Differentiating $\sigma$-fields for Gaussian and shifted Gaussian processes

arXiv:math/0701910v1 [math.PR] 31 Jan 2007

Dierentiating

σ

-elds for Gaussian and shifted

Gaussian pro esses

Sébastien Darses

∗

, Ivan Nourdin

†

and Giovanni Pe ati

‡

February 2, 2008

Abstra t

We study the notions of dierentiating and non-dierentiating

σ

-elds in thegeneralframeworkof(possiblydrifted)Gaussianpro esses,and hara terize their invarian e propertiesunderequivalent hangesofprobabilitymeasure. As anappli ation,weinvestigatethe lassofsto hasti derivativesasso iatedwith shifted fra tional Brownian motions. We nally establish onditions for the existen e of a jointly measurable version of the dierentiated pro ess, and we outline ageneral frameworkfor sto hasti embedded equations.

1 Introdu tion

Let

X

be the solution of the sto hasti dierential equation

X

t

= X

0

+

R

t

0

σ(X

s

)dB

s

+

R

t

0

b(X

s

)ds, t ∈ [0, T ]

,where

σ, b : R → R

aresuitablyregularfun tions and

B

is a standard Brownian motion, and denote by

P

X

t

the

σ

-eld generated by {

X

s

,

s ∈ [0, t]

}. Then,the following quantity:

h

−1

E

f(X

t+h

) − f (X

t

)|P

t

X



(1)

∗

LPMA, UniversitéParis6,Boîte ourrier 188,4Pla e Jussieu, 75252ParisCedex 05, Fran e, sedarses r.jussieu.fr

†

LPMA, UniversitéParis6,Boîte ourrier 188,4Pla e Jussieu, 75252ParisCedex 05, Fran e, nourdin r.jussieu.fr

‡

onverges (in probability and for

h ↓ 0

) for every smooth and bounded fun tion

f

. This existen e result is the key to dene one of the entral operators in the theory of diusion pro esses: the innitesimal generator

L

of

X

, whi h is given by

Lf (x) = b(x)

_dx

df

(x) +

1

₂

σ(x)

2 d

2

f

dx

2

(x)

(the domain of

L

ontains allregular fun tions

f

as above). Note that the limit in (1) is taken onditionallyto the past of

X

before

t

; however, due to the Markov property of

X

, one may as well repla e

P

X

t

with the

σ

-eld

σ{X

t

}

generatedby

X

t

. Onthe otherhand,underrathermild onditionson

b

and

σ

,one an take

f = Id

in(1), sothat the limitstillexists and oin ideswith the natural denition of the mean velo ity of

X

at

t

(the reader is referred to Nelson's dynami altheory of Brownian diusions,asdeveloped e.g. in [8℄, formore results in this dire tion see also[1℄for a re ent survey).

In this paper we are on erned with the following question: is it possible to obtain the existen e, and to study the nature, of limits analogous to (1), when

X

is neither a Markov pro ess nor a semimartingale? We will mainly fo us on the ase where

X

is a (possibly shifted) Gaussian random pro ess and

f = Id

(the ase of a non-linear and smooth

f

will be investigated elsewhere). The subtleties of the problemarebetterappre iatedthroughanexample. Considerforinstan eafra tional Brownian motion (fBm)

B

of Hurst index

H ∈ (1/2, 1)

, and re all that

B

is neither Markoviannorasemimartingale(seee.g. [9℄). Then,thequantity

h

−1

_E[B

t+h

−B

t

|B

t

]

onvergesin

L

2

_(Ω)

(as

h ↓ 0

),whilethequantity

h

−1

_E[B

t+h

− B

t

|P

t

B

]

doesnotadmit alimitinprobability. More tothe point,similarproperties an beshowntoholdalso for suitably regularsolutions of sto hasti dierentialequations driven by

B

(see [3℄ forpre ise statementsand proofs).

To address the problem evoked above, we shall mainly use the notion of dif-ferentiating

σ

-eldintrodu edin [3℄: if

Z

is apro ess dened on a probability spa e

(Ω, F , P)

, wesay that a

σ

-eld

G

⊂ F

is dierentiating for

Z

at

t

if

h

−1

E [Z

t+h

− Z

t

|G ]

(2)

onverges in some topology, when

h

tends to

0

. When it exists, the limit of (2) is noted

D

G

Z

t

,anditis alledthesto hasti derivative of

Z

at

t

withrespe tto

G

. Note thatifa sub-

σ

-eld

G

of

F

isnot dierentiating,one an implementtwostrategies tomake (2) onverge: either one repla es

G

with a dierentiatingsub-

σ

-eld

H

, or one repla es

h

−1

with

h

−α

totheir invarian e properties under equivalent hangesof probability measure. The paper is organized as follows. In Se tions 2 and 3 we introdu e several notions related to the on ept of dierentiating

σ

-eld, and give a hara terization of dierentiating and non dierentiating

σ

-elds in a Gaussian framework. In Se -tion4weprovesomeinvarian epropertiesof dierentiating

σ

-eldsunderequivalent hanges of probability measure. Notably, we will be able to write an expli it rela-tion between the sto hasti derivatives asso iated with dierent probabilities. We will illustrate our results by onsidering the example of shifted fra tional Brownian motions, and we shall pinpoint dierent behaviors when the Hurst index is, respe -tively,in

(0, 1/2)

and in

(1/2, 1)

. In Se tion5 weestablish fairlygeneral onditions, ensuring the existen e of a jointly measurable version of the dierentiated pro ess indu ed by a olle tion of dierentiating

σ

-elds. Finally, in Se tion 6 we outline a generalframework forembedded ordinary sto hasti dierentialequations (asdened in[2℄) and we analyze a simpleexample.

2 Preliminaries on sto hasti derivatives

Let

(Z

t

)

t∈[0,T ]

beasto hasti pro essdened onaprobabilityspa e

(Ω, F , P)

. Inthe sequel, we willalways assume that

Z

t

∈ L

2

_{(Ω, F , P)}

for every

t ∈ [0, T ]

. It willalso be impli it that ea h

σ

-eld we onsider is a sub-

σ

-eld of

F

; analogously, given a

σ

-eld

H

, the notation

G

⊂ H

will mean that

G

is a sub-

σ

-eld of

H

. For every

t ∈ (0, T )

and every

h 6= 0

su h that

t + h ∈ (0, T )

, we set

∆

h

Z

t

=

Z

t+h

− Z

t

h

.

For the rest of the paper, we will use the letter

τ

as a generi symbol to indi ate a topology on the lass of real-valued and

F

-measurable random variables. For instan e,

τ

an be the topologyindu ed either by the a.s. onvergen e, or by the

L

p

onvergen e (

p > 1

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

L

p

onvergen es, in whi h ases we shall write, respe tively,

τ = a.s.,

τ = L

p

,

τ = proba,

τ = L

p

⋆ a.s..

Notethat,whennofurtherspe i ationisprovided,any onvergen eista itlydened with respe t tothe referen e probability measure

P

.

Denition 1 Fix

t ∈ (0, T )

andlet

G

⊂ F

. Wesay that

G

τ

-dierentiates

Z

at

t

if

Inthis ase, we dene the so- alled

τ

-sto hasti derivative of

Z

w.r.t.

G

at

t

by

D

_τ

G

Z

t

= τ

-

lim

h→0

E[∆

h

Z

t

| G ].

(4) If the limit in (3) does not exist, we say that

G

does not

τ

-dierentiate

Z

at

t

. If there is no risk of ambiguity on the topology

τ

, we will write

D

G

Z

t

instead of

D

G

τ

Z

t

to simplifythe notation.

