getdoc6608. 199KB Jun 04 2011 12:04:27 AM

in PROBABILITY

A CLARK-OCONE FORMULA IN UMD BANACH SPACES

JAN MAAS1

Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands.

email: J.Maas@tudelft.nl

JAN VAN NEERVEN2

Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands.

email: J.M.A.M.vanNeerven@tudelft.nl

Submitted February 13, 2008, accepted in final form February 20, 2008 AMS 2000 Subject classification: Primary: 60H07; Secondary: 46B09, 60H05

Keywords: Clark-Ocone formula, UMD spaces, Malliavin calculus, Skorokhod integral, γ-radonifying operators,γ-boundedness, Stein inequality

Abstract

LetH be a separable real Hilbert space and letF= (F_{t)t∈[0,T]be the augmented filtration}

gen-erated by anH-cylindrical Brownian motion (WH(t))t∈[0,T] on a probability space (Ω,F,P). We prove that ifEis a UMD Banach space, 1≤p <∞, andF ∈D1,p_{(Ω;E) isF}

T-measurable,

then

F=E(F) +

Z T

0

P_F(DF)dWH,

whereDis the Malliavin derivative ofF andP_F is the projection onto theF-adapted elements in a suitable Banach space ofLp_{-stochastically integrableL(H, E)-valued processes.}

1

Introduction

A classical result of Clark [5] and Ocone [17] asserts that ifF= (F_{t)t∈[0,T] is the augmented}

filtration generated by a Brownian motion (W(t))t∈[0,T]on a probability space (Ω,F,P), then everyF_{T-measurable random variableF ∈D}1,p_{(Ω), 1< p <∞, admits a representation}

F =E(F) +

Z T

0

E(DtF|Ft)dWt,

whereDtis the Malliavin derivative ofF at timet. An extension toFT-measurable random

variables F ∈ D1,1_{(Ω) was subsequently given by Karatzas, Ocone, and Li [10]. The} Clark-1_{RESEARCH SUPPORTED BY ARC DISCOVERY GRANT DP0558539.}

2_{RESEARCH SUPPORTED BY VIDI SUBSIDY 639.032.201 AND VICI SUBSIDY 639.033.604 OF THE}

NETHERLANDS ORGANISATION FOR SCIENTIFIC RESEARCH (NWO).

Ocone formula is used in mathematical finance to obtain explicit expressions for hedging strategies.

The aim of this note is to extend the above results to the infinite-dimensional setting using the theory of stochastic integration of L₍H_{, E)-valued processes with respect to} H_-cylindrical

Brownian motions, developed recently by Veraar, Weis, and the second named author [15]. Here,H _{is a separable Hilbert space andE is a UMD Banach space (the definition is recalled}

below).

For this purpose we study the properties of the Malliavin derivative D of smooth E-valued random variables with respect to an isonormal process W on a separable Hilbert space H. As it turns out, D can be naturally defined as a closed operator acting from Lp_{(Ω;E) to}

Lp_{(Ω;γ(H, E)), whereγ(H, E) is the operator ideal ofγ-radonifying operators from a Hilbert}

spaceH toE. Via trace duality, the dual object is the divergence operator, which is a closed operator acting from Lp(Ω;γ(H, E)) toLp(Ω;E). In the special case whereH=L2(0, T;H₎

for another Hilbert space H_{, the divergence operator turns out to be an extension of the}

UMD-valued stochastic integral of [15].

The first two main results, Theorems 6.6 and 6.7, generalize the Clark-Ocone formula for Hilbert spaces E and exponent p= 2 as presented in Carmona and Tehranchi [4, Theorem 5.3] to UMD Banach spaces and exponents 1< p <∞. The extension top= 1 is contained in our Theorem 7.1.

Extensions of the Clark-Ocone formula to infinite-dimensional settings different from the one considered here have been obtained by various authors, among them Mayer-Wolf and Zakai [13, 14], Osswald [18] in the setting of abstract Wiener spaces and de Faria, Oliveira, Streit [7] and Aase, Øksendal, Privault, Ubøe [1] in the setting of white noise analysis. Let us also mention the related papers [11, 12].

Acknowledgment– Part of this work was done while the authors visited the University of New South Wales (JM) and the Australian National University (JvN). They thank Ben Goldys at UNSW and Alan McIntosh at ANU for their kind hospitality.

2

Preliminaries

We begin by recalling some well-known facts concerning γ-radonifying operators and UMD Banach spaces.

