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A class of anticipative tangent processes on the Wiener space
C. R. Acad. Sci. Paris, t. 333, Série I, p. 353–358, 2001 Probabilités/Probability Theory
A class of anticipative tangent processes on the Wiener space Ana-Bela CRUZEIRO a , Paul MALLIAVIN b a b
Grupo de Física-Matemática, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal 10, rue S.-Louis-en-l’Ile, 75004 Paris, France
(Reçu le 14 mai 2001, accepté le 5 juin 2001)
Abstract.
We prove a representation and an integration by parts formula for a class of anticipative tangent processes on the Wiener space and give some applications. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Une classe de processus tangents anticipatifs sur l’espace de Wiener Résumé.
On démontre une formule de représentation et une formule d’intégration par parties pour une classe de processus tangents anticipatifs sur l’espace de Wiener. Quelques applications sont mentionnées. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Version française abrégée Dans cette Note nous démontrons une formule de représentation de la dérivée sur l’espace de Wiener par rapport à un « processus tangent anticipatif », c’est-à-dire, un processus qui s’écrit comme la somme d’une rotation et d’un processus de variation bornée, pas nécessairement adaptés, généralisant ainsi des résultats obtenus dans [3]. De cette formule on déduit un théorème d’intégration par parties sur l’espace de Wiener et une nouvelle preuve de l’extension au cas non adapté de la formule de Bismut sur l’espace de chemins d’une variété riemannienne.
1. Introduction In stochastic calculus of variations on the Wiener space one usually considers the Cameron–Martin space (the space of trajectories with L2 derivatives in time) as the natural tangent space: indeed Cameron–Martin theorem tells us that translations of the Brownian motion with respect to such trajectories are those which preserve Wiener measure. The study of the calculus of variations on the path space of a manifold (cf. [1,3,6, 7], among many others) showed the need to extend this tangent space and include rotations of the Brownian Note présentée par Paul M ALLIAVIN. S0764-4442(01)02055-9/FLA 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés
353
A.-B. Cruzeiro, P. Malliavin
trajectories, on one hand, and, on the other, to develop anticipative stochastic calculus [3]. For those reasons it seems of interest to consider anticipative rotations on the Wiener space, as it is done in [8]. In the present work we consider a class of such anticipative processes and give a representation formula for the derivative along such processes. As a consequence we obtain a corresponding integration by parts theorem. 2. Anticipative stochastic calculus Let X be the Wiener space of the Brownian motion in Rd and µ the Wiener measure. We consider on X the anticipative stochastic calculus in the sense of Nualart–Pardoux–Zakai, following the notations of [10]. Let L2,1 be the class of processes u ∈ L2 ([0, 1] × X) such that, for a.a. τ , uτ ∈ W12 and 1 1 E 0 |Dσ uτ |2 dσ dτ < ∞. The Skorohod integral of u ∈ L2,1 , that we denote by 0 u dx, is defined as the limit, when the mesh of the partition goes to zero, of the sums c E ∆k (Mk (u)).(x(σk+1 ) − x(σk )), σk+1
k c
where Mk (u) = uτ dτ and E ∆k is the conditional expectation consisting in averaging σk relatively to the σ-field generated by the complement of x(τ ) − x(σk ), τ ∈ (σk , σk+1 ). 1 TheSkorohod–Stratonovich integral of a process u, denoted by 0 u◦ dx, is defined as the limit of the sums k (Mk (u)).(x(σk+1 ) − x(σk )). For such integral to be well defined one needs to assume that u not only belongs to L2,1 but also that Dσ uτ satisfies some uniform continuity and boundedness properties near the diagonal of [0, 1]2 (cf. [10]). These two integrals are related by 1 1 1 1 + u dx = u◦ dx − (Dτ + Dτ− ).uτ dτ 2 0 0 0 1 σk+1 −σk
