Table Of ContentChaotic extensions and the lent particle method for
Brownian motion
Nicolas BOULEAU
2
1
Ecole des Ponts, Paris Tech, Paris-Est
0
6 Avenue Blaise Pascal
2
77455 Marne-La-Vallée Cedex 2- FRANCE
n e-mail: [email protected]
a
J
Laurent DENIS
6
1
Département de Mathématiques
Laboratoire Analyse et Probabilités
]
R Université d’Evry-Val-d’Essonne
23 Boulevard de France
P
91037 EVRY Cedex -FRANCE
.
h e-mail: [email protected]
t
a
Abstract: In previous works, we have developed a new Malliavin calculus on the
m
Poisson space based on the lent particle formula. The aim of this work is to prove
[ that, on the Wiener space for the standard Ornstein-Uhlenbeck structure, we also
have such a formula which permits to calculate easily and intuitively the Malliavin
1
v derivativeofafunctional.Ourapproachuseschaosextensionsassociatedtostationary
2 processes of rotations of normal martingales.
2
AMS 2000 subject classifications: Primary 60H07; 60H15; 60G44; 60G51
3
3 .
. Keywordsandphrases:Malliavincalculus,chaoticextensions,normalmartingales.
1
0
2
1. Introduction
1
:
v
i When a measurable space with a σ-finite measure ν is equipped on L2(ν) with a local
X
Dirichlet form with carré du champ γ, the associated Poisson space, i.e. the probability
r
a space of a random Poisson measure with intensity ν, may itself be endowed with a local
Dirichlet structure with carré du champ Γ (cf. [16], [17]). If a gradient ♭ has been chosen
associated with the operator γ, a gradient ♯ associated with Γ may be constructed on the
Poissonspace(cf[1],[10],[14],[2])andwehaveshown[2],[3],thatsuchagradient isprovided
by the lent particle formula which amounts to add a point to the configuration, to derive
with respect to this point, and then to take back the point before integrating with respect
to a random Poisson measure variant of the initial one.
On the example of a Lévy process, in order to find the gradient of the functional V =
t
ϕ(Y )dY , this method consists in adding a jump to the process Y at time s and then
0 u− u
deriving with respect to the size of this jump.
R
IfwethinktheBrownian motionasaLévyprocess,thisaddressesnaturally thequestion
of knowing whether to obtain the Malliavin derivative of a Wiener functional we could add
∗Theworkofthesecondauthorissupportedbythechairrisquedecrédit,FédérationbancaireFrançaise
1
N. Bouleau and L. Denis/ 2
a jump to the Brownian path and derive with respect to the size of the jump, in other
words whether we have, denoting D F the Malliavin derivative of F
s
1
D F = lim (F(ω+a1 ) F(ω)). (1)
s a→0 a {.>s} −
1 1
Formula (1) is satisfied in the case F = Φ( h dB,..., h dB) with Φ regular and h
0 1 0 n i
continuous. But this formula has no sense in general, since the mapping t 1 does
R R 7→ {t>s}
not belong to the Cameron-Martin space.
We tackle this question by means of the chaotic extension of a Wiener functional to
a normal martingale weighted combination of a Brownian motion and a Poisson process,
and we show that the gradient and its domain are characterized in terms of derivative of
a second order stationary process.
We show that a formula similar to (1) is valid and yields the gradient if F belongs to
the domain of the Ornstein-Uhlenbeck Dirichlet form, but whose meaning and justification
involve chaotic decompositions. This gives rise to a concrete calculus allowing 1 changes
C
of variables.
Let us also mention the works of B. Dupire ([8]), R. Cont and D.A. Fournié ([5]) which
use an idea somewhat similar but in a completely different mathematical approach and
context.
2. The second order stationary process of rotations of normal martingales.
Let B be a standard one-dimensional Brownian motion defined on the Wiener space Ω
1
under the Wiener measure P .
1
In this section, we consider N˜ a standard compensated Poisson process independent of B.
We denote by P the law of the Poisson process N and P = P P .
2 1 2
×
Let us point out that in the next sections, we shall replace N˜ by any normal martingale.
