Differentiating sigma-fields for Gaussian and shifted Gaussian processes

0.34 MB·English

by  Sébastien Darses

σ
Di(cid:27)erentiating -(cid:28)elds for Gaussian and shifted
Gaussian pro
esses
∗ † ‡
7 Sébastien Darses, Ivan Nourdin and Giovanni Pe

ati
0
0
February 2, 2008
2

n
a
J

1
3 Abstra
t
σ

] We study the notions of di(cid:27)erentiating and non-di(cid:27)erentiating -(cid:28)elds in
R
thegeneralframeworkof(possiblydrifted)Gaussianpro
esses,and
hara
terize
P
their invarian
e properties under equivalent
hanges of probability measure. As
.
h an appli
ation, we investigate the
lass of sto
hasti
derivatives asso
iated with
t
a shifted fra
tional Brownian motions. We (cid:28)nally establish
onditions for the
m
existen
e of a jointly measurable version of the di(cid:27)erentiated pro
ess, and we
[ outline a general framework for sto
hasti
embedded equations.

1
v
0
1
9
1
0
1 Introdu
tion
7
0
X X = X +
h/ tσ(XL)edtB +bteb(tXhe)dsosl,utti∈on[0,oTf ]the sto
σh,abst:iR
d→i(cid:27)Rerential equation t 0
at 0 Bs s 0 s , where PareXsuitabσlyregularfun
tions
m Rand is a staRndard Brownian motion, and denote by t the -(cid:28)eld generated by
X s ∈ [0,t]
s
: { , }. Then, the following quantity:
v
i h−1E f(X )−f(X )|PX
X t+h t t (1)
r
a ∗ (cid:2) (cid:3)
LPMA, Université Paris 6, Boîte
ourrier 188, 4 Pla
e Jussieu, 75252 Paris Cedex 05, Fran
e,
sed†arses

r.jussieu.fr
LPMA, Université Paris 6, Boîte
ourrier 188, 4 Pla
e Jussieu, 75252 Paris Cedex 05, Fran
e,
nou‡rdin

r.jussieu.fr
LSTA, Université Paris 6, Boîte
ourrier 188, 4 Pla
e Jussieu, 75252 Paris Cedex 05, Fran
e,
giovanni.pe

atigmail.
om
1
h ↓ 0

onverges (in probability and for ) for every smooth and bounded fun
tion
f
. This existen
e result is the key to de(cid:28)ne one of the
entral operators in the
L X
Lthfe(oxr)y=ofbd(xi(cid:27))udfsi(oxn)+pro1
σe(sxs)e2sd:2fth(xe)in(cid:28)nitesimal genLerator of , whi
h is given bfy
dx 2 dx2 (the domain of
ontains all regular fun
tions
X
ats above). Note that the limit in (1) is takXen
onditionally to the pastPoXf before
t
; however, due to the Markov property of , one may as well repla
e with the
σ σ{X } X b
t t
-(cid:28)eld generated by . On the other hand, under rather mild
onditions on
σ f = Id
and , one
an take in (1), so that the limit still exists and
oin
ides with the
X t
natural de(cid:28)nition of the mean velo
ity of at (the reader is referred to Nelson's
dynami
al theory of Brownian di(cid:27)usions, as developed e.g. in [8℄, for more results in
this dire
tion (cid:21) see also [1℄ for a re
ent survey).
In this paper we are
on
erned with the following question: is it possible to
X
obtain the existen
e, and to study the nature, of limits analogous to (1), when
is neither a Markov pro
ess nor a semimartingale? We will mainly fo
us on the
X f = Id

ase where is a (possibly shifted) Gaussian random pro
ess and (the
ase
f
of a non-linear and smooth will be investigated elsewhere). The subtleties of the
problemarebetterappre
iatedthroughanexample. Considerforinstan
eafra
tional
B H ∈ (1/2,1) B
Brownian motion (fBm) of Hurst index , and re
all that is neither
h−1E[B −B |B ]
t+h t t
Markovian noLr2(aΩs)emimhar↓ti0ngale(see e.g. [9℄). Thhe−n1,Et[hBe qua−nBtit|yPB]