+(−)
where Dτ .uτ = limσ→τ +(−) Dτ .uσ . β α If ξ is a process on X that can be written as dξ α (τ ) = aα β dx (τ )+ c dτ , we consider the generalizations of the anticipative integrals with respect to Brownian motion, resp. the Skorohod and the Skorohod– Stratonovich integral relative to ξ, 1 1 uτ dξ(τ ) and uτ ◦ dξ(τ ) 0
0
which are defined respectively as the limit, when it exists, of the sums c E ∆k (Mk (u)).∆k (ξ) and (Mk (u)).∆k (ξ), k
k
where ∆k (ξ) = ξ(σk+1 ) − ξ(σk ). In [3] (cf. also the appendix of [5]) we have shown that when ξ is a semi-martingale, the derivatives with respect to ξ can be represented as 1 1 α aβ Dτ,α F dxβ (τ ) + cα Dτ,α F dτ. Dξ F = 0
α
α
0
Here we prove that this formula can be extended to the case where a is not adapted. 3. Anticipative tangent processes We shall consider the class of processes ξ on the Wiener space X of the form β α dξ α (τ ) = aα β dx (τ ) + c dτ,
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A class of anticipative tangent processes on the Wiener space
1 2 β α α where aα β = −aα , aβ (0) = 0, c (0) = 0, E 0 |c| dτ < ∞ and the (non necessarily adapted) functionals aα β are such that, besides its representation by a Skorohod stochastic integral, ξ can be also represented in terms of a Stratonovich–Skorohod one. We call these processes anticipative tangent processes. Given a smooth cylindrical function on X, namely F (x) = f (x(τ1 ), . . . , x(τm )), the derivative of F with respect to ξ is defined as Dξ F = dk f, ξ(τk ) . k
In this paragraph we shall extend the results on tangent processes proved in [3] and [5] by proving the 1 1 2 β α T HEOREM 1. – Let dξ α (τ ) = aα β dx (τ ) + c dτ with 0 a(τ ) 1,p dτ < ∞ for all p and E 0 |c| dτ < ∞. Then W2,p ∈ Dom(Dξ ),
∀p > 1,
and the following representation formula holds: 1 α aβ Dτ,α F dxβ (τ ) + Dξ F = 0
α
α
1
cα Dτ,α F dτ
0
for F ∈ W2,p . From last theorem an integration by parts formula for anticipative tangent processes follows. C OROLLARY 1 (Integration by parts). – If ξ is an anticipative tangent process and F a functional satisfying the assumptions in Theorem 1, we have 1 cα dxα (τ ) . E(Dξ F ) = E F 0
α
Proof of Theorem 1. – We shall prove Theorem 1 for cylindrical functionals F , the operator Dξ being defined by closure. We also assume c = 0 since the result for the bounded variation term is known. If {σ1 , . . . , σr } is a subdivision of the interval [0, 1] that we choose to be finer than {τ1 , . . . , τm }, we have r−1 1 (1σl+1 <τk − 1σl−1 <τk ) dk f, ξ(σl ) dk f, ξ(τk ) = − lim 2 k k l=2 1
1 = lim Dτ,α F ◦ dξ α (τ ) Dσl F | ξ(σl+1 ) − ξ(σl−1 ) = 2 0
Dξ F =
l
the limit being taken when the mesh of the partition goes to zero. Let us start by considering a of the form aα aα β (x)(τ ) = φ(x)˜ β (τ ) with a ˜ adapted. We have 1 α β
Dξ F = Dτ,α F ◦ φ˜ aβ dx = lim Mσk (D. ,α F ) 0
=
k
α β Dτ,α F φ ◦(˜ aβ dx ) − lim Mσk (D. ,α F )
= 0
1
α β
(Dτ,α F φ)◦ a ˜β dx −
β (φ˜ a)α β dx
σk
1
0
σk+1
k
σk+1
Ds,β φ˜ aα β ds
σk
1
Dτ,α F Dτ,β φ˜ aα β dτ. 0
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A.-B. Cruzeiro, P. Malliavin
The difference between the Stratonovich–Skorohod integral 1 (Dτ,α F φ)◦ dξ α (τ ) 0
and the corresponding Skorohod one is given by the limit of the sums
c Mk (D . ,α F φ) − E ∆k Mk (D . ,α F φ) ξ α (∆k ). k
By the Clark–Ocone formula, this expression is equal to σk+1
γ Pλ E Mk Dλ,γ (Dτ,α F φ) dx (λ) ξ α (∆k ). σk
k
Using the quadratic variation of the Brownian motion the limit is σk+1 σk+1 2
α lim Mk Dλ,γ; F φ a ˜ dτ + lim Mk (Dτ,α F Dλ,γ φ)˜ aα . ,α γ γ dτ. k
σk
k
σk
a ˜..