Remark. When

τ = a.s.

(i.e., when

τ

isthe topologyindu ed by a.s. onver-gen e),equation(3)mustbeunderstoodinthefollowingsense(notethat,in(3),

t

a ts asaxed parameter): there exists a jointly measurableappli ation

(ω, h) 7→ q(ω, h)

, from

Ω × (−ε, ε)

to

R

,su hthat(i)

q(·, h)

isaversionof

E[∆

h

Z

t

| G ]

foreveryxed

h

, and (ii) there exists a set

Ω

′

_{⊂ Ω}

, of

P

-probability one, su h that

q(ω, h)

onverges, as

h → 0

, for every

ω ∈ Ω

′

. An analogous remark applies to the ase

τ = L

p

_{⋆ a.s.}

(

p > 1

).

The set of all

σ

-elds that

τ

-dierentiate

Z

at time

t

is denoted by

M

(t),τ

Z

.

Intuitively, one an say that the more

M

(t),τ

Z

is large, the more

Z

is regularat time

t

. For instan e, one has learly that

{∅, Ω} ∈ M

(t),τ

Z

if, and only if, the appli ation

s 7→ E(Z

s

)

is dierentiable at time

t

. On the other hand,

F

∈ M

(t),τ

Z

if, and onlyif, the randomfun tion

s 7→ Z

s

is

τ

-dierentiable attime

t

.

Before introdu ing some further denitions, we shall illustrate the above no-tionsbyasimpleexampleinvolvingthe

L

2

_⋆a.s.

-topology. Assumethat

Z = (Z

t

)

t∈[0,T ]

is a Gaussian pro ess su h that

Var(Z

t

) 6= 0

for every

t ∈ (0, T ]

. Fix

t ∈ (0, T )

and take

G

to be the present of

Z

at a xed time

s ∈ (0, T ]

, that is,

G

= σ{Z

s

}

is the

σ

-eld generated by

Z

s

. Sin e one has, by linear regression,

E[∆

h

Z

t

| G ] =

Cov(∆

h

Z

t

, Z

s

)

Var(Z

s

)

Z

s

,

weimmediatelydedu e that

G

dierentiates

Z

at

t

if, and only if,

d

du

Cov(Z

u

, Z

s

)|

u=t

exists (seealsoLemma1). Now, let

H

be a

σ

-eld su hthat

H

⊂ G

. Owingtothe proje tion prin iple, one an write:

E[∆

h

Z

t

|H ] =

Cov(∆

h

Z

t

, Z

s

)

Var(Z

s

)

E[Z

s

|H ],

(A) If

G

dierentiates

Z

at

t

, then it isalsothe ase for any

H

⊂ G

.

(B) If

G

doesnotdierentiates

Z

at

t

,thenany

H

⊂ G

eitherdoesnotdierentiates

Z

at

t

, or (when

E[Z

s

|H ] = 0

)dierentiates

Z

at

t

with

D

H

Z

t

= 0

The phenomenon appearing in (A) is quite natural, not only in a Gaussian setting, anditisduetothewell-knownpropertiesof onditionalexpe tations: seeProposition 1below. On the other hand, (B) seems strongly linked tothe Gaussian assumptions we made on

Z

. We shall use ne arguments to generalize (B) to a non-Gaussian framework, see Se tions 3 and 4 below.

This examplenaturally leads tothe subsequent denitions.

Denition 2 Fix

t ∈ (0, T )

and let

G

⊂ F

. If

G

τ

-dierentiates

Z

at

t

and if we have

D

G

τ

Z

t

= c a.s.

for a ertain real

c ∈ R

, we say that

G

τ

-degenerates

Z

at

t

. We say that a random variable

Y τ

-degenerates

Z

at

t

if the

σ

-eld

σ{Y }

generated by

Y τ

-degenerates

Z

at

t

.

If

D

G

τ

Z

t

∈ L

2

(for instan e when we hoose

τ = L

2

, or

τ = L

2

_{⋆ a.s.}

, et .), the onditionon

D

G

τ

Z

t

inthepreviousdenitionisobviouslyequivalentto

Var(D

G

τ

Z

t

) = 0

. Forinstan e, if

Z

is apro ess su hthat

s → E(Z

s

)

isdierentiablein

t ∈ (0, T )

then

{∅, Ω}

degenerates

Z

at

t

.

Denition 3 Let

t ∈ (0, T )

and

G

⊂ F

. Wesaythat

G

reallydoesnot

τ

-dierentiate

Z

at

t

if

G

does not

τ

-dierentiate

Z

at

t

and ifany

H

⊂ G

either

τ

-degenerates

Z

at

t

, or does not

τ

-dierentiate

Z

at

t

.

Consider e.g. the phenomenon des ribed at point (B) above: the

σ

-eld

G

_{, σ{Z}

_s

_}

really does not dierentiate the Gaussian pro ess

Z

at

t

whenever

d

du

Cov(Z

u

, Z

s

)|

u=t

doesnot exist,sin e every

H

⊂ G

either doesnot dierentiate or degenerates

Z

at

t

. It is for instan e the ase when

Z = B

is a fra tional Brownian motion with Hurst index

H < 1/2

and

s = t

, see Corollary 2. Another interesting example is given by the pro ess

Z

t

= f

1

(t)N

1

+ f

2

(t)N

2

, where

f

1

, f

2

: [0, T ] → R

aretwodeterministi fun tionsand

N

1

, N

2

aretwo enteredandindependentrandom variables. Assumethat

f

1

isdierentiableat

t ∈ (0, T )

but that

f

2

isnot. Thisyields that

G

, σ{N

1

, N

2

}

does not dierentiates

Z

at

t

. Moreover, one an easily show that

H

, σ{N

1

} ⊂ G

dierentiates

Z

at

t

with

D

H

In this se tion we mainly fo us on Gaussian pro esses, and we shall systemati ally workwiththe

L

2

-orthe

L

2

_{⋆ a.s.}

-topology,whi hare quitenaturalinthisframework. In the sequel we will also omit the symbol

τ

in (4), as we will always indi ate the topology weare working with.

Our aim is to establish several relationships between dierentiating and (re-ally)nondierentiating

σ

-eldsunderGaussian-typeassumptions. However, ourrst resultpinpointsageneralsimplefa t,whi halsoholds inanon-Gaussianframework, that is: any sub-

σ

-eld of a dierentiating

σ

-eld is alsodierentiating.

Proposition 1 Let

Z

be a sto hasti pro ess (not ne essarily Gaussian) su h that

Z

t

∈ L

2

(Ω, F , P)

for every

t ∈ (0, T )

. Let

t ∈ (0, T )

be xed, and let

G

⊂ F

. If

G

L

2

-dierentiates

Z

at

t

, thenany

H

⊂ G

also

L

2

-dierentiates

Z

at

t

. Moreover, we have

D

H

Z

t

= E[D

G

Z

t

|H ].

(5)

Proof: We an write, by the proje tion prin ipleand Jensen inequality:

E

h

E(∆

h

Z

t

| H ) − E(D

G

Z

t

| H )

2

i

= E

E[E(∆

h

Z

t

− D

G

Z

t

| G ) | H ]

2



≤ E

h

E(∆

h

Z

t

| G ) − D

G

Z

t

2

i

.

So, the

L

2

- onvergen e of

E(∆

h

Z

t

| H )

to

E(D

G

Z

t

| H )

as

h → 0

is obvious.