onverges in (as ), while the quantity t+h t t does not admit
a limitin probability. More to the point, similarproperties
an be shown to hold also
B
for suitably regular solutions of sto
hasti
di(cid:27)erential equations driven by (see [3℄
for pre
ise statements and proofs).
To address the problem evoked above, we shall mainly use the notion of dif-
σ Z
(feΩr,eFnti,aPti)ng -(cid:28)eld introdσu
ed inG[3⊂℄: Fif is a pro
ess de(cid:28)nedZon atprobability spa
e
, we say that a -(cid:28)eld is di(cid:27)erentiating for at if
h−1E[Z −Z |G]
t+h t
(2)
h 0

onverDgeGsZin some topology, when tends to . WheZn it etxists, the limit Gof (2) is
t
noted ,σanditiGs
alleFdthesto
hasti
derivative of at withrespe
t to . Note
that if a sub- -(cid:28)eld of is not di(cid:27)erentGiating, one
an implement twσo (cid:16)stratHegies(cid:17)
to make (2)
onverge: either one repla
es with a di(cid:27)erentiating sub- -(cid:28)eld , or
h−1 h−α 0 < α < 1
one repla
es with with σ . GIn parti
ular, the se
ond strategGy pays
dividends when a non-di(cid:27)erentiating -(cid:28)eld is too poor, in the sense that does
σ
not
ontain su(cid:30)
ieBntly good di(cid:27)Here<nt1ia/t2ing -(cid:28)eGlds. We will see tBhat this is exsa>
tl0y
s
the
ase for a fBm with index , when is generated by for some .
Theaimofthispaperistogiveapre
ise
hara
terizationofthe
lassesofdi(cid:27)er-
σ
entiatingand non di(cid:27)erentiating -(cid:28)elds forGaussianand shiftedGaussian pro
esses.
2
We will systemati
ally investigate their mutual relations, and pay spe
ial attention
to their invarian
e properties under equivalent
hanges of probability measure.
The paper is organized as follows. In Se
tions 2 and 3 we introdu
e several
σ
notions related to the
on
ept of di(cid:27)erentiating -(cid:28)eld, and give a
hara
terization
σ
of di(cid:27)erentiating and non di(cid:27)erentiating -(cid:28)elds in a Gaussian framework. In Se
-
σ
tion 4 we prove some invarian
e properties of di(cid:27)erentiating -(cid:28)elds under equivalent

hanges of probability measure. Notably, we will be able to write an expli
it rela-
tion between the sto
hasti
derivatives asso
iated with di(cid:27)erent probabilities. We
will illustrate our results by
onsidering the example of shifted fra
tional Brownian
motions, and we shall pinpoint di(cid:27)erent behaviors when the Hurst index is, respe
-
(0,1/2) (1/2,1)
tively, in and in . In Se
tion 5 we establish fairly general
onditions,
ensuring the existen
e of a jointly measurable version of the di(cid:27)erentiated pro
ess
σ
indu
ed by a
olle
tion of di(cid:27)erentiating -(cid:28)elds. Finally, in Se
tion 6 we outline a
general framework for embedded ordinary sto
hasti
di(cid:27)erential equations (as de(cid:28)ned
in [2℄) and we analyze a simple example.
2 Preliminaries on sto
hasti
derivatives
(Z ) (Ω,F,P)
t t∈[0,T]
Let be a sto
hasti
pro
essZde∈(cid:28)nLe2d(Ωo,nFa,pPro)babilityspta
∈e[0,T] . In the
t
sequel, we will alwaysσassume that σ for everFy . It will also
σbe implHi
it that ea
h -(cid:28)Geld⊂wHe
onsider is a subG- -(cid:28)eld of σ ; analogHously, given a
-(cid:28)eld , the notation will mean that is a sub- -(cid:28)eld of . For every
t ∈ (0,T) h 6= 0 t+h ∈ (0,T)
and every su
h that , we set
Z −Z
t+h t
∆ Z = .
h t
h
τ
For the rest of the paper, we will use the leFtter as a generi
symbol to indi
ate
a topology on the
lass of real-valued and -measurable random variables. For
τ Lp
instan
e,
an be the topology indu
ed either by the a.s.
onvergen
e, or by the
p > 1 Lp

