Weak calculus of variations for functionals of laws of semi-martingales
aa r X i v : . [ m a t h . P R ] J a n WEAK CALCULUS OF VARIATIONS FOR FUNCTIONALS OF LAWS OFSEMI-MARTINGALES
R´EMI LASSALLE AND ANA BELA CRUZEIRO
Abstract.
We develop a non-anticipating calculus of variations for functionals on a space oflaws of continuous semi − martingales, which extends the classical one. We extend Hamilton’s leastaction principle and Noether’s theorem to this generalized stochastic framework. As an applicationwe obtain, under mild conditions, a stochastic Euler − Lagrange condition and invariants for thecritical points of recent problems in stochastic control, namely for the semi-martingale optimaltransportation problems.
Keywords :
Stochastic analysis, Least action principle, Stochastic control, Semi-martingale optimaltransportation problems;
Mathematics Subject Classification :
Introduction
In this paper we formulate a weak calculus of variations which extends the classical one. Roughlyspeaking this enables to perform a calculus on functions defined on laws of semi-martingales. Weapply this calculus to obtain a stochastic extension of Hamilton’s least action principle. Recallthat the classical version of this principle provides a characterization of the paths satisfying theEuler − Lagrange condition as critical points of a functional which is called an action. Here wewill characterize laws of semi-martingales which satisfy a constraint that extends the classical one.Namely these laws will be proved to be critical points of a stochastic action. Once this extension isachieved we use it to relate some invariance properties of the critical processes to the symmetriesof the corresponding Lagrangian; in other words, we derive a stochastic extension of Noether’stheorem. Finally we consider applications to stochastic control, in particular to some semi-martingaleoptimal transportation problems. These problems were recently introduced in [22] with applicationto financial mathematics.As a warm up, let us give, in an informal way, some details on our motivation and on thedifficulties we overcome with our approach. The first motivation lies in classical mechanics. Inclassical mechanics one usually considers paths sufficiently regular to model the kinematics of asystem. In particular if one describes the trajectory of a classical particle by a path q : [0 , → R one will usually ask q to be sufficiently regular in order to provide a realistic description of theobservation. Namely it will be often assumed to be C for both its speed ˙ q t (:= dq t dt ) and itsacceleration ¨ q t to be defined. Thus, for the sake of simplicity let us first consider the space Ω , ofthe C paths q : [0 , → R as being the set of the paths providing an admissible description of thekinematics. The possibility to make predictions i.e., to be able to estimate the configuration of thesystem ( q t , ˙ q t ) at time t from the initial conditions, relies on the existence of a dynamics which is of physical origin. This latter is expressed in the model by a further constraint on the q paths whichinvolves a function L : ( x, v ) ∈ R × R → L ( x, v ) ∈ R where x (resp. v ) may stand for the position (resp. the speed). This function L , which is calleda Lagrangian, contains all the physics of the model, and the related constraint which is called theEuler − Lagrange condition (see [1], [2], [8]) reads(0.1) ddt ∂ v L ( q t , ˙ q t ) = ∂ x L ( q t , ˙ q t )Integrating in time, it becomes(0.2) ∂ v L ( q t , ˙ q t ) − Z t ∂ x L ( q s , ˙ q s ) ds = c where c is some constant. Under mild conditions on L the paths q ∈ Ω , satisfying the Euler − Lagrangecondition can be characterized as critical points of a functional S path which is called the action ofthe system (see [1]). It is defined by S path ( q ) = Z L ( q t , ˙ q t ) dt and q is said to be critical if for all h ∈ Ω , satisfying h = h = 0 ddǫ S path ( q ǫ ) | ǫ =0 = 0where for ǫ ∈ R , q ǫ := q + ǫh is a perturbation of the path. The theorem which states the equivalencefor a path q ∈ Ω , to satisfy the Euler − Lagrange condition (0.1) or to be a critical point of theaction is called Hamilton’s least action principle (see [1], [2], [8]). One of the goals of this paper isto extend this result to some stochastic framework.Let us denote by S the set of laws of continuous semi-martingales such that for ν ∈ S , the canonicalprocess satisfies ν − a.s. for all t ∈ [0 , W t = W + Z t v νs ds + M νt where ( M νt ) is some (local) martingale on the probability space ( C ([0 , , R ) , B ( C ([0 , , R )) ν , ν ) forthe filtration ( F νt ) (which denotes the ν − usual augmentation of the filtration generated by theevaluation process), where ( < M ν > t ) is assumed to be absolutely continuous with a derivative( α νt ). Setting ν = δ Diracq for q ∈ Ω , we have ν − a.s. for all t ∈ [0 , W t = W + Z t v νs ds where λ ⊗ ν − a.s. v νt = ˙ q t i.e. ν ∈ S and M ν = 0. Thus, let us regard S as an extension of the set of the paths describing admis-sible kinematics in an extended stochastic context. In this paper we will consider a constraint, whichwe call the stochastic Euler − Lagrange condition, that extends on S the classical Euler − Lagrangecondition ; in particular it is a natural way to introduce some dynamics in a stochastic framework (see
EAK CALCULUS OF VARIATIONS 3 also [12]). Namely, given some suitable function L : ( t, x, y, a ) ∈ [0 , × R × R × R → L t ( x, y, a ) ∈ R a law ν ∈ S will be said to satisfy the stochastic Euler − Lagrange condition if(0.4) ∂ v L t ( W t , v νt , α νt ) − Z t ∂ x L s ( W s , v νs , α νs ) ds = N νt for some ( F νt ) − martingale ( N νt ) on ( C ([0 , , R ) , B ( C ([0 , , R )) ν , ν ). Indeed by taking ν = δ Diracq for q ∈ Ω , and a Lagrangian L not depending on a and t (0.4) is equivalent to (0.2). By extendingHamilton’s least action principle to S we will relate the dynamical condition (0.4) to recent problemsof stochastic control which is our second motivation.Consider the variational problems of the form(0.5) inf ( {S ( ν ) : ν ∈ S , Law ( W ) = ν , Law ( W ) = ν } )where(0.6) S ( ν ) := E ν (cid:20)Z L ( W s , v νs , α νs ) ds (cid:21) . Such problems (extending those considered in [18], [19]) have been recently investigated in [22]; oneminimizes among laws of semi-martingales with fixed initial (resp. final) marginal law ν (resp. ν ).As a matter of fact they extend the so-called Sch¨odinger problem (see [5] and [13]), which can bewritten as an entropy minimization problem. In this latter case, where the optimal processes maybe computed explicitly, it was noticed by J.C. Zambrini (see [12] for instance) that the optimumsolves a stochastic Euler − Lagrange condition (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 ofBernstein’s processes) of ( W , W ) to be equal to a given Borel probability γ on R d × R d ,(0.7) inf (cid:18)(cid:26) E ν (cid:20)Z L ( W s , v νs , α νs ) ds (cid:21) : ν ∈ S , Law ( W , W ) = γ (cid:27)(cid:19) It is not convenient to use explicit formulae: in this paper we rather state a stochastic least actionprinciple which extends the classical one, proving that the optimum of these problems of stochasticcontrol are critical points of a stochastic action. In the classical Hamilton’s principle the pathssatisfying 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 forproblems of the form (0.7). Actually, as it will be pointed out on examples on the classical Wienerspace 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 theyfocus on Bernstein’s processes. Our results may be compared to those. We also show that in somecases (0.4) is related to systems of coupled stochastic differential equation and to PDEs (such asNavier − Stokes equations).Finally, let us add some comments concerning technical issues. When one expresses the proof ofthe least action principle using probabilities by(0.8) Ω , ֒ → δ Dirac S we set ν ǫ = δ Diracq ǫ and differentiate ddǫ S ( ν ǫ ) | ǫ =0 . R´EMI LASSALLE AND ANA BELA CRUZEIRO where ν ǫ = ( I W + ǫh ) ⋆ ν i.e. the variation becomes the image of ν by the measurable mapping τ ǫh : ω ∈ W → ω + ǫh for h ∈ Ω , . In a stochastic framework one will have to consider more general perturbations of theform τ k : ω ∈ C ([0 , , R ) → ω + k ( ω ) ∈ C ([0 , , R )where k := R . ˙ k s ds is now random and adapted to the canonical filtration. Setting(0.9) ν ǫ := ( I W + ǫk ) ⋆ ν we realize that some essential properties will not necessarily hold. We do not have that τ k is invertible(even almost surely) in general, and most of all in general we do not have a.e. v ν ǫ t ( ω + ǫk ( ω )) = v νt ( ω ) + ǫ ˙ k t ( ω ) . 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 aderivative on S and to obtain a necessary and sufficient condition for (0.4) on S by means of a leastaction principle.The structure of the paper is the following. In Section 1 we fix the notations of the wholepaper and we recall the variational characterization of martingales, as well as some results abouttransformation of measures preserving the filtration. In the following section we define the variationprocesses and state their main properties, namely that they form a dense vector subspace of thespace of the adapted shifts of finite energy. In Section 3 we compute the changing formula ofthe characteristics of a ν ∈ S given several particular transformations of measure (which will beused to compute explicitly the differential of actions on S ). We also lift transformations of spacedepending on the time to transformations on S . In Section 4 we define the differential of functionalsdefined on S in such a way that extends the usual calculus of variations by (0.8). We note that thedefinition extends directly to Borel probabilities on the space of continuous functions. In Section 5we state precisely the definition of the laws satisfying the stochastic Euler-Lagrange condition, ourhypothesis on the Lagrangian (we call it regular) and we prove the stochastic least action principleTheorem 5.1, which is our main result. Then, in Section 6, we generalize Noether’s theorem (such asit is formulated by [2]) to this general framework (Theorem 6.1). Section 7 is devoted to applicationsin stochastic control and in particular to the problems considered in [22]. Namely we obtain someinformation on the optimum of variational problems by using the stochastic least action principleand Noether’s theorem. Finally in the last section we illustrate the content of (0.4) in the case of theclassical action defined on S and we investigate the corresponding critical processes. In this case werelate the results to systems of stochastic differential equations and provide some explicit examplesand counterexamples. EAK CALCULUS OF VARIATIONS 5 Preliminaries and notations
The path spaces and their stochastic counterparts.
In the whole paper (Ω , A , P ) willalways denote a complete probability space and ( A t ) a filtration on Ω satisfying the usual conditions(i.e. right continuous and complete) such that for all t ∈ [0 , A t ⊂ A . Under these hypothesis,following [7], we call (Ω , A , ( A t ) t ∈ [0 , , P ) a complete stochastic basis. We emphasize that all theseassumptions are crucial for our results to hold. The most convenient way to handle transformationsof laws of stochastic processes whose trajectories are sufficiently regular is to consider them asrandom trajectories. Thus consider the space W = C ([0 , , R d ) of continuous functions on [0 , R d . 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 | . | W of the uniformconvergence ( | ω | W := sup t ∈ [0 , | ω ( t ) | R d ). We can consider the related Borel sigma field B ( W ),which turns W into a measurable space. Within this perspective, we consider a continuous stochasticprocess ( X t ) t ∈ [0 , as a A / B ( W ) − measurable mapping X : Ω → W .We denote by P W the set of Borel probabilities on W , which are the laws of the continuousprocesses seen as random trajectories. We denote f ⋆ P the image of a measure P by a measurablemapping f : Ω → e Ω where ( e Ω , e A ) is some other measurable space.In the sequel we shall work under the usual conditions that insure existence of sufficiently regularmodifications of martingales. Therefore we will always work on complete probability spaces withfiltrations satisfying the usual conditions (i.e. complete and right continuous). Taking this intoaccount we introduce the following notations. If η ∈ P W and G is a sigma-field such that G ⊂ B ( W ) G η will denote the η − completion of G i.e. the smallest sigma field which contains all the elementsof G and all the η − negligible sets. The unique extension of η ∈ P W to B ( W ) η will be still denotedby η . We denote by ( W t ) the evaluation process defined by W t : ω ∈ W → W t ( ω ) ∈ R d for t ∈ [0 , η ∈ P W , ( W t ) defines a process on the probability space ( W, B ( W ) η , η ) : it is howwe will consider it in the sequel. The corresponding measurable mapping is the identity I W : ω ∈ W → ω ∈ W which is Borel measurable (and thus B ( W ) η / B ( W ) − measurable).By considering a path ω ∈ W , and denoting by δ Diracω ∈ P W the Dirac measure concentrated on ω (i.e. δ Diracω ( A ) = I A ( ω ), A ∈ B ( W )) we obtain an embedding W ֒ → δ Dirac P W . In this sense any path can be seen as a stochastic process, and the weak calculus of variations we willintroduce below is such that, through this embedding, it extends the classical one. More generallytransformations of measures can be formalized by transference plans (Borel probabilities of W × W ).In this work we shall not need this generality: transformations of measure will be merely achieved byimages of probabilities induced by measurable mappings. More precisely we will handle equivalenceclasses of mappings. For (Ω , A , P ) a complete probability space, M P ((Ω , A ) , ( W, B ( W )) will denotethe set obtained by identifying A / B ( W ) − measurable mappings f : Ω → W which are P − a.s. equal.Following [16] we will sometimes call the elements of this space morphisms of probability spaces. If
R´EMI LASSALLE AND ANA BELA CRUZEIRO U ∈ M P ((Ω , A ) , ( W, B ( W )) and f : Ω → W is a A / B ( W ) − measurable mapping we will note P − a.s.U = f to denote that the P− equivalence class associated to f is U (i.e. the P− equivalence class U can be seen as the set of the A / B ( W ) − measurable mappings g : Ω → W such that P − a.s. f = g ).Similarly if V ∈ M P ((Ω , A ) , ( W, B ( W )) we will note P − a.s. U = V to denote that U and V arethe same P− equivalence class.We introduce the Hilbert space of the absolutely continuous functions on [0 ,
1] vanishing at t = 0with a square integrable derivative H := (cid:26) h : [0 , → R d , h := Z . ˙ h s ds, Z | ˙ h s | R d ds < ∞ (cid:27) (the so-called Cameron-Martin space) and we note < ., . > H (resp. | . | H ) the corresponding Hilbertproduct (resp. norm). Then we denote by W abs the subset of W whose elements are absolutelycontinuous functions (i.e. the set of ω ∈ W of the form ω := R . ˙ ω s ds ) and by H , the subset of H given by(1.10) H , := { h ∈ H : h = 0 } . Note that by definition of H for h ∈ H , we have h = h = 0. In the classical setting this set isthe 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 theirvalues in such spaces.When E is a Borel measurable subset of W , let us denote by L ( P , E ) the space of the P− equivalenceclasses of mappings u (i.e. u ∈ M P ((Ω , A ) , ( W, B ( W ))) such that P − a.s. u ∈ E . To control theintegrability within this stochastic context, we also need the space L ( P , H ) (resp. L ( P , H , )) ofthe functions u ∈ L ( P , H ) (resp. in L ( P , H , )) such that E P [ | u | H ] < ∞ i.e. E P (cid:20)Z | ˙ u s | R d ds (cid:21) < ∞ where P − a.s. u = R . ˙ u s ds . Similarly L ∞ ( P , W ) (resp. L ∞ ( P , H )) will denote the set of u ∈ L ( P , W ) for which there exists a K u > P − a.s. | u | W < K u (resp. | u | H < K u ). Oneof the main differences with respect to the classical case is that our variations need to preservethe filtrations, and our processes will be adapted. If (Ω , A , ( A t ) t ∈ [0 , , P ) is a complete stochasticbasis (see above) we denote by L a ( P , H ) (similarly for the other L p ( P , H ) and L p ( P , H , ) spaces)the subspace of u ∈ L ( P , H ) such that ( t, ω ) → u t ( ω ) ∈ R d is ( A t ) − adapted for any (and thenall) continuous processes whose P− equivalence class is u . For u ∈ L a ( P , W abs ) we can alwaysfind R . ˙ u s ds in the equivalence class of u so that ( t, ω ) → ˙ u t is ( A t ) − predicable: we choose suchmodifications of the derivative unless expressively stated. In particular, for η ∈ P W , ( F ηt ) t ∈ [0 , willdenote the η − usual augmentation of the filtration generated by the evaluation process ( W t ), withthe convention F η = B ( W ) η . The space L a ( η, H ) (similarly for the other L pa spaces) will denote theset L a ( P , H ) for (Ω , A , P ) = ( W, B ( W ) η , η ) and the filtration ( F ηt ). In the whole paper λ will denotethe Lebesgue measure on [0 , u := R . ˙ u s ds ∈ L a ( P , W abs ),and τ is an ( A t ) − stopping time we note π τ u := Z . ∧ τ ˙ u s ds the process stopped by τ . EAK CALCULUS OF VARIATIONS 7
Martingales by duality.
