Weak characterizations of stochastic integrability and Dudley's theorem in infinite dimensions

0.31 MB·English

by  Martin Ondrejat

WEAK CHARACTERIZATIONS OF STOCHASTIC INTEGRABILITY AND
DUDLEY’S THEOREM IN INFINITE DIMENSIONS
MARTINONDREJÁTANDMARKVERAAR
3
1 Abstract. In this paper we consider stochastic integration with respect to cylindrical Brow-
0 nian motion in infinite dimensional spaces. We study weak characterizations of stochastic in-
2 tegrability and present a natural continuation of results of van Neerven, Weis and the second
named author. The limitation of weak characterizations will be demonstrated with a nontriv-
n
ial counterexample. The second subject treated in the paper addresses representation theory
a
J for random variables in terms of stochastic integrals. In particular, we provide an infinite di-
mensionalversionofDudley’srepresentation theoremforrandomvariablesandanextensionof
0
Doob’srepresentation formartingales.
3

]
R
P 1. Introduction
.
h Representations of martingales and random variables as stochastic integrals with respect to
t
a Wiener processes (or terminal values thereof) have been a classical issue of stochastic analy-
m sis. Representation results show the remarkable universality of the Brownian motion among all
[ continuous local martingales and have a variety of applications in stochastic analysis including

backward stochastic differential equations (e.g. [25] and [44]) and mathematical finance (e.g. in
2
hedging, nonlinear pricing, continuous trading or optimal portfolio, see [8], [15], [26], [45]).
v
8 To be more precise, let (Ω,A,P) be a probability space with filtration F = (Ft)t 0 and W
3 be a Brownian motion with respect to F. One typically considers an F -measurab≥le random
4 variable ξ and asks whether there exists a jointly measurable adapted pr∞ocess φ with paths in
.5 L2(R+)suchthat 0∞φ(s)dW(s)=ξ. Letusrefertothisastothe “terminalvaluerepresentation
5 problem” hereafterR. Often one also considers a continuous local F-martingale M with M(0)=0
0 and of a given quadratic variation of the form M = t φ(s)2ds and asks whether there exists
2 h it 0 | |
1 a Brownian motion W with respect to F such that MRt = 0tφ(s)dW(s) for every t ≥ 0. Let us
: refer to this as to the “martingale representation problem”.R One may also consider a third kind
v
of problems (which we will not pursue in this paper), when M is a local FW-martingale with
i
X
M(0) = 0 and the question is, whether there exists a stochastically integrable process φ such
r that M = tφ(s)dW(s) for every t 0. Let us refer to this as to the “Brownian martingale
a t 0 ≥
representatioRnproblem” and let us remark that this issue goes historically hand in hand with the
terminal value representationproblem.
As far as the terminal value representation problem is concerned, in the scalar case, when
the terminal random variable is FW-measurable, consider the mapping I : Lp (Ω;L2(R ))
Lp(Ω,FW,P) defined by ∞ FW + →
∞
∞
Iφ= φdW.
Z
0
Then it is known that
(i) I isanisomorphismontothespaceofcenteredterminalstatesinLp(Ω,FW,P)forp (1, ),
(ii) I is a surjection when 0 p<1, ∞ ∈ ∞
≤
2000 Mathematics Subject Classification. Primary: 60H05Secondary: 28C20,60B11,60F99.
Keywordsandphrases. StochasticintegrationinBanachspaces,UMDBanachspaces,cylindricalBrownianmo-
tion,randomvariableswithpolynomialtails,almostsurelimittheorems,Dudleyrepresentationtheorem,universal
representation theorem, weakcharacterization ofstochasticintegrability, Doobrepresentation theorem.
The first named author is supported by the GAČR grant P201/10/0752. The second author is supported by
VENIsubsidy639.031.930oftheNetherlands OrganisationforScientificResearch(NWO).
1
2 MARTINONDREJÁTANDMARKVERAAR
(iii) I is an isomorphism onto the Hardy space H1 in the sense of Garcia [20] when p=1.