onvergen
e ( ), or by the
onvergen
e in probability, or by both a.s. and

onvergen
es, in whi
h
ases we shall write, respe
tively,
τ = a.s., τ = Lp, τ = proba, τ = Lp ⋆a.s..
Notethat,whennofurtherspe
i(cid:28)
ationisprovided, any
onvergen
e ista
itlyde(cid:28)ned
P
with respe
t to the referen
e probability measure .
t ∈ (0,T) G ⊂ F G τ Z t
De(cid:28)nition 1 Fix and let . We say that -di(cid:27)erentiates at if
E[∆ Z |G] τ h → 0
h t

onverges w.r.t. when . (3)
3
τ Z G t
In this
ase, we de(cid:28)ne the so-
alled -sto
hasti
derivative of w.r.t. at by
DGZ = τ limE[∆ Z |G].
τ t -h→0 h t (4)
G τ Z t
If the limit in (3) does not exist, we say that does not -di(cid:27)Gerentiate at .G If
τ D Z D Z
there is no risk of ambiguity on the topology , we will write t instead of τ t
to simplify the notation.
τ = a.s. τ
Remark. When (i.e., when is the topology indu
ed by a.s.
onver-
t
gen
e),equation(3)mustbeunderstoodinthefollowingsense(notethat,in(3), a
ts
(ω,h) 7→ q(ω,h)
as a (cid:28)Ωxe×d(−paεr,aεm)eteRr): there exists aq(j·o,ihn)tly measurableEap[∆pli
Zat|iGon] h,
h t
from to , su
h that (i) is a version of for every (cid:28)xed ,
Ω′ ⊂ Ω P q(ω,h)
and (ii) there exists a set , of -probability one, su
h that
onverges,
h → 0 ω ∈ Ω′ τ = Lp ⋆ a.s.
as , for every . An analogous remark applies to the
ase
p > 1
( ).
σ τ Z t M(t),τ
The set of all -(cid:28)elds that -di(cid:27)erentiate at time is denoted by Z .
M(t),τ Z
Intuitively, one
an say that the more Z is large, the more is regular at time
t {∅,Ω} ∈ M(t),τ
. For instan
e, one has
learly that Z if, and only if, the appli
ation
s 7→ E(Z ) t F ∈ M(t),τ
s is di(cid:27)erentiable at time . On the other hand, Z if, and only if,
s 7→ Z τ t
s
the random fun
tion is -di(cid:27)erentiable at time .
Before introdu
ing some further de(cid:28)nitions, we shall illustrate the above no-
L2⋆a.s. Z = (Z )
t t∈[0,T]
tionsby a simpleexampleinvolvingthe -topology. Assume that
Var(Z ) 6= 0 t ∈ (0,T] t ∈ (0,T)
t
is a GGaussian pro
ess su
h thZat fosr e∈ve(r0y,T] . FGix= σ{Z } and
s
take to be the present of at a (cid:28)xed time , that is, is the
σ Z
s
-(cid:28)eld generated by . Sin
e one has, by linear regression,
Cov(∆ Z ,Z )
E[∆ Z |G] = h t s Z ,
h t s
Var(Z )
s
G Z t
we immediately dedu
e that di(cid:27)erentiates at if, and only if,
d
Cov(Z ,Z )|
u s u=t
du
H σ H ⊂ G
exists (see also Lemma 1). Now, let be a -(cid:28)eld su
h that . Owing to the
proje
tion prin
iple, one
an write:
Cov(∆ Z ,Z )
E[∆ Z |H ] = h t s E[Z |H ],
h t s
Var(Z )
s
and we
on
lude that
4
G Z t H ⊂ G
(A) If di(cid:27)erentiates at , then it is also the
ase for any .
G Z t H ⊂ G
(B) ZIf dtoesnotdi(cid:27)eEre[nZti|aHtes] =a0t ,thenany Z teitherdDoHesZnot=d0i(cid:27)erentiates
s t
at , or (when ) di(cid:27)erentiates at with
The phenomenon appearing in (A) is quite natural, not only in a Gaussian setting,
anditisduetothewell-known properties of
onditionalexpe
tations: see Proposition
1 below. On the other hand, (B) seems strongly linked to the Gaussian assumptions
Z
we made on . We shall use (cid:28)ne arguments to generalize (B) to a non-Gaussian
framework, see Se
tions 3 and 4 below.
This example naturally leads to the subsequent de(cid:28)nitions.
t ∈ (0,T) G ⊂ F G τ Z t
De(cid:28)nDitGiZon=2cFai.xs. and let c ∈ R . If -di(cid:27)Gerτentiates at Zandtif we
τ t
have for a
ertain real , we say that -degenerates at . We
Y τ Z t σ σ{Y}
say that a random variable -degenerates at if the -(cid:28)eld generated by
Y τ Z t
-degenerates at .
DGZ ∈ L2 τ = L2 τ = L2 ⋆a.s.
τ t
If G (for instan
e when we
hoose , or ,Get
.), the
D Z Var(D Z ) = 0