On the other hand, a non dierentiating

σ

-eld may ontain a dierentiating

σ

-eld (forinstan e, whenthe nondierentiating

σ

-eld isgenerated both bydierentiating and non dierentiating randomvariables).

Wenowprovidea hara terizationofthereallynon-dierentiating

σ

-eldsthat are generated by some subspa e of the rst Wiener haos asso iated with a entered Gaussianpro ess

Z

,noted

H

1

(Z)

. Were allthat

H

1

(Z)

isthe

L

2

- losedlinearve tor spa egenerated by randomvariables of the type

Z

t

,

t ∈ [0, T ]

.

Theorem 1 Let

I = {1, 2, . . . , N}

, with

N ∈ N

∗

_{∪ {+∞}}

and let

Z = (Z

t

)

t∈[0,T ]

be a entered Gaussian pro ess. Fix

t ∈ (0, T )

, and onsider a subset

{Y

i

}

i∈I

of

H

1

(Z)

su hthat, forany

n ∈ I

, the ovarian ematrix

M

n

of

{Y

i

}

1≤i≤n

isinvertible. Finally, note

Y

= σ{Y

i

, i ∈ I}

. Then:

1. If

Y

L

2

-dierentiates

Z

at

t

, then, for any

i ∈ I

,

Y

i

L

2

-dierentiates

Z

at

t

. If

2. Suppose

N < +∞

. Then

Y

really does not

L

2

_{⋆ a.s.}

-dierentiate

Z

at

t

if, and only if, any nite linear ombination of the

Y

i

's either

L

2

_{⋆ a.s.}

-degenerates or does not

L

2

_{⋆ a.s.}

-dierentiate

Z

at

t

.

3. Suppose that

N = +∞

and that the sequen e

{Y

i

}

i∈I

is i.i.d.. Write moreover

R(Y )

to indi ate the lass of all the sub-

σ

-elds of

Y

that are generated by re tangles of the type

A

1

× · · · × A

d

, with

A

i

∈ σ{Y

i

}

, and

d > 1

. Then, the previous hara terization holds in a weak sense: if

Y

really does not

L

2

_{⋆ a.s.}

-dierentiate

Z

at

t

, then every nite linear ombination of the

Y

i

's either

L

2

_⋆

a.s.

-degenerates or does not

L

2

_{⋆ a.s.}

-dierentiate

Z

at

t

; on the other hand, if every nitelinear ombinationofthe

Y

i

'seither

L

2

_{⋆ a.s.}

-degeneratesordoes not

L

2

_{⋆ a.s.}

-dierentiate

Z

at

t

, then any

G

∈ R(Y )

either

L

2

_{⋆ a.s.}

-degenerates or does not

L

2

_{⋆ a.s.}

-dierentiate

Z

at

t

.

The lass

R

(Y )

ontains for instan e the

σ

-elds of the type

G

_{= σ{f}

₁

_(Y

₁

_{), ..., f}

_d

_(Y

_d

_)},

where

d > 1

. When

N = 1

, the se ond point of Theorem 1 an be reformulated as follows (see also the examples dis ussed inSe tion 2above).

Corollary 1 Let

Z = (Z

t

)

t∈[0,T ]

be a entered Gaussian pro essand let

H

1

(Z)

be its rst Wiener haos. Fix

t ∈ (0, T )

, as wellas

Y ∈ H

1

(Z)

, and set

Y

= σ{Y }

. Then,

Y

does not

L

2

-dierentiate

Z

at

t

(resp.

L

2

_{⋆ a.s.}

) if, and only if,

Y

really does not

L

2

-dierentiate

Z

at

t

(resp.

L

2

_{⋆ a.s.}

).

In parti ular, when

Z = B

is a fra tional Brownian motion with Hurst index

H ∈

(0, 1/2) ∪ (1/2, 1)

,

t

is a xed time in

(0, T )

and

Y

= σ{B

t

}

is the present of

B

at time

t

, weobserve two distin tbehaviors, a ording tothe dierent values of

H

:

(a) If

H > 1/2

,then

Y

L

2

_{⋆ a.s.}

-dierentiates

B

at

t

and itis alsothe ase forany

Y

₀

_{⊂ Y}

.

(b) If

H < 1/2

, then

Y

really does not

L

2

_{⋆ a.s.}

-dierentiate

B

at

t

.

Indeed, (a) and (b) are dire t onsequen es of Proposition 1, Corollary 1 and the equality

E[∆

h

B

t

|B

t

] =

(t + h)

2H

_{− t}

2H

_{− |h|}

2H

2 t

2H

_h

B

t

,

whi h isimmediatelyveried by a Gaussianlinear regression.

Notethat [3, Theorem 22℄ generalizes (a) to the ase of fra tional diusions. Inthesubsequent se tions,wewillproposeageneralizationof(a)and(b) tothe ase of shifted fra tionalBrownianmotions see Proposition2.

Lemma 1 Let

Z = (Z

t

)

t∈[0,T ]

be a entered Gaussian pro ess, and let

H

1

(Z)

be its rst Wiener haos. Fix

Y ∈ H

1

(Z)

and

t ∈ (0, T )

. Then, the following assertions are equivalent:

(a)

Y a.s.

-dierentiates

Z

at

t

. (b)

Y L

2

-dierentiates

Z

at

t

. ( )

d

ds

Cov(Z

s

, Y )|

s=t

exists and is nite.

If either (a), (b) or ( ) are veried and

P (Y = 0) < 1

, one has moreoverthat

D

Y

Z

t

=

Y

Var(Y )

.

d

ds

Cov(Z

s

, Y )|

s=t

.

(6)

Inparti ular, for every

s, t ∈ (0, T )

, we have:

Z

s

L

2

_{⋆ a.s.}

-dierentiates

Z

at

t

if, and only if,

u 7→ Cov(Z

s

, Z

u

)

is dierentiableat

u = t

.

On the other hand, suppose that

Y ∈ H

1

(Z)

is su h that: (i)

P (Y = 0) < 1

, and(ii)

Y

doesnot

L

2

_⋆a.s.

-dierentiate

Z

at

t ∈ (0, T )

. Then, forevery

H

⊂ σ{Y }

, either

H

does not

L

2

_{⋆ a.s.}

-dierentiate

Z

at

t

, or

H

issu hthat

E [Y | H ] = 0

and

D

H

Z

t

= 0

. Proof: If

Y ∈ H

1

(Z) \ {0}

, we have

E [∆

h

Z

t

| Y ] =

Cov(∆

h

Z

t

, Y )

Var(Y )

Y.

The on lusions follow.

We nowturn to the proof of Theorem 1:

Proof: Sin e

M

n

isan invertiblematrix for any

n ∈ I

, the Gram-S hmidt orthonor-malizationpro edure an be applied to

{Y

i

}

i∈I

. Forthis reason we may assume,for therestoftheproofandwithoutlossofgenerality,thatthefamily

{Y

i

}

i∈I

is omposed of i.i.d. random variables with ommonlaw

N

(0, 1)

.

1. The rst impli ation is an immediate onsequen e of Proposition 1. Assume now that

N < +∞

and that any

Y

i

,

i = 1, . . . , N

,

L

2

-dierentiates

Z

at

t

. By Lemma 1, wehave inparti ular that

d

exists for any

i = 1, . . . , N

. Sin e

E [∆

h

Z

t

| Y ] =

N

X

i=1

Cov(∆

h

Z

t

, Y

i

) Y

i

(7) wededu e that

Y

L

2

-dierentiates

Z

at

t

. 2. By denition, if

Y

reallydoes not

L

2

_{⋆ a.s.}

-dierentiate

Z

at

t

, then any nite linear ombinationof the

Y

i

's either

L

2

_{⋆ a.s.}

-degenerates, ordoes not

L

2

_{⋆ a.s.}

-dierentiate

Z

at

t

.