The antisymmetry of and the symmetry of the second derivatives are responsible for the vanishing of the second order derivatives and the difference reduces to 1 Dτ,α F Dτ,β φ˜ aα β dτ. 0
The result follows from the equality (cf. [5], A2 ) 1 1 α α (Dτ,α F φ) dξ (τ ) = φ˜ aβ Dτ,α F dxβ (τ ). 0
1
0
α
For general a satisfying 0 a(τ ) 1,p dτ < ∞ we can approximate the matrix-valued functional in the corresponding Sobolev norms by a sequence of functionals of the form φ˜ a, with a ˜ adapted. Finally, the operator Dξ is closable in W2,p : this follows from the identity Dξ (F G) = F Dξ G + GDξ F which holds for cylindrical functionals F and G by the corresponding representation formulae. C OROLLARY 2. – Let ξ be an anticipative tangent process as in Theorem 1 and φ be a real valued Wiener functional in W1,p . We have: Dφξ F = φDξ F for every F ∈ W2,p . Proof. – This is a consequence of the representation formula of Theorem 1 and and the following identity for Skorohod integrals: φ a dx = (φa) dx + (Dτ φ)a dτ. 4. An integration by parts formula on the path space of a Riemannian manifold On a compact d-dimensional Riemannian manifold M let p(τ ), τ ∈ [0, 1], denote the Brownian motion associated to the Laplace–Beltrami operator and Pm0 (M ) the space of continuous paths on M starting at m0 at time zero endowed with the corresponding Wiener measure σ. We consider the calculus of variations on the path space (cf. [3,7]). If F is a cylindrical functional on Pm0 (M ), namely F (p) =
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A class of anticipative tangent processes on the Wiener space
f (p(τ1 ), . . . , p(τm )), the derivatives Dτ,α F , τ ∈ [0, 1], α = 1, . . . , d, are defined by
(Dτ,α F )(p) = 1τ <τi tp0←τi (∂τi F ) | α , i p
where t denotes the (Levi-Civita) parallel transport constructed by Itô over the Brownian paths, α-canonical vector in Rd . Consider the following norms 1 2 DF (p) = [Dτ,α F ]2 dτ α
DF qLq
α
the
0
and = E( DF ). The operator D is closable in Lq , as a consequence of Bismut’s integration by parts formula on the path space. We denote by W1q (Pm0 (M )) its domain. If Z is a tangent vector field on the path space, namely if Zp (τ ) belongs to the tangent space to M at the point p(τ ), and if Zp (τ ) = tp0←τ (z(τ )) with z a Cameron–Martin space valued vector field on the Wiener space, the derivative on the path space with respect to Z is defined by 1 z˙ α (τ )Dτ,α F dτ. DZ F = q
α
0
Then the Bismut integration by parts formula reads [1]: 1 1 Eσ (DZ F ) = Eµ F ◦I z˙ + RM z dx 2 0 where RτM = tp0←τ ◦Riccip(τ ) ◦tpτ ←0 and where I is the Itô map. This map, introduced in [9], is a map from the Wiener space to the path space defined in terms of the parallel transport over Brownian paths and its lift to the orthonormal frame bundle over M and provides an isomorphism of measures; in particular we can write Eσ (DZ F ) = Eµ ((DZ F )◦I). But it does not provide an isomorphism of tangent spaces because the Itô map is not Cameron–Martin differentiable: in fact, we have the following result (cf. [3,6,7]): T HEOREM 2. – A smooth functional f defined on the path space Pm0 (M ) is differentiable along a tangent vector field Z if and only if f ◦I is differentiable on the Wiener space along the tangent process dξ = z˙ dτ + ρ dx(τ ) where dρ(τ ) = Ω(◦ dx, z), Ω being the curvature tensor of the underlying manifold M . In fact Bismut’s integration by parts formula can be directly derived from this result and from the characterization of the divergence on the Wiener space, since dρ.dx = 12 RM dτ . Bismut’s integration by parts formula has been generalized for anticipative tangent vector fields on the path space in [4] and in [11]. Here we derive the formula from Theorem 1 (we remark that the expressions obtained in the last references may look quite different but are equivalent). We first observe that Theorem 2 holds for anticipative tangent vector fields Z on the path space as long as we interpret the stochastic integrals in the formulae in the sense of Stratonovich–Skorohod and assume that such integrals are well defined. If ψ is a real-valued functional defined on the path space, we have (DψZ f )◦I = (ψDZ f )◦I = (ψ◦I)Dξ (f ◦I) ˜ ξ (f ◦I) = D ˜ (f ◦I). Now By Corollary 2, and writting ψ˜ = ψ◦I, we have ψD ψξ ˜ = ψ˜z˙ dτ + (ψρ)◦ ˜ d(ψξ) dx(τ ),
˜ ˜ d(ψρ)(τ ) = Ω(◦ dx, (ψz))
and the result follows for general anticipative vector fields Z by approximation.