The variational characterization of martingales is a result of sto-chastic control (see [4] and the references therein) which relies on duality. Since it will play a centralrole in this paper we provide here a precise statement of this result.Let (Ω , A , ( A t ) t ∈ [0 , , P ) be a complete stochastic basis. The mapping r : β ∈ L ( P , R d ) → Z . E ν [ β |F νs ] ds ∈ L a ( P , H )defines a linear operator which is continuous by Jensen’s inequality. Its adjoint is given by theoperator q : u ∈ L a ( P , H ) → u := Z ˙ u s ds ∈ L ( P , R d )which is also linear and continuous. Indeed from the definitions we obtain directly(1.11) E P [ < q ( u ) , β > R d ] = E P [ < u, r ( β ) > H ]for any β ∈ L ( P , R d ) and u ∈ L a ( P , H ). By a classical result of functional analysis (see [21]Chapter VI Lemma 6 for instance) the orthogonal of the kernel of q (i.e. q − ( { L ( P , R d ) } )) in theHilbert space L a ( P , H ) coincides with the closure of the range of r in L a ( P , H ). As a matterof fact, by a stopping argument, it is straightforward to see that this latter space is the space ofmaps u ∈ L a ( P , H ) with a martingale derivative. A precise statement of this result is the followingorthogonal decomposition of L a ( P , H ) which immediatly yields the variational characterization ofthe martingale: Proposition 1.1.
For any complete stochastic basis (Ω , A , ( A t ) t ∈ [0 , , P ) we have (1.12) L a ( P , H ) = M a ( P , H ) ⊕ ⊥ L a ( P , H , ) where M a ( P , H ) is the set of u ∈ L a ( P , H ) for which there exists a c`adl`ag ( A t ) − martingale ( M t ) t ∈ [0 , such that P− a.s. u = Z . M s ds and where (1.13) L a ( P , H , ) := { h ∈ L a ( P , H ) : P − a.s. h = 0 } In particular, for α ∈ L ( P , R d ) , if C ( α ) := (cid:8) u ∈ L a ( P , H ) , P − a.s. u = α (cid:9) and I : α ∈ L ( P , R d ) → I ( α ) ∈ R ∪ {∞} is defined by I ( α ) := inf (cid:0)(cid:8) E P (cid:2) | u | H (cid:3) : u ∈ C ( α ) (cid:9)(cid:1) , for any α ∈ D I := { α ∈ L ( P , R d ) : I ( α ) < ∞} the infimum is attained by a unique element u ⋆ ( α ) ∈ C ( α ) , which is the orthogonal projection of any (and then of all) element(s) of C ( α ) on M a ( P , H ) . Conversely if a u ∈ C ( α ) is an element of M a ( P , H ) it attains the infinimum of I ( α ) . R´EMI LASSALLE AND ANA BELA CRUZEIRO
Remark 1.2.1.
For convenience of notations we considered R d − valued processes in the proofs ofProposition 1.1 and Proposition 1.2 but the result also holds for processes with values in any separableHilbert space, as it is well known. Moreover by taking some trivial probability space one obtains asa particular case that the orthogonal to H , := { h ∈ H : h = 0 } in H is the set of h ∈ H such thatthere exists a c h ∈ R d with a.s. for all s ∈ [0 ,
1] ˙ h s = c h . The following result is dual to Proposition 1.1 :
Proposition 1.2.
Let (Ω , A , ( A t ) t ∈ [0 , , P ) be a complete stochastic basis and let L ( P , R d ; A − ) bethe set of the α ∈ L ( P , R d ) such that α is A − measurable. Then the set of α ∈ L a ( P , R d ) that canbe attained by an adapted shift (i.e. such that there exists a u ∈ L a ( P , H ) with P − a.s. u = α ) isa dense subspace of L ( P , R d ; A − ) for the L ( P , R d ) topology. Proof:
First note that the set of α ∈ L ( P , R d ) that can be attained by an adapted shift coincideswith the range q ( L a ( P , H )) of q . Hence, if we denote by q ( L a ( P , R d )) the closure of q ( L a ( P , H )), wehave to prove that q ( L a ( P , R d )) = L ( P , R d ; A − ). By continuity q ( L a ( P , R d ) ⊂ L ( P , R d ; A − ),and since this latter space is closed we obtain(1.14) q ( L a ( P , R d )) ⊂ L ( P , R d ; A − )We now prove the converse inclusion. By duality, the closure of q ( L a ( P , H )) is the orthogonal in L a ( P , H ) to the kernel r − ( { } ) of r , which is given by r − ( { } ) := (cid:26) α ∈ L ( P , R d ) : P − a.s. Z . E P [ α |A s ] ds = 0 (cid:27) . By considering a right continuous modification of ( E P [ α |A t ]) t ∈ [0 , , the martingale convergencetheorem yields(1.15) r − ( { } ) = {P − a.s. E P [ α |A − ] = 0 } . Let X ∈ L ( P , R d ; A − ) and α ∈ r − ( { } ). Then, by definition, E P [ < X, α > R d ] = E P [ < X, E P [ α |A − ] > R d ] = 0 . Hence L ( P , R d ; A − ) ⊂ r − ( { } ) ⊥ = q ( L a ( P , R d ))Together with (1.14) we obtain the desired result.1.3. Transformations of measure preserving the filtration.
In this section we introduce iso-morphisms of a filtered probability space, which are usually used to perform transformations of mea-sure preserving the filtrations, in particular in Malliavin calculus. Here we will handle morphisms ofprobability spaces (see above). Indeed the results we use only provide existence of equivalence classesof mappings measurable with respect to completed sigma fields. Recall that M P ((Ω , A ) , ( W, B ( W ))denotes the set of P− equivalence class of A / B ( W ) − measurable mappings f : Ω → W . To avoidheavy notations, whenever we handle a property which does not depend on the element in theequivalence class, we implicitly denote with the same letter U ∈ M P ((Ω , A ) , ( W, B ( W )) and a A / B ( W ) − measurable mapping in this class. However within this whole subsection we will makethe difference, in order to avoid any ambiguity on the notations. The main properties related EAK CALCULUS OF VARIATIONS 9 to transformations of measure preserving the filtrations concern their inverse images and pull-backs. If G is a sigma-field and U is a P− equivalence class of A / B ( W ) − measurable mappings(i.e. U ∈ M P ((Ω , A ) , ( W, B ( W ))), we denote by U − ( G ) the P− completion of f − ( G ) for any (andthen all) A / B ( W ) − measurable f : Ω → W such that P − a.s. U = f (i.e. U is the P− equivalenceclass of f , see above) and we call it the inverse image of G by U . This name is justified by itsbehaviour by pullback which we now recall.Given η ∈ P W , and U (resp. X ) a η − equivalence class of B ( W ) η / B ( W ) − measurable mappings(resp. a P− equivalence class of A / B ( W ) − measurable mappings), under the assumption that X ⋆ P is absolutely continuous w.r.t. η (i.e. X ⋆ P << η ) we have P − a.s.f U ◦ g X = f e U ◦ g e X for any measurable f U , f e U : W → W (resp. g X , g e X : Ω → W ) in the η − equivalence class U (resp.in the P− equivalence class X ), where f U ◦ g X : ω ∈ Ω → f U ( g X ( ω )) ∈ W (similarly for f e U ◦ g e X ).We denote by U ◦ X the P− equivalence class of the A / B ( W ) measurable mapping f U ◦ g X for any(and then all) such f U and g X . Then, for all sigma field G of W (1.16) ( U ◦ X ) − ( G ) = X − ( U − ( G )) . This is related to adapted processes in the following way. Denote by ( B t ( W )) t ∈ [0 , the filtrationgenerated by the evaluation process on W i.e. for all t ∈ [0 , B t ( W ) := σ ( W s , s ≤ t ) , Since we shall deal with progressively measurable processes and c`adl`ag modifications of martingales,for η ∈ P W we will consider its usual augmentation ( F ηt ) t ∈ [0 , (under η ). We recall that F ηt := B t + ( W ) η for t ∈ [0 , t = 1 the usual augmentation is just thecompletion and that B ( W ) := B ( W ). Similarly, for U ∈ M P ((Ω , A ) , ( W, B ( W )), we will need toconsider the following filtration generated by U . To any f : Ω → W which is A / B ( W ) − measurablefor all t ∈ [0 ,
1] denote G ft := σ ( f s , s ≤ t )where ( f t ) is the measurable process associated to f by( f t ) : ( t, ω ) ∈ [0 , × W → f t ( ω ) := W t ( f ( ω )) ∈ R d . Note that by definition we also have for all t ∈ [0 , G ft = f − ( B t ( W )), and that it is elementary tocheck that G ft + = f − ( B t + ( W )). Then, if (Ω , A , ( A t ) t ∈ [0 , , P ) is a complete stochastic basis we saythat U is ( A t ) − adapted if and only if any (and then all) A / B ( W ) − measurable f : Ω → W such that P − a.s. U = f is ( A t ) − adapted i.e. for all t ∈ [0 , G ft ⊂ A t . We define the filtration generatedby U , which we note ( G Ut ), to be the usual augmentation with respect to P of the filtration ( G ft ) forany (and then all) A / B ( W ) − measurable f such that P − a.s. U = f . In particular for all t ∈ [0 , G Ut = U − ( B t + ( W )) = U − ( F U ⋆ P t )and that, due to our hypothesis on ( A t ), U is ( A t )-adapted if and only if for all t ∈ [0 , G Ut ⊂ A t . Thus, by (1.16) if U is ( A t )-adapted and X is ( F ηt ) adapted U ◦ X is also ( A t )-adapted. Conversely,an easy criterion for the existence of an adapted pullback is the following Proposition. We emphasizethat it only yields the existence of a measurable function which is measurable w.r.t. the completedspace with equality up to negligible sets. Proposition 1.3.