Conversely,assumethatanynitelinear ombinationof the

Y

i

'seither

L

2

_{⋆ a.s.}

-degenerates or does not

L

2

_{⋆ a.s.}

-dierentiate

Z

at

t

. Let

G

⊂ Y

. By the proje tion prin iple, we an write:

E [∆

h

Z

t

| G ] =

X

i∈I

Cov (∆

h

Z

t

, Y

i

) E [Y

i

| G ] .

(8)

Let us assume that

G

L

2

_{⋆ a.s.}

-dierentiates

Z

at

t

. By (8) this implies in parti ular that, for almost allxed

ω

0

∈ Ω

,

E [∆

h

Z

t

| G ] (ω

0

) = Cov

∆

h

Z

t

,

N

X

i=1

a

i

(ω

0

) Y

i

!

,

onvergesas

h → 0

,where

a

i

(ω

0

) = E [Y

i

| G ] (ω

0

)

. DuetoLemma1,wededu e that

X

(ω

0

)

, P

N

i=1

a

i

(ω

0

) Y

i

L

2

⋆ a.s.

-dierentiates

Z

at

t

for almost all

ω

0

∈ Ω

. By hypothesis,wededu e that

X

(ω

0

)

_L

2

_{⋆ a.s.}

-degenerates

Z

at

t

foralmost all

ω

0

∈ Ω

. But, byLemma 1,the sto hasti derivative

D

X

(ω0)

Z

t

ne essarilywrites

c(ω

0

)X

(ω

0

)

with

c(ω

0

) ∈ R

. Sin e

X

(ω

0

)

is entered and

Var(D

X

(ω0)

Z

t

) = 0

, we dedu e that

D

X

(ω0)

Z

t

= 0

. Thus

lim

h→0

Cov(∆

h

Z

t

, X

(ω

0

)

_{) = lim}

h→0

E [∆

h

Z

t

| G ] (ω

0

) = 0

for almost all

ω

0

∈ Ω

. Thus

G

a.s.

-degenerates

Z

at

t

. Sin e

G

also

L

2

-dierentiates

Z

at

t

,we on lude that

G

L

2

_{⋆ a.s.}

-degenerates

Z

at

t

. The proof that

Y

really doesnot

L

2

_{⋆ a.s.}

-dierentiate

Z

at

t

is omplete. 3. Againby denition, if

Y

reallydoes not

L

2

_{⋆ a.s.}

-dierentiate

Z

at

t

,then any nite linear ombination of the

Y

i

's either

L

2

_{⋆ a.s.}

-degenerates, or does not

L

2

_{⋆ a.s.}

-dierentiate

Z

at

t

. Weshall now assumethat every nite linear om-binationof the

Y

i

'seither

L

2

_{⋆ a.s.}

-degeneratesordoesnot

L

2

_{⋆ a.s.}

Z

at

t

. Let

(J

m

)

m∈N

be the in reasing sequen e given by

J

m

= {1, . . . , m}

, so that

∪

m∈N

J

m

= I = N

.

Suppose that

G

∈ R(Y )

and that

G

L

2

_{⋆ a.s.}

-dierentiates

Z

at

t

. By Propo-sition 1,

G

i

_{, G ∩ σ{Y}

i

} L

2

-dierentiates

Z

at

t

,for any

i ∈ N

. But

E[∆

h

Z

t

|G

i

] = Cov(∆

h

Z

t

, Y

i

)E[Y

i

|G

i

].

So, forany

i ∈ N

:

either

lim

h→0

Cov(∆

h

Z

t

, Y

i

)

exists, or

E[Y

i

|G

i

_{] = 0.}

(9)

Set

G

m

, G ∩ σ(Y

j

, j ∈ J

m

)

, and observe that,if

G

∈ R(Y )

, then

E[Y

i

|G

i

] = E[Y

i

|G

m

]

for every

i = 1, ..., m

. We have

E [∆

h

Z

t

| G

m

] =

X

i∈J

m

Cov (∆

h

Z

t

, Y

i

) E

Y

i

| G

i

 .

(10)

By (9), and sin e

J

m

is nite,we dedu e that

G

m

L

2

_{⋆ a.s.}

-dierentiates

Z

at

t

. By the same proof as instep (a) for

G

m

insteadof

G

and using (10) insteadof (8), wededu e that

X

(t)

m

, D

G

m

Z

t

= 0.

But, from Proposition 1, wehave:

D

G

m

Z

t

= E[D

G

Z

t

|G

m

],

m > 1.

Thus

{X

(t)

m

, m ∈ N}

is a (dis rete) square integrable martingale w.r.t. the ltration

{G

m

, m ∈ N}

. So we on lude that

D

G

Z

t

= lim

m→∞

X

(t)

m

= 0 a.s.

In other words,

G

L

2

_{⋆ a.s.}

-degenerates

Z

at

t

. Therefore,

Y

really does not

L

2

_{⋆ a.s.}

-dierentiate

Z

at

t

.

Counterexample. In what follows we show that, if

N = +∞

, the onverse of the rst point in the statement of Theorem 1 does not hold in general. Indeed, let

{ξ

i

: i > 1}

be an innite sequen e of i.i.d. entered standard Gaussian random variables. Let

{f

i

: i > 1}

be a olle tion of deterministi fun tions belonging to

L

2

_{([0, 1] , dt)}

 for every

i > 1

,

f

i

(t)

isdierentiablein

t

for every

t ∈ [0, 1]

;  there exists

A ∈ (0, +∞)

su h that, for every

t ∈ [0, 1]

,

P

+∞

i=1

f

i

(t)

2

< A

.

Then,wemay applytheIt-Nisiotheorem(see[6℄) todedu e thatthereexists aGaussian pro ess

{Z

t

: t ∈ [0, 1]}

su h that,a.s.-

P

,

lim

N →+∞

_t∈[0,1]

sup











Z

t

−

N

X

i=1

ξ

i

f

i

(t)











= 0

.

Now suppose that the paths of

Z

are a.s. not-dierentiable for every

t

. Then, by setting

Y

= σ(ξ

i

, i > 1)

,we obtainthat

Y

doesnot

L

2

_{⋆ a.s.}

dierentiate

Z

atevery

t

, although, for every

i > 1

and every

t ∈ [0, 1]

,

ξ

i

L

2

_{⋆ a.s.}

dierentiates

Z

at

t

. As anexample,one an onsider the ase

f

i

(t) =

Z

t

0

e

i

(x) dx

,

i > 1

,

where

{e

i

: i > 1}

isanyorthonormalbasis of

L

2

_{([0, 1] , dx)}

, sothatthe limitpro ess

Z

is a standard Brownian motion. See also Kadota [7℄ for several related results, on erning the dierentiability of sto hasti pro esses admitting a Karhunen-Loève type expansion.