2.1. The chaotic extension
For real θ, let us consider the normal martingale
Xθ = B cosθ+N˜sinθ.
t t
If f is a symmetric function of L2(Rn,λ ), we denote I (f ) the Brownian multiple
n n n n
stochastic integral and Iθ(f ) the multiple stochastic integral with respect to Xθ. We have
n n
classically cf [6]
I (f ) 2 = Iθ(f ) 2 = n! f 2 .
k n n kL2(P1) k n n kL2(P) k nkL2(λn)
It follows that if F L2(P ) has the expansion on the Wiener chaos
1
∈
∞
F = I (f )
n n
n=0
X
the same sequence f defines a chaotic extension of F : Fθ = ∞ Iθ(f ).
n n=0 n n
P
N. Bouleau and L. Denis/ 3
Remark 1. Let us emphasize that the chaotic extension F Fθ is not compatible with
7→
composition of applications : Φ Ψθ = (Φ Ψ)θ except obvious cases as seen by taking
◦ 6 ◦
Φ(x) = x2, Ψ = I (f) and θ = π/2. Thus it is important that the sequence (f ) appears
1 n n
in the notation : we will use the "short notation" of [6].
We denote (resp. (t)) the set of finite subsets of ]0, [ (resp. ]0,t]). We write A =
P P ∞
s < < s for the current element of and dA for the measure whose restriction to
1 n
{ ··· } P
each simplex is the Lebesgue measure, cf [6] p201 et seq.
If F L2(P ) expans in F = ∞ I (f ) we denote f L2( ) the sequence f =
∈ 1 n=0 n n ∈ P
(n!f ) and
n n∈N
P
Iθ(f) = f(A)dXθ
P A
R
= f( )+ f(s ,...,s )dXθ dXθ .
∅ n>0 s1<···<sn 1 n s1··· sn
Thus we have f( )= E[F], F =PI0(f)Rand Iπ/2(f)= f(A)dN˜ = Iπ/2(f ).
∅ P A n n n
In all the paper we confond the stochastic integrals H dXθ and H dXθ thanks to
R s− s P s s
the fact that Xθ is normal.
R R
Proposition 1. Let be f and g L2( ), and h = f +ig L2( ). The random variable
∈ P ∈ C P
Hθ = h(A)dXθ defines a second order stationary process continuous in L2(P).
P A C
Proof.RLet us denote similarly Fθ = f(A)dXθ and Gθ = g(A)dXθ. It is enough to
P A P A
show that Fθ and Gθ are measurable, second order stationary and stationary correlated.
R R
Using the chaos expansions Fθ+ϕ = Iθ+ϕ(f ) and Gθ = Iθ(g ) that comes from
n n n n n n
the fact that the bracket of the martingales Xθ+ϕ and Xθ is
P P
Xθ+ϕ,Xθ = tcosϕ (2)
t
h i
So E[Inθ+ϕ(fn)Inθ(gn)] = n!hfn,gniL2(λn)cosnϕ and E[Imθ+ϕ(fm)Inθ(gn)] = 0 if m is different
of n.
It follows that the stationary process Hθ possesses a spectral representation
Hθ = c einθξ (3)
n n
n∈Z
X
where the c are real > 0 and the ξ are orthonormal in L2(P). The norm Hθ 2 which
n n C k k
doesn’t depend on θ is the total mass of the spectral measure c2.
n
P
The c are linked with the norms of the components of H on the chaos by formulas
n
involving Bessel functions. In the case where H is an exponential vector,
∞ h⊗k 1
H = I ( )= exp[ hdB h2dt]
k
k! − 2
k=0 Z Z
X
with h = f +ig what implies H 2 = exp h 2, we have by (2) the covariance
k kL2C k k
1
E[Hθ+ϕHθ] = h 2k coskϕ = exp( h 2cosϕ)
k!k kL2C(λ1) k k
k
X
N. Bouleau and L. Denis/ 4
which is the Fourier transform of the spectral measure hence equal to c2einϕ.
n n
By the relation defining the Bessel functions J (formula of Schlömilch)
n
P
exp(izsinϕ) = einϕJ (z) z C, z = 0,
n
∈ 6
n∈Z
X
it comes c2 = inJ ( i h 2) and for n > 0 (cf [15])
n n − k k
∞ 1 h 2
c2 = c2 = (k k )2k+n.
n −n k!(n+k)! 2
k=0
X
The variables c ξ may be also expressed in terms of Bessel functions using the expression
n n
of exponential vectors for Xθ, cf formula (5) below.
2.2. Chaotic structure of L2(P).
This part is independent of the rest of the paper. It is devoted to the study of chaotic
representations for Xθ.