Assume that
Y, X are two P− equivalence classes of A / B ( W ) − measurable map-pings (i.e. two elements of M P ((Ω , A ) , ( W, B ( W )) . Then the following assertions are equivalent(i) Y is adapted to the filtration generated by X i.e. for all t ∈ [0 , G Yt ⊂ G Xt where ( G Xt ) (resp. ( G Yt ) ) is the P− usual augmentation of the filtration generated by any A / B ( W ) mesurable f X : Ω → W (resp. g Y : Ω → W ) whose P− equivalence class is X (resp. Y ).(ii) There exists a F ∈ M X ⋆ P (( W, B ( W ) X ⋆ P ) , ( W, B ( W )) which is ( F X ⋆ P t ) − adapted such that P − a.s. Y = F ◦ X where F ◦ X denotes the pullback defined above, and ( F X ⋆ P t ) − is the X ⋆ P -usual augmentationof the filtration generated by the evaluation process. In particular F ⋆ ( X ⋆ P ) = Y ⋆ P Moreover the two following assertions are equivalent :(1) X and Y generate the same filtrations i.e. for all t ∈ [0 , G Yt = G Xt (2) There exists a F ∈ M X ⋆ P (( W, B ( W ) X ⋆ P ) , ( W, B ( W )) which is ( F X ⋆ P t ) − adapted and a G ∈ M Y ⋆ P (( W, B ( W ) Y ⋆ P ) , ( W, B ( W )) which is ( F Y ⋆ P t ) − adapted such that P − a.s.Y = F ◦ X and X = G ◦ Y Moreover X ⋆ P − a.s. G ◦ F = I W and Y ⋆ P − a.s. F ◦ G = I W Proof:
Similar to the proof of the Yamada-Watanabe criterion (see [6]).The isomorphisms of filtered probability spaces play a key role in Malliavin’s work. We now statetheir definition. First note that whenever A is complete, f : Ω → W is A / B ( W ) − measurable ifand only if it is A / B ( W ) f ⋆ P measurable. This ensures that the pullbacks below are well defined.Let η, ν ∈ P W . We say that U ∈ M P (( W, B ( W ) η ) , ( W, B ( W )) with U ⋆ η = ν is an isomorphismof filtered probability spaces on ( W, F η. , η ) to ( W, F ν. , ν ) if U is ( F ηt ) − adapted and if there exists e U ∈ M ν (( W, B ( W ) ν ) , ( W, B ( W )) which is ( F νt ) − adapted and such that η − a.s. e U ◦ U = I W EAK CALCULUS OF VARIATIONS 11 and ν − a.s. U ◦ e U = I W where I W : ω ∈ W → ω ∈ W . In this case e U is unique and we call it the inverse of U . Note thatby (1.16) and (1.17) we have, G Ut = F ηt and G e Ut = F νt for all t ∈ [0 , f U (resp. f e U ) B ( W ) η / B ( W ) (resp. B ( W ) ν / B ( W )) measurable whose equivalenceclass is U (resp. e U ) the equality f U ◦ f e U = I W (resp. f e U ◦ f U = I W ) is merely assumed tohold η − a.s. (resp. ν − a.s. ). In particular this definition doesn’t claim the existence of a Borelmeasurable bijection (invertible everywhere) in the equivalence class of U . Note also that any such U induces an obvious isometric identification of the L p spaces of η and U ⋆ η (and of the L pa ( η, H ) and L pa ( U ⋆ η, H )) which is sometimes used as an alternative definition (just consider k ∈ L pa ( U ⋆ η, H ) → k ◦ U ∈ L pa ( η, H ) for instance). Useful characterizations to handle isomorphisms of filtered probabilityspaces are provided by the following proposition: Proposition 1.4.
Let η ∈ P W and U ∈ M P (( W, B ( W ) η ) , ( W, B ( W )) . Then the following areequivalent(i) U is an isomorphism of filtered probability spaces on ( W, F η. , η ) to ( W, F U ⋆ η. , U ⋆ η ) (ii) For all t ∈ [0 , G Ut = F ηt (iii) (1.19) holds for all t ∈ [0 , and, for all X, Y ∈ M P ((Ω , A ) , ( W, B ( W )) defined on the samecomplete space (Ω , A , P ) such that X ⋆ P = Y ⋆ P = η, we have P − a.s. U ( X ) = U ( Y ) = ⇒ P − a.s. X = Y (iv) (1.19) holds for all t ∈ [0 , and, for every complete probability space (Ω , A , P ) and for all Y ∈ M P ((Ω , A ) , ( W, B ( W )) such that Y ⋆ P = U ⋆ η , there exists a X ∈ M P ((Ω , A ) , ( W, B ( W )) with X ⋆ P = η such that P − a.s. Y = U ◦ X Moreover, in the case where one of the above assumptions is satisfied, X in ( iv ) is unique. Proof:
Similar to the proof of the Yamada-Watanabe criterion (see [6]).
Remark 1.3.1.
In practice ( ii ) is useless to obtain ( i ) ; since one may use ( iii ) to prove that G U = F η = B ( W ) η , ( iii ) is the most efficient criterion to obtain ( i ) . In the sequel we will need to control the initial behaviour of isomorphisms, namely we will needthem to preserve the initial information. For this reason we set the following definition.
Definition 1.1.
For η ∈ P W , we denote by I f ( η ) the set of isomorphisms of filtered probabilityspaces U ∈ M η (( W, B ( W ) η ) , ( W, B ( W )) on ( W, F η. , η ) to ( W, F U ⋆ η. , U ⋆ η ) which further satisfy σ ( W ) η = σ ( U ) η Remark 1.3.2.
Note that by Dynkin’s lemma an isomorphism of probability spaces on ( W, F η. , η ) to ( W, F U ⋆ η. , U ⋆ η ) is an element of I f ( η ) if and only if there exists a B ( R d ) U ⋆ η / B ( R d ) − measurablefunction f : R d → R d and a B ( R d ) W ⋆ η / B ( R d ) − measurable function g : R d → R d satisfying U ⋆ η − a.s. (resp. W ⋆ η − a.s. ) g ◦ f = I R d (resp. f ◦ g = I R d ) such that η − a.s. U = g ( W ) and U ⋆ η − a.s. e U = f ( W ) where e U denotes the inverse of U . In particular for ν ∈ P W , if U is an isomorphism of probability spaces on ( W, F η. , η ) to ( W, F ν. , ν ) then U ∈ I f ( η ) if and onlyif e U ∈ I f ( ν ) . Using this it is straightforward to check that if U ∈ I f ( η ) and T ∈ I f ( U ⋆ η ) then T ◦ U ∈ I f ( η ) . Some spaces of laws of continuous semi-martingales.
The space S . Within our stochastic extensions, the space S will play a role analogous to thepath space Ω , of the R d -valued C − functions on [0 ,
1] in the classical calculus of variations. Inthe whole paper S will denote the space of the Borel probabilities ν ∈ P W for which there exist(i) A continuous ( F νt ) − local martingale ( f M νt ) and a ( F νt ) − predicable process ( e v νt ) defined onthe space ( W, B ( W ) ν , ν ) such that ν − a.s. for any t ∈ [0 , W t = W + f M νt + Z t e v νs ds (ii) Two M d ( R ) valued ( F νt ) predicable processes ( e α νt ) and ( e σ νt ) related by e α νt := e σ νt ( e σ νt ) † such that Z | ( e α νt ) ij | dt < ∞ Z | ( e σ νt ) ij | dt < ∞ for all i, j ∈ [1 , d ] and ν − a.s.< ( f M ν ) i , ( f M ν ) j > = Z . ( e α νs ) ij ds where M d ( R ) denotes the set of d × d matrices endowed with its usual topology, and where < ., . > denotes the predicable quadratic co-variation process.Note that for ν ∈ S the continuous local martingale ( f M νt ) and the finite variation term ( R . e v νt dt )are unique up to a ν -evanescent set. Hence to ν we associate canonically its martingale part (resp.its finite variation part) which is defined to be M ν (resp. b ν ), the element of L a ( ν, W ) (resp. L a ( ν, W abs )) such that ν − a.s. for all t , W t ◦ M ν = f M νt (and ν − a.s. b ν = R . e v νt dt ). On theother hand we note ( v νt ) (resp. ( α νt )) the equivalence classes of ( F νt ) − optional processes which are λ ⊗ ν equal to ( e v νt ) (resp. to ( e α νt )). Actually we can always chose a ( F νt ) − predicable process in theequivalence class of ( v νt ) (resp. of ( α νt )) and we will do this, unless it is explicitly stated that wetake it optional, or more precisely (when such a modification exists) that we take it right continuousand ( F νt ) − adapted. EAK CALCULUS OF VARIATIONS 13
Definition 1.2.
For ν ∈ S we note M ν (resp. b ν ), both elements of M ν (( W, B ( W ) ν ) , ( W, B ( W )) ,the martingale part (resp. the finite variation part) of ν and we call ( R . v νt dt ) (resp. ( R . α νt dt ) ), suchthat ν − a.s. < M ν > = Z . α νt dt and ν − a.s. b ν = Z . v νt dt the characteristics of ν . Variation processes
Vector space of variation processes.Definition 2.1.
For ν ∈ P W we denote by V ν the set of h ∈ L a ( ν, H ) such that for any U ∈ I f ( ν ) (see Definition 1.1) we also have U h ∈ I f ( ν ) where (2.21) U h := U + h Moreover V ∞ ν (resp. V ν , resp. V , ∞ ν ) will denote the linear subspace of V ν defined by V ∞ ν := V ν ∩ L ∞ ( ν, W ) V ν := V ν ∩ L ( ν, H , ) V , ∞ ν := V ∞ ν ∩ V ν We say that h ∈ V ν is the variation process of the curve ( τ ǫh⋆ ν ) ǫ ∈ R ⊂ P W at ν where for any k ∈ L a ( ν, H ) , τ k denotes τ k := I W + k Proposition 2.1.
For any ν ∈ P W , V ν is a vector subspace of L a ( ν, H ) . Proof:
Consider ν ∈ P W , an element h ∈ V ν , ǫ ∈ R . We first prove that ǫh ∈ V ν . Let U ∈ I f ( η ).For ǫ ∈ R / { } set U ǫh = U + ǫh We have to prove that U ǫh ∈ I f ( η ). Note that(2.22) U ǫh = ǫ ( U ǫ + h )where U ǫ := 1 ǫ U Since U ∈ I f ( ν ) by Proposition 1.4 we also have U ǫ ∈ I f ( ν ). Hence by the definition of V ν , U ǫ + h ∈ I f ( ν ). Similarly, by (2.22) we have U ǫh ∈ I f ( ν ). Therefore, if h ∈ V ν and ǫ ∈ R , ǫh ∈ V ν .We take h, k ∈ V ν and we want to derive h + k ∈ V ν . For U ∈ I f ( ν ), we need to prove that U k + h := U + h + k ∈ I f ( ν ) . Since k ∈ V ν , U k := U + k ∈ I f ( ν ). On the other hand since h ∈ V ν and U h + k = U k + h , thedefinition of V ν yields U h + k ∈ I f ( ν ). Therefore V ν is a vector space. Remark 2.1.1.
First note that for ν ∈ P W , by Proposition 1.4, it is straightforward to check that H ֒ → V ∞ ν ⊂ V ν ⊂ L a ( ν, H ) and H , ֒ → V ∞ , ν ⊂ V ν ⊂ L a ( ν, H , ) . In the case where ν = δ Dirach (Dirac measure concentrated on h ∈ H ) we have L a ( ν, H ) ≃ H and L a ( ν, H , ) ≃ H , so that all the inclusions become equalities. On the other hand if we take ν ∈ S to be the law of the solution to Tsirelson’s equation (see [24] or [6] ) which we note dX t = dB t + v t ( X ) dt then the ν -equivalence class of I W is an isomorphism of probability spaces on ( W, F ν. , ν ) and R . v s ds ∈ L a ( ν, H ) but I W − R . v s ds is not an isomorphism of filtered probability spaces (see [10] , [25] , [11] ).By localization one can build examples of probabilities in S where V ν is a proper linear subspace of L a ( ν, H , ) . However we shall see that fortunately for any ν ∈ P W these injections (except those of H and H , ) are always dense in the topology of L ( ν, H ) for any ν ∈ S . The following Proposition shows that the variation processes are invariant by isomorphisms. Sincewe will not use it in the sequel, it can be skipped in a first reading.
Proposition 2.2.
For any ν ∈ P W and any U ∈ I f ( ν ) we have V U ⋆ ν ≃ j U V ν More precisely the mapping j U defined by j U : h ∈ V U ⋆ ν → h ◦ U ∈ V ν is a bijection (and an isometry)of V U ⋆ ν onto V ν whose inverse is given by j e U : h ∈ V ν → h ◦ e U ∈ V U ⋆ ν where e U is the inverse of U . Proof:
Consider ν ∈ P W and U ∈ I f ( ν ), whose inverse is denoted by e U ∈ I f ( U ⋆ ν ). By symmetry,to prove the result it is sufficient to swhow that for h ∈ V U ⋆ ν we have j U ( h ) ∈ V ν . Hence we consider T ∈ I f ( ν ), whose inverse is denoted by e T ∈ I f ( T ⋆ ν ) and we prove that T j U ( h ) := T + j U ( h )is an element of I f ( ν ). To see this, note that T ◦ e U ∈ I f ( U ⋆ ν ) (with inverse U ◦ e T ∈ I f ( T ⋆ ν )). Since h ∈ V U ⋆ ν , we have T h := T ◦ e U + h ∈ I f ( U ⋆ ν )We denote by S h ∈ I f (( T h ◦ U ) ⋆ ν ) the inverse of T h . Finally, by definition, ν − a.s.T j U ( h ) = T h ◦ U so that T j U ( h ) ∈ I f ( ν ) with inverse e U ◦ S h . This achieves the proof. EAK CALCULUS OF VARIATIONS 15
Density of variation processes.Proposition 2.3.