4 Invarian e properties of dierentiating

σ

-elds and sto hasti deriva-tives under equivalent hanges of probability

Let

Z

bea Gaussianpro ess, and let

G

⊂ F

be dierentiatingfor

Z

. In this se tion we establish onditions on

Z

and

G

, ensuring that

G

is still dierentiating for

Z

after an equivalent hange of probability measure. As anti ipated, this result will beused to study the lass ofdierentiating

σ

-elds asso iatedwith drifted Gaussian pro esses. Roughlyspeaking,wewillshowthatunderadequate onditionsone an study the sto hasti derivativesof adrifted Gaussianpro ess by rst eliminatingthe driftthrough a Girsanov-type transformation. We on entrate on

σ

-elds generated byasingle randomvariable. Toa hieve ourgoals wewilluse several te hniques from Malliavin al ulus, as for instan e those developed by H. Föllmer(see [5, Se . 4℄) in orderto ompute the ba kward driftof a non-MarkovianBrownian diusion.

Let

Z = (Z

t

)

t∈[0,T ]

be a square integrable sto hasti pro ess dened on a probabilityspa e

(Ω, F , P)

. Weassumethat,underanequivalentprobability

Q

∼ P

,

Z

isa entered Gaussianpro ess (sothat,inparti ular,

Z

t

∈ L

2

_{(P) ∩ L}

2

_(Q)

forevery

under

Q

(this means that the losure is in

L

2

_(Q)

), anoni ally represented as an isonormal Gaussian pro ess with respe t to a separable Hilbert spa e

(H, h·, ·i

H

)

. In parti ular: (i) the spa e

H

ontains the set

E

of step fun tions on

[0, T ]

, (ii) the ovarian e fun tion of

Z

under

Q

is given by

ρ

Q

(s, t) = h1

[0,s]

, 1

[0,t]

i

H

, and (iii) the s alarprodu t

h·, ·i

H

veries the general relation:

∀ h, h

′

∈ H,

hh, h

′

i

H

= E

Q

[Z(h)Z(h

′

)]

(11)

(notethat,given

Z

,theproperties(i)-(iii) ompletely hara terizethepair

(H, h·, ·i

H

)

). Wedenoteby

D

the Malliavinderivativeasso iated withthe pro ess

Z

under

Q

(the readerisreferred to [9℄ for moredetails about these notions). The followingresult is an extension of Theorem 22 in [3℄ to a general Gaussian setting. Note that, in the following statements, we willex lusively referto the

L

2

topology.

Theorem 2 Fix

t ∈ (0, T )

and sele t

g ∈ H

su h that

h1

[0,t]

, gi

H

6= 0

. We write

η

to indi ate the Radon-Nikodym derivative of

Q

with respe t to

P

(that is,

dQ = η dP

), and we assume that

η

has the form

η = exp(−ζ)

, for some random variable

ζ

for whi h

Dζ

exists. Suppose that

µ

t

, lim

h→0

h

−1

_h1

[t,t+h]

, Dζi

H

exists in the

L

2

topology

.

(12)

Then,

Z(g) L

2

-dierentiates

Z

at

t

under

P

if, and only if,

Z(g) L

2

-dierentiates

Z

at

t

under

Q

, that is, if, and only if,

d

du

hg, 1

[0,u]

i

H

|

u=t

=

d

du

Cov

Q

(Z(g), Z

u

)|

u=t

exists. (13) Moreover, 1. If

Z(g) L

2

-dierentiates

Z

at

t

under

Q

, then

D

_P

Z(g)

Z

t

=

|g|

2

H

E

P

[Z

t

− h1

[0,t]

, Dζi

H

|Z(g)]

Z(g) hg, 1

[0,t]

i

H

D

_Q

Z(g)

Z

t

+ E

P

[µ

t

|Z(g)].

(14) 2. If

Z(g)

does not

L

2

-dierentiate

Z

at

t

under

Q

, then

H

⊂ σ{Z(g)}

dier en-tiates

Z

at

t

with respe t to

P

if, and only if,

E

P

_[Z

t

− h1

[0,t]

, Dζi

H

|H ] = 0.

In this ase,

D

H

Remark. Sin e

Z

is Gaussian under

Q

, Corollary 1implies that

Z(g)

is not dierentiating for

Z

at

t

w.r.t.

Q

if, and only if,

Z(g)

is really not dierentiating w.r.t.

Q

. Point2 inTheorem 2 shows that this double impli ationdoes not hold, in general,undertheequivalentprobability

P

. Indeed,evenif

Z(g)

doesnotdierentiate

Z

under

P

(and thereforeunder

Q

),one may have that thereexists adierentiating

H

_{⊂ σ{Z(g)}}

su h that

D

H

P

Z

t

is non-deterministi . Observe, however, that

D

H

P

Z

t

isfor ed to have the parti ularform

D

H

P

Z

t

= E

P

[µ

t

|H ]

. Proof: Let

ξ ∈ L

2

_{(P) ∩ L}

2

_(Q)

and

A ∈ G ⊂ F

. The relation

Z

A

ξdP =

Z

A

E

P

_{[ξ|G ]dP}

implies

Z

A

E

Q

_[ξη

−1

_{|G ]dQ =}

Z

A

ξη

−1

dQ =

Z

A

E

P

_{[ξ|G ]η}

−1

_dQ

=

Z

A

E

P

_{[ξ|G ]E}

Q

_η

−1

_|G

_{ dQ.}

Thus:

E

P

_{[ξ|G ] =}

E

Q

[ξη

−1

|G ]

E

Q

_[η

−1

_{|G ]}

,

(15)

from whi h we dedu e that the study of

E

P

_[∆

h

Z

t

|Z(g)]

an be redu ed to that of

E

Q

_[η

−1

_∆

h

Z

t

|Z(g)]

. Let

φ ∈ C

1

b

(R)

. We have

E

Q

_[(Z

t+h

− Z

t

)η

−1

φ(Z(g))] = E

Q

[h1

[t,t+h]

, D(η

−1

φ(Z(g)))i

H

]

= E

Q

_[φ(Z(g))η

−1

_h1

[t,t+h]

, Dζi

H

]

+h1

[t,t+h]

, gi

H

E

Q

[η

−1

φ

′

(Z(g))].

Byusing ananalogous de ompositionfor

E

Q

_[Z

t

η

−1

φ(Z(g))]

, we an alsowrite:

whereas

E

P

_[∆

h

Z

t

|Z(g)]

equals the following expression:

E

P

_[Z

t

− h1

[0,t]

, Dζi

H

|Z(g)]

1

[t,t+h]

, g



H

hh1

[0,t]

, gi

H

+ h

−1

1

[t,t+h]

, E

P

[Dζ|Z(g)]



H

.

(17)

Now, by assumption (12) and thanks toProposition 1,we have that

lim

h→0

h

−1

₁

[t,t+h]

, E

P

[Dζ|Z(g)]



H

= E

P

[µ

t

|Z(g)]

inthe

L

2

topology

.

Notemoreover that

P(E

P

_[Z

t

− h1

[0,t]

, Dζi

H

|Z(g)] = 0) < 1

. Indeed, if it was not the ase, one would have (

δ

standsfor the Skorohod integral)

0 = E

Q

_[(Z

t

η

−1

− h1

[0,t]

, Dη

−1

i

H

)Z(g)] = E

Q

[δ(1

[0,t]

η

−1

)Z(g)]

= E

Q

_[η

−1

_]h1

[0,t]

, gi

H

= h1

[0,t]

, gi

H

6= 0

whi h is learly a ontradi tion. As a onsequen e, we dedu e from (17) that

Z(g)

L

2

-dierentiates

Z

at

t

under

P

if, and onlyif,

d

du

hg, 1

[0,u]

i

H

|

u=t

exists. ByLemma1, this last ondition is equivalent to

Z(g)

being

L

2

-dierentiating for

Z

at

t

under

Q

. We an thereforededu e (14) from (17) and (6).