See Garling [18] for p [0, ) 1 and [2], [13], [22], [28], [40], [46] for p = 1. The case p = 2
∈ ∞ \{ }
is due to Itô [27] and Kunita-Watanabe [32] and p = 0 to Dudley [14]. The terminal value
representation problem and the Brownian martingale representation problem have been recently
studied and resolvedfor E-valued terminal conditions ξ Lp(Ω,FW;E) for p (1, ) in [38] by
van Neerven, Weis and the second named author under∈the assum∞ption that E∈is a∞umd Banach
space (see explanation below). As a consequence, in full generality of the umd -setting it holds
that if terminalrandomvariables canbe representedthen localFW-martingaleshave continuous
modifications.
The martingale representation problem was first solved by Doob [12] in one dimension (exten-
sionstofinitedimensionsarenowadaysstandard). Generalizationstocontinuouslocalmartingales
in Hilbert spaces have been given in [34] and [43] by Lépingle and Ouvrard, to complete nuclear
spaces in [30] by Körezlioğlu and Martias and to Banach spaces in [11] by Dettweiler and in [42]
by the first named author. The martingale representation theorem has plenty of applications in
the theory of stochastic equations, e.g. in the proof of coincidence of weak solutions of stochastic
evolution equations and of solutions of the associated martingale problem of Stroock and Varad-
han and in proofs of existence of weak solutions of stochastic evolutionequations by compactness
methods, see e.g. [3], [7], [17] or [21].
In the setting of infinite dimensional spaces, i.e. when the terminal values or martingales take
values in a Hilbertspace ora Banachspace, representationproblems become moredifficult and it
turns out that they are closely connected with the problem of weak characterizationof stochastic
integrability (which do not appear when working in Rd case). Let us demonstrate this with the
following example. Let E be a Hilbert space, W a standard Brownian motion, p,q (0, ), φ a
strongly F-progressively measurable E-valued process such that ∈ ∞
p
1 2
(1.1) E φ(s),x 2ds < , x E
(cid:18)Z |h i| (cid:19) ∞ ∈
0
and ξ Lq(Ω;E) satisfies, for all x E, almost surely
∈ ∈
1
(1.2) φ(s),x dW(s)= ξ,x .
Z h i h i
0
1
Does this imply thatφ is stochasticallyintegrablein E and,in particular,that φ(s)dW(s)=ξ
0
almost surely? Surprisingly this is true for p = q > 1 as proved in [38] iRn a more general
setting which will be explained below. In particular, for p = q > 1, (1.1) and (1.2) imply that
φ Lp(Ω;L2(0,1;E)) and then also 1φ(s)dW(s) = ξ (thus weak=strong). We will show that
∈ 0
there is an extension to p = q = 1 ofRthis results. Moreover, we complete the picture by giving a
counterexample in the case where p < 1 and q is arbitrary. Hence, the only cases which remain
unanswered are p [1, ) and q (0,1].
∈ ∞ ∈
In the alreadymentionedwork [38], a theory of stochastic integrationfor processeswith values
inaumdBanachspaceE hasbeendevelopedusingideasof[19]byGarlingand[35]byMcConnell.
The class of umd Banach space have been extensively studied by Burkholder in the eighties (see
[6]andreferencestherein). ThespacesLp andSobolevspacesWs,p withp (1, )ands Rare
examplesofspaceswhichhaveumd. As arule ofthumbone couldsaythat∈all“c∞lassical” r∈eflexive
spaceshaveumd. TheLp andWs,p-spacesplayacentralroleinthetheoryofstochasticevolution
equations. In [39] it has been shown how the stochastic integration theory can be combined with
functional calculus tools to obtain maximalregularityresults for largeclasses ofstochastic PDEs.
Solutions to stochastic evolution equations and in particular stochastic PDEs are often formu-
lated in a weak sense (see [10] and [31]). Therefore, weak characterizations of stochastic integrals
canbehelpfulinobtainingstochasticintegrability. Theweakcharacterizationsof[38]havealready
played an important role in proving vector-valuedversions of Itô’s formula in [4]. Moreover,for a
concreteSPDE,stochasticintegrabilityhasbeencheckedusingthistechniquein[5]. Finally,letus
mention that weak identities for stochastic integrals appear in the setting of martingale solutions
to stochastic evolution equations (see [33] and [41]).
