l y d e f i n e d a.s. and LI i s
assumed t o be bounded. Then t h e measure LI i s o p t i o n a l . F o r l e t N be a n e g l i g i b l e s e t such t h a t , f o r
w
4 N,
B . ( w ) i s i n c r e a s i n g , f i n i t e and
r i g h t c o n t i n u o u s . We s e t
As
LI
i s a l s o t h e measure a s s o c i a t e d w i t h t h e i n t e g r a b l e i n c r e a s i n g
process A, Theorem 11 t e l l s us t h a t u i s indeed o p t i o n a l . Conversely, l e t u be a p r e d i c t a b l e o r o p t i o n a l bounded
12
IP-measure on
IR,
x
n; we wish t o r e p r e s e n t i t u s i n g a process w i t h
p r o p e r t i e s as p r e c i s e as p o s s i b l e . THEOREM.
IP-measure LI has a r e p r e s e n t a t i o n
The p r e d i c t a b l e ( o p t i o n a l )
u(X) =
(12.1)
IE[/
XSdBSl LO,-[
where B i s a p r e d i c t a b l e ( o p t i o n a l ) process a l l o f whose paths a r e i n c r e a s i n g and r i g h t continuous, w i t h t h e p o s s i b l e e x c e p t i o n o f a s i n g l e value o f t f o r which Bt <
m
and Bt+ =
-.
Moreover, B can be r e p r e s e n t e d as a sum zn Bn o f t r u e predictable
( o p t i o n a l ) i n t e g r a b l e i n c r e a s i n g processes.
P r o o f . We b e g i n w i t h t h e p r e d i c t a b l e case. The measure
u i s predictable
r e l a t i v e t o ( F t ) and hence has a r e p r e s e n t a t i o n (VI.52)
where cx i s an i n t e g r a b l e i n c r e a s i n g process which i s z e r o a t 0, adapted t o t h e f a m i l y (Ft) and continuous, t h e A,, a r e p o s i t i v e constants and t h e Tn a r e p r e d i c t a b l e times o f t h e f a m i l y ( F t ) . We a l s o i n t r o d u c e t h e s t o p p i n g t i m e s o f t h e f a m i l y (Ft)
APPENDIX I
406
sn
=
STRONG SUPERMARTINGALES
infit: a
>
t -
nl
which a r e predictable ( S n i s t h e debut of a r i g h t closed p r e d i c t a b l e s e t ) , and the increasing processes bounded by 1
We now regularize these processes. For B~ we choose a predictable time
-
Tn of the family
(Fi)
such t h a t
Tn
=
T n a . s . (IV.78) and we s e t
For a n we choose f o r r r a t i o n a l a r . v .
s:
equal a . s . t o a: and then f o r t real
0 we s e t
and then f o r t real
2
>
which i s F;+-measurable
and
0
an increasing process which i s adapted t o ( F i + ) , bounded by 1 and indistinguishable from an. We then denote t h e continuous p a r t o f r;" by -n~1 ( a t t h e same time removing the j u m p a t 0); t h i s increasing n process i s s t i l l indistinguishable from ~1 , s t i l l adapted to ( F { + ) and, as i t i s continuous, i t i s predictable r e l a t i v e t o ( F ; ) . remains t o s e t
I t only
(12.2) This process i s indistinguishable from A , i t i s p r e d i c t a b l e with , r e s p e c t t o ( F : ) and i t s paths a r e a l l increasing. I f B t ( w ) < m y t h e s e r i e s ( 1 2 . 2 ) i s uniformly convergent on CO, t l and hence the path B { w ) i s r i g h t continuous on COY t C , hence B can have only the type of r i g h t discontinuity indicated in the statement. As lELB-1 c m y note a l s o t h a t t h i s lack o f r i g h t continuity can occur only on a s e t o f measure zero.
We now come t o the optional case, which i s as usual a l i t t l e
DUAL P ROJ E CTI ONS
407
more d e l i c a t e . L e t p be a bounded IP-measure on u(X) = p('X) as a sum
IR+
x
n such t h a t
f o r e v e r y bounded measurable process X . We decompose u
p = v
+
IT
o f two p o s i t i v e measures w i t h t h e f o l l o w i n g
properties i s c a r r i e d by a c o u n t a b l e u n i o n o f graphs o f s t o p p i n g times o f (F:Iy v has no mass on any graph o f a s t o p p i n g t i m e
( i t i s s u f f i c i e n t t o define
as t h e l e a s t upper bound o f t h e measures
IT
bounded above by V. c a r r i e d by a c o u n t a b l e u n i o n o f graphs o f s t o p p i n g times and t o s e t u = p
-
I t i s immediately v e r i f i e d t h a t t h e r e
IT).
e x i s t s a s e t H which i s a c o u n t a b l e u n i o n o f graphs o f s t o p p i n g times such t h a t
IT
=
IH*p;
hence
i s o p t i o n a l and the same i s t r u e o f v by
IT
taking the difference.