If

H

⊂ σ{Z(g)}

, the proje tion prin ipleand (17) yieldthat

E

P

_[∆

h

Z

t

|H ]

equals

E

P

[Z

t

− h1

[0,t]

, Dζi

H

|H ]

1

[t,t+h]

, g



H

hh1

[0,t]

, gi

H

+ h

−1

1

[t,t+h]

, E

P

[Dζ|H ]



H

.

When

d

du

h1

[0,s]

, 1

[0,u]

i

H

|

u=t

does not exist, we dedu e that

H

dierentiates

Z

at

t

if, and only if,

E

P

_[Z

t

− h1

[0,t]

, Dζi

H

|H ] = 0

. If this ondition is veried, we then have

D

H

P

Z

t

= E

P

[µ

t

|H ],

again by Proposition 1.

Asanappli ationofTheorem2,weshall onsiderthe asewheretheisonormal pro ess

Z

in (11) is generated by a fra tional Brownian motion of Hurst index

H ∈

(0, 1/2) ∪ (1/2, 1)

(see also [3, Theorem 22℄, for related results on erning the ase

H ∈ (1/2, 1)

).

We briey re all some basi fa ts about sto hasti al ulus with respe t to a fra tional Brownian motion. We refer the reader to [10℄ for any unexplained notion orresult. Let

B = (B

t

)

t∈[0,T ]

be a fra tionalBrownian motionwith Hurst parameter

Wedenoteby

E

the set ofall

R

−

valued stepfun tionson

[0,T ]

. Let

H

bethe Hilbert spa edened asthe losureof

E

with respe t to the s alar produ t

1

[0,t]

, g



H

= R

H

(t, s),

and denoteby

| · |

H

the asso iate norm. The mapping

1

[0,t]

7→ B

t

an beextended to anisometrybetween

H

and theGaussianspa e

H

1

(B)

asso iatedwith

B

. Wedenote this isometryby

ϕ 7→ B(ϕ)

. Re all that the ovarian ekernel

R

H

(t, s)

introdu edin (18) an be writtenas

R

H

(t, s) =

Z

s∧t

0

K

H

(s, u)K

H

(t, u)du,

where

K

H

(t, s)

is the square integrablekernel dened, for

s < t

, by

K

H

(t, s) = Γ(H +

1

2

)

−1

_{(t − s)}

H−

1

2

F H −

1

2

,

1

2

− H, H +

1

2

, 1 −

t

s

,

(19) where

F (a, b, c, z)

isthe lassi al Gauss hypergeometri fun tion. By onvention, we set

K

H

(t, s) = 0

if

s ≥ t

. We dene the operator

K

H

on

L

2

_{([0, T ])}

as

(K

H

h)(t) =

Z

t

0

K

H

(t, s)h(s)ds.

Let

K

∗

H

: E → L

2

([0, T ])

bethe linear operator dened as:

K

∗

_H

1

[0,t]

= K

H

(t, ·).

The following equalityholds for any

φ, ψ ∈ E

hφ, ψi

H

= hK

_H

∗

φ, K

_H

∗

ψi

L

2

_{([0,T ])}

= E (B(φ)B(ψ)) ,

implying that

K

∗

H

is indeed an isometry between the Hilbert spa es

H

and a losed subspa e of

L

2

_{([0, T ])}

. Now onsider the pro ess

W = (W

t

)

t∈[0,T ]

dened as

W

t

= B (K

H

∗

)

−1

(1

[0,t]

)
,

and observe that

W

is a standard Wiener pro ess, and also that the pro ess

B

has anintegral representation of the type

B

t

=

Z

t

0

K

H

(t, s)dW

s

,

sothat, for any

φ ∈ H

,

We willalsoneed the fa t that the operator

K

H

an beexpressed interms of fra tionalintegrals asfollows:

(K

H

h)(s) = I

0+

2H

s

1

2

−H

I

1

2

−H

0+

s

H−

1

2

h(s),

if

H < 1/2,

(20)

(K

H

h)(s) = I

0+

1

s

H−

1

2

I

H−

1

2

0+

s

1

2

−H

h(s),

if

H > 1/2,

(21) for every

h ∈ L

2

_{([0, T ])}

. Here,

I

α

0+

f

denotes the left fra tional Riemann-Liouville integralof order

α

of

f

, whi h is dened by

I

₀₊

α

f (x) =

1

Γ(α)

Z

x

0

(x − y)

α−1

f (y)dy.

Let

Υ

H

bethe set of the so- alled shifted fBm

Z = (Z

t

)

t∈[0,T ]

dened by

Z

t

= x

0

+ B

t

+

Z

t

0

b

s

ds,

t ∈ [0, T ],

(22)

where

b

runs over the set of adapted pro esses (w.r.t. the natural ltration of

B

) havingintegrabletraje tories.

We alsoneed tointrodu ea te hni al assumption. Dene

a

r

=



K

−1

_H

Z

·

0

b

s

ds



(r);

(23)

inwhat follows weshall always assumethat (H1)

a

is bounded a.s.,

(H2)

Φ

dened by

Φ(s) =

R

T

0

D

s

a

r

δB

r

exists and belongs in

L

2

_{([0, T ])}

a.s..

First,letus onsiderthe ase

H > 1/2

. We suppose moreoverthat the traje -toriesof

b

are a.s. Hölder ontinuous oforder

H − 1/2 + ε

, forsome

ε > 0

. Then,the fra tionalversionoftheGirsanovtheorem (see[11,Theorem2℄)applies,yieldingthat

Z

is a fra tional Brownian motion of Hurst parameter

H

under the new probability

Q

dened by

dQ = ηdP

,where

η = exp



−

Z

T

0

K

−1

H

Z

·

0

b

r

dr
(s)dW

s

−

1

2

Z

T

0

K

−1

H

Z

·

0

b

r

dr

2

(s)ds



.

(24)

We an now state the following extension of Theorem 22in[3℄:

Corollary 2 Let

Z ∈ Υ

H

with

H > 1/2

and

s, t ∈ (0, T )

. Then

Z

s

L

2

-dierentiates

ments rehearsed beloware onlysket hed, sin e they are analogous tothose involved inthe proof of [3,Theorem 22℄. Letus onsider

ζ =

Z

T

0

a

s

dW

s

+

1

2

Z

T

0

a

2

_s

ds,

where

a

isdeneda ordingto(23). Weshall showthat(12) holds. We an ompute (see the proof of [3,Theorem 22℄)

h1

[t,t+h]

, Dζi

H

=

Z

t+h

t

b

r

dr + (K

H

Φ)(t + h) − (K

H

Φ)(t),

(25) where

Φ(s) =

R

T

0

D

s

a

r

δB

r

, see (H2). Sin e, in the ase where

H > 1/2

,

K

H

Φ

is dierentiable at

t

(see for instan e (21)) we dedu e that (12) holds. Moreover, one an easilyprovethat (13) also holds,so that the proof is on luded.

Nowwe onsider the ase

H < 1/2

. Weassumemoreover that

R

T

0

b

2

r

dr < +∞

a.s.. Then, the fra tional version of the Girsanov theorem (see [11, Theorem 2℄) holds again,implying that

Z

isa fra tional Brownian motion of Hurst parameter

H

underthe new probability

Q

dened by

dQ = ηdP

,with

η

given by (24). Notethat, when

H < 1/2

, we annot apply Theorem 2, sin e (12) does not hold in general. The reason is that

K

H

Φ

is no more dierentiable at

t

, see (20). In order to make

h

−1

_E[Z

t+h

− Z

t

|Z(g)]

onverge, we have to repla e

h

−1

with

h

−2H

(we only onsider the ase where

h > 0

). This fa tis made pre iseby the followingresult.