STOCHASTIC INTEGRABILITY IN INFINITE DIMENSIONS 3
Overview: The aim of this paper is to study weak characterizations of stochastic integrabil-
ity in respect of representation theorems in Banach spaces. We prove positive results on weak
characterizations in Theorem 4.1 and Corollary 4.2 and a negative result in Theorem 2.2. We
also provein Theorem3.1 a generalterminal value representationtheorem in every Banachspace
and for every filtration (i.e. not necessarily for the Brownian one) and we give an example of a
universal indefinite stochastic integral whose set of L0-accumulation points at infinity coincides
with L0(Ω;E), which is new even in the scalar setting. Finally, we prove a combination of a
martingale representationtheorem and a weak characterizationof stochastic integrability in umd
BanachspacesinTheorem5.1. Backgroundmaterialonstochasticintegrationandpathregularity
of martingales in Banach spaces is collected in the Appendix.
Convention: Below all vector spaces will be considered over the real scalar field. As complex
vector spaces can be viewed as real vector spaces, this is not really a restriction. If not stated
otherwise (Ω,A,P) is a probability space with a complete filtration F = (F ) . Moreover,
t t 0
W and W will be a Brownian motion and a cylindrical Brownian motion with≥respect to F
H
respectively. Here H is a separable real Hilbert space.
Let E be a Banach space. We write L0(Ω;E) for the space of strongly measurable mappings
ξ : Ω E. Note that L0(Ω;E) is a complete metric space with translation invariant metric
ξ → =E( ξ 1). Moreover, convergence in L0(Ω;E) coincide with convergence in proba-
L0(Ω;E)
k k k k∧
bility.
Let L(H,E) denote the bounded linear operators from H into E. Let T (0, ]. A process
Φ : [0,T] Ω L(H,E) is called H-strongly measurable if for all h H, ∈(t,ω)∞ Φ(t,ω)h is
× → ∈ 7→
stronglymeasurable. OnedefinesthenotionofH-stronglyadaptednessandH-stronglyprogressive
measurabilityinasimilarway. Letp [0, ). AnH-stronglymeasurableprocessΦ:[0,T] Ω
L(H,E) is said to be weakly Lp(Ω;∈L2(0,∞T;H)) if for all x E , the process (Φ x )(t,×ω) :→=
∗ ∗ ∗ ∗
Φ(t,ω) x belongs to Lp(Ω;L2(0,T;H)). In the special case t∈hat H = R, we identify L(H,E)
∗ ∗
with E and in that case we also write Φ,x instead of Φ x .
∗ ∗ ∗
h i
2. A counterexample to weak characterizations of stochastic integrability
In our first result we give a counterexample to weak characterizationsof stochastic integrals in
a Hilbert space. Counterexamples in more general Banachspace will be discussed in Remark 2.8.
Positive results for p 1, will be presented in Section 4.
≥
Remark 2.1. A general definition of the stochastic integral can be found in Definition A.1. In
the special case that E is a Hilbert spaces and H = R and φ : [0,T] Ω E is an adapted
× →
and strongly measurable process, the definition reduces to the classical setting and is equivalent
to φ L2(0,T;E) almost surely. Indeed, this follows from the standard fact that for a sequence
∈
of adapted step processes (cid:16) 0·φndW(cid:17)n 1 convergesin L0(Ω;Cb([0,T];E)) if and only if (φn)n≥1
converges in in L0(Ω;L2(0,TR;E)) (see [≥29, Proposition 17.6] for the scalar case).
Theorem 2.2. Let E be an infinite dimensional Hilbert space. There exists a strongly progressive
process φ :[0,1] Ω E which is weakly in Lp(Ω;L2(0,1)) for all p [0,1) and satisfies for all
× → ∈
x E,
∈
1
(2.1) φ(t),x dW(t) =0, almost surely,
Z h i
0
but φ = almost surely. In particular, φ is not stochastically integrable.
L2(0,1;E)
k k ∞
Some details on stochastic integration theory of vector-valued processes can be found in Ap-
pendix A where one should take H =R, and W =W.