Let (Ω , A , ( A t ) t ∈ [0 , , P ) be a complete stochastic basis and let p n be the linearoperator defined by (2.23) p n : u ∈ L a ( P , H ) → p n ( u ) := Z . n − X k =2 n [ kn , k +1 n ) ( s ) (cid:16) u k − n − u k − n (cid:17) ds ∈ L a ( P , H ) Then for any u ∈ L a ( P , H ) ( p n ( u )) converges to u strongly in L a ( P , H ) . Moreover it satisfies P − a.s. (2.24) | p n ( u ) | W ≤ n − | u | W and (2.25) | p n ( u ) | H ≤ | u | H Proof:
Inequality (2.24) follows from the definition. On the other hand (2.25) directly follows fromJensen’s inequality so that p n ( u ) is also a contraction of L ( P , H ) i.e. | p n ( u ) | L ( P ,H ) ≤ | u | L ( P ,H ) .For ǫ > β be the primitive of an elementary predicable process such that | u − β | L ( P ,H ) < ǫ β is of the form(2.26) β := n X k =0 Z . ( t k ,t k +1 ] ( s ) α k ds where n ∈ N , ( t k ) k ∈ [ | ,n +1 | ] ⊂ [0 ,
1] is increasing, and where for any k , α k is an element of L a ( P , R d )which is A t k measurable. Such a β always exists by a well known result (See [6]). Then one can seethat there exists N β ∈ N and a constant C β ∈ [0 , ∞ ) such that, for any n > N β , E P (cid:2) | p n ( β ) − β | R d (cid:3) ≤ n C β ( n + 1) max k ∈ [ | ,n +1 | ] E P (cid:2) | α k | R d (cid:3) This shows that ( p n ( β )) converges to β for any simple process β . Together with the fact that forany n ∈ N p n is a linear contraction this yields | p n ( u ) − u | L ( P ,H ) ≤ | p n ( u − β ) | L ( P ,H ) + | p n ( β ) − β | L ( P ,H ) + | β − u | L ( P ,H ) ≤ | u − β | L ( P ,H ) + | p n ( β ) − β | L ( P ,H ) ≤ ǫ + | p n ( β ) − β | L ( P ,H ) By using the convergence of ( p n ( β )) to β which is a simple process we finally getlim sup | p n ( u ) − u | L ( P ,H ) ≤ ǫ Since ǫ > p n ( u )) converges to u strongly in L ( P , H ). Proposition 2.4.
For any ν ∈ P W , V ∞ ν ∩ L ∞ ( ν, H ) (and then V ∞ ν and V ν ) is dense (strongly) in L a ( ν, H ) . Proof:
First, note that the space L ∞ ( ν, H ) is dense in L ( ν, H ) for the topology of L ( ν, H ). Indeedor u ∈ L ( ν, H ) by taking(2.27) τ n := inf( { t : | π t u | H > n } ) ∧ u n := π τ n u := R . [0 ,τ n ] ( s ) ˙ u s ds , the dominated convergence theorem yields the convergence of u n to u in L ( ν, H ). Hence it is sufficient to prove that V ν ∩ L ∞ ( ν, H ) is dense in L ∞ ( ν, H ) for the L ( ν, H ) topology. To prove this we set V delν := { p n ( u ) , u ∈ L ∞ a ( ν, H ) , n ∈ N } where p n is the operator defined in Proposition 2.3. To prove the density of V ν ∩ L ∞ ( ν, H ) it issufficient to prove that(2.28) V delν ⊂ V ν ∩ L ∞ ( ν, H )and that V delν is dense in L ∞ ( ν, H ). First note that by (2.25) we already know that V delν ⊂ L ∞ ( ν, H ).Hence the density follows directly by the definition of V delν together with Proposition 2.3. So wejust have to prove (2.28): we consider a h ∈ V delν and set(2.29) U h := U + h where U ∈ I f ( ν ). We consider two measurable mappings X : Ω → W and Y : Ω → W defined onthe same probability space (Ω , A , P ) such that X ⋆ P = Y ⋆ P = ν and we assume that(2.30) U h ( X ) = U h ( Y )By Proposition 1.4, to obtain that U h ∈ I f ( ν ) it is sufficient to prove that we necessarily have P − a.s. X = Y , and that for all t ∈ [0 , G U h t = F νt We postpone (2.31) to the end of the proof and we set τ := inf( { t : X t = Y t } ) ∧ U ∈ I f ( ν ) we also have(2.32) τ = inf ( { t : U t ( X ) = U t ( Y ) } ) ∧ U ∈ I f ( ν ) there exists a B ( R d ) U ⋆ η / B ( R d ) − measurable mapping f : R d → R d sothat ν − a.s. W = f ( U )by (2.30), since ν − a.s. h = h = 0 it yields P − a.s. (2.33) X = Y On the other hand (see Proposition 2.3) there exists a λ > h is adapted to the filtration( H λt ) where for t ≥ λ (resp. t < λ ) H λt = F νt − λ (resp. H t = B ( W ) ν ). Using this together with (2.33)we obtain P − a.s. inf { t : h t ( X ) = h t ( Y ) } ≥ ( τ + λ ) ∧ τ ≥ ( τ + λ ) ∧ EAK CALCULUS OF VARIATIONS 17 so that
P − a.s. τ = 1 and X = Y . Hence to show that U h ∈ I f it is now sufficient to prove (2.31)for all t ∈ [0 , t ∈ [0 , F νt ⊂ G U h t We choose N λ ∈ N such that λN λ > t i = iN λ for a i ∈ [0 , N λ − ∩ N . For t ∈ [ t , t )we have ν − a.s. h t = 0 for all t ∈ [ t , t ) so that σ ( U s , s ≤ t ) ν = σ ( U hs , s ≤ t ) ν . Since U ∈ I f ( ν ) wehave G U h t = G Ut = F νt for t ∈ [ t , t ). Finally we assume that (2.34) holds for any t ∈ [ t , t i ) and wetake t ∈ [ t i , t i +1 ). Almost surely with respect to ν , we have U t = U ht − h t . But h t is F νt − λ measurableand therefore F νt i − measurable so that by the induction hypothesis it is also G U h t i − measurable andhence G U h t − measurable. Thus, for all t ∈ [ t i , t i +1 )(2.35) σ ( U s , s ≤ t ) ⊂ G U h t Using the fact that U ∈ I f ( ν ) together with the right continuity of ( G U h t ), (2.35) yields F νt = G Ut ⊂ G U h t for all t ∈ [ t i , t i +1 ), and by induction for all t ∈ [0 , h ∈ V ν ∩ L ∞ ( ν, H ) sothat (2.28) holds. The proof is complete.2.3. Variation processes with vanishing endpoints.Proposition 2.5.
For any ν ∈ P W the space L ∞ ( ν, W ) ∩ L a ( ν, H , ) is dense in L a ( ν, H , ) for thetopology of L ( ν, H ) . Proof:
For u := R . ˙ u s ds ∈ L a ( ν, H , ) and τ a ( F νt ) − stopping time we set(2.36) k τ [ u ] := Z . ∧ τ ˙ u s ds − Z . ( τ, ( s )1 − τ u τ ds By Jensen’s inequality (2.36) defines a linear and continuous operator k τ : L a ( ν, H , ) → L a ( ν, H , ).More precisely for u ∈ L a ( ν, H , ), since ν − a.s. u = 0, Jensen’s inequality yields ν − a.s. (2.37) | k τ [ u ] | H ≤ | u | H and(2.38) | k τ [ u ] | W ≤ t ≤ τ | u t | = 2 | π τ u | W Since | . | W ≤ | . | H by (2.38) we have, ν − a.s., (2.39) | k τ [ u ] | W ≤ | π τ u | H For any n ∈ N , let ( τ n ) be the sequence of stopping time associated to u by (2.27). We define asequence ( u n ) by setting u n := k τ n [ u ]. For all n ∈ N u n ∈ L a ( ν, H , ) and by (2.39) u n ∈ L ∞ ( ν, W ).Hence ( u n ) n ∈ N ⊂ L a ( ν, H , ) ∩ L ∞ ( ν, W ). Finally by (2.37) the dominated convergence theoremyields the convergence of ( u n ) to u and therefore the density of L ∞ ( ν, W ) ∩ L a ( ν, H , ). Lemma 2.1.
For any ν ∈ P W , V , ∞ ν is dense in L a ( ν, H , ) . Proof:
For n ∈ N , n ≥ q n : u ∈ L a ( ν, H ) → q n ( u ) := Z . [1 − n , ( s ) nu − n ds ∈ L a ( ν, H )and(2.41) r n := p n − q n where p n is defined in Proposition 2.3. In particular for any u ∈ L a ( ν, H ) ν − a.s. (2.42) | q n ( u ) | W ≤ | u | W We set V ,delν := { r n ( u ) : u ∈ L a ( ν, H , ) ∩ L ∞ ( ν, W ) , n ∈ N } By Proposition 2.5 it is sufficient to prove that V ,delν is a dense subset of L ∞ ( ν, W ) ∩ L a ( ν, H , )for the L ( ν, H ) topology and that(2.43) V ,delν ⊂ V , ∞ ν The density follows by Jensen’s inequality. Indeed for u ∈ L ∞ ( ν, W ) ∩ L a ( ν, H , ) since ν − a.s.u = 0, Jensen’s inequality yields ν − a.s. , | q n ( u ) | H ≤ Z − n | ˙ u s | ds and | q n [ u ] | L ( ν,H ) ≤ √ | ( I H − π − n ) u | L ( ν,H ) Bby the dominated convergence theorem q n ( u ) converges strongly to 0 L ( ν,H ) in L ( ν, H ). Henceby Proposition 2.3, ( r n ( u )) converges to u . On the other hand by (2.24) and (2.42)(2.44) V ,delν ⊂ L ∞ ( ν, W )and by (2.23), (2.40) and (2.41) we have ν − a.s. r n (1) = 0 so that we also have(2.45) V ,delν ⊂ L a ( ν, H , )Finally for a U ∈ I f ( ν ) and h ∈ V ,delν , similarly to the proof of Proposition 2.4 we obtain that U + h ∈ I f ( ν ) i.e. V ,delν ⊂ V ν . Together with (2.44) and (2.45) we derive (2.43), by which theresult follows. 3. Transformations of S Transformation formulas.
The aim of this subsection is to prove Proposition 3.3 and Lemma 3.1which are the only results of this subsection that will be used in the sequel. Before we prove Propo-sition 3.1 in order to derive Proposition 3.2. Both results can be skipped; however this latterproposition justifies the definitions of the set V ν . Despite their apparent generality Proposition 3.1and Proposition 3.2 assume some further integrability assumptions due to the fact that their proofsinvolve the dual predicable projection. In the particular case of Lemma 3.1 and Proposition 3.3 westress that the use of isomorphisms of filtered probability spaces allows us to drop this condition. EAK CALCULUS OF VARIATIONS 19
Proposition 3.1.
Given a complete stochastic basis (Ω , A , ( A t ) t ∈ [0 , , P ) , let U : Ω → W be a A / B ( W ) -measurable mapping such that P -a.s. for all t ∈ [0 , U t = U + M ut + Z t ˙ u s ds where ( t, ω ) ∈ [0 , × Ω → ˙ u t ∈ R d is ( A t ) − predicable with, for all T < , (3.47) E P "Z T | ˙ u s | R d ds < ∞ and where ( M ut ) is a continuous R d − valued ( A t ) − local martingale such that P− a.s. for all i, j ∈ [1 , d ](3.48) < ( M u ) i , ( M u ) j > = Z . ( α us ) ij ds for some M d ( R ) − valued ( A t ) − predicable process ( α us ) . By setting ν := U ⋆ P we have ν ∈ S . Moreover P − a.s. (3.49) Z . α νt ◦ U dt = Z . α ut dt and P − a.s. (3.50) Z . v νt ◦ U dt = Z . E P [ ˙ u t |G Ut ] dt where ( E P (cid:2) ˙ u t (cid:12)(cid:12) G Ut (cid:3) ) t ∈ [0 , denotes the optional projection of ( ˙ u t ) on ( G Ut ) t ∈ [0 , , the usual augmen-tation of the filtration generated by ( U t ) t ∈ [0 , . Proof:
Let ( b u t ) be the dual predicable projection of ( u t ) (whose variations are locally integrableby (3.47)) on ( G Ut ). In particular(3.51) b u = Z . E P (cid:2) ˙ u t (cid:12)(cid:12) G Ut (cid:3) dt Then by setting
P − a.s. for all t ∈ [0 , c M ut := M ut + u t − b u t we have that P − a.s. for all t ∈ [0 , U t − U − Z t E P (cid:2) ˙ u s (cid:12)(cid:12) G Us (cid:3) ds = c M ut so that ( c M ut ) is ( G Ut ) − adapted. Let b (resp. M ) be the U ⋆ P− equivalence class of B ( W ) U ⋆ P / B ( W )-measurable mappings ( F U ⋆ P t ) − adapted such that P − a.s. (3.54) b ◦ U = b u resp. such that P − a.s. (3.55) M ◦ U = c M u whose existence is insured by Proposition 1.3. Since P − a.s. b u is absolutely continuous, by (3.54), U ⋆ P − a.s. b is absolutely continuous. Therefore, there exists a ( F U ⋆ P t ) − predicable process ( v t ) on( W, B ( W ) U ⋆ P , U ⋆ P ) so that P − a.s. b = Z . v s ds and by (3.47) and (3.54) we have, for all T < E U ⋆ η "Z T | v t | R d dt < ∞ Note that the ( G Ut ) − local martingale ( c M ut ) is reduced by the sequence of ( G Ut ) stopping times(3.56) b τ n := inf( { t ∈ [0 ,
1] : | c M ut | > n } )Indeed by the stopping theorem and the dominated convergence theorem (3.47) easily implies thatthe process ( M ut ) is reduced to ( A t ) − martingale by the sequence ( b τ n ) n ∈ N , while by (3.55) and (3.56)( b τ n ) is also a family of ( G Ut ) stopping times. Therefore together with (3.52) and using the inclusion( G U. ) ⊂ A . we obtain that ( b τ n ) also reduce ( c M ut ) to ( G Ut ) martingales. By Dynkin’s Lemma andsince ω → b τ n ( ω ) is G U measurable, for n ∈ N we denote by τ n the ( F U ⋆ P t ) − stopping time such that P − a.s. b τ n = τ n ◦ U Since c M u is a ( G Ut ) − local martingale reduced by b τ n , it is straightforward to check that M is a( F U ⋆ P t ) local martingale under U ⋆ P (reduced by ( τ n )). Indeed for n ∈ N and s ≤ t we have P− a.s. E P h c M ut ∧ b τ n |G Us i = E P (cid:2) ( M t ∧ τ n ) ◦ U (cid:12)(cid:12) G Us (cid:3) = E U ⋆ P (cid:2) M t ∧ τ n (cid:12)(cid:12) F U ⋆ P s (cid:3) ◦ U so that P − a.s. E U ⋆ P (cid:2) M t ∧ τ n (cid:12)(cid:12) F U ⋆ P s (cid:3) ◦ U = E P h c M ut ∧ b τ n (cid:12)(cid:12)(cid:12) G Us i = c M us ∧ b τ n = M s ∧ τ n ◦ U and U ⋆ P − a.s. E U ⋆ P (cid:2) M t ∧ τ n (cid:12)(cid:12) F U ⋆ P s (cid:3) = M s ∧ τ n By writing the Dolean’s approximations of the predicable quadratic co-variation process as a limitof finite sums we obtain
P − a.s. for all t ∈ [0 , < M i , M j > t ◦ U = < ( M u ) i , ( M u ) j > t since the process of the right hand term is absolutely continuous, the process of the left handterm too and we have the existence of a ( F U ⋆ P t ) − predicable process ( α t ) on the probability space( W, F U ⋆ P t , U ⋆ P ) such that U ⋆ P − a.s.,< M i , M j > = Z . ( α s ) ij ds Finally by reporting (3.54) and (3.55) in (3.53) we obtain, U ⋆ P − a.s. for all t ∈ [0 , W t − W − Z t v s ds = M t by which the result follows. EAK CALCULUS OF VARIATIONS 21
Proposition 3.2.