357
A.-B. Cruzeiro, P. Malliavin
T HEOREM 3 (Integration by parts formula on the path space). – Let Z be a tangent vector field on 1 2 δ γ Pm0 (M ), z(τ ) = tp0←τ Z(τ ), with E 0 |z| ˙ dτ < ∞ and such that dρα (τ ) = Ωα γ,δ,β z ◦ dx (τ ) is well 1 1 τ β + − defined and satisfies E 0 ρ 1,p < ∞. Assume also that E 0 | β 0 Ω(◦ dx, (Dτ,β + Dτ,β )z)|2 dτ < ∞. Then, for every cylindrical functional f on the path space we have 1
+ 1 τ 1 − Eσ (DZ f ) = Eµ (f ◦I) z .dx(τ ) . Ω ◦ dx, Dτ,β + Dτ,β z(τ ˙ ) + RM z (τ ) + 2 2 0 0 β
Proof. – We have shown that derivating f on the path space with respect to Z corresponds to derivating β f ◦I with respect to the anticipative tangent process dξ α (τ ) = z˙ α dτ + ρα β ◦ dx (τ ). In order to apply Corollary 1. we only need to compute the difference between the Stratonovich–Skorohod and the Skorohod integral of ρ. This difference is equal to: τ
α +
α
1 1 + − − ρβ (τ ) dτ = Dτ,β + Dτ,β Ωγ,δ,β z δ ◦ dxγ (s) Dτ,β + Dτ,β dτ 2 2 0 β γ,δ,β 1 + Ricci(z)α dτ. 2 The derivatives of Ω vanish for s < τ , therefore only the derivatives of z contribute to the final expression. We remark that, when z is adapted, we recover Bismut’s formula since in this case the derivatives of z(s) also vanish for s < τ . 5. Flows associated to anticipative tangent processes The construction of flows associated to tangent processes in the adapted case was made in [2]. The proof relies on finite dimensional approximations of the process as well as of its derivatives and divergence. We can therefore perform the same construction for anticipative tangent processes. For such type of results see also [8]. References [1] Bismut J.-M., Large Deviations and the Malliavin Calculus, Birkhäuser, 1984. [2] Cipriano F., Cruzeiro A.B., Flows associated to tangent processes on the Wiener space, J. Funct. Anal. 166 (1999) 310–331. [3] Cruzeiro A.B., Malliavin P., Renormalized differential geometry on path space, structural equation, curvature, J. Funct. Anal. 139 (1996) 119–181. [4] Cruzeiro A.B., Malliavin P., Energy identities and estimates for anticipative stochastic integrals on a Riemannian manifold, in: Stochastic Anal. and Rel. Topics, Vol. 42, Birkhäuser, 1998, pp. 221–234. [5] Cruzeiro A.B., Malliavin P., Frame bundle of Riemannian path space and Ricci tensor in adapted differential geometry, J. Funct. Anal. 177 (2000) 219–253. [6] Driver B., A Cameron–Martin type quasi-invariance theorem for Brownian motion on a compact manifold, J. Funct. Anal. 110 (1992) 272–376. [7] Fang S., Malliavin P., Stochastic analysis on the path space of a Riemannian manifold, J. Funct. Anal. 118 (1993) 249–274. [8] Hu Y., Ustunel A.S., Zakai M., Tangent processes on Wiener space, Preprint. [9] Malliavin P., Formule de la moyenne, calcul de perturbations et théorème d’annulation pour les formes harmoniques, J. Funct. Anal. 1 (1974) 274–291. [10] Nualart D., Pardoux E., Stochastic calculus with anticipating integrands, Prob. Th. and Rel. Fields (1988) 535–581. [11] Prat J.-J., Privault N., Explicit stochastic analysis of Brownian motion and point measures on Riemannian manifolds, J. Funct. Anal. 167 (1999) 201–242.