Let (γn)n≥1be sequence of independent standard Gaussian random variables on a probability space (Ω,F_{,P) and let H be a separable real Hilbert space. A bounded linear operator}

R : H → E is called γ-radonifying if for some (equivalently, for every) orthonormal basis (hn)n≥1 the Gaussian sumPn≥1γnRhn converges in L2(Ω;E). Here, (γn)n≥1 is a sequence of independent standard Gaussian random variables on (Ω,F_{,P). Endowed with the norm}

kRkγ(H,E):=

³

E°°°X

n≥1 γnRhn

° °2´12

,

the space γ(H, E) is a Banach space. ClearlyH⊗E ⊆γ(H, E), and this inclusion is dense. We have natural identificationsγ(H,R) =H andγ(R, E) =E.

For all finite rank operators T :H →E andS:H →E∗ _{we have}

|tr(S∗_{T)| ≤ kTk}

Since the finite rank operators are dense inγ(H, E) andγ(H, E∗_{), we obtain a natural} con-tractive injection

γ(H, E∗_{)֒→(γ(H, E))}∗_{. (1)} Let 1< p <∞. A Banach space E is called a UMD(p)-space if there exists a constantβp,E

such that for every finiteLp_{-martingale difference sequence (d}

j)nj=1with values inEand every

Using, for instance, Burkholder’s goodλ-inequalities, it can be shown that ifE is a UMD(p) space for some 1< p < ∞, then it is a UMD(p)-space for all 1< p <∞, and henceforth a space with this property will simply be called aUMD space.

Examples of UMD spaces are all Hilbert spaces and the spaces Lp_{(S) for 1 < p < ∞ and}

σ-finite measure spaces (S,Σ, µ). If E is a UMD space, then Lp_{(S;E) is a UMD space for}

1< p <∞. For an overview of the theory of UMD spaces and its applications in vector-valued stochastic analysis and harmonic analysis we recommend Burkholder’s review article [3]. Below we shall need the fact that if E is a UMD space, then trace duality establishes an isomorphism of Banach spaces

γ(H, E∗_{)≃(γ(H, E))}∗ .

As we shall briefly explain, this is a consequence of the fact that every UMD isK-convex. Let (γn)n≥1be sequence of independent standard Gaussian random variables on a probability space (Ω,F_{,P). For a random variableX ∈L}2_{(Ω;E) we define}

It is easy to see thatEisK-convex if and only its dualE∗ _{isK-convex, in which case one has} K(E) =K(E∗_{). For more information we refer to the book of Diestel, Jarchow, Tonge [8].} The next result from [19] (see also [9]) shows that ifEisK-convex, the inclusion (1) is actually an isomorphism:

Proposition 2.1. IfE isK-convex, then trace duality establishes an isomorphism of Banach spaces

γ(H, E∗_{)≃(γ(H, E))}∗_.

The main step is to realize that K-convexity implies that the ranges of πE _{and π}E∗

are canonically isomorphic as Banach spaces. This isomorphism is then used to represent elements of (γ(H, E))∗ _{by elements ofγ(H, E}∗_).

3

The Malliavin derivative

Throughout this note, (Ω,F_{,P) is a complete probability space,H is a separable real Hilbert}

space, and W : H → L2_{(Ω) is an isonormal Gaussian process, i.e., W is a bounded linear} operator from H to L2_{(Ω) such that the random variables W(h) are centred Gaussian and} satisfy

E(W(h1)W(h2)) = [h1, h2]H, h1, h2∈H.

A smooth random variableis a functionF : Ω→Rof the form F =f(W(h1), . . . , W(hn))

withf ∈C∞

b (Rn) andh1, . . . , hn∈H. Here,Cb∞(Rn) denotes the vector space of all bounded real-valued C∞_{-functions on R}n _{having bounded derivatives of all orders. We say that F}

is compactly supported if f is compactly supported. The collections of all smooth random variables and compactly supported smooth random variables are denoted byS_{(Ω) and}S_c(Ω),

respectively.

Let E be an arbitrary real Banach space and let 1≤p <∞. Noting that S_{c(Ω) is dense in}

Lp_{(Ω) and thatL}p_{(Ω)⊗E is dense inL}p_{(Ω;E), we see:}

Lemma 3.1. S_{c(Ω)⊗E is dense in L}p_(Ω;E).