onditionon τ t inthepreviousde(cid:28)nitionisobviouslyequivalentto τ t .
Z s → E(Z ) t ∈ (0,T)
s
For instan
e, if is a pro
ess su
h that is di(cid:27)erentiable in then
{∅,Ω} Z t
degenerates at .
t ∈ (0,T) G ⊂ F G τ
ZDe(cid:28)ntitioGn 3 Let τ and Z .tWesaythatHre⊂alGlydoesnoτt -di(cid:27)erentiaZte
at if does not -di(cid:27)erentiate at and if any either -degenerates
t τ Z t
at , or does not -di(cid:27)erentiate at .
σ
G , σC{oZns}ider e.g. the phenomenon des
ribed at point (B) abZove: tthe -(cid:28)eld
s
d Cov(Z ,Z )r|eally does not di(cid:27)erentiate thHe G⊂auGssian pro
ess at whenever
du u s u=t does not exist, sin
e every either does not di(cid:27)erentiate or
Z t Z = B
degenerates at . It is for instan
e the
ase when is a fra
tional Brownian
H < 1/2 s = t
motion with Hurst index Z a=ndf (t)N, +seefC(to)rNollary 2. Afn,ofthe:r[i0n,tTer]e→stinRg
t 1 1 2 2 1 2
example is given by the pro
ess , where
N ,N
1 2
are two deterministi
fun
tions and are two
entered and independent random
f t ∈ (0,T) f
1 2
variabGle,s. Aσ{ssNum,Ne th}at is di(cid:27)erentiable at Z t but that is not. This yields
1 2
that H , σ{N } ⊂dGoes not di(cid:27)erentZiates t at .DMHoZreo=verf,′(otn)eN
an easily show
1 t 1 1
that di(cid:27)erentiateGs at with Z t , whi
h is not

