Weak calculus of variations for functionals of laws of semi-martingales

0.43 MB·English

by  Ana Bela Cruzeiro

WEAK CALCULUS OF VARIATIONS FOR FUNCTIONALS OF LAWS OF
SEMI-MARTINGALES
5
1 RE´MILASSALLEANDANABELACRUZEIRO
0
2
Abstract. We develop a non-anticipating calculus of variations for functionals on a space of

n lawsofcontinuoussemi−martingales,whichextendstheclassicalone. WeextendHamilton’sleast
a
actionprincipleandNoether’stheoremtothisgeneralizedstochasticframework. Asanapplication
J
we obtain, under mild conditions, a stochastic Euler−Lagrange condition and invariants for the
1
critical points of recent problems in stochastic control, namely for the semi-martingale optimal
2
transportationproblems.

]
R
P
.
h Keywords: Stochasticanalysis,Leastactionprinciple,Stochasticcontrol,Semi-martingaleoptimal
t
a transportation problems;
m
Mathematics Subject Classification : 60H30, 93E20
[

1
v Introduction
4
3 Inthispaperweformulateaweakcalculusofvariationswhichextendsthe classicalone. Roughly
1
speaking this enables to perform a calculus on functions defined on laws of semi-martingales. We
5
0 apply this calculus to obtain a stochastic extension of Hamilton’s least action principle. Recall
.
1 that the classical version of this principle provides a characterization of the paths satisfying the
0
Euler Lagrange condition as critical points of a functional which is called an action. Here we
5 −
1 will characterize laws of semi-martingales which satisfy a constraint that extends the classical one.
:
v Namely these laws will be provedto be critical points of a stochastic action. Once this extension is
i achieved we use it to relate some invariance properties of the critical processes to the symmetries
X
of the corresponding Lagrangian; in other words, we derive a stochastic extension of Noether’s
r
a
theorem. Finallyweconsiderapplicationstostochasticcontrol,inparticulartosomesemi-martingale
optimal transportationproblems. These problems were recently introduced in [22] with application
to financial mathematics.
As a warm up, let us give, in an informal way, some details on our motivation and on the
difficulties we overcome with our approach. The first motivation lies in classical mechanics. In
classical mechanics one usually considers paths sufficiently regular to model the kinematics of a
system. In particular if one describes the trajectory of a classical particle by a path q : [0,1] R
→
one will usually ask q to be sufficiently regular in order to provide a realistic description of the
observation. Namely it will be often assumed to be C2 for both its speed q˙ (:= dqt) and its
t dt
accelerationq¨ to be defined. Thus, for the sake of simplicity let us first consider the space Ω2 of
t [0,1]
the C2 paths q :[0,1] R as being the set of the paths providing an admissible description of the
→
kinematics. The possibility to make predictions i.e., to be able to estimate the configuration of the
system (q ,q˙ ) at time t fromthe initial conditions, relies on the existence of a dynamics which is of
t t
1
2 RE´MILASSALLEANDANABELACRUZEIRO
physical origin. This latter is expressed in the model by a further constraint on the q paths which
involves a function
:(x,v) R R (x,v) R
L ∈ × →L ∈
where x (resp. v) may stand for the position (resp. the speed). This function , which is called
L
a Lagrangian, contains all the physics of the model, and the related constraint which is called the
Euler Lagrangecondition (see [1], [2], [8]) reads
−
d
(0.1) ∂ (q ,q˙ )=∂ (q ,q˙ )
v t t x t t
dt L L
Integrating in time, it becomes
t
(0.2) ∂ (q ,q˙ ) ∂ (q ,q˙ )ds=c
v t t x s s
L − L
Z0
wherecissomeconstant. Undermildconditionson thepathsq Ω2 satisfyingtheEuler Lagrange
L ∈ [0,1] −
condition can be characterized as critical points of a functional which is called the action of
path
S
the system (see [1]). It is defined by
1
(q)= (q ,q˙ )dt
path t t
S L
Z0
and q is said to be critical if for all h Ω2 satisfying h =h =0
∈ [0,1] 0 1
d
(qǫ) =0
path ǫ=0
dǫS |
whereforǫ R,qǫ :=q+ǫhisaperturbationofthepath. Thetheoremwhichstatestheequivalence
∈
for a path q Ω2 to satisfy the Euler Lagrange condition (0.1) or to be a critical point of the
∈ [0,1] −
action is called Hamilton’s least action principle (see [1], [2], [8]). One of the goals of this paper is
to extend this result to some stochastic framework.