TheMalliavin derivativeof anE-valued smooth random variable of the form F =f(W(h1), . . . , W(hn))⊗x

with f ∈ C∞

b (Rn), h1, . . . , hn ∈ H and x ∈ E, is the random variable DF : Ω → γ(H, E)

defined by

DF =

n

X

j=1

∂jf(W(h1), . . . , W(hn))⊗(hj⊗x).

Here, ∂j denotes thej-th partial derivative. The definition extends toS(Ω)⊗E by linearity.

For h∈H we defineDF(h) : Ω→ E by (DF(h))(ω) := (DF(ω))h. The following result is the simplest case of the integration by parts formula. We omit the proof, which is the same as in the scalar-valued case [16, Lemma 1.2.1].

Lemma 3.2. For all F ∈S_{(Ω)⊗E andh∈H we have E(DF(h)) =E(W(h)F).}

A straightforward calculation shows that the following product rule holds for F ∈S_(Ω)⊗E

andG∈S_(Ω)⊗E∗_:

DhF, Gi=hDF, Gi+hF, DGi. (2)

On the left hand sideh·,·idenotes the duality betweenEandE∗_{,which is evaluated pointwise} on Ω. In the first term on the right hand side, the H-valued pairingh·,·i between γ(H, E) and E∗ _{is defined by hR, x}∗_{i := R}∗_x∗_{. Similarly, the second term contains the H-valued} pairing betweenE and γ(H, E∗_{),which is defined byhx, Si:=S}∗_{x,thereby consideringxas} an element ofE∗∗_.

For scalar-valued functionsF ∈S_{(Ω) we may identifyDF ∈L}2_{(Ω;γ(H,R)) with the classical} Malliavin derivativeDF ∈L2_{(Ω;H).Using this identification we obtain the following product} rule forF ∈S_{(Ω) andG∈}S_(Ω)⊗E:

An application of Lemma 3.2 to the product hF, Giyields the following integration by parts formula forF∈S_{(Ω)⊗E andG∈}S_(Ω)⊗E∗_:

EhDF(h), Gi=E(W(h)hF, Gi)−EhF, DG(h)i. (4)

From the identity (4) we obtain the following proposition.

Proposition 3.3. For all 1 ≤p <∞, the Malliavin derivative D is closable as an operator fromLp_{(Ω;E) intoL}p_{(Ω;γ(H, E)).}

Proof. Let (Fn) be a sequence inS(Ω)⊗E be such that Fn→0 inLp(Ω;E) andDFn→X

inLp_{(Ω;γ(H, E)) asn→ ∞. We must prove thatX= 0.}

Fixh∈H and define

Vh:={G∈S(Ω)⊗E∗: W(h)G∈S(Ω)⊗E∗}.

We claim that Vh is weak∗-dense in (Lp(Ω;E))∗. Let 1_p +1_q = 1. To prove this it suffices

to note that the subspace{G∈S_{(Ω) :W(h)G∈}S_{(Ω)} is weak}∗_{-dense in L}q_{(Ω) and that}

Lq_(Ω)⊗E∗ _{is weak}∗_{-dense in (L}p_(Ω;E))∗_.

Fix G ∈ Vh. Using (4) and the fact that the mapping Y 7→ EhY(h), Gi defines a bounded

linear functional onLp_{(Ω;γ(H, E)) we obtain}

EhX(h), Gi= lim

n→∞EhDFn(h), Gi= limn→∞E(W(h)hFn, Gi)−EhFn, DG(h)i.

Since W(h)G and DG(h) are bounded it follows that this limit equals zero. Since Vh is

weak∗_{-dense in (L}p_(Ω;E))∗_{, we obtain thatX(h) vanishes almost surely. Now we choose an} orthonormal basis (hj)j≥1ofH.It follows that almost surely we haveX(hj) = 0 for allj≥1.

Hence,X= 0 almost surely.

With a slight abuse of notation we will denote the closure ofD again by D.The domain of this closure in Lp_{(Ω;E) is denoted by D}1,p_{(Ω;E). This is a Banach space endowed with the}

norm

kFkD1,p_(Ω;E):= (kFkp

Lp_(Ω;E)+kDFk

p

Lp_{(Ω;γ(H,E))})

1

p.

We writeD1,p_{(Ω) :=D}1,p_(Ω;R).

As an immediate consequence of the closability of the Malliavin derivative we note that the identities (2), (3), (4) extend to larger classes of functions. This fact will not be used in the sequel.

Proposition 3.4. Let 1≤p, q, r <∞such that 1_p+1

q =

1

r.