As t h e s e t where
and 'X
OX
d i f f e r i s a countable u n i o n o f
graphs o f s t o p p i n g times, we have v ( X ) = .(OX)
= v ( ~ X ) ; hence u i s
p r e d i c t a b l e and we can a p p l y t h e f i r s t p a r t o f t h e p r o o f t o i t . Hence i t remains t o c o n s i d e r IT.
We r e p r e s e n t H as a u n i o n o f d i s j o i n t graphs o f s t o p p i n g t i m e s Un. Then
IT
= I H * u i s t h e sum o f t h e measures
i t i s s u f f i c i e n t t o r e p r e s e n t each measure
IT
n
.
As
I UUnll - u
IT^
IT^
=
and
i s c a r r i e d by
t h e graph o f Un, we can w r i t e ."X)
IE[X
=
'n
6 ]
where 6n i s an i n t e g r a b l e p o s i t i v e r . v . On t h e o t h e r hand
rrn
is
o p t i o n a l ; t a k i n g X = 0 o u t s i d e t h e graph o f Un, we see t h a t 6 n i s a.s. equal t o a p o s i t i v e F" -measurable r . v . Un 6
n
=
zm
X
I
I t o n l y remains t o w r i t e
nm Hnm
a.s
and t h e n t o where t h e Xnm a r e c o n s t a n t s and t h e tinm elements o f f " Un s e t Tnm = U n on Hnm and = +m on H i m y t o o b t a i n t h e r e p r e s e n t a t i o n
and hence t h e " i n c r e a s i n g process" a s s o c i a t e d w i t h n, which i s
APPENDIX I - STRONG SUPERMARTINGALES
408
'nm A nm 1 I t r T n m l ' REMARK.
R e t u r n i n g t o t h e r e s u l t s o f nos. 11 and 12, we o b t a i n a
r e s u l t which i s o f i n t e r e s t i n i t s e l f : i f A i s an i n t e g r a b l e i n c r e a s i n g process which i s i n d i s t i n g u i s h a b l e from a p r e d i c t a b l e process, t h e n
B which i s i n d i s t i n g u i s h a b l e from A and i s a
t h e r e e x i s t s a process
sum o f p r e d i c t a b l e i n t e g r a b l e i n c r e a s i n g processes ( a l l t h e p r o p e r t i e s h o l d s i m u l t a n e o u s l y f o r each term i n t h e sum). S i m i l a r l y f o r t h e o p t i o n a l case.
RETURN TO STRONG SUPERMARTINGALES
13
L e t X be a s t r o n g s u p e r m a r t i n g a l e which f o r s i m p l i c i t y we assume t o be indexed by L O , m l ( r i g h t c l o s e d ) . We a s s o c i a t e w i t h i t a measurable process ,X,
which i s i n d i s t i n g u i s h a b l e from t h e process o f
r i g h t l i m i t s o f X ( f o r example t h e v e r s i o n c o n s t r u c t e d i n no. 5 ( b ) , b u t t h e r e i s no need t o be so p r e c i s e ) , and a p r e d i c t a b l e process X-, which i s i n d i s t i n g u i s h a b l e f r o m t h e process o f l e f t l i m i t s o f X . The r e l a t i o n s (5.2)
can be w r i t t e n
x
(13.1)
2
px,
x
r "(X,).
As we d i d under t h e usual c o n d i t i o n s ( V I . , p X = X- ( i n c l u d i n g a t
m
V I I . ) we c a l l X r e g u l a r i f
i n t h e s i t u a t i o n i n which we a r e i n t e r e s t e d ) .
I f X belongs t o c l a s s ( 0 ) t h i s means, as i n no. VII.10, t h a t X i s continuous i n e x p e c t a t i o n : i f (T,) i s an i n c r e a s i n g sequence o f
s t o p p i n g times and T = l i m n Tn, then IEIXTl = l i m n I E I X T second i n e q u a l i t y ( 5 . 2 )
i n t r o d u c e s a new n o t i o n :
n
left
1. B u t t h e
LEMMA. With t h e above n o t a t i o n , t h e f o l l o w i n g p r o p e r t i e s a r e equivalent: (a)
x
= "(x,).
( b ) F o r e v e r y s t o p p i n g t i m e T, IEIXT] = l i m IEIXT+l,,nl. n ( c ) F o r e v e r y s t o p p i n g t i m e T and e v e r y d e c r e a s i n g sequence o f s t o p p i n g times Tn
J.
T,
IEIXTl = l i m n lE[X
I.
Tn I f these p r o p e r t i e s h o l d , X i s c a l l e d r i g h t continuous i n e x p e c t a t i o n .
P r o o f . C l e a r l y ( c ) a ( b ) . Since the r . v . XT+l,n
are u n i f o r m l y i n t e g r a b l e
RETURN TO STRONG SUPERMARTINGALES ( b ) can be w r i t t e n as IEIXTl = IE [ X T + l ;
(V.30),
XT
IE [XT+
2
I FT-I , whence
on t h e o t h e r hand
I FTl .
t h e equal it y XT = IE [XTt
f o r e v e r y s t o p p i n g t i m e T, we see t h a t ( b )
3
409
(a).