For ν ∈ S whose characteristics are denoted ( R . v νt dt, R . α νt dt ) (see Definition 1.2),let ν − a.s. h := R . ˙ h s ds ∈ L a ( ν, W abs ) where ( ˙ h s ) is a ( F νt ) -predicable process and let ( θ s ) be a M d ( R ) -valued ( F νt ) -predicable process on ( W, B ( W ) ν , ν ) such that for all T < E ν "Z T | θ s v νs + ˙ h s | R d ds < ∞ and ν − a.s. for all i, j (3.59) Z ( θ i,js ) α νs jj ds < ∞ Furthermore define R ∈ L ( ν, W ) by ν − a.s. for all t ∈ [0 , R t = f ( W ) + Z t θ s dW s where f : R d → R d is any B ( R d ) W ⋆ν / B ( W ) − measurable function, and U ∈ L ( ν, W ) by (3.61) U = R + h i.e. ν − a.s. for all t ∈ [0 , U t := f ( W ) + Z t θ s dW s + Z t ˙ h s ds Then U ⋆ ν ∈ S and we have, ν − a.s. , (3.63) Z . v U ⋆ νs ◦ U ds = Z . E ν h θ s v νs + ˙ h s (cid:12)(cid:12)(cid:12) G Us i ds and ν − a.s. (3.64) Z . α U ⋆ νs ◦ U ds = Z . θ s α νs θ † s ds where (cid:16) E ν h θ t v νt + ˙ h t (cid:12)(cid:12)(cid:12) G Ut i(cid:17) t ∈ [0 , denotes the optional projection of the process ( θ t v νt + ˙ h t ) t ∈ [0 , on the filtration ( G Ut ) . In particular if U ∈ I f ( ν ) , for all ǫ ∈ R ν − a.s. (3.65) Z . v U ⋆ νs ◦ U ds = Z . (cid:16) θ s v νs + ǫ ˙ h s (cid:17) ds Proof:
Denote M ν the martingale part of ν . Then ν − a.s. for all t ∈ [0 , R t = f ( W ) + M ut + Z t θ s v νs ds and U t = f ( W ) + M ut + Z t ( θ s v νs + ˙ h s ) ds where ( M ut ) is the process defined on ( W, B ( W ) ν , ν ) by ν − a.s. for all t ∈ [0 , M ut := Z t θ s dM νs By (3.59) ( M ut ) t ∈ [0 , is a continuous ( F νt )-local martingale on ( W, B ( W ) ν , ν ) and ν − a.s. for all t ∈ [0 , < ( Z . θ s dM νs ) i , ( Z . θ s dM νs ) j > t = X l,m Z t θ ils θ jms ( α νs ) lm dt for i, j ∈ [1 , d ]. Hence by applying Proposition 3.1 to U on ( W, B ( W ) ν , ν ) with the filtration ( F νt )we obtain U ⋆ ν ∈ S and (3.63) and (3.64). The end of the claim follows from the definition of I f ( ν ). Proposition 3.3.
For ν ∈ S , let ν − a.s. k u := R . ˙ u s ds ∈ L a ( ν, W abs ) and let ( M ut ) be a continuous ( F νt ) local martingale on ( W, B ( W ) ν , ν ) vanishing at t = 0 , such that < ( M u ) i , ( M u ) j > = Z . ( α us ) i,j ds for some R d (resp. M d ( R ) ) valued ( F νt ) predicable process ( ˙ u s ) (resp. ( α us ) ) on ( W, B ( W ) ν , ν ) .Moreover consider a continuous process defined by ν − a.s. for all t ∈ [0 , U t := f ( W ) + M ut + k ut where f : R d → R d is as in Proposition 3.2. Assume that U ∈ I f ( ν ) . Then U ⋆ ν ∈ S Moreover ν − a.s. (3.70) Z . v U ⋆ νs ◦ U ds = Z . ˙ u s ds and ν − a.s. (3.71) Z . α U ⋆ νs ◦ U ds = Z . α us ds Proof:
We denote e U ∈ I f ( U ⋆ ν ) the inverse of U and we set η := U ⋆ ν . Using this we denote f M (resp. e b ) the elements of L a ( η, W ) (resp. of L a ( η, W abs )) defined by η − a.s. (3.72) f M := M u ◦ e U and by η − a.s. (3.73) e b := k u ◦ e U Denote ( τ n ) (resp. ( e τ n )) a sequence of ( F νt ) stopping times reducing M u to a ( F νt )-martingale (resp.,since G e U. = F η. , of ( F ηt )-stopping times defined for all n ∈ N by η − a.s. e τ n := τ n ◦ e U ), similarly tothe proof of Proposition 3.1 and using both that U preserves the filtration and that e U ⋆ U ⋆ ν = ν it isstraightforward to obtain that, for all n ∈ N , ( e τ n ) reduces f M to a ( F ηt ) martingale on ( W, B ( W ) η , η ).Since U ∈ I f ( ν ) by (3.69) η − a.s. ,(3.74) W = U ◦ e U = f ( e U )By (3.69), (3.72), (3.73) and (3.74) since η − a.s. U ◦ e U = I W we obtain η − a.s. for all t ∈ [0 , W t = W + f M t + e b t EAK CALCULUS OF VARIATIONS 23 so that η ∈ S . Then the result follows similarly to the proof of Proposition 3.1 by the hypothesis U ∈ I f ( ν ). Indeed by (3.75) and (3.73) we obtain ν − a.s. b η ◦ U = k u which yields (3.70) while (3.71)follows similarly. Lemma 3.1.
For ν ∈ S whose characteristics are denoted ( R . v νt dt, R . α νt dt ) (see Definition 1.2),let ν − a.s. h := R . ˙ h s ds ∈ L a ( ν, W abs ) and let ( θ s ) be a M d ( R ) -valued predicable process on ( W, B ( W ) ν , ν ) such that ν − a.s. (3.59) holds for all i, j ∈ [1 , d ] and ν − a.s. , (3.76) Z | θ s v νs | R d ds < ∞ For f : R d → R d as in Proposition 3.2, let R ∈ L ( ν, W ) be given by (3.60) and U ∈ L ( ν, W ) by (3.61). If we further assume that U ∈ I f ( ν ) then U ⋆ ν ∈ S and ν − a.s. we have (3.64) and (3.65).In particular for h ∈ V ν , denoting ν − a.s. τ h := I W + h where I W denotes the ν − equivalence class of mappings ν − a.s. equal to the identity map, we havefor all ǫ ∈ R , τ ǫh⋆ ν ∈ S . Moreover ν − a.s. , (3.77) Z . v τ ǫh⋆ νs ◦ τ ǫh ds = Z . ( v νs + ǫ ˙ h s ) ds and (3.78) Z . α τ ǫh⋆ νs ◦ τ ǫh ds = Z . α νs ds Proof:
Denote M ν the martingale part of ν . Then ν − a.s. for all t ∈ [0 , R t is given by (3.66)and U t by(3.79) U t = f ( W ) + M ut + k ut where k u := Z . ( θ s v νs + ˙ h s ) ds and where ( M ut ) is the process defined on ( W, B ( W ) ν , ν ) by (3.67). By (3.59), ( M ut ) t ∈ [0 , is acontinuous ( F νt )-local martingale on ( W, B ( W ) ν , ν ) and ν − a.s. we also have (3.68). Since U ∈ I f ( ν )the result follows by Proposition 3.3. Moreover since I W ∈ I f ( ν ) by definition of V ν , τ h ∈ V ν sothat (3.77) and (3.78) follows as a particular case.3.2. Lift of transformations on the space to transformations of S . The next proposition isgeneral, however it is formulated to provide an insight on the behaviour of transformations whichare close to the identity
Lemma 3.2.
Let h : ( t, x ) ∈ [0 , × R d → h ( t, x ) ∈ R d be a C , function ( C in t , C in x ), andset u ( t, x ) := h ( t, x ) − x . Assume also that, for any t ∈ [0 , , h t : x ∈ R d → h t ( x ) := h ( t, x ) ∈ R d is an homeomorphism, whose inverse j t is such that ( t, x ) ∈ [0 , × R d → j t ( x ) ∈ R d is continuous.Denote G : W → W (resp. e G : W → W ) the mapping defined for all ( t, ω ) ∈ [0 , × W by G ( ω )( t ) := h ( t, ω ( t )) = ω ( t ) + u ( t, ω ( t )) (resp. e G ( ω )( t ) := j t ( ω ( t )) ). For any ν ∈ S , G induces an element Γ of I f ( ν ) ( Γ is the element of L ( ν, W ) such that ν − a.s. Γ = G ) whose inverse e Γ ∈ I f (Γ ⋆ ν ) is induced by e G . Then, for all ν ∈ S we have Γ ⋆ ν ∈ S Moreover ν − a.s. Z . v Γ ⋆ νt ◦ Γ dt = Z . ∂ t h ( t, W t ) + ( v νt . ∇ ) h ( t, W t ) + X i,j α νt i,j ∂ i,j h ( t, W t ) dt = Z . v νt + ∂ t u t ( W t ) + ( v νt . ∇ ) u t ( W t ) + X i,j α νt i,j ∂ i,j u ( t, W t ) dt and ν − a.s. Z . α Γ ⋆ νt ◦ Γ dt = Z . (cid:0) ( ∇ h )( t, W t ) . ( α νt ) . (( ∇ h ) † )( t, W t ) (cid:1) dt = Z . (cid:0) α νt + ( ∇ u t )( W t ) .α νt + α νt . ( ∇ u t ) † )( W t ) + ( ∇ u t )( W t ) .α νt . ( ∇ u t ) † ( W t ) (cid:1) dt where ( ∇ h ) i,j ( x ) = ∂h i ∂x j ( t, x ) (similarly for u ) Proof:
By definition for all t ∈ [0 ,
1] and for all x ∈ R d j t ( h t ( x )) = h t ( j t ( x )) = x Hence for all ω ∈ W e G ◦ G ( ω )( t ) = j t ( G t ( ω )) = j t ( h t ( W t ( ω )) = ω ( t )so that G (resp. e G ) induces an isomorphism of probability spaces (resp. its inverse). On the otherhand both Γ and e Γ are adapted to the respective canonical filtrations. This proves that Γ ∈ I f ( ν ).Set k u := Z . ∂ t h ( t, W t ) + ( v νt . ∇ ) h ( t, W t ) + X i,j α νt i,j ∂ i,j h ( t, W t ) dt and M u := d X i =1 (cid:18)Z . ∂ j h i ( t, W t ) d ( M νt ) j (cid:19) e i where ( e i ) i =1 ,d is the canonical orthogonal basis of R d and set P − a.s. for all t ∈ [0 , U t := h ( W ) + M ut + k ut On the other hand ( M ut ) is a ( F tν ) local martingale and < ( M u ) i , ( M u ) j > = Z . X l,m ∂ l h i ( t, W t ) ∂ m h j ( t, W t )( α νt ) l,m dt By Itˆo’s formula, ν − a.s. for all t ∈ [0 , W t ◦ Γ = U t Therefore the result follows from Proposition 3.3
EAK CALCULUS OF VARIATIONS 25
Definition 3.1.