(i) For allF ∈D1,p_{(Ω;E)andG∈D}1,q_(Ω;E∗_{)we havehF, Gi ∈}

D1,r_(Ω)and

DhF, Gi=hDF, Gi+hF, DGi.

(ii) For allF ∈D1,p_{(Ω)andG∈D}1,q_{(Ω;E) we haveF G∈D}1,r_(Ω;E)and

D(F G) =F DG+DF⊗G.

(iii) For allF ∈D1,p_{(Ω;E), G∈D}1,q_(Ω;E∗_{)andh∈H we havehDF(h), Gi ∈L}r_{(Ω) and}

4

The divergence operator

In this section we construct a vector-valued divergence operator. The trace inequality (1) implies that we have a contractive inclusion γ(H, E)֒→ (γ(H, E∗₎₎∗_{. Hence for 1< p < ∞} and 1_p +1_q = 1, we obtain a natural embedding

Lp(Ω;γ(H, E))֒→(Lq_{(Ω;γ(H, E}∗₎₎₎∗ .

For the moment letDdenote the Malliavin derivative onLq_(Ω;E∗_{), which is a densely defined} closed operator with domainD1,q_(Ω;E∗_{) and taking values inL}q_{(Ω;γ(H, E}∗_{)). Thedivergence} operator δ is the part of the adjoint operator D∗ _{in L}p_{(Ω;γ(H, E)) mapping into L}p_(Ω;E).

Explicitly, the domain domp(δ) consists of thoseX ∈Lp_{(Ω;γ(H, E)) for which there exists an}

FX ∈Lp(Ω;E) such that

EhX, DGi=EhFX, Gi for allG∈D1,q(Ω;E∗).

The function FX, if it exists, is uniquely determined, and we define

δ(X) :=FX, X∈domp(δ).

The divergence operatorδis easily seen to be closed, and the next lemma shows that it is also densely defined.

Lemma 4.1. We have S_{(Ω)⊗γ(H, E)⊆domp(δ) and}

δ(f⊗R) =X

j≥1

W(hj)f⊗Rhj−R(Df), f ∈S(Ω), R∈γ(H, E).

Here (hj)j≥1 denotes an arbitrary orthonormal basis ofH.

Proof. Forf ∈S_{(Ω),R∈γ(H, E), andG∈}S_(Ω)⊗E∗ _{we obtain, using the integration by} parts formula (4) (or Proposition 3.4(iii)),

Ehf⊗R, DGi=X

j≥1

Ehf ⊗Rhj, DG(hj)i

=X

j≥1

E(W(hj)hf⊗Rhj, Gi)−Eh[Df, hj]H⊗Rhj, Gi

=E¿X

j≥1

W(hj)f ⊗Rhj−

X

j≥1

[Df, hj]H⊗Rhj, G

À

=E¿X

j≥1

W(hj)f ⊗Rhj−R(Df), G

À

.

The sumP_j≥1W(hj)f⊗Rhjconverges inLp(Ω;E).This follows from the Kahane-Khintchine

inequalities and the fact that (W(hj))j≥1 is a sequence of independent standard Gaussian variables; note that the functionf is bounded.

Using an extension of Meyer’s inequalities, for UMD spacesE and 1< p <∞it can be shown that δextends to a bounded operator fromD1,p_{(Ω;γ(H, E)) toL}p_{(Ω;E). For details we refer}

5

The Skorokhod integral

We shall now assume thatH =L2_{(0, T;H), whereT is a fixed positive real number andH is} a separable real Hilbert space. We will show that if the Banach spaceEis a UMD space, the divergence operatorδ is an extension of the stochastic integral for adaptedL₍H_{, E)-valued}

processes constructed recently in [15]. Let us start with a summary of its construction. LetWH = (WH(t))_t∈[0,T] be anH-cylindrical Brownian motion on (Ω,F,P), adapted to a

filtrationF= (F_{t)t∈[0,T]satisfying the usual conditions. The Itˆo isometry defines an isonormal}

processW :L2(0, T;H_)→L2_{(Ω) by}

W(φ) :=

Z T

0

φ dWH, φ∈L2(0, T;H).