onstant in general. Then, although does not di(cid:27)erentiate at , it does not meet
the requirements of De(cid:28)nition 3.
5
3 Sto
hasti
derivatives and Gaussian pro
esses
In this se
tion we mainly fo
us on Gaussian pro
esses, and we shall systemati
ally
L2 L2⋆a.s.
work with the - or the -topology, whi
h are quite natural in this framework.
τ
In the sequel we will also omit the symbol in (4), as we will always indi
ate the
topology we are working with.
Our aim is to establish several relationships between di(cid:27)erentiating and (re-
σ
ally)non di(cid:27)erentiating -(cid:28)elds under Gaussian-type assumptions. However, our (cid:28)rst
result pinpoints a general simple fa
t, whi
h also holds in a non-Gaussian framework,
σ σ
that is: any sub- -(cid:28)eld of a di(cid:27)erentiating -(cid:28)eld is also di(cid:27)erentiating.
Z
ZPro∈pLo2s(iΩti,oFn,1P)Let be ats∈to
(h0a,sTti)
pro
ets∈s ((n0,oTt )ne
essarily GaussiGan⊂) sFu
h thaGt
t
L2 Zfor tevery H ⊂. LGet L2 be (cid:28)xed, aZnd lett . If
-di(cid:27)erentiates at , then any also -di(cid:27)erentiates at . Moreover, we
have DHZ = E[DGZ |H ].
t t
(5)
Proof: We
an write, by the proje
tion prin
iple and Jensen inequality:
E E(∆ Z |H )−E(DGZ |H ) 2 = E E[E(∆ Z −DGZ |G)|H ]2
h t t h t t
h(cid:0) (cid:1) i ≤ E(cid:2) E(∆ Z |G)−DGZ 2 . (cid:3)
h t t
h i
(cid:0) (cid:1)
L2 E(∆ Z |H ) E(DGZ |H ) h → 0
h t t
So, the -
onvergen
e of to as is obvious.
σ σ
On the other hand, a non di(cid:27)erentiating -(cid:28)eld may
ontain a di(cid:27)erentiating -(cid:28)eld
σ
(for instan
e, when the non di(cid:27)erentiating -(cid:28)eld is generated both by di(cid:27)erentiating
and non di(cid:27)erentiating random variables).
σ
Wenowprovidea
hara
terizationofthereallynon-di(cid:27)erentiating -(cid:28)eldsthat
are generated by some subspa
e of the (cid:28)rst Wiener
haos asso
iated with a
entered
Z H (Z) H (Z) L2
1 1
Gaussian pro
ess , noted . We re
all that is the -
losed linear ve
tor
Z t ∈ [0,T]
t
spa
e generated by random variables of the type , .
I = {1,2,...,N} N ∈ N∗ ∪{+∞} Z = (Z )
t t∈[0,T]
Theorem 1 Let , with and let be
t ∈ (0,T) {Y } H (Z)
i i∈I 1
a
entered Gaussian pro
ess. Fix , and
onsider a subset of
n ∈ I M {Y }
n i 1≤i≤n
su
h tYha=t, fσo{rYan,iy∈ I} , the
ovarian
e matrix of is invertible. Finally,
i
note . Then:
Y L2 Z t i ∈ I Y L2 Z t
i
1. If -di(cid:27)erentiates at , then, for any , -di(cid:27)erentiates at . If
N < +∞
, the
onverse also holds.
6
N < +∞ Y L2⋆a.s. Z t
2. Suppose . Then really does not -di(cid:27)erentiate at if, and
Y L2 ⋆a.s.
i
only if, any (cid:28)nite linear
ombination of the 's either -degenerates or
L2 ⋆a.s. Z t
does not -di(cid:27)erentiate at .
N = +∞ {Y }
i i∈I
3. RSu(pYpo)se that and that the sequenσ
e Yis i.i.d.. Write moreover
to indi
ate the
lass of all the sub- -(cid:28)elds of that are generated by
A × · · · × A A ∈ σ{Y } d > 1
1 d i i
re
tangles of the type , with Y , and . TLh2en⋆,a.tsh.e
previous
hara
terization holds in a weak sense: if really does not -
Z t Y L2 ⋆
i
di(cid:27)erentiate at , then every (cid:28)nite linear
ombination of the 's either
a.s. L2 ⋆a.s. Z t
-degenerates or does not -di(cid:27)erentiate at ; on the other hand, if
Y L2⋆a.s.
i
eLv2e⋆rya.(cid:28)s.nite linear
omZbinattion of the G's∈eiRth(eYr ) -dLe2g⋆enae.sr.ates or does not
-di(cid:27)erentiate at , then any either -degenerates or
L2 ⋆a.s. Z t
does not -di(cid:27)erentiate at .
R(Y ) σ
The
lass
ontains for instan
e the -(cid:28)elds of the type
G = σ{f (Y ),...,f (Y )},
1 1 d d
d > 1 N = 1
where . When , the se
ond point of Theorem 1
an be reformulated as
follows (see also the examples dis
ussed in Se
tion 2 above).
Z = (Z ) H (Z)
t t∈[0,T] 1
Corollary 1 Let t ∈ (0,T)be a
enteredYG∈auHssi(aZn)pro
ess anYd l=etσ{Y} be its
1
(cid:28)Yrst WienerL
2haos. Fix Z ,tas wellLa2s⋆a.s. , and set Y . Then,
does not -di(cid:27)erentiate at (resp. ) if, and only if, really does not
L2 Z t L2 ⋆a.s.
-di(cid:27)erentiate at (resp. ).
Z = B H ∈
(In0,p1/a2rt)i
∪u(la1r/,2w,1h)ent is a fra
tio(n0a,Tl B) rownYian=mσot{iBon}with Hurst index B
t
, is a (cid:28)xed time in and is the present of at
t H
time , we observe two distin
t behaviors, a