Since t h i s h o l d s
F i n a l l y , i n the
iC~T+~,
n o t a t i o n o f ( c ) , l i m n IE[X I = I E [ l i m n x I = ~ X ~ I + P Tn Tn where A i s t h e s e t 13: n : Tn = T I , which belongs t o F;. Given ( a ) , t h i s sum i s equal t o IEIXTl and the l a s t i m p l i c a t i o n ( a )
( c ) i s proved.
Under the usual c o n d i t i o n s , p r o p e r t i e s ( a ) , ( b ) , ( c ) mean s i m p l y t h a t X i s r i g h t continuous b u t t h i s i s n o t t r u e i n general, as t h e example o f m a r t i n g a l e s immediately shows. The f o l l o w i n g n o t i o n i s due t o Bismut 131 (who c a l l s i t a
14
quasi-stopping time). It i s e s p e c i a l l y useful f o r studying strong
s upermart ngales w i t h o u t t h e usual c o n d i t i o n s , b u t i s a l s o o f i n t e r e s t i n " c l a s s c a l " s i t u a t i o n s . A g e n e r a l i z a t i o n can be found i n Maingueneau [ll. A s p l i t stopping time (abbreviated s.s.t.)
i s an ordered
H , T), where T i s a s t o p p i n g t i m e and H an element o f F; such t h a t TH i s p r e d i c t a b l e (hence H i n f a c t belongs t o F;-). T i s pair
T
=
c a l l e d bounded, f i n i t e ,
...
i f T i s bounded, f i n i t e ,
...
The i n t u i t i v e
i d e a o f such an o r d e r e d p a i r i s t h e f o l l o w i n g : d i v i d e each p o i n t x i n
Rt
i n t o two p o i n t s , one again denoted x and t h e o t h e r x - and extend
t h e o r d e r r e l a t i o n by t h e conventions xy
5
x and y s x- i f and o n l y i f y
f u n c t i o n w i t h values i n
<
Et doubled
<
x . Then
x, y- s x- i f and o n l y i f T
can be considered as a
i n t h i s way, which takes t h e value
T- on H and t h e v a l u e T on HC. I f X i s a s t r o n g s u p e r m a r t i n g a l e or more g e n e r a l l y a process which a . s . admits l e f t l i m i t s , we s e t (14.1)
X
T
=X
w i t h t h e c o n v e n t i o n t h a t X,wise ( i f 10,
T
=
X,
X
i f X,-
C
=XTonH
has n o t been s p e c i f i e d o t h e r -
i s n o t a . s . f i n i t e , i t i s necessary f o r X t o be indexed by
ml).
I f X i s optional,
o - f i e l d F:, < T
XT
i s measurable w i t h r e s p e c t t o t h e
consisting o f the A L e t u = (G, S ) and
u
onH,
T-
E
T =
i f S s T and H n { S = T I
c
F y such t h a t A n H belongs t o F;-. (H, T) be two s . s . t . ;
we w r i t e
G n IS = T I , conforming w i t h t h e
o r d e r r e l a t i o n d e f i n e d above on t h e s p l i t h a l f - l i n e . I t i s very
APPENDIX I - STRONG SUPERMARTINGALES
41 0
easy t o check t h a t F; c F;. With every stopping time T can be associated the s . s . t .
(a,
T ) which we i d e n t i f y with T . I f T i s p r e d i c t a b l e , we can a s s o c i a t e w i t h i t the s . s . t . ( n , T ) , which we denote by T - . Then t h e fol 1 owing theorem general i zes t h e two stopping theorems VI.10 and VI.14. Let X be an optional strong supermartingale indexed by
15 THEOREM. C O Y -1.
We adopt the convention X,-
be two s . s . t . such t h a t u s
T.
=
X,.
KU = (G,
S ) pnd
T
= (H, T )
Then -
with e q u a l i t y i f X i s an optional strong martingale on LO, m l Proof. Suppose f i r s t t h a t X i s an optional strong martingale. Then i t i s immediate t h a t X T = I E I X m I F ~ l ,Xu = IEIXmIFil and (15.1) with e q u a l i t y follows immediately. martingale positive.
We now consider the general case. Subtracting the strong IEIXmIFil from X , we a r e reduced t o the case where X i s We wish t o e s t a b l i s h the i n e q u a l i t y
We begin by t r e a t i n g the case where A = n . We a r e f r e e t o truncate X a t n and then take t h e limit,and hence we may assume t h a t X i s bounded. Let (R,) be a sequence which a . s . f o r e t e l l s SG (IV.77) and l e t Sn = Rn A S ; then Xu = lirn S a . s . and hence i t i s s u f f i c i e n t t o n Sn check t h a t
j xs
n
IP
2
jXTIP.