Following [15] we say that a process X : (0, T)×Ω → γ(H_{, E) is an elementary adapted}

process with respect to the filtrationFif it is of the form

X(t, ω) =

m

X

i=1

n

X

j=1

1(ti−1,ti](t)1Aij(ω)

l

X

k=1

hk⊗xijk, (5)

where 0≤t0<· · ·< tn≤T, the setsAij∈Fti−1 are disjoint for eachj, andhk, . . . , hk ∈H

are orthonormal. The stochastic integral with respect toWH of such a process is defined by

I(X) :=

Z T

0

X dWH :=

m

X

i=1

n

X

j=1

l

X

k=1

1Aij(WH(ti)hk−WH(ti−1)hk)⊗xijk,

Elementary adapted processes define elements ofLp_(Ω;γ(L2_{(0, T;H), E)) in a natural way.} Their closure inLp_(Ω;γ(L2_{(0, T;H), E)) is denoted byL}p

F(Ω;γ(L2(0, T;H), E)).

Proposition 5.1 ([15, Theorem 3.5]). Let E be a UMD space and let 1 < p < ∞. The stochastic integral uniquely extends to a bounded operator

I:Lp_F(Ω;γ(L2(0, T;H_{), E))→L}p_(Ω;E).

Moreover, for allX ∈Lp_F(Ω;γ(L2_{(0, T;H), E))we have the two-sided estimate} kI(X)kLp_(Ω;E)hkXk_Lp_(Ω;γ(L2(0,T;H),E)),

with constants only depending onpandE.

A consequence of this result is the following lemma, which will be useful in the proof of Theorem 6.6.

Lemma 5.2. Let E be a UMD space and let 1 < p, q < ∞ satisfy 1_p + 1_q = 1. For all X ∈Lp_F(Ω;γ(L2_{(0, T;H), E))andY ∈L}q

F(Ω;γ(L2(0, T;H), E

∗_{))we have} EhI(X), I(Y)i=EhX, Yi.

In the next approximation result we identifyL2(0, t;H_{) with a closed subspace ofL}2_{(0, T;H).} The simple proof is left to the reader.

Lemma 5.3. Let 1 ≤ p < ∞ and 0 < t ≤ T be fixed, and let (ψn)n≥1 be an orthonormal basis of L2(0, t;H_{). The linear span of the functions f(W(ψ1), . . . , W(ψn))⊗(h⊗x), with}

f ∈S_{(Ω),h∈H,x∈E, is dense inL}p_(Ω,F

t;γ(H, E)).

The next result shows that the divergence operatorδis an extension of the stochastic integral I.This means thatδ is a vector-valued Skorokhod integral.

Theorem 5.4. LetEbe a UMD space and fix1< p <∞.The spaceLp_F(Ω;γ(L2(0, T;H_{), E))}

is contained indomp(δ)and

δ(X) =I(X) for all X ∈Lp_F(Ω;γ(L2_{(0, T;H), E)).}

Proof. Fix 0< t≤T, let (hk)k≥1be an orthonormal basis ofH, and putX := 1APnk=1hk⊗

xk with A ∈ Ft and xk ∈ E for k = 1, . . . , n. Let (ψj)j≥1 be an orthonormal basis of L2_{(0, t;H). By Lemma 5.3 we can approximate X in L}p_(Ω,F

t;γ(H, E)) with a sequence

(Xl)l≥1 inS(Ω, γ(H, E)) of the form

Xl:= Ml

X

m=1

flm(W(ψ1), . . . , W(ψn))⊗(hm⊗xlm)

withxlm∈E.

Now let 0< t < u≤T. Fromψm⊥1(t,u]⊗hinL2(0, T;H) it follows thatDXl(1(t,u]⊗h) = 0 for allh∈H_{. By Lemma 4.1,}

1_(t,u]⊗Xl= Ml

X

m=1

flm(W(ψ1), . . . , W(ψn))⊗((1(t,u]⊗hm)⊗xlm)

belongs to domp(δ) and

δ(1_(t,u]⊗Xl) = Ml

X

m=1

W(1_(t,u]⊗hm)flm(W(ψ1), . . . , W(ψn))⊗xlm=I(1(t,u]⊗Xl).

Noting that1(t,u]⊗Xl→1(t,u]⊗X inLp(Ω;γ(L2(0, T;H), E)) as l→ ∞, the closedness of δ implies that1(t,u]⊗X ∈domp(δ) and

δ(1(t,u]⊗Xl) =I(1(t,u]⊗Xl).

By linearity, it follows that the elementary adapted processes of the form (5) with t0>0 are contained in domp(δ) and thatI andδcoincide for such processes.