For ν ∈ S and h : ( t, x ) ∈ [0 , × R d → h t ( x ) ∈ R d which satisfy the hypothesisof Lemma 3.2 we call the isomorphism of filtered probability spaces Γ ∈ I f ( ν ) associated to h byLemma 3.2 the lift of h on S at ν . The following Proposition directly follows from Lemma 3.2 and Itˆo’s formula. It characterizesmartingales in terms of the invariance of the finite variation part of the law of processes by theassociated lifted transformations of space depending on time:
Proposition 3.4.
Let ν ∈ S and assume that u : ( t, x ) ∈ [0 , × R d → u t ( x ) ∈ R d is such that themapping h := I R d + u satisfy the hypothesis of Lemma 3.2. Denote by Γ the lift of h (see Definition 3.1) on S at ν . Thenthe following assertions are equivalent :(i) ( t, ω ) → u ( t, W t ( ω )) is a ( F νt ) − martingale(ii) ν − a.s. Z . ∂ t u t ( W t ) + ( v νt . ∇ ) u t ( W t ) + X i,j α νt i,j ∂ i,j u ( t, W t ) dt = 0 (iii) We have ν − a.s. (3.80) Z . v Γ ⋆ νt ◦ Γ dt = Z . v νt dt Differential calculus associated to the variation processes
For ν ∈ P ( W ) and k ∈ V ν set τ k := I W + k where I W stands for the ν − equivalence class of mappings ν − a.s. equal to the identity on W . For ǫ ∈ R set ν kǫ := τ ǫk⋆ ν = ( I W + ǫk ) ⋆ ν thus defining a path ǫ ∈ R → ν kǫ ∈ P ( W )As we have seen (Lemma 3.1) if ν k ∈ S then for all ǫ ∈ R , ν kǫ ∈ S so that we have a path ǫ ∈ R → ν kǫ ∈ S on S . Moreover if b ǫ denotes the finite variation part of ν kǫ ∈ S (see Definition 1.2), then b ǫ is simplyrelated to b (the finite variation part of ν ) by b ǫ = b + ǫk Finally, since τ ǫk is an isomorphism of filtered probability spaces, we can identify isometrically thespaces L pa ( ν ǫ , H ) along any such path ( ν ǫ ) as well as the vector spaces of variations processes (seeProposition 2.2). This motivates the following : Definition 4.1.
Given a mapping F : ν ∈ S → F ( ν ) ∈ R ∪ { + ∞} and ν ∈ S such that F ( ν ) < ∞ , F will be said to be L a ( ν, H , ) -differentiable (resp. L a ( ν, H ) -differentiable) at ν if for all k ∈ V , ∞ ν (resp. k ∈ V ∞ ν ) ddǫ F ( ν kǫ ) | ǫ =0 exists where for all ǫ ∈ R and k ∈ V , ∞ ν (resp. k ∈ V ∞ ν ), ν kǫ := ( τ ǫk ) ⋆ ν := ( I W + ǫk ) ⋆ ν, and if, in addition, there exists ξ ∈ L a ( ν, H ) such that for all k ∈ V , ∞ ν (resp. k ∈ V ∞ ν ) (4.81) ddǫ F ( ν kǫ ) | ǫ =0 = E ν [ < ξ, k > H ] In this case, by Lemma 2.1 (resp. Proposition 2.4) ξ is the unique element of L a ( ν, H , ) (resp. in L a ( ν, H ) ), that we note grad ν F (resp. grad ν F ), such that (4.81) holds for any k ∈ V , ∞ ν (resp.in V ∞ ν ), namely the orthogonal projection on L a ( ν, H , ) of any (and then all) (resp. the uniqueelement) ξ satisfying (4.81) for all k ∈ V , ∞ ν (resp. k ∈ V ∞ ν ). We denote by δF ν the linearcontinuous form defined by (4.82) δF ν : h ∈ L a ( ν, H , ) → δF ν ( h ) := E ν [ < grad ν F, h > H ] ∈ R (resp. by the same symbol δF ν : h ∈ L a ( ν, H ) → δF ν ( h ) := E ν [ < grad ν F, h > H ] ∈ R ) Remark 4.0.1. If F is L a ( ν, H ) − differentiable at some ν ∈ S , then for all k ∈ L a ( ν, H , ) we have E ν [ < grad ν F, k > H ] = E ν [ < grad ν F, k > H ] Indeed for k ∈ L a ( ν, H , ) take ( k n ) ⊂ V , ∞ ν a sequence which converges to h . We then have, for all n ∈ N , E ν [ < grad ν F, k n > H ] = ddǫ F (( I W + ǫk n ) ⋆ ν ) | ǫ =0 = E ν [ < grad ν F, k n > H ] so that by continuity E ν [ < grad ν F, k > H ] = lim n →∞ E ν [ < grad ν F, k n > H ] = lim n →∞ E ν [ < grad ν F, k n > H ] = E ν [ < grad ν F, k > H ] In particular grad ν F is the orthogonal projection of grad ν F on L a ( ν, H , ) and it is meaningful todenote by the same symbol δF ν the two linear forms. The stochastic extension of Hamilton’s least action principle
Regular Lagrangians and their actions.Definition 5.1.
A Borel measurable mapping L : ( t, x, v, a ) ∈ [0 , × R d × R d × M d ( R ) → L t ( x, v, a ) ∈ R ∪ { + ∞} will be called a Lagrangian . We denote its domain by
Dom ( L ) := { ( t, x, v, a ) : L t ( x, v, a ) < ∞} And we define the action of L on S to be the mapping S : ν ∈ S → S ( ν ) ∈ R ∪ { + ∞} EAK CALCULUS OF VARIATIONS 27 defined by S ( ν ) := E ν (cid:20)Z L t ( W t , v νt , α νt ) dt (cid:21) if E ν hR |L t ( W t , v νt , α νt ) | dt i < ∞ and by S ( ν ) = ∞ otherwise. The following conditions will be used to ensure the least action principle :
Definition 5.2.
A Lagrangian L (see Definition 5.1) will be said to be regular if it satisfies thefollowing assumptions(i) The domain of L is the whole space i.e. Dom ( L ) = [0 , × R d × R d × M d ( R ) (ii) For all ( t, x, v, a ) ∈ Dom ( L ) , the mapping L ( t, x, v, a ) : ( e x, e v ) ∈ R d × R d → L t ( x + e x, v + e v, a ) ∈ R is Fr´echet differentiable at (0 R d , R d ) and we denote by (5.83) D L t,x,v,a : ( e x, e v ) ∈ R d × R d → < ( ∂ x L t )( x, v, a ) , e x > R d + < ( ∂ v L t )( x, v, a ) , e v > R d its derivative which the linear operator defined by (5.84) D L t,x,v,a [ e x, e v ] := ddǫ L t ( x + ǫ e x, v + ǫ e v, a ) | ǫ =0 (iii) The mappings ( t, x, v, a ) ∈ Dom ( L ) → ∂ x L t ( x, v, a ) ∈ R d and ( t, x, v, a ) ∈ Dom ( L ) → ∂ v L t ( x, v, a ) ∈ R d are Borel measurable. The stochastic least action principle and the related Euler-Lagrange condition.Definition 5.3.
Let L : ( t, x, v, a ) ∈ [0 , × R d × R d × M d ( R ) → L t ( x, v, a ) ∈ R be a regularLagrangian (see Definition 5.2). A probability ν ∈ S such that for all T < , ν − a.s. , Z T | ∂ x L s ( W s , v νs , α νs ) | R d ds < ∞ is said to satisfy the stochastic Euler-Lagrange condition if and only if there exists a R d − valuedc`adl`ag ( F νt ) − martingale ( N νt ) t ∈ [0 , on the probability space ( W, B ( W ) ν , ν ) such that ν ⊗ λ − a.s. (5.85) ∂ v L t ( W t , v νt , α νs ) − Z t ∂ x L s ( W s , v νs , α νs ) ds := N νt where ( v νt ) (resp. ( α νt ) ) are the derivatives of the characteristics of ν (see Definition 1.2). Theorem 5.1.
Let L be a regular Lagrangian whose associated action on S is noted S (see Defini-tions 5.1 and 5.2). Assume also the existence of a strictly positive continuous function f : R d → R + and p , p ≥ such that (5.86)lim sup | ǫ |↓ sup ( t,x,v,a, e x, e v ) ∈ Dom ( L ) × R d × R d |L t ( x + ǫ e x, v + ǫ e v, a ) − L t ( x, v, a ) − ǫD L t,x,v,a [ e x, e v ] | ǫf ( e x ) (cid:0) | e v | R d + G ( t, x, v, a ) (cid:1) ! = 0 holds, where G ( t, x, v, a ) := |L t ( x, v, a ) | + | ∂ x L ( x, v, a ) | p R d + | ∂ v L ( x, v, a ) | p R d Then for any ν ∈ S satisfying (5.87) S ( ν ) + E ν (cid:20)Z | ∂ x L ( W s , v νs , α νs ) | p R d dt (cid:21) + E ν (cid:20)Z | ∂ v L ( W s , v νs , α νs ) | p R d dt (cid:21) < ∞ we have that S is L a ( ν, H , ) − differentiable (see subsection 4) at ν . Moreover ν satisfies the sto-chastic Euler − Lagrange condition (see Definition 5.3) if and only if δ S ν ( h ) = 0 for all h ∈ L a ( ν, H , ) i.e. if and only if for all h ∈ V , ∞ ν , d S ( τ ǫh⋆ ν ) dǫ | ǫ =0 = 0 where for h ∈ V , ∞ ν τ h := I W + h Proof:
By (5.87) since p , p ≥ E ν (cid:20)Z | ∂ v L t ( W t , v νt , α νt ) | dt (cid:21) + E ν (cid:20)Z | ∂ x L t ( W t , v νt , α νt ) | dt (cid:21) < ∞ Define ˙ ξ t := ∂ v L t ( W t , v νt , α νt ) − Z t ∂ x L s ( W s , v νs , α νs ) ds and ξ := Z . ˙ ξ t dt. From (5.88) and Jensen’s inequality, ξ ∈ L a ( ν, H ). Take h ∈ V , ∞ ν and set A ǫ := (cid:12)(cid:12)(cid:12)(cid:12) S ( τ ǫh⋆ ν ) − S ( ν ) ǫ − E ν [ < h, ξ > H ] (cid:12)(cid:12)(cid:12)(cid:12) where τ ǫh⋆ ν is the image of the probability ν by the mapping τ ǫh := I W + ǫh . We want to show that A ǫ converges to 0. By Lemma 3.1 we have S ( τ ǫh⋆ ν ) = E τ ǫh⋆ ν (cid:20)Z L s ( W s , v τ ǫh⋆ νs , α τ ǫh⋆ νs ) ds (cid:21) = E ν (cid:20)Z L s ( W s + ǫh s , v τ ǫh⋆ νs ◦ τ ǫh , α τ ǫh⋆ νs ◦ τ ǫh ) ds (cid:21) = E ν (cid:20)Z L s ( W s + ǫh s , v νs + ǫ ˙ h s , α νs ) ds (cid:21) so that we first obtain, for ǫ ∈ R , A ǫ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ν "Z L s ( W s + ǫh s , v νs + ǫ ˙ h s , α νs ) − L s ( W s , v νs , α νs ) ǫ − < ˙ h s , ˙ ξ s > ! ds On the other hand, since ν − a.s. h = h = 0, an integration yields A ǫ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ν "Z L s ( W s + ǫh s , v νs + ǫ ˙ h s , α νs ) − L s ( W s , v νs , α νs ) ǫ − D L t ( W s , v νs , α νs )[ h s , ˙ h s ] ! ds where ( t, x, v, a ) → D L t ( x, v, a ) is given by (5.83). By the hypothesis0 ≤ E ν (cid:20)Z G ( t, W t , v νt , α νt ) dt (cid:21) < ∞ EAK CALCULUS OF VARIATIONS 29
For ǫ > α > ǫ ∈ R / { } with | ǫ | < α , the followinginequality holds everywhere (cid:12)(cid:12)(cid:12)(cid:12) L t ( x + ǫ e x, v + ǫ e v, a ) − L t ( x, v, a ) ǫ − D L t ( x, va )[ e x, e v ] (cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ B f ( e x )(1 + | e v | R d + G ( t, x, v, a ))where B := ( sup z ∈ B (0 R d ,R ) f ( z )) (cid:18) | h | L ( ν,H ) + E ν (cid:20)Z G ( t, W t , v νt , α νt ) dt (cid:21)(cid:19) and R := | h | L ∞ ( µ,W ) + 1 < ∞ Since h ∈ V , ∞ ν we have ν − a.s. for all t ∈ [0 , | h t ( ω ) | R d ≤ | h | L ∞ ( ν,W ) . Thus by the continuity of f , for any ǫ such that | ǫ | < α we obtain A ǫ ≤ ǫ B E ν (cid:20)Z f ( h s )(1 + | ˙ h s | R d + G ( W s , v νs , α νs )) ds (cid:21) ≤ ǫ B E ν "Z sup z ∈ B (0 R d ,R ) f ( z ) ! (1 + | ˙ h s | R d + G ( W s , v νs , α νs )) ds ≤ ǫ B sup z ∈ B (0 R d ,R ) f ( z ) ! E ν (cid:20)Z (1 + | ˙ h s | R d + G ( W s , v νs , α νs )) ds (cid:21) ≤ ǫ Hence we have lim sup | ǫ |↓ A ǫ ≤ ǫ Since this inequality holds for any ǫ >
0, we conclude that lim sup | ǫ |↓ A ǫ = 0. Using the definitionof A ǫ we get that for all h ∈ V , ∞ ν ddǫ S ( τ ǫh⋆ ν ) | ǫ =0 exists and(5.89) δ S ν [ h ] = ddǫ S ( τ ǫh⋆ ν ) | ǫ =0 = E ν [ < ξ, h > H ]In particular, S is L a ( ν, H , ) − differentiable (see subsection 4). Note that by Proposition 1.1 andby definition of ξ , ν satisfies the stochastic Euler − Lagrange condition if and only if ξ is orthogonalto L a ( ν, H , ). On the other hand, by (5.89) ν satisfies the least action principle for S if and onlyif for all h ∈ V , ∞ ν E ν [ < h, ξ > H ] = 0while by Lemma 2.1 this latter condition is also satisfied if and only if ξ is orthogonal to L a ( ν, H , ).This achieves the proof. Remark 5.2.1. • Whenever ν satisfies the stochastic Euler-Lagrange condition it also satis-fies the following averaged stochastic Euler-Lagrange condition (5.90) (5.90) ddt E ν [ ∂ v L t ( W t , v νt , α νt )] = E ν [ ∂ x L s ( W t , v νt , α νt )] Moreover the left hand side of (5.90) is well defined and it is trivial to check that (5.90)holds if and only if for any h ∈ H , we have (5.91) δ S ν [ h ] = 0 • We refer to [12] for an Hamiltonian point of view on the stochastic Euler − Lagrange condi-tion. • Assume that L is a regular Lagrangian which does not depends on ( α νs ) . For x ∈ R d and u ∈ H let γ ∈ W be defined by γ t := x + u t for all t ∈ [0 , , and denote by δ Diracγ theDirac measure concentrated on γ , namely δ Diracγ ( A ) = I A ( γ ) for A ∈ B ( W ) . Then ( F νt ) isthe filtration constant equal to the set of the subsets of W , so that the martingales can beidentified with the constants, and V , ∞ ν = V ν = L a ( ν, H , ) ≃ H , . Moreover in this casefor h ∈ V ν , ǫ ∈ R , we have τ ǫh⋆ ν = δ γ + ǫh so that τ ǫh⋆ ν ∈ S has a martingale part equal to and τ ǫh⋆ ν a.s. for all t ∈ [0 , , W t = x + γ t + ǫh t In particular ˙ W t exists a.s. and τ ǫh⋆ ν a.s. v τ ǫh⋆ νt ( ω ) = ˙ W t = ˙ γ t + ǫ ˙ h t . Hence we obtain S ( τ ǫh⋆ ν ) = R L ( γ s + ǫh s , ˙ γ s + ǫ ˙ h s ) ds and the stochastic least action principle reads ddǫ | ǫ =0 Z L ( γ s + ǫh s , ˙ γ s + ǫ ˙ h s ) = 0 for all h ∈ H , , while the stochastic Euler-Lagrange condition holds if and only if thereexists c ∈ R such that λ − a.s.∂ v L ( γ t , ˙ γ t ) − Z t ∂ x L ( γ s , ˙ γ s ) ds = c or if and only if t → ∂ v L ( t, γ t , ˙ γ t ) is differentiable a.e. and λ − a.e.. (5.92) ddt ∂ v L ( γ t , ˙ γ t ) = ∂ x L ( γ t , ˙ γ t ) Moreover, if we assume also that ( x, v ) ∈ R d × R d → ∂ x L ( x, v ) ∈ R d is continuous (resp. ( x, v ) ∈ R d × R d → ∂ v L ( x, v ) ∈ R d is C ), and that t ∈ [0 , → γ t ∈ R d is C , then (5.92)holds for all t ∈ [0 , . In this case we recover the least action principle of classical mechanics. Invariances and Noether’s theorem
Definition 6.1.