Let us first remark that the above considerations dont use the chaotic representation
property for Xθ which is false if sinθcosθ = 0, as it is well known cf for instance [7]. Let
6
us denote L2(P ) the vector space of σ(Xθ)-measurable random variables belonging to
Xθ
L2(P). That means that, if sinθcosθ = 0, the vector space C(Xθ) = Fθ : F L2(P )
1
6 { ∈ }
which is closed in L2(P ) has a non empty complement.
Xθ
If we consider the simplest example of the square of a functional of the first Brownian
chaos F = hdB with h L1 L∞, we have Fθ = hdXθ and by Ito formula
∈ ∩
R t R
(Fθ)2 = 2 h(s)dXθ h(t)dXθ + h2ds+sin2θ h2dN˜
s t
Z Z0 Z Z
since N˜ = sinθXθ +cosθXθ+π/2, we see that
Fθ =U +sin2θcosθ h2dXθ+π/2
Z
with U C(Xθ) and h2dXθ+π/2 orthogonal to C(Xθ).
∈
It follows that for k L2(R ), kdXθ+π/2 L2(P ) and this implies
R∈ + ∈ Xθ
Proposition 2. Let us suppose siRnθcosθ = 0. The σ-fields generated by Xθ and Xθ+π/2
6
are the same. The spaces L2(P ) do not depend on θ and are equal to L2(P).
Xθ
The intuitive meaning of this proposition is that on a sample path of Xθ it is possible
to measurably detect the underlying Brownian and Poisson paths.
The multiple stochastic integrals w.r. to Xθ are not enough to fill L2(P ). In view of
Xθ
the previous example, we may think to add the stochastic integrals w.r. to Xθ+π/2, i.e. to
add C(Xθ+π/2), which is orthogonal to C(Xθ) and included in L2(P ). But this is not
Xθ
sufficient, we must add also the crossed chaos in the following manner:
N. Bouleau and L. Denis/ 5
LetusconsiderthevectormartingaleXθ = (Xθ,Xθ+π/2).Forh= (h ,h ) L2(R ,R2)
1 2 +
∈
we may consider the stochastic integral h.dXθ, and more generally (notation of [4] Chap
II §2)
R
f.d(n)Xθ (4)
Z∆n
for f L2(∆ ,(R2)⊗n). And using (2) for ϕ = π/2 we have
n
∈
k f.d(n)XθkL2(P) =kfkL2(∆n,(R2)⊗n).
Z∆n
These stochastic integals define orthogonal sub-spaces of L2(P ) = L2(P). Now consider-
Xθ
ing the exponential vector
1
θ(h ,h ) = h h dXα1 dXαn
E 1 2 n! i1 ⊗···⊗ in ···
Xn ik∈X{1,2}Z
whereα = θorθ+π/2accordingtoi = 1or2,andputting θ(h ,h )for θ(h 1 ,h 1 ),
k k Et 1 2 E 1 [0,t] 2 [0,t]
we see that the following SDE is satisfied
t
θ(h ,h )= 1+ θ (h ,h )(h dXθ +h dXθ+π/2)
Et 1 2 Es− 1 2 1 s 2 s
Z0
what gives
Etθ(h1,h2) = eVt−12[V,V]ct (1+∆Vs)e−∆Vs (5)
s6t
Y
with V = th dXθ + th dXθ+π/2. We obtain
t 0 1 0 2
PropositiRon 3. For aRny θ, the stochastic integrals (4) define a complete orthogonal de-
composition of L2(P).
Proof. a) Let us suppose first sinθcosθ = 0. Starting with (5) an easy computation yields
6
that θ(h ,h ) is equal up to a multiplicative constant to
exp[ t(hEtcos1θ 2h sinθ)dB+ tudN˜] where we have taken eu 1= h sinθ+h cosθ.
0 1 − 2 0 − 1 2
If we take a step function ξ L2(R ) and choose h and h such that
+ 1 2
R ∈R
h sinθ+h cosθ = eξsinθ 1
1 2
−
h cosθ h sinθ = ξcosθ
1 2
−
we obtain that exp[ ξdXθ] belongs to the space generated by the chaos, and the result
follows.
R
b) Now if θ = 0, Xθ = (B,N˜). The above argument is still valid and
t 1 t t t
0(h ,h ) = exp[ h dB h2ds+ u dN˜ + u ds]
Et 1 2 1 − 2 1 2 2
Z0 Z0 Z0 Z0
with h = eu2 1. That gives easily the result and the same in the other cases where
2
−
sinθcosθ = 0.
6
InotherwordsL2(P)isisomorphictothesymmetricFockspaceFock(L2(R ,R2)).This
+
implies the predictable representation property with respect to Xθ.