To show that this equality extends to all X ∈ Lp_F(Ω;γ(L2_{(0, T;H), E)) we take a sequence} Xn of elementary adapted processes of the above form converging toX.SinceI is a bounded

operator fromLp_F(Ω;γ(L2_{(0, T;H), E)) intoL}p_{(Ω;E),it follows thatδ(X}

n) =I(Xn)→I(X)

6

A Clark-Ocone formula

Our next aim is to prove that the spaceLp_F(Ω;γ(L2_{(0, T;H), E)), which has been introduced} in the previous section, is complemented in Lp_(Ω;γ(L2_{(0, T;H), E)). For this we need a} number of auxiliary results. Before we can state these we need to introduce some terminology. Let (γj)j≥1 be a sequence of independent standard Gaussian random variables. Recall that a collectionT _⊆L_{(E, F) of bounded linear operators between Banach spacesE andF is said}

to beγ-boundedif there exists a constantC >0 such that

E°°°

Proposition 6.1. Let T _{be a γ-bounded subset of} L_{(E, F) and let H be a separable real}

Hilbert space. For eachT ∈T _letTe_∈L_{(γ(H, E), γ(H, F))be defined by T R}e _{:=T◦R. The}

This proves the inequalityγ(Tf_)≤γ(T_{). The reverse inequality holds trivially.}

The next proposition is a result by Bourgain [2], known as the vector-valued Stein inequality. We refer to [6, Proposition 3.8] for a detailed proof.

Proposition 6.2. Let E be a UMD space and let(F_t)

t∈[0,T] be a filtration on(Ω,F,P). For all 1 < p < ∞ the conditional expectations {E(·|F_{t) : t ∈ [0, T]} define a γ-bounded set in} L_(Lp_(Ω;E)).

Since Rf determinesf uniquely almost everywhere, in what follows we shall always identify

Rf and f.

Proposition 6.3. Let E and F be real Banach spaces and letM : (0, T)→ L_{(E, F) have}

γ-bounded range{M(t) :t∈(0, T)}=:M_{. Assume that for allx∈E,t7→M(t)xis strongly}

measurable. Then the mapping M :f 7→[t 7→M(t)f(t)] extends to a bounded operator from γ(L2_{(0, T;H), E)toγ(L}2_{(0, T;H), F)of norm kMk ≤γ(M).}

Here we identifiedM(t)∈L_{(E, F) withM}]_(t)∈L_(γ(H_{, E), γ(}H_{, F)) as in Proposition 6.1.}

The next result is taken from [15].

Proposition 6.4. LetH be a separable real Hilbert space and let1≤p <∞. Thenf 7→[h7→ f(·)h]defines an isomorphism of Banach spaces

Lp(Ω;γ(H, E))≃γ(H, Lp_(Ω;E)).

After these preparations we are ready to state the result announced above. We fix a filtration F= (F_{t)t∈[0,T] and define, for step functionsf : (0, T)→γ(}H_{, L}p_(Ω;E)),

(PFf)(t) :=E(f(t)|Ft), (6)

whereE(·|F_{t) is considered as a bounded operator acting onγ(}H_{, L}p_{(Ω;E)) as in Proposition}

6.1.

Lemma 6.5. Let E be a UMD space, and let 1< p, q <∞satisfy 1_p+1

q = 1.

(i) The mappingPF extends to a bounded operator onγ(L2(0, T;H), Lp(Ω;E)).

(ii) As a bounded operator onLp_(Ω;γ(L2_{(0, T;H), E)),P}

F is a projection onto the subspace

Lp_F(Ω;γ(L2_{(0, T;H), E)).}

(iii) For allX ∈Lp_(Ω;γ(L2_{(0, T;H), E))andY ∈L}q_(Ω;γ(L2_{(0, T;H), E}∗_{))we have} EhX, P_FYi=EhP_FX, Yi.

(iv) For allX ∈Lp_(Ω;γ(L2_{(0, T;H), E))we haveEP}

FX =EX.

Proof. (i): From Propositions 6.1 and 6.2 we infer that the collection of conditional expecta-tions{E(·|F_{t) :t∈[0, T]} isγ-bounded in} L_(γ(H_{, L}p_{(Ω;E))). The boundedness ofP}

F then

follows from Proposition 6.3. For step functions f : (0, T)→γ(H_{, L}p_{(Ω;E)) it is clear from}

(6) that P_F2f =P_Ff, which means thatP_F is a projection.