Proposition 2 Let

Z ∈ Υ

H

with

H < 1/2

and

s, t ∈ (0, T )

. Then,

lim

h↓0

h

−2H

_E[Z

t+h

− Z

t

|Z

s

]

exists in the

L

2

-topology

.

Bysetting

φ(s) = s

1

2

−H

I

1

2

−H

0+

s

H−

1

2

Φ(s)

, we have

K

H

Φ(t + h) − K

H

Φ(t) = I

0+

2H

φ(t + h) − I

0+

2H

φ(t)

=

1

Γ(2H)

Z

t

0

(t + h − y)

2H−1

− (t − y)

2H−1

_φ(y)dy

+

1

Γ(2H)

Z

t+h

t

(t + h − y)

2H−1

φ(y)dy

=

1

Γ(2H)

Z

t

0

(y + h)

2H−1

− y

2H−1

φ(t − y)dy

+

1

Γ(2H)

Z

h

0

y

2H−1

φ(t + h − y)dy

=

h

2H

Γ(2H)

Z

t/h

0

(y + 1)

2H−1

_{− y}

2H−1

_{φ(t − hy)dy}

+

h

2H

Γ(2H)

Z

1

0

y

2H−1

φ(t + h − hy)dy.

We dedu e that

h

−2H

K

H

Φ(t + h) − K

H

Φ(t)
−→ c

H

φ(t),

as

h → 0,

where

c

H

=

1

Γ(2H)

Z

+∞

0

(y + 1)

2H−1

− y

2H−1

dy +

1

Γ(2H)

Z

1

0

y

2H−1

dy < +∞.

Thus, by using the notations adopted in (the proof of) Theorem 2, one dedu es an analogueof (12), obtained by repla ing

h

−1

with

h

−2H

,that is:

˜

µ

t

, lim

h→0

h

−2H

_h1

[t,t+h]

, Dζi

H

exists in the

L

2

topology

.

Moreover, it is easily shown that

lim

h→0

h

−2H

_h1

[t,t+h]

, 1

[0,s]

i

H

exists. By using (17), weobtain the desired on lusion.

5 Dierentiating olle tions of

σ

-elds and the asso- iated dierentiated pro ess

properties of those

σ

-eld that are dierentiating for some pro esses at a xed time

t

. We will now on entrate on olle tions of dierentiating

σ

-elds indexed by the wholeinterval

(0, T )

.

Denition 4 Wesaythata olle tion

(A

t

₎

t∈(0,T )

of

σ

-elds

τ

-dierentiates

Z

if, for any

t ∈ (0, T )

,

A

t

_τ

-dierentiates

Z

at

t

.

Adierentiating olle tionof

σ

-eldsneed not bea ltration(see e.g. se tion 5 in [3℄). Nevertheless, we an asso iate to ea h

τ

-dierentiating olle tion

A

=

(A

t

₎

t∈(0,T )

for

Z

altration

A = (A

t

)

t∈(0,T )

, obtained by setting:

A

_t

₌

_

0<s6t

A

t

_,

_{t ∈ (0, T ).}

The olle tion of r.v.

(D

A

t

Z

t

)

t∈(0,T )

is a

A

-adapted pro ess [12, Denition 27.1℄, in the sense that for all

t ∈ (0, T )

,

D

A

t

Z

t

is

A

t

-measurable. We all it the dierentiated pro ess of

Z

w.r.t.

A

,and wedenote it by

D

A

Z

.

Inordertousesu hapro ess insto hasti analysis,one shouldknowwhether it admits a measurable version, that is, whether there exists a pro ess

Y

whi h is

B

_{(0,T )}

_{⊗ F}

-measurable and su h that for all

t

,

Y

t

= D

A

t

Z

t

a.s.. Our aim in this se tion is to obtain a su ient ondition for the existen e of a measurable version. Tothis end, weintrodu e the following

Denition 5 Let

A

= (A

t

₎

t

bea olle tionof

σ

-eldsand

Z

beameasurable sto has-ti pro ess. Wesaythat

A

isregularfor

Z

ifforall

n ∈ N

,

i ∈ {1, · · · , n}

,

t

i

∈ [0, T ]

,

φ

i

∈ C

0

∞

(R

d

)

, the pro ess

t 7→ E[φ

1

(Z

t

1

) · · · φ

n

(Z

t

n

)|A

t

_]

has a measurableversion.

If

A

is a ltration,then

A

is regular for any pro ess. For Gaussian pro esses and most of drifted Gaussian pro esses

X

, the olle tion

A

= (σ{X

t

})

t

is a regular olle tionfor

X

.

Theorem 3 Let

X

be a

B

(0,T )

⊗ F

-measurablesto hasti pro essdened on a om-plete probability spa e

(Ω, F , P)

, and assume that

F

= σ{X}

. Let

A

be a regular

L

1

-dierentiating olle tion for

X

. Then, there exists a measurable version of the dierentiated pro ess

D

A

_X

. This versionisalso adapted tothe ltration generatedby

A

.

Proof: Fix

ε > 0

, and let

(h

k

)

bea sequen e onverging to

0

and

Z

k

be the pro ess dened by

Z

_t

k

=

X

t+h

k

− X

t

h

k

.

Sin e

X

ismeasurable,soisthepro ess

Z

. Then,by[4,Theorem30p.158℄,thereexist elementarypro esses

U

n

k

t

su hthat, forall

t ∈ (0, T )

and every

k

,

E|Z

k

t

− U

n

k

t

| < ε/2

. Theseelementarypro esses havethe form:

U

n

k

t

=

X

i

1

_A

n

k

i

(t)H

n

k

i

,

where

(A

n

k

i

)

i

is a nite partition of

(0, T )

and

H

n

k

i

are

F

-measurable random vari-ables. We have

E[U

n

k

t

|A

t

] =

X

i

1

_A

nk

i

(t)E[H

n

k

i

|A

t

].

Sin e ylindri alfun tionalsof

X

aredensein

L

1

_{(Ω, F )}

,wededu efromtheregularity onditionthat the pro esses

t 7→ E[H

n

k

i

|A

t

]

admits a

B

(0,T )

⊗ F

-measurable modi- ationandalso,bylinearity,thesame on lusionholdsforthepro ess

t 7→ E[U

n

k

t

|A

t

]

. Moreover,

E





E[Z

_t

k

|A

t

] − E[U

n

k

t

|A

t

]





< ε/2.

Sin e

A

is a

L

1

-dierentiating olle tion for

X

, we dedu e that there exists

k

su h that

E





D

A

t

X

t

− E[Z

t

k

|A

t

]





 < ε/2,

and therefore

E





D

A

t

X

t

− E[U

t

n

k

|A

t

]





< ε

forevery

t

. We nowdedu e that the map

t 7→ [D

A

t

X

t

]

ismeasurable, where

[·]

denotes the lass of a pro ess in

L

1

_(Ω)

redu ed by null sets. Indeed, it is the limit in the Bana h spa e

L

1

_(Ω)

(when

k

goesto innity) of the measurable map

t 7→ E[U

n

k

t

|A

t

]

. Sin e

L

1

_(Ω)

isseparable, we again dedu e from [4,Theorem 30 p.158℄ that

D

A

The last se tion of the paper is devoted to the outline of a general framework for sto hasti embeddingproblems (introdu edin[2℄)relatedtoordinarydierential equa-tions. As we willsee, this notioninvolvesthe sto hasti derivativeoperators that we have dened and studied in the previous se tions. Roughly speaking, the aim of a sto hasti embeddingpro edureistowritea"sto hasti equation"whi hadmitsboth sto hasti and deterministi solutions,insu ha way that the deterministi solutions alsosatisfy axed ordinarydierentialequation(see [2℄). It follows that the embed-ded sto hasti equationisa genuine extensionof the underlyingordinary dierential equationto asto hasti framework.