N. Bouleau and L. Denis/ 6
3. Derivative in θ and gradient of Malliavin.
We come back to the setting of subsection 2.1 with stochastic integrals with respect to the
real process Xθ. We want to study the behavior near θ = 0 using the fact that X0 = B.
But since we deal no more with chaotic representation we may replace N˜ by any normal
martingale M independent of B (for instance a centered normalized Lévy process) define
under a probability that we still denote P and as in subsection 2.1, P = P P . Let us
2 1 2
×
put
Yθ = B cosθ+M sinθ
t t t
and consider the chaotic extensions F Fθ with respect to Yθ i.e. if F = ∞ I (f ),
then Fθ = ∞ Iθ(f ), where from n7→ow on, Iθ denotes the multiple stochastni=c0inntegnral
n=0 n n n P
with respect to Yθ.
P
To see the connection with the Brownian chaos expansion, let us remark that – as in
the preceding part – for any θ R, the pair (Y ,Y ) = (Yθ,Yθ+π/2) is a vector normal
1 2
∈
martingale in the sense of [6] i.e.
Y ,Y = δ t (6)
i j t ij
h i
this allows to prove the following property
Proposition 4. For any F with finite Brownian chaos expansion, F = n I (f ), f
k=0 n n n
symmetric,
n P
d
E[( Fθ)2]= n!n f 2
dθ k nkL2
k=0
X
Proof. Our notation is Fθ = n Iθ(f ) and F0 = F = n I (f ).
k=0 k k k=0 k k
Let us consider first Iθ(f ) in the case of an elementary tensor f = g g . We
n nP P n 1 ⊗···⊗ n
can write this multiple integral (with the notation of Bouleau-Hirsch [4] p79)
t
Iθ(f )= n! f d(n)Yθ =n! [ g g d(n−1)Yθ]g (s)dYθ
n n n 1⊗···⊗ n−1 n s
Z∆n Z0 Z∆n−1(s)
so that
d Iθ(f ) = n! t d [ g g d(n−1)Yθ]g (s)dYθ
dθ n n 0 dθ ∆n−1(s) 1⊗···⊗ n−1 n s
R R
+n! t[ g g d(n−1)Yθ]g (s)dYθ+π/2
0 ∆n−1(s) 1⊗···⊗ n−1 n s
R R
hence by (6)
E[( d Iθ(f ))2] = (n!)2 tE[( d [ g g d(n−1)Yθ])2]g2(s)ds
dθ n n 0 dθ ∆n−1(s) 1⊗···⊗ n−1 n
R R
+(n!)2 tE[( g g d(n−1)Yθ)2]g2(s)ds
0 ∆n−1(s) 1⊗···⊗ n−1 n
R R
what gives, running the induction down,
d
E[( Iθ(f ))2] = n(n!)2 (g g g )2dλ = n.n! f 2.
dθ n n 1 ⊗···⊗ n−1⊗ n n k nk
Z∆n
N. Bouleau and L. Denis/ 7
This extends to general tensors and similarly we can show that if k = ℓ
6
d d
E[( Iθ(f ))( Iθ(f ))] = 0
dθ k k dθ ℓ ℓ
what yields the proposition.
We denote by D, the domain of the Ornstein-Uhlenbeck form. We recall that an element
F = +∞ I (f ) L2(P ) belongs to D iff
n=0 n n ∈ 1
P +∞
nn! f 2 < + .
k n kL2 ∞
n=0
X
Let us take now an F D, the random variables n Iθ(f ) converge in L2(P) to Fθ
∈ k=0 k k
uniformly in θ. Their derivatives – because F D – form a Cauchy sequence and converge
∈ P
also uniformly in θ. This implies that Fθ is differentiable and that the derivatives of the
n Iθ(f ) converge to the derivative of Fθ. So we have
k=0 k k
PProposition 5. F D the process θ Fθ is differentiable in L2(P) and
∀ ∈ 7→
d
θ E[( Fθ)2]= n2c2 = EΓ[F]= 2 [F]
∀ dθ n E
n∈Z
X
where is the Ornstein-Uhlenbeck form and Γ the associated carré du champ operator.
E
We also have the converse property:
Proposition 6. Let F L2(P ). If the map θ Fθ is differentiable in L2(P) at a certain
1
∈ 7→
point θ R then F belongs to D.