H
Remark 2.3. If φ is weakly L1(Ω;L2(0,1)) and satisfies (2.1), then φ=0. Indeed, it follows from
(2.1) and the martingale property of the stochastic integral that for all x E and all t [0,1],
∈ ∈
t φ(s),x dW(s) = E 1 φ(s),x dW(s) F = 0 almost surely. Therefore, the Burkholder-
RD0ahvis-Gunidyinequality(cid:16)(Rs0eeh[29,Priopositio(cid:12)(cid:12)(cid:12)n1t7(cid:17).7])yieldsthatforallx∈E,khφ,xikL1(Ω;L2(0,1)) =
0 and hence φ=0 a.s.
4 MARTINONDREJÁTANDMARKVERAAR
From[14],onecandeducethatthereexistsaprogressivelymeasurableprocessψ L0(Ω;L2(0,1))
∈
which satisfies (see also [48, p. 196])
1/2 1
ψdW =W(1/2) and ψdW =0.
Z Z
0 0
This process plays an important role in the construction in Theorem 2.2. Before we turn to the
proof we present two lemmas. In the first lemma, we calculate the distribution of ξ = ψ 2
k kL2(0,1)
using arguments of [14], where the tail of ξ was estimated from above.
Let the function f : [0,b) R be given by f (x) = (b x) 1 if 0 < b < and f (x) = 1 if
b b − b
→ − ∞
b= .
∞
Lemma 2.4. Let 0 a < b . Let η :Ω R be F -measurable. Let Y : [a,b) Ω R, the
a
stopping time τ and≤h:[a,b)≤ ∞R be defined →by × →
→
r r
Y(r)= f (s)dW(s), τ =inf r a:Y(r)=η , and h(r)= f (s)2ds
b b
Za n ≥ o Za
Then a τ <b a.s. and
≤
2/(πe)Emin η /√t,1 P(h(τ)>t) Emin η /√t,1 , t>0.
{| | }≤ ≤ {| | }
p
Proof. One has that τ is a stopping time with a τ < b a.s. (see [48, p. 192]). As in [48,
≤
p. 192] one sees that Y and (W(h(t))) are identically distributed. Moreover, η and Y are
t [a,b)
∈
independent. Let µ be the distribution function of η. It follows that
η
P(h(τ)>t)=P(τ >h 1(t))
−
=P Y(s)=η for all s [a,h 1(t)]
−
6 ∈
=P(cid:0)Y(h 1(s))=η for all s [0,t](cid:1)
−
6 ∈
(cid:0) (cid:1)
= ∞ P Y(h 1(s))=x for all s [0,t] dµ (x)
− η
Z 6 ∈
(cid:0) (cid:1)
−∞
= ∞ P W(s)=x for all s [0,t] dµ (x)= ∞ P(τ >t)dµ (x)
η x η
Z 6 ∈ Z
(cid:0) (cid:1)
−∞ −∞
where τ = inf s > 0 : W(s) = x for x R. The random variable τ has density f (s) =
x x x
{ } ∈
(2π) 1/2s 3/2 x exp( x2/(2s)) (see [46, Section III.3]) for x=0 and τ =0 a.s. It follows that
− − 0
| | − 6
P(h(τ)>t)=(2π)−1/2 ∞ ∞s−3/2 x exp( x2/(2s))dsdµη(x)
Z Z | | −
t
−∞
x/√t
=21/2π 1/2 ∞ | | exp( r2/2)drdµ (x)
− η
Z Z −
0
−∞
21/2π 1/2 ∞ min x/√t,(π/2)1/2 dµ (x)
− η
≤ Z {| | }
−∞
and
P(h(τ)>t) 21/2π 1/2e 1/2 ∞ min x/√t,1 dµ (x).
− − η
≥ Z {| | }
−∞
Clearly, the result follows from these two estimates. (cid:3)
Remark 2.5. It follows from the proof of Lemma 2.4 that if η is a centered Gaussian random
variable with Eη2 =σ2, then the assertion of the lemma can be strengthened to
P(h(τ)>t)=π 1σ 1 ∞ ∞s 3/2xexp( x2/(2s))dse x2σ−2/2dx
− − − −
Z Z −
0 t
=π 1σ 1 ∞s 3/2 1 ds=2/π σ/√t 1 du=2/πarctan(σ/√t).