ording to the di(cid:27)erent values of :
H > 1/2 Y L2⋆a.s. B t
(a) IYf ⊂ Y , then -di(cid:27)erentiates at and it is also the
ase for any
0
.
H < 1/2 Y L2 ⋆a.s. B t
(b) If , then really does not -di(cid:27)erentiate at .
Indeed, (a) and (b) are dire
t
onsequen
es of Proposition 1, Corollary 1 and the
equality (t+h)2H −t2H −|h|2H
E[∆ B |B ] = B ,
h t t 2t2Hh t
whi
h is immediately veri(cid:28)ed by a Gaussian linear regression.
Note that [3, Theorem 22℄ generalizes (a) to the
ase of fra
tional di(cid:27)usions.
In the subsequent se
tions, we will propose a generalization of (a) and (b) to the
ase
of shifted fra
tional Brownian motions (cid:21) see Proposition 2.
In order to prove Theorem 1, we state an easy but quite useful lemma:
7
Z = (Z ) H (Z)
t t∈[0,T] 1
Lemma 1 Let be a
entered Gaussian pro
ess, and let be its
Y ∈ H (Z) t ∈ (0,T)
1
(cid:28)rst Wiener
haos. Fix and . Then, the following assertions
are equivalent:
Y a.s. Z t
(a) -di(cid:27)erentiates at .
Y L2 Z t
(b) -di(cid:27)erentiates at .
d Cov(Z ,Y)|
(
) ds s s=t exists and is (cid:28)nite.
P(Y = 0) < 1
If either (a), (b) or (
) are veri(cid:28)ed and , one has moreover that
Y d
DYZ = . Cov(Z ,Y)| .
t s s=t
Var(Y) ds (6)
s,t ∈ (0,T) Z L2⋆a.s. Z t
s
In parti
ular, for every , we have: -di(cid:27)erentiates at if, and
u 7→ Cov(Z ,Z ) u = t
s u
only if, is di(cid:27)erentiable at .
Y ∈ H (Z) P(Y = 0) < 1
1
OYn the otherLh2a⋆nad.s,.suppose that Z t ∈ (0,Tis)su
h that: (i) H ⊂ σ{Y},
and (iiH) does notL2⋆a.s.-di(cid:27)erentiateZ att H . Then, forEev[eYry| H ] = 0 ,
eitHher does not -di(cid:27)erentiate at , or is su
h that and
D Z = 0
t
.
Y ∈ H (Z)\{0}
1
Proof: If , we have
Cov(∆ Z ,Y)
h t
E[∆ Z | Y] = Y.
h t
Var(Y)
The
on
lusions follow.
We now turn to the proof of Theorem 1:
M n ∈ I
n
Proof: Sin
e is an invertible matrix for any , the Gram-S
hmidt orthonor-
{Y }
i i∈I
malization pro
edure
an be applied to . For this reason we may assume, for
{Y }
i i∈I
therestoftheproofandwithoutlossofgeneraNlit(y0,,t1h)atthefamily is
omposed
of i.i.d. random variables with
ommon law .
1. The (cid:28)rst impli
ation is an immediate
onsequen
e of Proposition 1. Assume
N < +∞ Y i = 1,...,N L2 Z t
i
now that and that any , , -di(cid:27)erentiates at . By
Lemma 1, we have in parti
ular that
d
Cov(Z ,Y )|
s i s=t
ds
8
i = 1,...,N
exists for any . Sin
e
N
E[∆ Z | Y ] = Cov(∆ Z ,Y )Y
h t h t i i
(7)
i=1
X
Y L2 Z t
we dedu
e that -di(cid:27)erentiates at .
Y L2 ⋆a.s. Z t
2. By de(cid:28)nition, if really does not -di(cid:27)erentiate at , then any (cid:28)nite
Y L2 ⋆a.s. L2 ⋆a.s.
i
linear
ombination of the 's either -degenerates, or does not -
Z t
di(cid:27)erentiate at .
Y L2⋆a.s.
i
Conversely, assume that aLn2y⋆(cid:28)nai.tse. linear
ombinZationtof the G's⊂eithYer -
degenerates or does not -di(cid:27)erentiate at . Let . By the
proje
tion prin
iple, we
an write:
E[∆ Z | G] = Cov(∆ Z ,Y )E[Y | G].
h t h t i i
(8)
i∈I
X
G L2 ⋆ a.s. Z t
Let us assume that -di(cid:27)erentiates at . By (8) this implies in
ω ∈ Ω
0
parti
ular that, for almost all (cid:28)xed ,
N
E[∆ Z | G](ω ) = Cov ∆ Z , a (ω )Y ,
h t 0 h t i 0 i
!
i=1
X
h → 0 a (ω ) = E[Y | G](ω )
i 0 i 0