Let (U,) be a sequence which a . s . f o r e t e l l s TH; the r e l a t i o n u implies t h a t t h e sequence T, = S, v (Urn A T ) i s then such t h a t
I T
XT a . s . ; on t h e o t h e r h a n d f o r a l l n and a l l m we have J X , IP n IP by the d e f i n i t i o n of s t r o n g supermartingales, whence the Tm required i n e q u a l i t y . X
Tm I X
-+
2
41 1
APPLICATION TO INEQUALITIES
To deal w i t h t h e general case, we proceed as f o r o r d i n a r y s t o p p i n g t i m e s : we a p p l y t h e above t o t h e s . s . t . -cA = ( H n A,
uA = ( G n A,
SA) and
TA).
APPLICATION TO INEQUALITIES We now s e t o u t t o extend t h e i n e q u a l i t i e s o f Chapter V I . § 3
16
t o v e r y general s i t u a t i o n s and a t t h e same t i m e t o u n i f y them.
two
We c o n s i d e r denote by A- and A+;
i n t e g r a b l e i n c r e a s i n g processes which we
t h e f i r s t i s p r e d i c t a b l e r e l a t i v e t o ( F i ) and
zero a t 0 b u t may jump a t i n f i n i t y , t h e second i s o p t i o n a l r e l a t i v e t o
(Fi),
may jump a t 0 b u t n o t a t i n f i n i t y and i s p u r e l y d i s c o n t i n u o u s .
+
+ A_,
We denote by A t h e p r e d i c t a b l e i n t e g r a b l e " i n c r e a s i n g " process A-
which i s n e i t h e r r i g h t n o r l e f t c o n t i n u o u s . The p o t e n t i a l Z generated by A i s by d e f i n i t i o n t h e o p t i o n a l p r o j e c t i o n o f t h e process ( A m - A t ) t y t h a t i s t h e process (16.1)
- At
Zt = E[A,IFi]
where t h e process lEIAmIFi] must be i n t e r p r e t e d as an o p t i o n a l s t r o n g m a r t i n g a l e . Our aim i s t o e s t i m a t e A,
as a f u n c t i o n o f Z. Under t h e
usual c o n d i t i o n s , t h e s i t u a t i o n i n c l u d e s t h e two cases o f no. VI.89: t h e p r e d i c t a b l e case corresponds t o a r i g h t continuous process A and t h e o p t i o n a l case t o a l e f t continuous process A. Under t h e "unusual" c o n d i t i o n s , o u r hypotheses a r e r a t h e r t o o r e s t r i c t i v e , b u t see no. 19. We saw i n no. V I I . 3 t h a t A determines A+ and A - . 17
We a p p l y formula (91.4) t o t h e r i g h t continuous process A+ = A-
+ A'.
The l a t t e r i n t e g r a l i s equal t o ~ ( ( A A ; ) ~- (AA:)~). We thus o b t a i n A:
=
\(Am
-
At)dAt+
+
((Am
-
A t - ) d A i + j(A,
-
At+)dA;
.
41 2
I -
APPENDIX
STRONG SUPERMARTINGALES
-
We now i n t e g r a t e ; we can r e p l a c e (A, (Zt),
(A,
-
At-)
by i t s p r e d i c t a b l e p r o j e c t i o n (Zt-)
by i t s o p t i o n a l p r o j e c t i o n (Ztt) (17.1)
At) b y i t s o p t i o n a l p r o j e c t i o n
IE[A:l
relative to (Fit).
lE[j ZsdAst
=
t
j ZS-dA;
- At.)
and (A, We o b t a i n
j ZStdA:l.
t
We thus g e n e r a l i z e t h e two "energy formulae" o f no. VI.94. 18
We s h a l l now g e n e r a l i z e t h e fundamental theorem V I .99. As noted i n no. 100, i t a l l amounts t o g e n e r a l i z i n g t h e "maximal lemma" (100.1) and t o t h i s end we argue d i r e c t l y , u s i n g t h e s p l i t s t o p p i n g times i n t r o d u c e d i n no. 14. I t w i l l be convenient t o assume t h a t A does n o t jump a t i n f i n i t y , which i s no r e s t r i c t i o n i n g e n e r a l i t y ( V I . 5 3 ( c ) ,
V I I . 15).
T h i s t i m e we s e t
at
t
A t t A-t; a i s a r i g h t continuous
=
o p t i o n a l i n c r e a s i n g process and the o p t i o n a l s e t J = { ( t , A}
i s a stochastic interval
at(.) CS-
z A}
[TT,
i s a stopping time o f
mu,
z
: at(")
whose debut T(w) = i n f { t : t
(Fi+).
s A 5 a, t h e s e t s { A 2 I } and { a
M)
Let H =
{O
: AT(w)
z
A};
? E
Q,
as
A } d i f f e r o n l y by a subset o f
t h e graph o f T and hence
i s a p r e d i c t a b l e s e t . On t h e o t h e r hand, TH i s a s t o p p i n g t i m e o f (Fit),
and hence I T H , m U
is
p r e d i c t a b l e and f i n a l l y
i s p r e d i c t a b l e , which means t h a t TH i s a p r e d i c t a b l e time (IV.69) and hence t h a t
T =
(H, T) i s a s . s . t .