358
A class of anticipative tangent processes on the Wiener space Ana-Bela CRUZEIRO a , Paul MALLIAVIN b a b
Grupo de Física-Matemática, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal 10, rue S.-Louis-en-l’Ile, 75004 Paris, France
(Reçu le 14 mai 2001, accepté le 5 juin 2001)
Abstract.
We prove a representation and an integration by parts formula for a class of anticipative tangent processes on the Wiener space and give some applications. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Une classe de processus tangents anticipatifs sur l’espace de Wiener Résumé.
On démontre une formule de représentation et une formule d’intégration par parties pour une classe de processus tangents anticipatifs sur l’espace de Wiener. Quelques applications sont mentionnées. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Version française abrégée Dans cette Note nous démontrons une formule de représentation de la dérivée sur l’espace de Wiener par rapport à un « processus tangent anticipatif », c’est-à-dire, un processus qui s’écrit comme la somme d’une rotation et d’un processus de variation bornée, pas nécessairement adaptés, généralisant ainsi des résultats obtenus dans [3]. De cette formule on déduit un théorème d’intégration par parties sur l’espace de Wiener et une nouvelle preuve de l’extension au cas non adapté de la formule de Bismut sur l’espace de chemins d’une variété riemannienne.
1. Introduction In stochastic calculus of variations on the Wiener space one usually considers the Cameron–Martin space (the space of trajectories with L2 derivatives in time) as the natural tangent space: indeed Cameron–Martin theorem tells us that translations of the Brownian motion with respect to such trajectories are those which preserve Wiener measure. The study of the calculus of variations on the path space of a manifold (cf. [1,3,6, 7], among many others) showed the need to extend this tangent space and include rotations of the Brownian Note présentée par Paul M ALLIAVIN. S0764-4442(01)02055-9/FLA 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés
353
A.-B. Cruzeiro, P. Malliavin
trajectories, on one hand, and, on the other, to develop anticipative stochastic calculus [3]. For those reasons it seems of interest to consider anticipative rotations on the Wiener space, as it is done in [8]. In the present work we consider a class of such anticipative processes and give a representation formula for the derivative along such processes. As a consequence we obtain a corresponding integration by parts theorem. 2. Anticipative stochastic calculus Let X be the Wiener space of the Brownian motion in Rd and µ the Wiener measure. We consider on X the anticipative stochastic calculus in the sense of Nualart–Pardoux–Zakai, following the notations of [10]. Let L2,1 be the class of processes u ∈ L2 ([0, 1] × X) such that, for a.a. τ , uτ ∈ W12 and 1 1 E 0 |Dσ uτ |2 dσ dτ < ∞. The Skorohod integral of u ∈ L2,1 , that we denote by 0 u dx, is defined as the limit, when the mesh of the partition goes to zero, of the sums c E ∆k (Mk (u)).(x(σk+1 ) − x(σk )), σk+1
k c
where Mk (u) = uτ dτ and E ∆k is the conditional expectation consisting in averaging σk relatively to the σ-field generated by the complement of x(τ ) − x(σk ), τ ∈ (σk , σk+1 ). 1 TheSkorohod–Stratonovich integral of a process u, denoted by 0 u◦ dx, is defined as the limit of the sums k (Mk (u)).(x(σk+1 ) − x(σk )). For such integral to be well defined one needs to assume that u not only belongs to L2,1 but also that Dσ uτ satisfies some uniform continuity and boundedness properties near the diagonal of [0, 1]2 (cf. [10]). These two integrals are related by 1 1 1 1 + u dx = u◦ dx − (Dτ + Dτ− ).uτ dτ 2 0 0 0 1 σk+1 −σk