LetusdenotebySthesetoflawsofcontinuoussemi-martingalessuchthatforν S,thecanonical
∈
process satisfies ν a.s. for all t [0,1],
− ∈
t
W =W + vνds+Mν
t 0 s t
Z0
where (Mν) is some (local) martingale on the probability space (C([0,1],R), (C([0,1],R))ν,ν) for
t B
the filtration ( ν) (which denotes the ν usual augmentation of the filtration generated by the
Ft −
evaluation process), where (< Mν > ) is assumed to be absolutely continuous with a derivative
t
(αν). Setting ν =δDirac for q Ω2 we have ν a.s. for all t [0,1]
t q ∈ [0,1] − ∈
t
(0.3) W =W + vνds
t 0 s
Z0
where λ ν a.s.
⊗ −
vν =q˙
t t
i.e. ν SandMν =0. Thus,letusregardSasanextensionofthesetofthepathsdescribingadmis-
∈
siblekinematicsinanextendedstochasticcontext. Inthispaperwewillconsideraconstraint,which
we call the stochastic Euler Lagrange condition, that extends on S the classical Euler Lagrange
− −
condition;inparticularitisanaturalwaytointroducesomedynamicsinastochasticframework(see
WEAK CALCULUS OF VARIATIONS 3
also[12]). Namely, givensome suitable function :(t,x,y,a) [0,1] R R R (x,y,a) R
t
L ∈ × × × →L ∈
a law ν S will be said to satisfy the stochastic Euler Lagrangecondition if
∈ −
t
(0.4) ∂ (W ,vν,αν) ∂ (W ,vν,αν)ds=Nν
vLt t t t − xLs s s s t
Z0
for some ( ν) martingale (Nν) on (C([0,1],R), (C([0,1],R))ν,ν). Indeed by taking ν = δDirac
Ft − t B q
forq Ω2 andaLagrangian notdependingonaandt(0.4)isequivalentto(0.2). Byextending
∈ [0,1] L
Hamilton’sleastactionprincipletoSwewillrelatethedynamicalcondition(0.4)torecentproblems
of stochastic control which is our second motivation.
Consider the variational problems of the form
(0.5) inf( (ν):ν S,Law(W )=ν ,Law(W )=ν )
0 0 1 1
{S ∈ }
where
1
(0.6) (ν):=E (W ,vν,αν)ds .
S ν L s s s
(cid:20)Z0 (cid:21)
Such problems (extending those consideredin [18], [19]) have been recently investigatedin [22]; one
minimizes amonglawsofsemi-martingaleswith fixedinitial(resp. final) marginallawν (resp. ν ).
0 1
As a matter of fact they extend the so-called Scho¨dinger problem (see [5] and [13]), which can be
written as an entropy minimization problem. In this latter case, where the optimal processes may
be computed explicitly, it was noticed by J.C. Zambrini (see [12] for instance) that the optimum
solves a stochastic Euler Lagrangecondition (0.4). On the other hand in the general case of (0.5),
−
or by considering even more general problems where one fixes the joint law (see [14] for the case of
Bernstein’s processes) of (W ,W ) to be equal to a given Borel probability γ on Rd Rd,
0 1
×
1
(0.7) inf E (W ,vν,αν)ds :ν S,Law(W ,W )=γ
ν L s s s ∈ 0 1
(cid:18)(cid:26) (cid:20)Z0 (cid:21) (cid:27)(cid:19)
It is not convenient to use explicit formulae: in this paper we rather state a stochastic least action
principle which extends the classical one, proving that the optimum of these problems of stochastic
control are critical points of a stochastic action. In the classical Hamilton’s principle the paths
satisfying Euler Lagrange conditions are critical points and not necessarily minimum. Similarly,
−
within our stochastic extension we also allow processes satisfying (0.4) which are not minimum for
problems of the form (0.7). Actually, as it will be pointed out on examples on the classical Wiener
space for a quadratic cost, the situation is more complicated in the stochastic case of (0.4).