The s e t {A, 2 A} = ( T < - } u { T = -, A, 2 A } belongs t o F Y and i t s i n t e r s e c t i o n w i t h H i s equal t o H and hence i s Fy--measurable; hence i t belongs t o FY. F i n a l l y , AT 5 A , whence t h e i n e q u a l i t y
by 15 a p p l i e d t o the s t r o n g m a r t i n g a l e (Zt by a s t r o n g m a r t i n g a l e
(IE [Mm I Fg] ) ,
+
At).
we deduce t h a t
I f Z i s bounded above
APPLICATION TO INEQUALITIES
41 3
(18.1) t h a t i s (100.1) and a l l t h e consequences o f t h i s i n e q u a l i t y proved i n Chapter V I remain t r u e i n t h e general case. 19
We r e t u r n t o t h e hypotheses 16: i n f a c t , i f we a r e w o r k i n g under t h e "unusual" c o n d i t i o n s , i t was p o i n t e d o u t e a r l i e r t h a t A+ and A- a r e o n l y v e r y r a r e l y t r u e o p t i o n a l o r p r e d i c t a b l e i n c r e a s i n g processes: they a r e c o n s t r u c t e d from IP-measures by t h e procedure o f no. 12, i n o t h e r words A- and A+ a r e b o t h processes w i t h i n c r e a s i n g paths, t h e f i r s t i s p r e d i c t a b l e and t h e second o p t i o n a l , b u t r i g h t c o n t i n u i t y does n o t h o l d i d e n t i c a l l y : t h e r e may e x i s t w s e t o f measure zero
-
f o r which A+(.)
o r A:(w)
-
forming a
i s n o t r i g h t continuous.
How do we overcome t h i s d i f f i c u l t y ? By no. 12, A
+
and A- a r e
o f analogous processes, b u t
which a r e i d e n t i c a l l y r i g h t continuous and hence A i t s e l f i s a sum o f the corresponding processes An and Z t h e sum o f t h e corresponding n . We s e t
processes Z
c i k = z k, A n,
5 k = z k, Z n.
I f Z i s bounded above by t h e s t r o n g m a r t i n g a l e
(iEM[ , I F ; ] ) ,
a fortiori
so i s gk and, s l i g h t l y m o d i f y i n g t h e i n t e g r a t i o n s e t s , we o b t a i n
then, l e t t i n g k t e n d t o i n f i n i t y ,
and f i n a l l y , t a k i n g l i m i t s gives i n e q u a l i t y (100.1) i n complete generality (19.1) Moreover t h e same r e s u l t s a r e v a l i d i f t h e hypotheses on A+ and Aa r e s l i g h t l y m o d i f i e d : i t can be assumed t h a t these two processes a r e t r u e i n t e g r a b l e i n c r e a s i n g processes which a r e i n d i s t i n g u i s h a b l e , the
APPENDIX I - STRONG SUPERMARTINGALES
414
f i r s t from an optional process a n d the second from a predictable process. To see t h i s , we must use t h e Remark of n o . 1 2 , which reduces these hypotheses t o the preceding ones.
MERTENS DECOMPOSITION
We s h a l l now generalize the r e s u l t s a t t h e beginning of Chapter VII. As in no, 19, the theorems a r e valid under the usual conditions and under the most general conditions, b u t t h e r e i s a degree of additional complication i n t h e general case, r e s u l t i n g from the more d e l i c a t e manipulation of the notion of an increasing process ( n o . 1 2 ) . To avoid a ponderous statement, we a d o p t t h e usual conditions and then in n o . 21 i n d i c a t e t h e modifications t o be introduced in the general case. 20
Here then i s the r e s u l t under the usual conditions, which i s due t o Mertens. THEOREM. Let Z be a p o s i t i v e optional strong supermartingale of c l a s s ( D ) (with t h e convention Z o - = Z,, Zm = 0 ) . Then t h e r e e x i s t s a
predictable process A with increasing ( b u t not necessarily r i g h t o r l e f t continuous) paths such t h a t (20.1) IE[A,l
<
-,
ZT
=
IEIAmIFTl - AT f o r every stopping time T .
A i s unique and the following inequality between processes holds:
In p a r t i c u l a r , A i s r i g h t continuous i f f Z i s r i g h t continuous and l e f t continuous i f f z = PZ ( i . e . i f z is r e g u l a r ) . Proof. The reader i s requested t o r e f e r t o no. VII.6 a n d t o follow the modifications. In n o . 6 i t i s s u f f i c i e n t t o suppress "cadlag"
1
Under the usual conditions "(Z+)
=
Z+.