6.1 General setting Let

χ : R

d

_{→ R}

d

(

d ∈ N

∗

)beasmoothve toreld. Considerthe ordinarydierential equation:

dx

dt

(t) = χ(x(t)),

t ∈ [0, T ].

(26)

Let

Λ

be a set of measurable sto hasti pro esses

X : Ω × [0, T ] → R

d

, where

(Ω, F , P)

is a xed probability spa e. In order to distinguish two dierent kinds of families of

σ

-elds, we shall adopt the following notation: (i) the symbol

A

0

=

(A

0

t

)

t∈[0,T ]

denotes a olle tion of

σ

-elds whose denition does not depend on the hoi e of

X

inthe lass

Λ

, and (ii)

A

= (A

t

X

)

X∈Λ,t∈[0,T ]

indi ates a generi family of

σ

-elds su h that, for every

t ∈ [0, T ]

and every

X ∈ Λ

,

A

t

X

⊂ P

T

X

. We introdu e the following naturalassumption:

(T ) Λ

ontains allthe deterministi dierentiable fun tions

f : [0, T ] → R

d

(viewed as deterministi sto hasti pro esses).

We now x a topology

τ

, and des ribe two sto hasti embedded equations asso iated with (26).

Denition 6 Fixa lassofsto hasti pro esses

Λ

on

(Ω, F , P)

,verifyingassumption (T).

(a)Given a family

A

0

= (A

t

0

)

t∈[0,T ]

of

σ

-elds, we say thatthe equation

X ∈ Λ,

D

A

0

t

X

t

= χ(X

t

)

for every

t ∈ [0, T ],

(27) isthe strong sto hasti embedding in

Λ

of the ODE (26) w.r.t.

A

0

.

(b) Givena family

A

= (A

t

X

)

X∈Λ,t∈[0,T ]

of

σ

-elds su hthat forall

X ∈ Λ

and

t ∈ [0, T ]

,

A

t

X

⊂ P

T

X

, we say that the equation

X ∈ Λ,

D

A

X

t

X

isthe weak sto hasti embedding in

Λ

of the ODE (26) w.r.t.

A

.

( ) Asolutionof (27) (resp. (28))isa sto hasti pro ess

X ∈ Λ

su hthat: ( -1) the pro ess

D

A

t

0

X

t

(resp.

D

A

t

X

X

t

) admits a jointly measurable version, and ( -2) the equation

D

A

t

0

X

_t

= χ(X

_t

)

(resp.

D

A

t

X

X

t

= χ(X

t

)

) is veried for every

t ∈ [0, T ]

.

Note that a solution of (26) is always a solution of (27) or (28). Observe also that if one wants to obtain "genuinely sto hasti " solutions of (26) (i.e. non deterministi ), the previous denition impli itly imposes some restri tions on the lass

Λ

. Namely, if

X ∈ Λ

is a solution of (27) (resp. (28)), then for any

t ∈ [0, T ]

,

A

t

0

(resp.

A

t

X

) is dierentiating for

X

at

t

with respe t to the topology

τ

and the random variable

χ(X

t

)

is

A

t

0

-measurable (resp.

A

t

X

-measurable) for every

t

. As an example,let

Γ

bethe set of allpro esses

X

with the form:

X

t

= X

0

+ σB

t

+

Z

t

0

b

r

dr,

t ∈ [0, T ]

(29)

where

σ ∈ R

,

B

isafBmofHurstindex

H ∈ (0, 1)

,and

b

runsoverthe setofadapted pro esses (w.r.t. the natural ltration of

B

H

) having a.s. integrable traje tories. Suppose that we seek for solutions with

σ 6= 0

of the weak sto hasti embedding of (26) given by

X ∈ Γ,

D

σ{X

t

}

X

t

= χ(X

t

),

(30)

Then, Corollary 2 and Proposition 2 imply that su h solutions must ne essarily be driven by afBmof Hurst index

H > 1/2

.

Sto hasti embedded equations may be useful in the following framework. Suppose that a physi al system is des ribed by (26), and that we want to enhan e thisdeterministi mathemati almodelinordertotakeintoa ountsome"sto hasti phenomenon" perturbing the system. Then, the embedded equations (27) or (28) maybe thekey todeneasto hasti modelinavery oherentway, inthe sensethat everysto hasti pro esssatisfying(27)or(28)isalso onstrainedbythephysi allaws (i.e. the ODE (26))dening the originaldeterministi des ription of the system.

6.2 A rst example

Considertheset

Λ

ofall ontinuouspro essesdenedontheprobabilityspa e

(Ω, F , P)

, aswellasthe" onstant" olle tionof

σ

-elds

(F

t

)

t∈[0,T ]

su hthat

F

t

= F

forevery

t

. Sin ethe sto hasti derivativew.r.t.

F

oin ideswith theusual pathwise derivative, the embedding problem

Notethat in this example the embedded dierentialequation produ es no other so-lutionthan those given by (26).

6.3 A more interesting example

Let

W

beaWienerpro ess on

[0, T ]

and onsiderthe set

Λ

ofdeterministi pro esses and of allsto hasti pro esses that an be expressed in terms of multiple sto hasti integrals with respe t to

W

. More pre isely, denote by

Λ

W

the set of pro esses

u ∈ L

2

_{(Ω, L}

2

_{([0, T ]))}

su h that, for every

t ∈ [0, T ]

,

u

t

=

X

n≥0

J

n

(f

n

(·, t))

where,for any

t ∈ [0, T ]

,the

f

n

(·, t)

'sverify:

X

n≥0

kf

n

(·, t)k

2

L

2

_(∆

n

[0,T ])

+

∂f

n

∂t

(·, t)

2

L

2

_(∆

_n

_[0,t])

!

< +∞.

Here

∆

n

[0, T ] = {(s

1

, . . . , s

n

) ∈ R

n

+

: 0 ≤ s

n

≤ . . . ≤ s

1

≤ T }

and,for

g ∈ L

2

_(∆

n

[0, T ])

,

J

n

(g) =

Z

∆

n

[0,T ]

g dW =

Z

T

0

dW

s

1

Z

s

1

0

dW

s

2

. . .

Z

s

n

−1

0

dW

s

n

g(s

1

, . . . , s

n

).

On

Λ

W

, we an onsider sto hasti derivatives of Nelson type (i.e. w.r.t. a xed ltration[8℄):

Lemma 2 Fix

t ∈]0, T [

and let

P

t

be the past before

t

, that is the

σ

-eld generated by

{W

s

, 0 ≤ s ≤ t}

. If

u ∈ Λ

W

then

D

P

t

u

t

exists and it isgiven by

D

P

t

u

t

=

X

n≥0

J

n

 ∂f

n

∂t

(·, t)1

∆

n

[0,t]



in the

L

2

sense

.

(32)

Proof: Obvious by proje tion.

As an example, onsider the ase where

χ

is given by

χ(x) = ax + b

with

a, b ∈ R

. In otherwords, wewant to solve the strong embedding

X ∈ Λ,

D

P

t