We then prove a Noether theorem, which we apply to the extremum of (0.5) and (0.7).
We found inspiration for applications to stochastic control essentially in [23],[26], where they
focus on Bernstein’s processes. Our results may be compared to those. We also show that in some
cases (0.4) is related to systems of coupled stochastic differential equation and to PDEs (such as
Navier Stokes equations).
−
Finally, let us add some comments concerning technical issues. When one expresses the proof of
the least action principle using probabilities by
(0.8) Ω2 ֒ S
[0,1] →δDirac
we set ν =δDirac and differentiate
ǫ qǫ
d
(ν ) .
ǫ ǫ=0
dǫS |
4 RE´MILASSALLEANDANABELACRUZEIRO
where
ν =(I +ǫh) ν
ǫ W ⋆
i.e. the variation becomes the image of ν by the measurable mapping τ : ω W ω +ǫh for
ǫh
∈ →
h Ω2 . In a stochastic framework one will have to consider more general perturbations of the
∈ [0,1]
form
τ :ω C([0,1],R) ω+k(ω) C([0,1],R)
k
∈ → ∈
where k := .k˙ ds is now random and adapted to the canonical filtration. Setting
0 s
R
(0.9) ν :=(I +ǫk) ν
ǫ W ⋆
werealizethatsomeessentialpropertieswillnotnecessarilyhold. Wedonothavethatτ isinvertible
k
(even almost surely) in general, and most of all in general we do not have a.e.
vνǫ(ω+ǫk(ω))=vν(ω)+ǫk˙ (ω).
t t t
As a consequence we cannot differentiate relevant functionals in all (adapted) random directions.
This is essentially due to the fact that such perturbations may not preserve the filtration. To over-
come these difficulties we build, for any ν S, some associated vector space of variation processes,
∈
which is roughly speaking the set directions towards which the variations of relevant functionals on
S can be handled as in the classical case. Then we prove that the space is wide enough to build a
derivative on S and to obtain a necessary and sufficient condition for (0.4) on S by means of a least
action principle.
The structure of the paper is the following. In Section 1 we fix the notations of the whole
paper and we recall the variational characterization of martingales, as well as some results about
transformationofmeasurespreservingthe filtration. Inthe followingsectionwedefinethe variation
processes and state their main properties, namely that they form a dense vector subspace of the
space of the adapted shifts of finite energy. In Section 3 we compute the changing formula of
the characteristics of a ν S given several particular transformations of measure (which will be
∈
used to compute explicitly the differential of actions on S). We also lift transformations of space
depending onthe time totransformationsonS. InSection4 wedefine the differentialoffunctionals
defined on S in such a way that extends the usual calculus of variations by (0.8). We note that the
definition extends directly to Borel probabilities on the space of continuous functions. In Section 5
we state precisely the definition of the laws satisfying the stochastic Euler-Lagrange condition, our
hypothesis on the Lagrangian(we call it regular) and we prove the stochastic least action principle
Theorem5.1,whichisourmainresult. Then,inSection6,wegeneralizeNoether’stheorem(suchas
itisformulatedby[2])tothisgeneralframework(Theorem6.1). Section7isdevotedtoapplications
in stochastic control and in particular to the problems considered in [22]. Namely we obtain some
information on the optimum of variational problems by using the stochastic least action principle
andNoether’stheorem. Finallyinthelastsectionweillustratethecontentof(0.4)inthecaseofthe
classicalactiondefined on S and we investigatethe correspondingcritical processes. In this case we
relate the results to systems of stochastic differential equations and provide some explicit examples
and counterexamples.