(ii): By the identification of Proposition 6.4, P_F acts as a bounded projection in the space Lp_(Ω;γ(L2_{(0, T;H), E)). For elementary adapted processes X ∈ L}p_(Ω;γ(L2_{(0, T;H), E))} we have PFX =X, which implies that the range ofPF containsLp_F(Ω;γ(L2(0, T;H), E)). To

prove the converse inclusion we fix a step functionX : (0, T)→γ(H_{, L}p_{(Ω;E)) and observe}

thatPFX is adapted in the sense that (PFX)(t) is stronglyFt-measurable for everyt∈[0, T].

As is shown in [15, Proposition 2.12], this implies that PFX ∈LpF(Ω;γ(L2(0, T;H), E)). By

(iii): Keeping in mind the identification of Proposition 6.4, for step functions with values in the finite rank operators fromH _{toE this follows from (6) by elementary computation. The}

result then follows from a density argument.

(iv): Identifying a step functionf : (0, T)→γ(H_{, L}p_{(Ω;E)) with the associated operator in}

γ(L2_{(0, T;H), L}p_{(Ω;E)) and viewingEas a bounded operator fromγ(L}2_{(0, T;H), L}p_(Ω;E))

toγ(L2_{(0, T;H), E), by (6) we have}

EP_Ff(t) =EE(f(t)|F_{t) =Ef(t).}

Thus EPFf = Ef for all step functions f : (0, T)→ γ(H, Lp(Ω;E)), and hence for all f ∈

γ(L2_{(0, T;H), L}p_{(Ω;E)) by density. The result now follows by an application of Proposition}

6.4.

Now letF = (F_{t)t∈[0,T] be the augmented filtration generated byW}H. It has been proved

in [15, Theorem 4.7] that if E is a UMD space and 1 < p < ∞, and if F ∈ Lp_{(Ω;E) is}

F_{T-measurable, then there exists a uniqueX ∈L}p F(Ω;γ(L

2_{(0, T;H), E)) such that}

F =E(F) +I(X).

The following two results give an explicit expression for X. They extend the classical Clark-Ocone formula and its Hilbert space extension to UMD spaces.

Theorem 6.6(Clark-Ocone representation, firstLp-version). Let E be a UMD space and let 1< p <∞.If F ∈D1,p_(Ω;E)isF

T-measurable, then

F =E(F) +I(PF(DF)).

Moreover,P_F(DF)is the uniqueY ∈Lp_F(Ω;γ(L2_{(0, T;H), E))satisfyingF =E(F) +I(Y).} Proof. We may assume thatE(F) = 0. LetX ∈L_Fp(Ω;γ(L2_{(0, T;H), E)) be such thatF =} I(X) =δ(X).Let 1_p+1_q = 1,and letY ∈Lq_(Ω;γ(L2_{(0, T;H), E}∗_{)) be arbitrary. By Lemma} 6.5, Theorem 5.4, and Lemma 5.2 we obtain

EhPF(DF), Yi=EhDF, PFYi=EhF, δ(PFY)i

=Ehδ(X), δ(P_FY)i=EhI(X), I(P_FY)i =EhX, PFYi=EhPFX, Yi=EhX, Yi.

Since this holds for all Y ∈ Lq_(Ω;γ(L2_{(0, T;H), E}∗_{)), it follows that X = P}

F(DF). The

uniqueness of P_F(DF) follows from the injectivity of I as a bounded linear operator from Lp_F(Ω;γ(L2_{(0, T;H), E)) toL}p_(Ω,F

T).

With a little extra effort we can prove a bit more:

Theorem 6.7 (Clark-Ocone representation, secondLp_{-version). Let E be a UMD space and}

let 1 < p < ∞. The operator P_F ◦D has a unique extension to a bounded operator from Lp_(Ω,F

T;E)to Lp_F(Ω;γ(L2(0, T;H), E)), and for all F ∈Lp(Ω,FT;E) we have the

repre-sentation

F =E(F) +I((PF◦D)F).

Proof. It follows from Theorem 6.6 that F 7→ I((PF◦D)F) extends uniquely to a bounded

operator onLp_(Ω,F

T;E), since it equalsF 7→F−E(F) on the dense subspaceD1,p(Ω,FT;E).

The proof is finished by recalling thatIis an isomorphism fromLp_F(Ω;γ(L2_{(0, T;H), E)) onto} its range inLp_(Ω,F

T).