onveXrg(eωs0)a,s N ,aw(ωher)eY L2 ⋆a.s. .ZDue tto Lemma1, weωded∈uΩ
e
that i=1 i 0 i -di(cid:27)erentiates at for almost all 0 .
X(ω0) L2 ⋆a.s. Z t
Bωy∈hyΩpothesisP, we dedu
e that -degeneDraXt(eωs0)Z at for almost all
0 t
c(ω )X.(ωB0)ut, bycL(eωm)m∈aR1, the sto
Xh(aωs0t)i
derivative Var(DnXe
(ωe0s)sZar)ily=w0rites
0 0 t
wDithX(ω0)Z = 0 . Sin
e is
entered and , we
t
dedu
e that . Thus
limCov(∆ Z ,X(ω0)) = limE[∆ Z | G](ω ) = 0
h t h t 0
h→0 h→0
ω ∈ Ω G a.s. Z t G L2
0
for almost allZ t . Thus G-deLg2e⋆near.as.tes at . ZSin
et also -
di(cid:27)erYentiates at , weL
2on⋆
alu.sd.e that Z -dtegenerates at . The proof
that really does not -di(cid:27)erentiate at is
omplete.
Y L2⋆a.s. Z t
3. Again by de(cid:28)nition, if really does not -di(cid:27)erentiate at , then any
Y L2 ⋆ a.s.
i
(cid:28)nite linear
ombination of the 's either -degenerates, or does not
L2⋆a.s. Z t
-di(cid:27)erentiate at . We shall now assume that every (cid:28)nite linear
om-
Y L2⋆a.s. L2⋆a.s.
i
bination of the 's either -degenerates or does not -di(cid:27)erentiate
9
Z t (Jm)m∈N Jm = {1,...,m}
at∪.m∈LNeJtm = I = Nbe the in
reasing sequen
e given by , so
that G ∈ R(Y. ) G L2 ⋆a.s. Z t
SupposeGthia,t G ∩σ{Y } La2nd that Z -dti(cid:27)erentiaties∈ Nat . By Propo-
i
sition 1, -di(cid:27)erentiates at , for any . But
E[∆ Z |Gi] = Cov(∆ Z ,Y )E[Y |Gi].
h t h t i i
i ∈ N
So, for any :
limCov(∆ Z ,Y ) E[Y |Gi] = 0.
h t i i
either h→0 exists, or (9)
G , G ∩σ(Y ,j ∈ J ) G ∈ R(Y )
m j m
Set , and observe that, if , then
E[Y |Gi] = E[Y |G ]
i i m
i = 1,...,m
for every . We have
E[∆ Z | G ] = Cov(∆ Z ,Y )E Y | Gi .
h t m h t i i
(10)
iX∈Jm (cid:2) (cid:3)
J G L2⋆a.s. Z t
m m
By (9), and sin
e is (cid:28)nite, we dedGu
e that G -di(cid:27)erentiates at .
m
By the same proof as in step (a) for instead of and using (10) instead of
(8), we dedu
e that X(t) , DGmZ = 0.
m t
But, from Proposition 1, we have:
DGmZ = E[DGZ |G ], m > 1.
t t m
{X(t), m ∈ N}
m
Thus {G , m ∈ N}is a (dis
rete) square integrable martingale w.r.t. the
m
(cid:28)ltration . So we
on
lude that
DGZ = lim X(t) = 0 a.s.
t m
m→∞
G L2 ⋆ a.s. Z t Y
In other words, -degenerates at . Therefore, really does not
L2 ⋆a.s. Z t
-di(cid:27)erentiate at .
N = +∞
Counterexample. In what follows we show that, if , the
onverse
of the (cid:28)rst point in the statement of Theorem 1 does not hold in general. Indeed,
{ξ : i > 1}
i
let be an in(cid:28)nite sequen
e of i.i.d.
entered standard Gaussian random
{f : i > 1}
i
variables. Let be a
olle
tion of deterministi
fun
tions belonging to
L2([0,1],dt)
, su
h that the following hold:
10