Let h : [0 , × R d → R d be a mapping which satisfies the hypothesis of Lemma 3.2.We say that h is an S − invariant transformation for L if for all ν ∈ S , ν − a.s all ω ∈ W wehave, λ − a.s. , (6.93) L t (Γ t ( ω ) , v Γ ⋆ νt ◦ Γ( ω ) , α Γ ⋆ νt ◦ Γ( ω )) = L t ( W t ( ω ) , v νt ( ω ) , α νt ( ω )) where Γ is the lift of h on S at ν (see Definition 3.1). Moreover we say that a family ( h ǫ ) ǫ ∈ R of S − invariant transformation for L is a differentiable family of S − invariant transformations for L , if ( t, x, ǫ ) ∈ [0 , × R d × R → h ǫ ( t, x ) ∈ R d is C in ǫ and h ( t, x ) = x for all ( t, x ) ∈ [0 , × R d . We recall that for two real valued c`adl`ag semi-martingales X and Y , their quadratic co-variationis the process ([ X, Y ] t ) defined by[ X, Y ] . = X . Y . − Z . X s − dY s − Z . Y s − dX s see [3] for more. EAK CALCULUS OF VARIATIONS 31
Theorem 6.1.
Let L be a regular Lagrangian which is C and assume that ν ∈ S satisfies thestochastic Euler-Lagrange conditions (see Definition 5.3) for L . Assume also that ( h ǫ ) ǫ ∈ R is adifferentiable family of S − invariant transformations for L . Let ( I t ) t ∈ [0 , be any optional process on ( W, F ν. , ν ) such that λ ⊗ ν − a.s. (6.94) I t := < ddǫ | ǫ =0 h ǫt ( W t ) , p νt > R d − X i (cid:20) ddǫ | ǫ =0 h ǫ. ( W . ) i , p ν. i (cid:21) t + Z t θ s ds where [ ., . ] denotes the quadratic co-variation of semi-martingales, ( p νt ) denotes a c`adl`ag modificationof the process ( ∂ v L t ( W t , v νt , α νt )) , and (6.95) θ s := X i,j κ i,js ∂ L ∂α i,j ( W s , v νs , α νs ) and where ( κ s ( ω )) is the M d ( R ) − valued process defined by (6.96) κ s ( ω ) := α νs . (cid:18)(cid:18) ∇ ddǫ h ǫ | ǫ =0 (cid:19) ( s, W s ) (cid:19) † + (cid:18)(cid:18) ∇ ddǫ h ǫ | ǫ =0 (cid:19) ( s, W s ) (cid:19) α νs Then ( I t ) t ∈ [0 , is a ( F νt ) − local martingale on the probability space ( W, B ( W ) ν , ν ) . Proof:
For all ǫ ∈ R we denote by Γ ǫ the lift of h ǫ on S at ν (see Definition 3.1). We set ν ǫ := Γ ǫ⋆ ν and e u ( t, x ) := ddǫ h ǫt ( x ) | ǫ =0 For
T <
1, by Lemma 3.2,(6.97) Z T L (Γ ǫt , v ν ǫ t ◦ Γ ǫ , α ν ǫ t ◦ Γ ǫ ) dt = Z T L ( h ǫ ( t, W t ) , m ǫt , a ǫt ) dt where m ǫt = ∂ t h ǫ ( t, W t ) + ( v νt . ∇ ) h ǫ ( t, W t ) + X i,j α νt i,j ∂ i,j h ǫ ( t, W t )and a ǫt = α νt . ( ∇ h ǫ ) † ( t, W t ) + ∇ h ǫ ( t, W t ) .α νt By differentiating (6.97), condition (6.93) yields ν − a.s. ,(6.98) 0 = Z T ( < e u ( t, W t ) , ∂ x L ( W t , v νt , α νt ) > + < Q t , ∂ v L ( W t , v νt , α νt ) > + θ t ) dt where(6.99) Q t := ∂ t e u ( t, W t ) + ( v νt . ∇ ) e u ( t, W t ) + X i,j α νt i,j ∂ i,j e u ( t, W t )and where ( θ s ) is given by (6.95) and κ by (6.96). Since ν satisfies the stochastic Euler − Lagrangecondition there exists a ( F νt ) c`adl`ag martingale ( N νt ) such that ν ⊗ λ − a.s. (6.100) ∂ v L t ( W t , v νt , α νt ) = N νt + Z t ∂ x L s ( W s , v νs , α νs ) ds Denote by ( p νt ) the process defined by the right hand term of (6.100). We have(6.101) Z T < e u ( t, W t ) , ∂ x L ( W t , v νt , α νt ) > dt = Z T < e u ( t, W t ) , dp νt > − Z T < e u ( t, W t ) , dN νt > We now compute the first term of the right hand side. Denoting by M ν the martingale part of ν byItˆo’s formula we obtain a.e. e u i ( t, W t ) = M u,it + A ti where M u,it = e u i (0 , W ) + Z t X j ( ∂ j e u i )( s, W s ) dM ν,js and where A it := Z t Q is ds with ( Q s ) given by (6.99). Since ( M ut ) is a continuous local martingale and ( A t ) is continuous andof finite variation, by Itˆo’s formula (by (18.1) and (19.2) VIII p.343 of [3]) we obtain(6.102) Z T < e u ( t, W t ) , dp νt > = < e u ( T, W T ) , p νT > R d − Z T < p νt − , dM ut > ...... − Z T < p νt , Q t > dt − X i [ M u,i , p νi ] T Putting (6.102) into (6.101) we derive(6.103) Z T < e u ( t, W t ) , ∂ x L ( W t , v νt , α νt ) > dt = < e u T ( W T ) , p νT > R d ...... − Z T < p νt , Q t > dt − X i [ M u,i , p νi ] T − f M T where f M t = Z T < p t − ν , dM ut > + Z T < e u ( t, W t ) , dN tν > By putting (6.103) into (6.98) we obtain(6.104) I T = f M T On the other hand by construction ( f M t ) is a c`adl`ag ( F νt )-local martingale so that the result followsby (6.104). Remark 6.0.2.
This theorem must be compared with the original theorem such as it is formulatedp.88 of [2] together with the following remark. Take c ∈ W to be such that t ∈ [0 , → c t ∈ R d issmooth, ν := δ c the associated Dirac measure and the probability space ( W, B ( W ) ν , ν ) . Consider aparticular h : ( t, x ) ∈ [0 , × R d → h ( x ) ∈ R d not depending on time and satisfying the hypothesisof Lemma 3.2. and let Γ denote the lift of h at ν . Then for this transformation, we have Γ ⋆ ν = δ e c where e c t = h ( c t ) for all t , and λ ⊗ ν − a.s.v Γ ⋆ νt ◦ Γ = dh ( W t ) dt The r.h.s. of this latter equation is nothing but the image of ˙ W t by h (noted h ⋆ ( ˙ W t ) in geometry). EAK CALCULUS OF VARIATIONS 33 Application to stochastic control
Definition 7.1.
A subset N of S is said to be V − stable if for any ν ∈ N any ǫ ∈ R and h ∈ V ν τ ǫh⋆ ν ∈ N where τ ǫh := I W + ǫh Note that in particular (see Lemma 3.1), S is V − stable. Theorem 7.1.
Let N be a V − stable subset of S (see Definition 7.1). Consider L a non nega-tive regular Lagrangian with associated action S . Assume also that there exists a strictly positivecontinuous function f : R d → R + and p , p ≥ such that (5.86) holds for L and (7.105) sup ( t,x,v,a ) ∈ Dom ( L ) (cid:18) | ∂ x L t ( x, v, a ) | p R d + | ∂ v L t ( x, v, a ) | p R d )1 + L t ( x, v, a ) (cid:19) < ∞ Consider the minimization problem (7.106) I F := inf ( {S ( ν ) : ν ∈ N } ) and assume that I F < ∞ . Then for any η ∈ N which attains the infimum of (7.106), η sat-isfies the stochastic Euler-Lagrange condition (5.85) for L . Moreover if L is C and ( h ǫ ) is adifferentiable family of S − invariant transformations for L (see Definition 6.1), the process definedon ( W, B ( W ) η , η ) by (6.94) is a ( F ηt ) − local martingale. Moreover the same statements hold if wechange the inf by sup , assuming it is attained. Proof:
First notice that by (7.105), since S ( η ) < ∞ , (5.87) is satisfied with p , p as (5.86) isassumed. Thus Theorem 5.1 applies to L at η and we first obtain that S is L a ( η, H , ) − differentiable.Notice that, for any h ∈ V , ∞ η , by Definition 7.1 we have τ h⋆ η ∈ N Therefore since η (it is a minimum of the action, it satisfies δ S η [ h ] := ddǫ S ( τ ǫh⋆ η ) | ǫ =0 = 0for all h ∈ V , ∞ η , so that by definition of δ S η we get δ S η = 0Thus, the result directly follows by Theorem 5.1 and Theorem 6.1. If we change the inf by sup theproof is similar.For γ ∈ P R d × R d a Borel probability on R d × R d (resp. ν , ν two Borel probabilities on R d ) weset S γ := { ν ∈ S : ( W × W ) ⋆ ν = γ } where ( W × W ) : ω ∈ W → ( W ( ω ) , W ( ω )) ∈ R d × R d i.e. S γ is the set of the ν ∈ S with a fixedjoint law γ for ( W , W ) Consider also S ν ,ν := { ν ∈ S : W ⋆ ν = ν , W ⋆ ν = ν } i.e. the set of ν ∈ S with initial marginal ν and final marginal ν . We also denote the set of the ν ∈ S whose martingale part is a Brownian motion by S B i.e. S B := { ν ∈ S : M ν⋆ ν = µ } where µ is the standard Wiener measure (the law of the Brownian motion, with µ − a.s. W = 0).We refer to [22] an to the proofs therein for sufficient condition for the existence to the followingminimizer. By assuming their existence we obtain : Corollary 7.1.