Remark 6.8. An extension of the Clark-Ocone formula to a class of adapted processes taking values in an arbitrary Banach spaceB has been obtained by Mayer-Wolf and Zakai [13, The-orem 3.4]. The setting of [13] is slightly different from ours in that the starting point is an arbitrary abstract Wiener space (W, H, µ), whereµ is a centred Gaussian Radon measure on the Banach space W and H is its reproducing kernel Hilbert space. The filtration is defined in terms of an increasing resolution of the identity on H, and a somewhat weaker notion of adaptedness is used. However, the construction of the predictable projection in [13, Section 3] as well as the proofs of [14, Corollary 3.5 and Proposition 3.14] contain gaps. As a consequence, the Clark-Ocone formula of [13] only holds in a suitable ‘scalar’ sense. We refer to the errata [13, 14] for more details.

7

Extension to

_L

1

We continue with an extension of Theorem 6.7 to random variables inL1_(Ω,F

T;E). As before,

F= (F_{t)t∈[0,T] is the augmented filtration generated by the} H_{-cylindrical Brownian motion}

WH.

We denote byL0_{(Ω;F) the vector space of all strongly measurable random variables with values} in the Banach space F, identifying random variables that are equal almost surely. Endowed with the metric

d(X, Y) =E(kX−Yk ∧1), L0_{(Ω;F) is a complete metric space, and we have lim}

n→∞Xn =X in L0(Ω;F) if and only if

limn→∞Xn=X in measure inF.

The closure of the elementary adapted processes in L0_(Ω;γ(L2_{(0, T;H), E)) is denoted by} L0

F(Ω;γ(L2(0, T;H), E)).By the results of [15], the stochastic integralIhas a unique extension

to a linear homeomorphism fromL0

F(Ω;γ(L2(0, T;H), E)) onto its image inL0(Ω,FT;E).

Theorem 7.1 (Clark-Ocone representation, L1_{-version). Let E be a UMD space. The} op-erator P_F ◦D has a unique extension to a continuous linear operator from L1(Ω,F_{T;E) to}

L0_F(Ω;γ(L2_{(0, T;H), E)), and for all F∈L}1_(Ω,F

T;E)we have the representation

F =E(F) +I((PF◦D)F).

Moreover, (PF ◦ D)F is the unique element Y ∈ LF0(Ω;γ(L2(0, T;H), E)) satisfying F =

E(F) +I(Y).

Proof. We shall employ the process ξX : (0, T)×Ω → γ(L2(0, T;H), E) associated with a

strongly measurable random variableX : Ω→γ(L2_{(0, T;H), E), defined by}

(ξX(t, ω))f := (X(ω))(1[0,t]f), f ∈L2(0, T;H).

Some properties of this process have been studied in [15, Section 4].

Let (Fn)n≥1be a sequence ofFT-measurable random variables inS(Ω)⊗E which is Cauchy

in L1_(Ω,F

such that for allδ >0 andε >0 and allm, n≥1,

P¡kP_F(DFn−DFm)kγ(L2_(0,T;H),E)> ε

¢

≤ Cδ 2

ε2 +P

¡

sup

t∈[0,T]

kI(ξPF(DFn−DFm)(t))k ≥δ

¢

(∗) = Cδ

2

ε2 +P

¡

sup

t∈[0,T]

kE(Fn−Fm|Ft)−E(Fn−Fm)k ≥δ

¢

(∗∗) ≤ Cδ

2

ε2 + 1

δEkFn−Fm−E(Fn−Fm)k. In this computation, (∗) follows from Theorem 6.6 which gives

E(F|F_{t)−E(F) =E}¡_I(PFDF)¯¯F_t¢_=E¡_I(ξP

FDF(T))

¯

¯F_t¢_=I(ξP

FDF(t)).

The estimate (∗∗) follows from Doob’s maximal inequality. Since the right-hand side in the above computation can be made arbitrarily small, this proves that (PF(DFn))n≥1 is Cauchy in measure inγ(L2_{(0, T;H), E).}

ForF ∈L1_(Ω,F

T;E) this permits us to define

(PF◦D)F := lim

n→∞PF(DFn),

where (Fn)n≥1 is any sequence of FT-measurable random variables in S(Ω)⊗E satisfying

limn→∞Fn =F in L1(Ω,FT;E). It is easily checked that this definition is independent of

the approximation sequence. The resulting linear operator P_F◦D has the stated properties. This time we use the fact that Iis a homeomorphism fromL0_F(Ω;γ(L2_{(0, T;H), E)) onto its} image inL0(Ω,F_{T;E); this also gives the uniqueness of (PF◦D)F.}

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