WEAK CALCULUS OF VARIATIONS 5
1. Preliminaries and notations
1.1. The path spaces and their stochastic counterparts. In the whole paper (Ω, , ) will
A P
alwaysdenoteacompleteprobabilityspaceand( )afiltrationonΩsatisfyingtheusualconditions
t
A
(i.e. right continuous and complete) such that for all t [0,1], . Under these hypothesis,
t
∈ A ⊂ A
following [7], we call (Ω, ,( ) , ) a complete stochastic basis. We emphasize that all these
t t∈[0,1]
A A P
assumptions arecrucialfor ourresults to hold. The most convenientwayto handle transformations
of laws of stochastic processes whose trajectories are sufficiently regular is to consider them as
random trajectories. Thus consider the space W = C([0,1],Rd) of continuous functions on [0,1]
with values in Rd. Processes will be often regarded as random elements taking their values in W,
and we will sometimes call the elements of W paths or trajectories.
We recall that W is a separable Banach space with respect to the norm . of the uniform
W
||
convergence (|ω|W := supt∈[0,1]|ω(t)|Rd). We can consider the related Borel sigma field B(W),
whichturnsW intoameasurablespace. Withinthisperspective,weconsideracontinuousstochastic
process (X ) as a / (W) measurable mapping X :Ω W.
t t∈[0,1]
A B − →
We denote by the set of Borel probabilities on W, which are the laws of the continuous
W
P
processes seen as random trajectories. We denote f the image of a measure by a measurable
⋆
P P
mapping f :Ω Ω where (Ω, ) is some other measurable space.
→ A
Inthe sequelweshallworkunderthe usualconditionsthatinsureexistenceofsufficientlyregular
e e e
modifications of martingales. Therefore we will always work on complete probability spaces with
filtrations satisfying the usual conditions (i.e. complete and right continuous). Taking this into
accountwe introduce the followingnotations. If η and is a sigma-fieldsuchthat (W)
W
∈P G G ⊂B
η will denote the η completion of i.e. the smallest sigma field which contains all the elements
G − G
of and all the η negligible sets. The unique extension of η to (W)η will be still denoted
W
G − ∈P B
by η. We denote by (W ) the evaluation process defined by
t
W :ω W W (ω) Rd
t t
∈ → ∈
for t [0,1]. For η , (W ) defines a process on the probability space (W, (W)η,η) : it is how
W t
∈ ∈P B
we will consider it in the sequel. The corresponding measurable mapping is the identity
I :ω W ω W
W
∈ → ∈
which is Borel measurable (and thus (W)η/ (W) measurable).
B B −
By considering a path ω W, and denoting by δDirac the Dirac measure concentrated on
∈ ω ∈PW
ω (i.e. δDirac(A)=I (ω), A (W)) we obtain an embedding
ω A ∈B
W ֒ .
δDirac W
→ P
Inthissenseanypathcanbeseenasastochasticprocess,andtheweakcalculusofvariationswewill
introduce below is such that, through this embedding, it extends the classical one. More generally
transformationsofmeasurescanbeformalizedbytransferenceplans(BorelprobabilitiesofW W).
×
Inthisworkweshallnotneedthisgenerality: transformationsofmeasurewillbemerelyachievedby
images of probabilities induced by measurablemappings. More precisely we will handle equivalence
classes of mappings. For (Ω, , ) a complete probability space, M ((Ω, ),(W, (W)) will denote
P
A P A B
thesetobtainedbyidentifying / (W) measurablemappingsf :Ω W whichare a.s.equal.
A B − → P−
Following [16] we will sometimes call the elements of this space morphisms of probability spaces. If
6 RE´MILASSALLEANDANABELACRUZEIRO
U M ((Ω, ),(W, (W))andf :Ω W isa / (W) measurablemappingwewillnote a.s.
P
∈ A B → A B − P−
U =f to denote that the equivalence class associated to f is U (i.e. the equivalence class U
P− P−
can be seen as the set of the / (W) measurable mappings g :Ω W such that a.s. f =g).
A B − → P−
Similarly if V M ((Ω, ),(W, (W)) we will note a.s. U = V to denote that U and V are
P
∈ A B P −
the same equivalence class.