0
∈
Proof. We write F = I (f ) and consider a sequence (θ ) which converges to θ
n n n k k>1 0
and such that θ = θ , for all k > 1. As
k 0
6 P
Fθk Fθ0
lim −
k→+∞ θk θ0
−
exists in L2(P) we deduce that there exists a constant C > 0 such that
Fθk Fθ0 2 Iθk(f ) Iθ0(f ) 2
k > 1, − = n n − n n 6 C.
∀ θ θ θ θ
(cid:13) k − 0 (cid:13)L2(P) n (cid:13) k − 0 (cid:13)L2(P)
(cid:13) (cid:13) X(cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
By the Fatou’s Lemm(cid:13)a and the pr(cid:13)evious Prop(cid:13)osition, we get (cid:13)
Iθk(f ) Iθ0(f ) 2
n k→lim+∞(cid:13) n θnk −−θn0 n (cid:13)L2(P) = n n!nkfnk2L2 6 C,
X (cid:13) (cid:13) X
(cid:13) (cid:13)
which yields the result. (cid:13) (cid:13)
This provides the following result :
N. Bouleau and L. Denis/ 8
Proposition 7. For all F D with chaotic representation F = f(A)dB
∈ P A
dFθ d R
= f(A)d(Bcosθ+Msinθ) = D F dM
θ=0 A θ=0 s s
dθ | dθ |
ZP Z
the righthand term is a gradient for the Ornstein-Uhlenbeck form that we may denote F♯,
so we have in the sense of L2(P) = L2(P P )
1 2
×
1
F♯ = lim (Fθ F).
θ→0θ −
Proof. Let be F D. We assess the distance between 1(Fθ F) and D F dM by steps
∈ θ − s s
: distance between 1(Fθ F) and 1(Fθ F ) ; between 1(Fθ F ) and D F dM ;
θ − θ n − n θ n − nR s n s
then between D F dM and D F dM .
s n s s s
R
By the preceding propositions
R R
1 1
(Fθ F) (Fθ F ) 2 6 F F 2
kθ − − θ n − n k k − nkD
We may choose n so that the first one and the third one be both small independently of
θ. And n being fixed we do θ 0 in the second one and apply the argument of the proof
→
of Proposition 4.
The classical integration by part formula, i.e. the property that the divergence opera-
tor, dual of D, can be expressed by a stochasitic integral on predictable processes, is a
consequence of propositions 5 and 7 by derivation in θ.
Indeed let us denote the closed sub-vector space of L2(P,L2(R ,dt)) generated by
+
A
the processes of the form f d(n)B with f L2(Rn).
∆(t) n n ∈
Lemma 8. For F L2(PR), G , we have
1
∈ ∈ A
E[Fθ GθdYθ+π/2] = 0.
t t
Z
Proof. By relation (6) the property is true if F has a finite expansion on the chaos hence
also if F L2.
∈
Let us denote now D the closed vector space generated by the processes f d(n)B
A ∆(t) n
with f L2(Rn) for the norm of D with values in L2(R ).
n ∈ + R
Proposition 9. Let be F D and G D , we have
A
∈ ∈
dFθ
E[Fθ GθdYθ] = E[ GθdYθ+π/2]
u u dθ u u
Z Z
so that taking θ = 0
E[F G dB ]= E[ D FdM G dM ] = E[ D F G du].
u u u u u u u u
Z Z Z Z
Proof. We differentiate Fθ GθdYθ+π/2 taking in account the lemma and the fact that
t t
Yθ+π = Yθ.
− R
N. Bouleau and L. Denis/ 9
Remark 2. Taking anew N˜ for M, we may apply the previous reasoning at the point
θ = π/2.Denoting D(N) theoperatorofNualart-Vivès [12]whichactsonthePoissonchaos
asDactsontheBrownianones,Proposition7saysthatfor(f )suchthat n!n f 2 <
n n
k k ∞
the Poisson functional F = Iπ/2(f ) is such that d Fθ = D(N)FdB.
n n dθ |θ=π/2 P
Andby Proposition 9 we obtain that the finite difference operator D(N) of the Ornstein-
P R
Uhlenbeck structure on the Poisson space satisfies an integration by part formula (cf Øk-
sendal and al [9] Thm 12.10) despite its non local character.