41 5
MERTENS DECOMPOSITION
t h r o u g h o u t and s i m i l a r l y i n no. 7 . I n no. 8 t h e p r o o f shows t h a t
A
i
= 0 i f f Z i s r i g h t continuous; i n t h e general case we have s i m p l y
which, when X i s t h e i n d i c a t o r o f UT, WE, g i v e s
t h a t i s ( 2 0 . 1 ) . No. V I I . 1 0 g i v e s w i t h o u t m o d i f i c a t i o n t h e f i r s t e q u a l i t y o f ( 2 0 . 2 ) and the second f o l l o w s s i m i l a r l y . F i n a l l y t h e uniqueness o f A f o l l o w s f r o m t h e f a c t t h a t b e i n g g i v e n A i s e q u i v a l e n t t o b e i n g g i v e n t h e o r d e r e d p a i r (A-, A')
b y V I I . 3 , which i s i t s e l f
e q u i v a l e n t t o b e i n g g i v e n t h e l i n e a r form J ( V I I . 2 ) and f i n a l l y t o b e i n g g i v e n Z (V11.6). 21
We now c o n s i d e r t h e general ("unusual " ) c o n d i t i o n s . F i r s t we must c a r e f u l l y r e - r e a d t h e p r o o f o f Theorem V I I . 2 : a l l goes w e l l where c o n s t r u c t i n g measures i s concerned, b u t when i t comes t o i n c r e a s i n g processes, nos. 11-12 o f t h i s appendix l e a v e us t h e c h o i c e between two possibilities:
-
t o t a k e A+ and A- t o be two w e i n t e g r a b l e i n c r e a s i n g
processes, b u t which a r e o n l y i n d i s t i n g u i s h a b l e f r o m o p t i o n a l o r p r e d i c t a b l e processes;
-
=
t o take A+ and A- t o be two processes which are r e s p e c t -
i v e l y o p t i o n a l and p r e d i c t a b l e , w i t h i n c r e a s i n g paths, b u t i d e n t i c a l l y r i g h t c o n t i n u o u s and f i n i t e .
The f i r s t c h o i c e leads t o the simp e r r e s u l t . F o r i t enables us t o s e t A = A- t Af ( a process w i t h f i n i t e values which i s i n d i s t i n g u i s h a b l e from a p r e d i c t a b l e process
and then A+ = A- + A+,
which i s r i g h t c o n t i n u o u s . Then t h e measurab l i t y c o n d i t i o n s on A can merely be expressed as f o l l o w s : A i s i n d i s t i n g u i s h a b l e f r o m a p r e d i c t a b l e process and A+ from an o p t i o n a l process. That a p a r t , t h e r e i s no i m p o r t a n t m o d i f i c a t i o n . Note however t h a t r i g h t c o n t i n u i t y o f A i s no l o n g e r e q u i v a l e n t t o r i g h t c o n t i n u i t y
APPENDIX I - STRONG SUPERMARTINGALES
41 6
of Z in expectation. We s h a l l give no more d e t a i l s , f o r everything would have t o be redone under the general conditions: local martingale theory, Doob decomposition of a r b i t r a r y supermartingales (VII . l Z ) , e t c . Frankly, such a systematic study does n o t seem worthwhile a t t h e moment.
THE SNELL ENVELOPE We s h a l l now i l l u s t r a t e the i n t e r e s t of strong supermartingales (even under the usual conditions!) by presenting the elements o f the theory o f t h e Snell envelope. A h i s t o r i c a l word i s not out of place here. The raison d'i2tre o f t h i s theory i s t h e problem of optimal stopping: given an a r b i t r a r y process X , f i n d a stopping time T f o r which IEIXT] i s as large as p o s s i b l e . I t was i n order t o study optimal stopping in t h e d i s c r e t e case t h a t Snell in [11 introduced the notion of a submartingale and the corresponding stopping theorem, along with t h e envelope which now bears h i s name. Snell ' s works were extended t o the continuous case by J . F . Mertens, whose a r t i c l e s 131, [41 a r e fundamental t o the notion o f a strong supermartingale (and Mertens decomposition) and the notion of a strongly supermedian function i n potential theory. Again q u i t e r e c e n t l y , t h e theory of the Snell envelope and optimal stopping was taken up i n a r t i c l e s by Bismut [31, Bismut-Skalli [11 and Maingueneau [ l ] (using s p l i t stopping times, which were introduced on t h i s occasion). We say nothing here on the theory of optimal stopping, the l i t e r a t u r e on t h i s s u b j e c t i s enormous and largely orientated towards a p p l i c a t i o n s ; the reader can consult S h i r i a i e v ' s book [ l l . We hope t o return t o i t l a t e r w h e n we study the notion of a r e d u i t e i n potential theory. 22
Let Y denote an optional process, indexed by [O, m[ a n d belonging t o c l a s s [Dl: the r.v. Y T , where T runs through the s e t o f a l l bounded stopping times, form a uniformly i n t e g r a b l e s e t . We adopt the convention Ym = 0. Then YT i s defined f o r every stopping time T , f i n i t e o r otherwise, and the s e t o f Y T (where T runs through the s e t I of a l l stopping times) i s a l s o uniformly i n t e g r a b l e . We now give the fundamental existence theorem f o r the Snell
41 7
THE SNELL ENVELOPE envelope. As we a r e w i t h o u t t h e usual c o n d i t i o n s , we do n o t h e s i t a t e t o use a " s l e d g e hammer" theorem. F o r an elementary p r o o f (under t h e usual c o n d i t i o n s ) see Maingueneau [ l ] , p. 458. THEOREM.