P−
We introduce the Hilbert space ofthe absolutely continuousfunctions on [0,1]vanishing at t=0
with a square integrable derivative
. 1
H := h:[0,1] Rd,h:= h˙ ds, h˙ 2 ds<
→ s | s|Rd ∞
(cid:26) Z0 Z0 (cid:27)
(the so-called Cameron-Martin space) and we note <.,.> (resp. . ) the corresponding Hilbert
H H
||
product (resp. norm). Then we denote by W the subset of W whose elements are absolutely
abs
.
continuous functions (i.e. the set of ω W of the form ω := ω˙ ds) and by H the subset of H
∈ 0 s 0,0
given by
R
(1.10) H := h H :h =0 .
0,0 1
{ ∈ }
Note that by definition of H for h H we have h = h = 0. In the classical setting this set is
0,0 0 1
∈
the set of variations. Our main task will be to build its counterpart in the stochastic framework,
and we will need to consider spaces of (equivalence classes) of mappings taking almost surely their
values in such spaces.
WhenEisaBorelmeasurablesubsetofW,letusdenotebyL0( ,E)thespaceofthe equivalence
P P−
classes of mappings u (i.e. u M ((Ω, ),(W, (W))) such that a.s. u E. To control the
P
∈ A B P − ∈
integrability within this stochastic context, we also need the space L2( ,H) (resp. L2( ,H )) of
0,0
P P
the functions u L0( ,H) (resp. in L0( ,H )) such that E [u2 ]< i.e.
∈ P P 0,0 P | |H ∞
1
E u˙ 2 ds <
P | s|Rd ∞
(cid:20)Z0 (cid:21)
where a.s. u = .u˙ ds. Similarly L∞( ,W) (resp. L∞( ,H)) will denote the set of u
P − 0 s P P ∈
L0( ,W) for which there exists a K > 0 such that a.s. u < K (resp. u < K ). One
P R u P − | |W u | |H u
of the main differences with respect to the classical case is that our variations need to preserve
the filtrations, and our processes will be adapted. If (Ω, ,( ) , ) is a complete stochastic
t t∈[0,1]
A A P
basis (see above) we denote by L2( ,H) (similarly for the other Lp( ,H) and Lp( ,H ) spaces)
a P P P 0,0
the subspace of u L2( ,H) such that (t,ω) u (ω) Rd is ( ) adapted for any (and then
t t
∈ P → ∈ A −
all) continuous processes whose equivalence class is u. For u L0( ,W ) we can always
P− ∈ a P abs
.
find u˙ ds in the equivalence class of u so that (t,ω) u˙ is ( ) predicable: we choose such
0 s → t At −
modifications of the derivative unless expressively stated. In particular, for η , ( η) will
R ∈PW Ft t∈[0,1]
denote the η usual augmentation of the filtration generated by the evaluation process (W ), with
t
−
the convention η = (W)η. Thespace L2(η,H)(similarly forthe otherLp spaces)willdenote the
F1 B a a
setL2( ,H)for(Ω, , )=(W, (W)η,η)andthefiltration( η). Inthewholepaperλwilldenote
a P A P B Ft
the Lebesgue measure on [0,1]. Finally, for convenience of notations, if u:= .u˙ ds L0( ,W ),
0 s ∈ a P abs
and τ is an ( ) stopping time we note
At − R
.∧τ
π u:= u˙ ds
τ s
Z0
the process stopped by τ.
WEAK CALCULUS OF VARIATIONS 7
1.2. Martingales by duality. The variational characterization of martingales is a result of sto-
chasticcontrol(see[4]andthereferencestherein)whichreliesonduality. Sinceitwillplayacentral
role in this paper we provide here a precise statement of this result.