+(−)
where Dτ .uτ = limσ→τ +(−) Dτ .uσ . β α If ξ is a process on X that can be written as dξ α (τ ) = aα β dx (τ )+ c dτ , we consider the generalizations of the anticipative integrals with respect to Brownian motion, resp. the Skorohod and the Skorohod– Stratonovich integral relative to ξ, 1 1 uτ dξ(τ ) and uτ ◦ dξ(τ ) 0
0
which are defined respectively as the limit, when it exists, of the sums c E ∆k (Mk (u)).∆k (ξ) and (Mk (u)).∆k (ξ), k
k
where ∆k (ξ) = ξ(σk+1 ) − ξ(σk ). In [3] (cf. also the appendix of [5]) we have shown that when ξ is a semi-martingale, the derivatives with respect to ξ can be represented as 1 1 α aβ Dτ,α F dxβ (τ ) + cα Dτ,α F dτ. Dξ F = 0
α
α
0
Here we prove that this formula can be extended to the case where a is not adapted. 3. Anticipative tangent processes We shall consider the class of processes ξ on the Wiener space X of the form β α dξ α (τ ) = aα β dx (τ ) + c dτ,
354
A class of anticipative tangent processes on the Wiener space
1 2 β α α where aα β = −aα , aβ (0) = 0, c (0) = 0, E 0 |c| dτ < ∞ and the (non necessarily adapted) functionals aα β are such that, besides its representation by a Skorohod stochastic integral, ξ can be also represented in terms of a Stratonovich–Skorohod one. We call these processes anticipative tangent processes. Given a smooth cylindrical function on X, namely F (x) = f (x(τ1 ), . . . , x(τm )), the derivative of F with respect to ξ is defined as Dξ F = dk f, ξ(τk ) . k
In this paragraph we shall extend the results on tangent processes proved in [3] and [5] by proving the 1 1 2 β α T HEOREM 1. – Let dξ α (τ ) = aα β dx (τ ) + c dτ with 0 a(τ ) 1,p dτ < ∞ for all p and E 0 |c| dτ < ∞. Then W2,p ∈ Dom(Dξ ),
∀p > 1,
and the following representation formula holds: 1 α aβ Dτ,α F dxβ (τ ) + Dξ F = 0
α
α
1
cα Dτ,α F dτ
0
for F ∈ W2,p . From last theorem an integration by parts formula for anticipative tangent processes follows. C OROLLARY 1 (Integration by parts). – If ξ is an anticipative tangent process and F a functional satisfying the assumptions in Theorem 1, we have 1 cα dxα (τ ) . E(Dξ F ) = E F 0
α
Proof of Theorem 1. – We shall prove Theorem 1 for cylindrical functionals F , the operator Dξ being defined by closure. We also assume c = 0 since the result for the bounded variation term is known. If {σ1 , . . . , σr } is a subdivision of the interval [0, 1] that we choose to be finer than {τ1 , . . . , τm }, we have r−1 1 (1σl+1 <τk − 1σl−1 <τk ) dk f, ξ(σl ) dk f, ξ(τk ) = − lim 2 k k l=2 1
1 = lim Dτ,α F ◦ dξ α (τ ) Dσl F | ξ(σl+1 ) − ξ(σl−1 ) = 2 0
Dξ F =
l
the limit being taken when the mesh of the partition goes to zero. Let us start by considering a of the form aα aα β (x)(τ ) = φ(x)˜ β (τ ) with a ˜ adapted. We have 1 α β
Dξ F = Dτ,α F ◦ φ˜ aβ dx = lim Mσk (D. ,α F ) 0
=
k
α β Dτ,α F φ ◦(˜ aβ dx ) − lim Mσk (D. ,α F )
= 0
1
α β
(Dτ,α F φ)◦ a ˜β dx −
β (φ˜ a)α β dx
σk
1
0
σk+1
k
σk+1
Ds,β φ˜ aα β ds
σk
1
Dτ,α F Dτ,β φ˜ aα β dτ. 0
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A.-B. Cruzeiro, P. Malliavin
The difference between the Stratonovich–Skorohod integral 1 (Dτ,α F φ)◦ dξ α (τ ) 0
and the corresponding Skorohod one is given by the limit of the sums
c Mk (D . ,α F φ) − E ∆k Mk (D . ,α F φ) ξ α (∆k ). k
By the Clark–Ocone formula, this expression is equal to σk+1
γ Pλ E Mk Dλ,γ (Dτ,α F φ) dx (λ) ξ α (∆k ). σk
k
Using the quadratic variation of the Brownian motion the limit is σk+1 σk+1 2
α lim Mk Dλ,γ; F φ a ˜ dτ + lim Mk (Dτ,α F Dλ,γ φ)˜ aα . ,α γ γ dτ. k
σk
k
σk
a ˜..