Remark 3.InthecaseofanotherstandardBrownian motion Bˆ forM,Proposition 7gives
exactly the derivation operator in the sense of Feyel-La Pradelle cf [4] Chap. II §2.
d
Fθ = F′ = D FdBˆ
θ=0 u u
dθ |
Z
In that case the situation is quite different from the one we had in Section 2. Indeed
Yθ = Bcosθ+Bˆsinθ does satisfy the chaotic representation property, so that the space
Fθ : F L2(P ) is L2(P ). It is not possible to measurably detect the paths of B
{ ∈ 1 } Yθ
and Bˆ on those of Yθ. But the concept of chaotic extension becomes simpler because it is
compatible with the composition of the functions. It is valid to write in this case
Fθ = F(Bcosθ+Bˆsinθ).
Indeed, it is correct for F = Φ( h dB,..., h dB) with Φ a polynomial by Ito formula
1 k
and induction (what was false in the case of the Poisson process), and then for general F
R R
in L2 by approximation. As a consequence, Proposition 7 gives a formula of Mehler type
without integration for the gradient
d
F D F′ = D FdBˆ = F(Bcosθ+Bˆsinθ) (7)
u u θ=0
∀ ∈ dθ |
Z
and with integration for the carré du champ
d
Γ[F]= Eˆ[( F(Bcosθ+Bˆsinθ))2 ] (8)
θ=0
dθ |
where Eˆ acts on Bˆ as usual. By the change of variable cosθ = e−t/2 this may be also
written in a form similar to Mehler formula:
1
F′ = lim (F(B√e−t+ 1 e−tBˆ) F(B)) (9)
t→0 √t − −
p
what gives denoting P the Ornstein-Uhlenbeck semi-group
t
1
Γ[F] = lim (P (F2) 2FP F +F2) (10)
t t
t→0 t −
well known formula when F and F2 are in the domain of the generator and that we obtain
here for F D.
∈
To our knowledge, formulae (7), (8) and (9) seem to be new under these hypotheses.
N. Bouleau and L. Denis/ 10
4. Functional calculus of class C1 ∩Lip.
Proposition 10. Let us suppose that the process H be in D (cf proposition 9) then
s A
( H dB )θ = (H )θdYθ.
s s s s
Z Z
Proof. The functional F = H dB is in D and has a chaotic expansion F = f(A)dB .
s s P A
Following the short notation of [6] (p203) if we put for E
R ∈ P R
f˙(E) = f(E t ) if E [0,t[, = 0 otherwise,
t
∪{ } ⊂
and if g = f˙(E)dB , then
t P t E
R
F = f(A)dB = f( )+ g dB .
A t t
∅
ZP Z
Hence we have Fθ = f(A)dYθ and (g )θ = f˙(E)dYθ and
P A t P t E
R R
Fθ = f( )+ (g )θdYθ,
∅ t t
Z
what proves the proposition.
Let be F = (F ,...,F ) Dk and Φ 1 Lip(Rk,R), where in a natural way
1 k
∈ ∈ C ∩
Lip(Rk,R) denotes the set of uniformly Lipschitz real-valued functions defined on Rk.
It comes from Proposition 7 and from the functional calculus in local Dirichlet structures
the following result
Proposition 11. When θ 0, we have in L2(P)
→
1
[(Φ(F ,...,F ))θ Φ(Fθ,...,Fθ)] 0.
θ 1 k − 1 k →
Proof. We have indeed in the sense of L2, the function Φ being Lipschitz and 1
C
lim 1[Φ(Fθ,...,Fθ) Φ(F ,...,F )] = k Φ′(F ,...,F )F♯
θ 1 k − 1 k i=1 i 1 k i
= lim 1[(Φ(F ,...,F ))θ Φ(F ,...,F )].
P θ 1 k − 1 k
Itfollowsthatwemayreplace(Φ(F ,...,F ))θ byΦ(Fθ,...,Fθ)inapplyingthemethod.
1 k 1 k
Let us define an equivalence relation denoted = in the set of functionals in L2(P) de-
∼
pending on θ and differentiable in L2 at θ = 0 by
d d
F(ω ,ω ,θ)= G(ω ,ω ,θ) if F = G and F =G .
1 2 ∼ 1 2 dθ |θ=0 dθ |θ=0 |θ=0 |θ=0
(cid:18) (cid:19)
Let us also define a weaker equivalence relation denoted for functionals in L0(P)
≃
depending on θ and differentiable in probability at θ = 0 by
d d
F(ω ,ω ,θ) G(ω ,ω ,θ) if F = G and F = G
1 2 1 2 θ=0 θ=0 θ=0 θ=0
≃ dθ | dθ | | |
(cid:18) (cid:19)
the limits in the derivations being in probability.