Let (Ω, ,( ) , ) be a complete stochastic basis. The mapping
t t∈[0,1]
A A P
.
r :β L2( ,Rd) E [β ν]ds L2( ,H)
∈ P → ν |Fs ∈ a P
Z0
defines a linear operator which is continuous by Jensen’s inequality. Its adjoint is given by the
operator
1
q :u L2( ,H) u := u˙ ds L2( ,Rd)
∈ a P → 1 s ∈ P
Z0
which is also linear and continuous. Indeed from the definitions we obtain directly
(1.11) EP[<q(u),β >Rd]=EP[<u,r(β)>H]
for any β L2( ,Rd) and u L2( ,H). By a classical result of functional analysis (see [21]
∈ P ∈ a P
Chapter VI Lemma 6 for instance) the orthogonal of the kernel of q (i.e. q−1({0L2(P,Rd)})) in the
Hilbert space L2( ,H) coincides with the closure of the range of r in L2( ,H). As a matter
a P a P
of fact, by a stopping argument, it is straightforward to see that this latter space is the space of
maps u L2( ,H) with a martingale derivative. A precise statement of this result is the following
∈ a P
orthogonal decomposition of L2( ,H) which immediatly yields the variational characterization of
a P
the martingale:
Proposition 1.1. For any complete stochastic basis (Ω, ,( ) , ) we have
t t∈[0,1]
A A P
(1.12) L2( ,H)= ( ,H) L2( ,H )
a P Ma P ⊕⊥ a P 0,0
where ( ,H) is the set of u L2( ,H) for which there exists a c`adla`g ( ) martingale
Ma P ∈ a P At −
(M ) such that a.s.
t t∈[0,1)
P−
.
u= M ds
s
Z0
and where
(1.13) L2( ,H ):= h L2( ,H): a.s. h =0
a P 0,0 { ∈ a P P − 1 }
In particular, for α L2( ,Rd), if
∈ P
(α):= u L2( ,H), a.s. u =α
C ∈ a P P − 1
(cid:8) (cid:9)
and
I :α L2( ,Rd) I(α) R
∈ P → ∈ ∪{∞}
is defined by
I(α):=inf E u2 :u (α) ,
P | |H ∈C
for any α D := α L2( ,Rd) : I((cid:0)α(cid:8)) <(cid:2) t(cid:3)he infimum(cid:9)(cid:1)is attained by a unique element
I
∈ { ∈ P ∞}
u⋆(α) (α), which is the orthogonal projection of any (and then of all) element(s) of (α) on
∈ C C
( ,H). Conversely if a u (α) is an element of ( ,H) it attains the infinimum of I(α).
a a
M P ∈C M P
8 RE´MILASSALLEANDANABELACRUZEIRO
Remark 1.2.1. For convenience of notations we considered Rd valued processes in the proofs of
−
Proposition 1.1andProposition 1.2buttheresultalsoholdsforprocesseswithvaluesinanyseparable
Hilbert space, as it is well known. Moreover by taking some trivial probability space one obtains as
a particular case that the orthogonal to H := h H :h =0 in H is the set of h H such that
0,0 1
{ ∈ } ∈
there exists a c Rd with a.s. for all s [0,1] h˙ =c .
h s h
∈ ∈
The following result is dual to Proposition 1.1 :
Proposition 1.2. Let (Ω, ,( ) , ) be a complete stochastic basis and let L2( ,Rd; ) be
t t∈[0,1] 1−
A A P P A
the set of the α L2( ,Rd) such that α is measurable. Then the set of α L2( ,Rd) that can
∈ P A1− ∈ a P
be attained by an adapted shift (i.e. such that there exists a u L2( ,H) with a.s. u =α) is
∈ a P P − 1
a dense subspace of L2( ,Rd; ) for the L2( ,Rd) topology.
1−
P A P
Proof: First note that the set of α L2( ,Rd) that can be attained by an adapted shift coincides
∈ P
withtherangeq(L2( ,H))ofq. Hence,ifwedenotebyq(L2( ,Rd))theclosureofq(L2( ,H)),we
a P a P a P
have to prove that q(L2( ,Rd)) = L2( ,Rd; ). By continuity q(L2( ,Rd) L2( ,Rd; ),
a P P A1− a P ⊂ P A1−
and since this latter space is closed we obtain
(1.14) q(L2( ,Rd)) L2( ,Rd; )
a P ⊂ P A1−
We now prove the converse inclusion. By duality, the closure of q(L2( ,H)) is the orthogonal in
a P
L2( ,H) to the kernel r−1( 0 ) of r, which is given by
a P { }
.
r−1( 0 ):= α L2( ,Rd): a.s. E [α ]ds=0 .