The antisymmetry of and the symmetry of the second derivatives are responsible for the vanishing of the second order derivatives and the difference reduces to 1 Dτ,α F Dτ,β φ˜ aα β dτ. 0
The result follows from the equality (cf. [5], A2 ) 1 1 α α (Dτ,α F φ) dξ (τ ) = φ˜ aβ Dτ,α F dxβ (τ ). 0
1
0
α
For general a satisfying 0 a(τ ) 1,p dτ < ∞ we can approximate the matrix-valued functional in the corresponding Sobolev norms by a sequence of functionals of the form φ˜ a, with a ˜ adapted. Finally, the operator Dξ is closable in W2,p : this follows from the identity Dξ (F G) = F Dξ G + GDξ F which holds for cylindrical functionals F and G by the corresponding representation formulae. C OROLLARY 2. – Let ξ be an anticipative tangent process as in Theorem 1 and φ be a real valued Wiener functional in W1,p . We have: Dφξ F = φDξ F for every F ∈ W2,p . Proof. – This is a consequence of the representation formula of Theorem 1 and and the following identity for Skorohod integrals: φ a dx = (φa) dx + (Dτ φ)a dτ. 4. An integration by parts formula on the path space of a Riemannian manifold On a compact d-dimensional Riemannian manifold M let p(τ ), τ ∈ [0, 1], denote the Brownian motion associated to the Laplace–Beltrami operator and Pm0 (M ) the space of continuous paths on M starting at m0 at time zero endowed with the corresponding Wiener measure σ. We consider the calculus of variations on the path space (cf. [3,7]). If F is a cylindrical functional on Pm0 (M ), namely F (p) =
356
A class of anticipative tangent processes on the Wiener space
f (p(τ1 ), . . . , p(τm )), the derivatives Dτ,α F , τ ∈ [0, 1], α = 1, . . . , d, are defined by
(Dτ,α F )(p) = 1τ <τi tp0←τi (∂τi F ) | α , i p
where t denotes the (Levi-Civita) parallel transport constructed by Itô over the Brownian paths, α-canonical vector in Rd . Consider the following norms 1 2 DF (p) = [Dτ,α F ]2 dτ α
DF qLq
α
the
0
and = E( DF ). The operator D is closable in Lq , as a consequence of Bismut’s integration by parts formula on the path space. We denote by W1q (Pm0 (M )) its domain. If Z is a tangent vector field on the path space, namely if Zp (τ ) belongs to the tangent space to M at the point p(τ ), and if Zp (τ ) = tp0←τ (z(τ )) with z a Cameron–Martin space valued vector field on the Wiener space, the derivative on the path space with respect to Z is defined by 1 z˙ α (τ )Dτ,α F dτ. DZ F = q
α
0
Then the Bismut integration by parts formula reads [1]: 1 1 Eσ (DZ F ) = Eµ F ◦I z˙ + RM z dx 2 0 where RτM = tp0←τ ◦Riccip(τ ) ◦tpτ ←0 and where I is the Itô map. This map, introduced in [9], is a map from the Wiener space to the path space defined in terms of the parallel transport over Brownian paths and its lift to the orthonormal frame bundle over M and provides an isomorphism of measures; in particular we can write Eσ (DZ F ) = Eµ ((DZ F )◦I). But it does not provide an isomorphism of tangent spaces because the Itô map is not Cameron–Martin differentiable: in fact, we have the following result (cf. [3,6,7]): T HEOREM 2. – A smooth functional f defined on the path space Pm0 (M ) is differentiable along a tangent vector field Z if and only if f ◦I is differentiable on the Wiener space along the tangent process dξ = z˙ dτ + ρ dx(τ ) where dρ(τ ) = Ω(◦ dx, z), Ω being the curvature tensor of the underlying manifold M . In fact Bismut’s integration by parts formula can be directly derived from this result and from the characterization of the divergence on the Wiener space, since dρ.dx = 12 RM dτ . Bismut’s integration by parts formula has been generalized for anticipative tangent vector fields on the path space in [4] and in [11]. Here we derive the formula from Theorem 1 (we remark that the expressions obtained in the last references may look quite different but are equivalent). We first observe that Theorem 2 holds for anticipative tangent vector fields Z on the path space as long as we interpret the stochastic integrals in the formulae in the sense of Stratonovich–Skorohod and assume that such integrals are well defined. If ψ is a real-valued functional defined on the path space, we have (DψZ f )◦I = (ψDZ f )◦I = (ψ◦I)Dξ (f ◦I) ˜ ξ (f ◦I) = D ˜ (f ◦I). Now By Corollary 2, and writting ψ˜ = ψ◦I, we have ψD ψξ ˜ = ψ˜z˙ dτ + (ψρ)◦ ˜ d(ψξ) dx(τ ),
˜ ˜ d(ψρ)(τ ) = Ω(◦ dx, (ψz))
and the result follows for general anticipative vector fields Z by approximation.