( a ) There e x i s t s a p o s i t i v e o p t i o n a l s t r o n g supermartingale
Z w i t h the f o l l o w i n g property: Z
2
Y and f o r e v e r y p o s i t i v e o p t i o n a l s t r o n g s u p e r m a r t i n g a l e
Z ' which bounds Y above, Z '
2
Z ( t h e i n e q u a l i t i e s h o l d as usual except
on evanescent s e t s ) . Z i s unique ( t o w i t h i n evanescent s e t s ) . I t i s c a l l e d t h e S n e l l envelope o f Y. Moreover, Z belongs t o c l a s s ( 0 ) .
( b ) For e v e r y s t o p p i n g t i m e T we have a.s. (22.1) P r o o f . We f i x T and c o n s i d e r t h e s e t o f r . v . through t h e s e t o f s t o p p i n g times S
t
E I Y S I F i l , where S runs
T. We make t h e fundamental
remark t h a t t h i s s e t i s d i r e c t e d and i n c r e a s i n g . For i f S , and S, s t o p p i n g times t T, l e t Ui U, v U,
=
=
IEIYsi I F T I ( i = 1, 2 ) ; we have
where R = S , i f U,
EIYRIFYl,
are
t
U,,
R = S, i f U,
<
U2.
Then l e t Z ( T ) denote t h e supess on t h e r i g h t hand s i d e o f ( 2 2 . 1 ) - we c a r e f u l l y w r i t e Z(T) and n o t ZT because a c l a s s o f r . v .
i s i n v o l v e d and
we do n o t y e t know whether we can "paste t o g e t h e r " t h e Z(T) i n t o a process Z. Note some p r o p e r t i e s o f these r . v . : ( a ) Z(T)
2
0 (take S =
( b ) For a l l A
E
jA
(22.2)
+m
i n (22.1));
Z(T)
j-s
Fy-measurable.
FY,
IP
= SUPStTj
A
ys IP
(an immediate consequence o f t h e f a c t t h a t t h e r . v . E I Y S I F y ] form an increasing directed set).
( c ) - I f -T and U a r e two s t o p p i n g times w i t h T
5
U, W Z ( T ) t
EIZ(U)IFoTl a.s. and Z(T) = Z(U) a . s . on I T = U3 ( a n easy consequence o f (22.2)).
APPENDIX I - STRONG SUPERMARTINGALES
418
( d ) The r . v .
M = sups I E I Y s l < P{Z(T)
2
cl
5
m;
Z(T) a r e u n i f o r m l y i n t e g r a b l e . For l e t
by (22.2) w i t h A = n, we have IEIZTl
M/c. Then by (22.2) again
!I Z ( T ) r c I Z(T) P As P { Z ( T ) 2 c l
M and hence
5
sup
5
S2T
IZ(T)tc)
Ys
P.
M/c i n d e p e n d e n t l y o f T and t h e r . v . Ys a r e u n i f o r m l y
i n egrable, the r i g h t hand s i d e tends t o 0 as c *
m,
u n i f o r m l y i n T and
S. Hence t h e s t a t e d r e s u l t .
And now we r e t u r n t o no. V I I . 6 . I f X i s a cadlag process wh ch i s bounded on 10, -31 and o f t h e form (6.1)
x
= XoIno
Y 1
t
...
+
xnlDTn,mn
we s e t
and we t a k e t h e whole o f t h e p r o o f o f t h e Mertens decomposition
( n o . 21 above). We o b t a i n a r e p r e s e n t a t i o n o f J i n t h e form
as i n no. 21, then, i f we denote by Z t h e process which i s t h e o p t i o n a l
p r o j e c t i o n o f t h e process (A,
t
+ A, -
At
-
t
At-),
which i s a p o s i t i v e
o p t i o n a l s t r o n g supermartingale, we f i n d t h a t t h e measure a s s o c i a t e d f o r e v e r y s t o p p i n g t i m e T. Thus
w i t h Z i s J , and hence Z(T) = ZT a.s.
we have e s t a b l i s h e d t h e e x i s t e n c e o f a p o s i t i v e o p t i o n a l s t r o n g supermartingale o f class
(D), which s a t i s f i e s (22.1)
The r e s t i s obvious. Taking S = T i n (22.1), we see t h a t
ZT
and 2
2 Y b y t h e o p t i o n a l c r o s s - s e c t i o n theorem. I f Z ' i s T another p o s i t i v e o p t i o n a l s t r o n g s u p e r m a r t i n g a l e bounding Y above, we
2
Y
have f o r every s t o p p i n g t i m e T and e v e r y s t o p p i n g t i m e S
and t a k i n g supess
i n S we o b t a i n
Zt
2
2
T
ZT a . s . and hence Z ' 2 Z by t h e
THE SNELL ENVELOPE
41 9
o p t i o n a l c r o s s - s e c t i o n theorem. F i n a l l y , t h e S n e l l envelope Z i s c h a r a c t e r i z e d u n i q u e l y as t h e s m a l l e s t p o s i t i v e o p t i o n a l s t r o n g s u p e r m a r t i n g a l e bounding Y above ( e x c e p t on evanescent s e t s ) . 23
REMARK.