P s
{ } ∈ P P − |A
(cid:26) Z0 (cid:27)
By considering a right continuous modification of (E [α ]) , the martingale convergence
P t t∈[0,1]
|A
theorem yields
(1.15) r−1( 0 )= a.s. E [α ]=0 .
P 1−
{ } {P − |A }
Let X L2( ,Rd; ) and α r−1( 0 ). Then, by definition,
1−
∈ P A ∈ { }
EP[<X,α>Rd]=EP[<X,EP[α 1−]>Rd]=0.
|A
Hence
L2( ,Rd; ) r−1( 0 )⊥ =q(L2( ,Rd))
P A1− ⊂ { } a P
Together with (1.14) we obtain the desired result.
1.3. Transformations of measure preserving the filtration. In this section we introduce iso-
morphismsofafilteredprobabilityspace,whichareusuallyusedtoperformtransformationsofmea-
surepreservingthefiltrations,inparticularinMalliavincalculus. Herewewillhandlemorphismsof
probabilityspaces(seeabove). Indeedtheresultsweuseonlyprovideexistenceofequivalenceclasses
of mappings measurable with respect to completed sigma fields. Recall that M ((Ω, ),(W, (W))
P
A B
denotes the set of equivalence class of / (W) measurable mappings f : Ω W. To avoid
P− A B − →
heavy notations, whenever we handle a property which does not depend on the element in the
equivalence class, we implicitly denote with the same letter U M ((Ω, ),(W, (W)) and a
P
∈ A B
/ (W) measurable mapping in this class. However within this whole subsection we will make
A B −
the difference, in order to avoid any ambiguity on the notations. The main properties related
WEAK CALCULUS OF VARIATIONS 9
to transformations of measure preserving the filtrations concern their inverse images and pull-
backs. If is a sigma-field and U is a equivalence class of / (W) measurable mappings
G P− A B −
(i.e. U M ((Ω, ),(W, (W))), we denote by U−1( ) the completion of f−1( ) for any (and
P
∈ A B G P− G
then all) / (W) measurable f : Ω W such that a.s. U = f (i.e. U is the equivalence
A B − → P − P−
class of f, see above) and we call it the inverse image of by U. This name is justified by its
G
behaviour by pullback which we now recall.
Given η , and U (resp. X) a η equivalence class of (W)η/ (W) measurable mappings
W
∈ P − B B −
(resp. a equivalence class of / (W) measurable mappings), under the assumption that X
⋆
P− A B − P
is absolutely continuous w.r.t. η (i.e. X <<η) we have a.s.
⋆
P P −
fU ◦gX =fUe ◦gXe
for any measurable fU,fUe : W → W (resp. gX,gXe : Ω → W) in the η−equivalence class U (resp.
in the P−equivalence class X), where fU ◦gX : ω ∈ Ω → fU(gX(ω)) ∈ W (similarly for fUe ◦gXe).
We denote by U X the equivalence class of the / (W) measurable mapping f g for any
U X
◦ P− A B ◦
(and then all) such f and g . Then, for all sigma field of W
U X
G
(1.16) (U X)−1( )=X−1(U−1( )).
◦ G G
This is related to adapted processes in the following way. Denote by ( 0(W)) the filtration
Bt t∈[0,1]
generated by the evaluation process on W i.e. for all t [0,1]
∈
0(W):=σ(W ,s t),
Bt s ≤
Sinceweshalldealwithprogressivelymeasurableprocessesandc`adla`g modificationsofmartingales,
for η we will consider its usual augmentation ( η) (under η). We recall that
∈PW Ft t∈[0,1]
η := 0 (W)η
Ft Bt+
for t [0,1]. Here we adopt the conventions that at t = 1 the usual augmentation is just the
∈
completion and that 0 (W) := (W). Similarly, for U M ((Ω, ),(W, (W)), we will need to
B1+ B ∈ P A B
consider the following filtration generated by U. To any f :Ω W which is / (W) measurable
→ A B −
for all t [0,1] denote
∈
f :=σ(f ,s t)
Gt s ≤
where (f ) is the measurable process associated to f by
t
(f ):(t,ω) [0,1] W f (ω):=W (f(ω)) Rd.