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T HEOREM 3 (Integration by parts formula on the path space). – Let Z be a tangent vector field on 1 2 δ γ Pm0 (M ), z(τ ) = tp0←τ Z(τ ), with E 0 |z| ˙ dτ < ∞ and such that dρα (τ ) = Ωα γ,δ,β z ◦ dx (τ ) is well 1 1 τ β + − defined and satisfies E 0 ρ 1,p < ∞. Assume also that E 0 | β 0 Ω(◦ dx, (Dτ,β + Dτ,β )z)|2 dτ < ∞. Then, for every cylindrical functional f on the path space we have 1
+ 1 τ 1 − Eσ (DZ f ) = Eµ (f ◦I) z .dx(τ ) . Ω ◦ dx, Dτ,β + Dτ,β z(τ ˙ ) + RM z (τ ) + 2 2 0 0 β
Proof. – We have shown that derivating f on the path space with respect to Z corresponds to derivating β f ◦I with respect to the anticipative tangent process dξ α (τ ) = z˙ α dτ + ρα β ◦ dx (τ ). In order to apply Corollary 1. we only need to compute the difference between the Stratonovich–Skorohod and the Skorohod integral of ρ. This difference is equal to: τ
α +
α
1 1 + − − ρβ (τ ) dτ = Dτ,β + Dτ,β Ωγ,δ,β z δ ◦ dxγ (s) Dτ,β + Dτ,β dτ 2 2 0 β γ,δ,β 1 + Ricci(z)α dτ. 2 The derivatives of Ω vanish for s < τ , therefore only the derivatives of z contribute to the final expression. We remark that, when z is adapted, we recover Bismut’s formula since in this case the derivatives of z(s) also vanish for s < τ . 5. Flows associated to anticipative tangent processes The construction of flows associated to tangent processes in the adapted case was made in [2]. The proof relies on finite dimensional approximations of the process as well as of its derivatives and divergence. We can therefore perform the same construction for anticipative tangent processes. For such type of results see also [8]. References [1] Bismut J.-M., Large Deviations and the Malliavin Calculus, Birkhäuser, 1984. [2] Cipriano F., Cruzeiro A.B., Flows associated to tangent processes on the Wiener space, J. Funct. Anal. 166 (1999) 310–331. [3] Cruzeiro A.B., Malliavin P., Renormalized differential geometry on path space, structural equation, curvature, J. Funct. Anal. 139 (1996) 119–181. [4] Cruzeiro A.B., Malliavin P., Energy identities and estimates for anticipative stochastic integrals on a Riemannian manifold, in: Stochastic Anal. and Rel. Topics, Vol. 42, Birkhäuser, 1998, pp. 221–234. [5] Cruzeiro A.B., Malliavin P., Frame bundle of Riemannian path space and Ricci tensor in adapted differential geometry, J. Funct. Anal. 177 (2000) 219–253. [6] Driver B., A Cameron–Martin type quasi-invariance theorem for Brownian motion on a compact manifold, J. Funct. Anal. 110 (1992) 272–376. [7] Fang S., Malliavin P., Stochastic analysis on the path space of a Riemannian manifold, J. Funct. Anal. 118 (1993) 249–274. [8] Hu Y., Ustunel A.S., Zakai M., Tangent processes on Wiener space, Preprint. [9] Malliavin P., Formule de la moyenne, calcul de perturbations et théorème d’annulation pour les formes harmoniques, J. Funct. Anal. 1 (1974) 274–291. [10] Nualart D., Pardoux E., Stochastic calculus with anticipating integrands, Prob. Th. and Rel. Fields (1988) 535–581. [11] Prat J.-J., Privault N., Explicit stochastic analysis of Brownian motion and point measures on Riemannian manifolds, J. Funct. Anal. 167 (1999) 201–242.
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