− − Z − 1 + 1 Z 1+u2
t σ2 s 0
STOCHASTIC INTEGRABILITY IN INFINITE DIMENSIONS 5
Inthenextlemmawestudysummabilitybehaviorofindependentsequencesofpositiverandom
variables with polynomial tails. In particular, the random variable h(τ)1/2 of Lemma 2.4 often
satisfies the conditions below.
Lemma 2.6. Let (ξ ) be a sequence of independent [0, )-valued random variables for which
n n 1
≥ ∞
there is a constant C >0 such that
(2.2) C 1(t+1) 1 P(ξ >t) C(t+1) 1, t>0
− − 1 −
≤ ≤
For a sequence (c ) in (0,1] one has sup c ξ < a.s. if and only if c < .
n n≥1 n≥1 n n ∞ n≥1 n ∞
P
Remark 2.7.
(1) TheequivalenceofLemma2.6holdsforanysequenceofpositivenumbers(c ) ,butthis
n n 1
≥
requires an additional argument and we will not need this generality. One can also show
that the assertions of Lemma 2.6 are equivalent to: for all p (1, ], (cnξn)n 1 ℓp <
∈ ∞ k ≥ k ∞
a.s. Moreover,this result is sharp in the sense that for p=1 there are counterexamples.
(2) For the case of equivalences of moments of random variables such as the above ones, we
refer to [23] for detailed discussions.
Proof of Lemma 2.6. Let p (λ)=P(ξ >c 1λ) for λ>0. Observe that for all λ>0 one has
n n −n
(2.3) P(supc ξ λ)= (1 p (λ))=exp log 1 p (λ) .
n n n n
n≥1 ≤ nY1 − (cid:16)nX1 (cid:0) − (cid:1)(cid:17)
≥ ≥
Now first assume c < . Let λ > 2C. Then 0 p (λ) C(c 1λ+1) 1 < 1/2 for all
n 1 n ∞ ≤ n ≤ −n −
n 1. Therefore, onePha≥s
≥
log 1 p (λ) 2p (λ) 2C(λc 1+1) 1 2Cλ 1c .
− − n ≤ n ≤ −n − ≤ − n
(cid:0) (cid:1)
It follows that
c
log 1 p (λ) λ 1 c = Kλ 1,
n − n −
− ≥−2 −
nX1 (cid:0) (cid:1) nX1
≥ ≥
where K is a constant not depending on λ. From (2.3) we obtain that
P(supcnξn λ) exp( Kλ−1).
≤ ≥ −
n 1
≥
Letting λ , one sees that P(sup c ξ < )=1 and the result follows.
Next as→sum∞e sup c ξ < alnm≥o1stnsunrely∞. Choose λ 1 such that P(sup c ξ < λ)
1/2. Then it followsnt≥h1atnthne se∞ries on the right-hand side of≥(2.3) converges andn≥m1orneonver ≥
log(1/2) log 1 p (λ) p (λ) C 1 (c 1λ+1) 1 C 1λ 1 c .
≤ − n ≤− n ≤− − −n − ≤− − − n
nX1 (cid:0) (cid:1) nX1 nX1 nX1
≥ ≥ ≥ ≥
Hence c Cλlog(2). (cid:3)
n 1 n ≤
≥
P
Next we turn to the proof of Theorem 2.2. The process φ will be constructed in such a way
that the interval (0,1) will be decomposed to an infinite number of random subintervals, and on
each of theses subintervals, φ will be a process so that the stochastic integral over (0,1) vanishes
and, simultaneously, the L2(0,1;E)-norm explodes.
Proof of Theorem 2.2. Let (I ) be nonempty open disjoint intervals in (0,1). Write I =
n n 1 n
≥
(s ,b ) and let a be the center of I for each n 1.
n n n n
Now for each n 1, let Y :[a ,b ) Ω R,≥the FW-stopping time τ and h :[a ,b ) R
n n n n n n n
≥ × → →
be defined as in Lemma 2.4 with a = a , b = b and η = W(a ) W(s ). Then a τ < b
n n n n n n n
− ≤
a.s. and by Remark 2.5 one has that
2
P(θ >t)=P(h (τ )>(a s )t)= arctan(1/√t), t>0,
n n n n n
− π
where θ = h (τ )/(a s ). Moreover, (θ ) is sequence of independent random variables.