Consider a probability γ ∈ P R d × R d (resp. ν , ν ∈ P R d ) and any non negativeLagrangian L which is regular and whose action is denoted by S (see Definition 5.2). Assumealso that hypothesis (7.105) and (5.86) are satisfied for some p , p ≥ and some strictly positivecontinuous mapping f : R d → R + . Consider the variational problems (7.107) I ν ,ν := inf ( S ( ν ) : ν ∈ S ν ,ν )(7.108) I γ := inf ( S ( ν ) : ν ∈ S γ )(7.109) I Bν ,ν := inf ( S ( ν ) : ν ∈ S B ∩ S ν ,ν ) and (7.110) I Bγ := inf ( S ( ν ) : ν ∈ S B ∩ S γ ) By assuming that I ν ,ν (resp. I γ , resp. I Bν ,ν , resp. I Bγ ) is finite (i.e. < ∞ ), any ν ∈ S ν ,ν (resp. S γ , resp. S ν ,ν ∩ S B , resp. S γ ∩ S B ) which attains the infimum of (7.107) (resp. of (7.108), resp. of(7.109), resp. of (7.110)), satisfies the stochastic Euler-Lagrange condition (5.85) for L . Moreover if L is C and ( h ǫ ) is a differentiable family of S − invariant transformations for L (see Definition 6.1)then the process defined on the probability space ( W, B ( W ) ν , ν ) by (6.94) is a ( F νt ) − local martingale.Moreover the same statements hold if we change the inf by sup , assuming it is attained. Proof:
Note that S γ is V − stable (see Definition 7.1). Indeed, for all ν ∈ S γ and h ∈ V ν we have, ν − a.s. h = h = 0so that ( W × W ) ⋆ ( τ h⋆ ν ) = (( W + h ) × ( W + h )) ⋆ ν = ( W × W ) ⋆ ν = γ Similarly S ν ,ν is V − stable. Thus Paul Levy’s criterion and (3.78) of Lemma 3.1 imply that S γ ∩ S B and S ν ,ν ∩ S B are also V − stable. Therefore the results directly follow by Theorem 7.1. For sup ,the proof is similar. EAK CALCULUS OF VARIATIONS 35 The critical processes of the classical Lagrangians
The classical Lagrangians.Definition 8.1.
Given a function V : [0 , × R d → R which is assumed to be measurable and C in x , we define classical Lagrangians of the form L V : ( t, x, v, a ) ∈ [0 , × R d × R d × M d ( R ) → | v | − V ( x ) ∈ R and denote by G V ( R d ) the set of ν ∈ S which satisfy the stochastic Euler-Lagrange condition (seeDefinition 5.3) for L V . The following is the counterpart to the Galilean invariance for the free particle of classical me-chanics in our stochastic framework :
Proposition 8.1.
A measure ν ∈ S belongs to G ( R d ) if and only if for any U ∈ I f ( ν ) of the form U := I W + h where h ∈ M a ( ν, H ) ∩ V ν (see Proposition 1.1 and Definition 2.1) we have U ⋆ ν ∈ G ( R d ) Proof:
For any k := R . ˙ k s ds ∈ L a ( U ⋆ ν, H , ), by Lemma 3.1 and Proposition 1.1 we obtain E U ⋆ ν (cid:20)Z < v U ⋆ νs , ˙ k s > ds (cid:21) = E ν (cid:20)Z < v U ⋆ νs ◦ U, ˙ k s ◦ U > (cid:21) = E ν (cid:20)Z < v νs + ˙ h s , ˙ k s ◦ U > ds (cid:21) = E ν (cid:20)Z < v νs , ˙ k s ◦ U > ds (cid:21) + E ν [ < h, k ◦ U > H ]= E ν (cid:20)Z < v νs , ˙ k s ◦ U > ds (cid:21)
On the other hand since U is an isomorphism of filtered probability spaces, we have k ∈ L a ( ν, H , )iff there exists a e k ∈ L a ( U ⋆ ν, H , ) such that ν − a.s. k = e k ◦ U . Hence the result follows fromProposition 1.1.Next proposition characterizes the measures in G V ( R d ). Proposition 8.2.
For ν ∈ G V ( R d ) we have, ν − a.s. , b ν = Z . E ν (cid:2) ξ Vt (cid:12)(cid:12) F νt (cid:3) dt where ( ξ Vt ) t ∈ [0 , is the stochastic process on ( W, B ( W ) ν , ν ) defined by ( ξ Vt ) t ∈ [0 , : ( t, ω ) ∈ [0 , × W → ξ Vt ( ω ) := ω − ω t − t + Z t (1 − s )(1 − t ) ∇ V ( ω s ) ds ∈ R d Proof:
By the stochastic Euler-Lagrange condition the process ( A t ) defined by A t ( ω ) := v νt + Z t ( ∇ V )( W s ) ds is a ( F νt ) − martingale so that we have ν ⊗ λ − a.s.A t ( ω ) = 11 − t E ν (cid:20)Z t A σ dσ (cid:12)(cid:12)(cid:12)(cid:12) F νt (cid:21) = E ν (cid:20) W − W t − t (cid:12)(cid:12)(cid:12)(cid:12) F νt (cid:21) + 11 − t E ν (cid:20)Z t (cid:18)Z s ( ∇ V )( W σ ) dσ (cid:19) ds (cid:12)(cid:12)(cid:12)(cid:12) F νt (cid:21) where the last line is obtained by noticing that, from the definition of v ν , we have E ν (cid:20) W − W t − t (cid:12)(cid:12)(cid:12)(cid:12) F νt (cid:21) = E ν " R t v νs ds − t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F νt Hence we obtain ν ⊗ λ − a.s. (8.111) v νt = E ν (cid:20) W − W t − t (cid:12)(cid:12)(cid:12)(cid:12) F νt (cid:21) − Z t ( ∇ V )( W s ) ds + 11 − t E ν (cid:20)Z t (cid:18)Z s ( ∇ V )( W σ ) dσ (cid:19) ds (cid:12)(cid:12)(cid:12)(cid:12) F νt (cid:21) and the result follows by integrating by parts.8.2. Critical processes of classical Lagrangians and systems of stochastic differentialequations.Theorem 8.1.
Let b η ∈ P R d be a Borel probability on R d . Assume that ( X, Y ) satisfies the system dX t = σ t ( X ) dB t + Y t dt (8.112) dY t = dZ t − ( ∇ V )( t, X t ) dt (8.113) Law ( X ) = b η (8.114) on some complete stochastic basis (Ω , A , ( A t ) t ∈ [0 , , P ) , where ( Z t ) is a c`adl`ag ( A t ) t ∈ [0 , − martin-gale, ( B t ) an ( A t ) − Brownian motion, and ( X t ) is ( A t ) − adapted. Then X ⋆ P ∈ G V ( R d ) and conversely if ν ∈ G V ( R d ) with W ⋆ ν = b η then ( W t , v νt ) satisfy a system of this form with ( σ t ) such that α ν. = σ . σ † . on the space ( W, B ( W ) ν , ν ) with the filtration ( F νt ) or on one of its extensions(see [6] ). Proof:
We assume that (
X, Y, Z ) is a solution to (8.112) (8.113) (8.114) on a probability space(Ω , A , P ) with a filtration ( A t ) t ∈ [0 , for a ( A t ) − Brownian motion B . We set ν := X ⋆ P By Proposition 3.1 condition (8.112) implies that ν ∈ S with Z . α νt dt = Z . σ t σ † t dt and ν − a.s. b ν ◦ X = Z . E P (cid:2) Y t (cid:12)(cid:12) G Xt (cid:3) dt EAK CALCULUS OF VARIATIONS 37 where b ν denotes the finite variation part of ν (see Definition 1.2), and where (cid:0) E P (cid:2) Y t (cid:12)(cid:12) G Xt (cid:3)(cid:1) denotesa c`adl`ag modification of the optional projection of ( Y t ) on the usual augmentation ( G Xt ) of thefiltration generated by X . We now take ( t, ω ) ∈ [0 , × W → v νt ∈ R d to be c`adl`ag , ( F νt )-adaptedand such that P − a.s. for all t ∈ [0 , v νt ◦ X = E P (cid:2) Y t (cid:12)(cid:12) G Xt (cid:3) and we set(8.115) N νt := v νt + Z t ( ∇ V )( σ, W σ ) dσ We want to prove that ( N νt ) is a ( F νt ) − martingale. First note that by definition ( Z t ) is a ( A t )martingale and that for any t ∈ [0 , G Xt ⊂ A t . Therefore, for any s ≤ t , P − a.s. (8.116) E P (cid:2) Z t (cid:12)(cid:12) G Xs (cid:3) = E P (cid:2) Z s (cid:12)(cid:12) G Xs (cid:3) On the other hand by (8.115) and (3.50) we have, for 0 ≤ s ≤ t < P − a.s.E ν [ N νt |F νs ] ◦ X = E P (cid:2) N νt ◦ X (cid:12)(cid:12) G Xs (cid:3) = E P (cid:20)(cid:18) E P [ Y t |G Xt ] + Z t ( ∇ V )( σ, X σ ) dσ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) G Xs (cid:21) = E P h e Z t (cid:12)(cid:12)(cid:12) G Xs i where e Z t := Z t − Z + Y . Together with (8.116) and since ( N νt ) is ( F νt ) adapted we derive, for s ≤ t P − a.s. E ν [ N νt |F νs ] ◦ X = E P h e Z t (cid:12)(cid:12)(cid:12) G Xs i = E P h e Z s (cid:12)(cid:12)(cid:12) G Xs i = N νs ◦ X so that ( N νt ) is a ( F νt ) − martingale on ( W, B ( W ) ν , ν ). The converse follows from the definition.Indeed take Z t = N νt , where N νt is the martingale of the stochastic Euler-Lagrange condition. Then dv νt = dZ t + ( ∇ V )( t, W t ) dt and on the other hand, ν − a.s. dW t = dM νt + v νt dt so that the result follows by the martingale representation theorem (see [6]) which can be appliedon ( W, B ( W ) ν , ν ) or on one of its extensions if σ ν is degenerated. Remark 8.2.1.
It is an interesting problem to determine conditions that assure the existence ofsolutions for a system of type (8.112)-(8.114). Adding a final condition for the process Y leads usto the study of forward-backward stochastic differential systems (we refer, for example to [15] ). The next result shows the existence of a probability satisfying the Euler-Lagrange condition fora classical Lagrangian with force V = 0. Proposition 8.3.
Let γ be a Borel probability on P R d × R d whose first marginal π ⋆ γ is denoted by ν ∈ P R d , and let µ ν := R R d ν ( dx ) µ x where µ x is the Wiener measure starting from x ∈ R d .Assume also that (8.117) l ( γ ) := inf( H ( ν | µ ν ) : ν ∈ P W , ( W × W ) ⋆ ν = γ ) < ∞ where H ( ν | µ ν ) := E ν (cid:20) ln dνdµ ν (cid:21) denotes the relative entropy. Then there exists a unique probability ν ⋆ which attains the infimumof (8.117) and ν ⋆ ∈ G ( R d ) . Proof:
By a classical application of the Dunford − Pettis theorem, the relative entropy with respectto ν has compact level sets (for the weak convergence in measure). Moreover it is strictly convex.Since { ν ∈ P W : ( W × W ) ⋆ ν } is closed (for the weak convergence in measure) and convex, assoon as l ( γ ) < ∞ the infimum is attained by a unique probability ν ⋆ . If H ( ν | µ ν ) < ∞ then inparticular ν << µ ν and by the Gisanov theorem ν ∈ S B (see Corollary 7.1). On the other hand bythe celebrated formula of [5] if H ( ν | µ ν ) < ∞ we have2 H ( ν | µ ν ) = E ν (cid:20)Z | v νs | ds (cid:21) Thus we obtain l ( γ ) = inf (cid:18)(cid:26) E ν (cid:20)Z | v νs | ds (cid:21) , ν ∈ S B ∩ S γ (cid:27)(cid:19) and the result follows by Corollary 7.1. Examples:
Let us mention the following examples and counterexamples which follow by simplecalculus :(i) For any x, y ∈ R d , µ x,y ∈ G ( R d ), where µ x,y denotes the law of the pinned Brownian motionsuch that W = x and W = y .(ii) For d = 1, let α ∈ [0 ,
1) and µ be the standard Wiener measure with µ − a.s. W = 0. Define ν to be the probability which is absolutely continuous with respect to µ with density given by dνdµ := ( W − W α ) − α Then Clark − Ocone formula of Malliavin calculus shows that ν ∈ G ( R d ) and that it is the lawof the non-Markovian stochastic differential equation dX t = dB t + 21 [ α, ( s ) X s − X α − s + ( X s − X α ) ; X = 0Now if we denote by ν ⋆ the probability defined by dν ⋆ dµ = dW ⋆ νdW ⋆ µ ( W )and since µ − a.s. W = 0, by Jensen’s inequality ν ⋆ solvesinf( H ( η | µ ) : ( W × W ) ⋆ η = ( W × W ) ⋆ ν )By the strict convexity of the entropy we obtain H ( ν ⋆ | µ ) < H ( ν | µ ) < ∞ This proves that, even for the Lagrangian L = | v | , there may exist several elements of G ( R )of finite entropy with the same joint laws for W and W .(iii) Assume that u : ( t, x ) ∈ [0 , × R d → R d is a C , function, which is essentially bounded with an essentially bounded gradient, and thatsatisfies the incompressible Navier-Stokes equation(8.118) ∂ t u + ( u. ∇ ) u = −∇ p + ∆ u , div u = 0 EAK CALCULUS OF VARIATIONS 39
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