( a ) I n s t e a d o f d e s c r i b i n g Z i n t h e statement as a p o s i t i v e
o p t i o n a l s t r o n g s u p e r m a r t i n g a l e , Z c o u l d be d e s c r i b e d as an o p t i o n a l s t r o n g s u p e r m a r t i n g a l e on LO, m l , bounding Y above on LO, m l . I n t h i s form t h e r e s u l t extends t o processes Y indexed by COY m l . I n s t e a d o f going through t h e whole argument again f o r t h i s case, t h e s i m p l e s t t h i n g t o do i s t o i n t r o d u c e t h e o p t i o n a l s t r o n g m a r t i n g a l e Mt =
$1,
iE[Ym/
t h e o p t i o n a l process Y t = Yt
i t s S n e l l envelope ( Z t ) ; bounding Y above on C O Y
-
Mt ( z e r o a t i n f i n i t y ) and
then the smallest strong supermartingale ml
is Z = Z
+ M.
( b ) The f o l l o w i n g i s a consequence o f t h e p r o o f o f 22. For e v e r y s t o p p i n g t i m e T 5 +m l e t Z(T) be a c l a s s under a.s. e q u a l i t y , b e l o n g i n g t o L1(F;),
such t h a t , f o r S
5
T, Z(S)
2
I E [ Z ( T ) ( F i I a.s. and
Z(S) = Z(T) a.s. on I S = TI. Then t h e r e e x i s t s a unique o p t i o n a l s t r o n g s u p e r m a r t i n g a l e Z such t h a t ZT = Z(T) a.s. f o r e v e r y s t o p p i n g time
T (reduce i t t o t h e p o s i t i v e case by s u b t r a c t i n g IE [ Z ( m ) IF;],
t r u n c a t e a t n t o reduce i t t o the bounded case, c o n s t r u c t t h e f u n c t i o n a l J and proceed as a t t h e end o f 22, then l e t n t e n d t o +-). There i s an analogous r e s u l t when l i m i t i n g t o bounded T. ( c ) We a d o p t t h e usual c o n d i t i o n s . I t i s n o t d i f f i c u l t t o check t h a t , f o r e v e r y f i n i t e s t o p p i n g t i m e T,
and t o deduce t h a t Z = Y v 2,.
I n p a r t i c u l a r i f Y i s r i g h t continuous,
Z i s cadlag. Here i s an i n t e r e s t i n g consequence o f t h e e x i s t e n c e o f S n e l l envelopes. I t i s due t o Mertens. 24
THEOREM.
Let Y -
be an o p t i o n a l process. F o r Y t o b e l o n g t o c l a s s ( D ) ,
i t i s necessary and s u f f i c i e n t t h a t t h e r e e x i s t an o p t i o n a l s t r o n g
m a r t i n g a l e Mt = lEIMmIFi] such t h a t sets).
IYI
5
M ( e x c e p t on evanescent
APPENDIX I - STRONG SUPERMARTINGALES
420
Proof. The condition i s obviously s u f f i c i e n t . To show t h a t i t i s necessary, we note t h a t the p o s i t i v e process I Y I belongs t o c l a s s ( D ) and hence has a Snell envelope which i t s e l f belongs to c l a s s ( 0 ) . Thus Z has a Mertens decomposition ZT = lEIAm/FT] - AT where A i s a predictable increasing process (with no r i g h t o r l e f t c o n t i n u i t y ) . I t i s then s u f f i c i e n t t o take M, = Am. 25
REMARK. Note t h a t IE[A,l = IE[Z,l = supT I E I I Y T I l and t h a t the martingale M i s the optional projection of the non-adapted process U t = Am such t h a t IE[U*l = IE[A,I. In [21 Emery gave another proof of Theorem 24 (under the usual conditions) and made i t more precise as fol 1 ows : Let Y be an optional process of c l a s s ( D ) . Then there e x i s t s a measurable process U such t h a t Y i s the optional projection of U & IE[U*l = s u p T I E [ / Y T I l .I f Y i s cadlag, U can be taken t o be cadlag. This r e s u l t may becompared t o a more d i f f i c u l t theorem of Bismut [ Z I : under the usual conditions,
E
>
Let Y be a regular cadlag process of c l a s s ( D ) . Then f o r a l l 0 there e x i s t s a measurable process U with continuous paths such
t h a t Y i s the optional projection o f U and IE[U*l
5
supTIEIIYTIl +
E.