t t t
∈ × → ∈
Notethatbydefinitionwealsohaveforallt [0,1], f =f−1( 0(W)), andthatitiselementaryto
∈ Gt Bt
check that f =f−1( 0 (W)). Then, if (Ω, ,( ) , ) is a complete stochastic basis we say
Gt+ Bt+ A At t∈[0,1] P
thatU is( ) adaptedifandonlyifany(andthenall) / (W) measurablef :Ω W suchthat
t
A − A B − →
a.s. U = f is ( ) adapted i.e. for all t [0,1], f . We define the filtration generated
P − At − ∈ Gt ⊂ At
by U, whichwe note ( U), to be the usualaugmentationwith respectto ofthe filtration( f) for
Gt P Gt
any (and then all) / (W) measurable f such that a.s. U =f. In particular for all t [0,1]
A B − P − ∈
it is elementary to check that with these definitions
(1.17) U =U−1( 0 (W))=U−1( U⋆P)
Gt Bt+ Ft
and that, due to our hypothesis on ( ), U is ( )-adapted if and only if for all t [0,1]
t t
A A ∈
(1.18) U .
Gt ⊂At
10 RE´MILASSALLEANDANABELACRUZEIRO
Thus,by(1.16)ifU is ( )-adaptedandX is( η)adaptedU X isalso( )-adapted. Conversely,
At Ft ◦ At
aneasycriterionfortheexistenceofanadaptedpullbackisthefollowingProposition. Weemphasize
that it only yields the existence of a measurable function which is measurable w.r.t. the completed
space with equality up to negligible sets.
Proposition 1.3. Assume that Y,X are two equivalence classes of / (W) measurable map-
P− A B −
pings (i.e. two elements of M ((Ω, ),(W, (W)). Then the following assertions are equivalent
P
A B
(i) Y is adapted to the filtration generated by X i.e. for all t [0,1]
∈
Y X
Gt ⊂Gt
where ( X) (resp. ( Y)) is the usual augmentation of the filtration generated by any
Gt Gt P−
/ (W) mesurable f : Ω W (resp. g : Ω W) whose equivalence class is X
X Y
A B → → P−
(resp. Y).
(ii) There exists a F ∈ MX⋆P((W,B(W)X⋆P),(W,B(W)) which is (FtX⋆P)− adapted such that
a.s.
P −
Y =F X
◦
where F X denotes the pullback defined above, and ( X⋆P) is the X -usual augmentation
◦ Ft − ⋆P
of the filtration generated by the evaluation process. In particular
F (X )=Y
⋆ ⋆ ⋆
P P
Moreover the two following assertions are equivalent :
(1) X and Y generate the same filtrations i.e. for all t [0,1]
∈
Y = X
Gt Gt
(2) There exists a F ∈ MX⋆P((W,B(W)X⋆P),(W,B(W)) which is (FtX⋆P)− adapted and a G ∈
MY⋆P((W,B(W)Y⋆P),(W,B(W)) which is (FtY⋆P)− adapted such that P −a.s.
Y =F X
◦
and
X =G Y
◦
Moreover X a.s.
⋆
P −
G F =I
W
◦
and Y a.s.
⋆
P −
F G=I
W
◦
Proof: Similar to the proof of the Yamada-Watanabe criterion (see [6]).
TheisomorphismsoffilteredprobabilityspacesplayakeyroleinMalliavin’swork. We nowstate
their definition. First note that whenever is complete, f : Ω W is / (W) measurable if
A → A B −
and only if it is / (W)f⋆P measurable. This ensures that the pullbacks below are well defined.
A B
Let η,ν . We say that U M ((W, (W)η),(W, (W)) with U η = ν is an isomorphism
W P ⋆
∈ P ∈ B B
of filtered probability spaces on (W, η,η) to (W, ν,ν) if U is ( η) adapted and if there exists
F. F. Ft −
U M ((W, (W)ν),(W, (W)) which is ( ν) adapted and such that η a.s.
∈ ν B B Ft − −
U U =I
e ◦ W
e