n n n n n n n 1
Let φ :R R be define−d by ≥
n
→
1
φ = 1 (s)+1 (s) .
n − [sn,an) [an,τn] b s
n
−
6 MARTINONDREJÁTANDMARKVERAAR
Then φ is FW-progressivelymeasurable, φ 2 =(a s )(1+θ ) and
n k nkL2(0,1) n− n n
1 1
φ dW = 1 (s)+1 (s)(b s) 1 dW(s)
Z0 n Z0 (cid:16)− [sn,an) [an,τn] n− − (cid:17)
= (W(a ) W(s ))+W(a ) W(s )=0.
n n n n
− − −
Letting ξ :=(1+θ )1/2, one sees that (ξ ) satisfies the conditions of Lemma 2.6 with α=1.
n n n n 1
Let c =n 1. Let (e ) be an orthonorma≥lsequence in E. Let φ:[0,1] E be defined by
n − n n 0
≥ →
φ= (an sn)−1/2cnφnen.
−
nX1
≥
Let p (0,1). We prove that φ is weakly Lp(Ω;L2(0,1)). By Jensen’s inequality it suffices to
∈
consider2/3<p<1. Sincethe(φ ) havedisjointsupports,wefindthatforallx E,(letting
n n 0
≥ ∈
x = x,e for n 1)
n n
h i ≥
1 2 1/2
φ,x = (a s ) 1/2x c φ dt
L2(0,1) n n − n n n
kh ik (cid:16)Z0 (cid:12)(cid:12)nX0 − (cid:12)(cid:12) (cid:17)
(cid:12) ≥ (cid:12)
1 1/2
(2.4) =(cid:16)nX0(an−sn)−1x2nc2nZ0 φ2ndt(cid:17)
≥
1/2 1/p
= x2c2ξ2 x p c pξp
(cid:16)nX0 n n n(cid:17) ≤(cid:16)nX0| n| | n| n(cid:17)
≥ ≥
where we used kykℓ2 ≤ kykℓp. Note that Cp = E(ξnp) = E(ξ1p) is finite. Taking p-th moments on
both sides of the above estimate it follows that
(2.5) E φ,x p C x p c p C x p (c ) p < ,
kh ikL2(0,1) ≤ pnX0| n| | n| ≤ pk kℓ2k n n≥1kℓpr ∞
≥
were we applied Hölder’s inequality with p + 1 =1, and we used pr= 2p >1. Moreover,
2 r 2 p
−
1 1 1
(2.6) φ,x dW = (a s ) 1/2x c φ dW = (a s ) 1/2c x φ dW =0.
n n − n n n n n − n n n
Z0 h i Z0 (cid:16)nX0 − (cid:17) nX0 − Z0
≥ ≥
Finally,
1 1 1
(2.7) φ 2 dt= (a s ) 1c2 φ 2dt= (a s ) 1c2 φ 2dt= c2ξ2
Z k kE Z n− n − n| n| n− n − nZ | n| n n
0 0 nX0 nX0 0 nX0
≥ ≥ ≥
and the latter is a.s. infinite by Lemma 2.6 in view of the embedding ℓ2 ֒ ℓ . (cid:3)
∞
→
Remark 2.8. The construction in Theorem 2.2 can be extended to the setting where E is a umd
Banachspacewithanormalizedunconditionalbasicsequence(e ) . ThestatementofTheorem
n n 1
≥
2.2 should then be replaced by: there is a progressively measurable φ : [0,1] Ω E which is
× →
weakly in Lp(Ω;L2(0,1)) for all p [0,1) and for all x E
∗ ∗
∈ ∈
1
φ(t),x∗ dW(t)=0, almost surely.
Z h i
0
and φ = , almost surely. In particular, φ is not stochastically integrable (in the
γ(L2(0,1),E)
k k ∞
sense of Appendix A). Not every Banach space contains an unconditional basic sequence (see
[24]). However, spaces such as E = Lw(S), where (S,Σ,µ) is a measure space and w (1, )
are umd. Moreover, if E = Lw(S) is infinite dimensional, it contains a normalized uncon∈ditio∞nal
basic sequence e =µ(S ) 1/w1 , with (S ) of finite measure and disjoint.
n n − Sn n n≥1
Let us briefly explain the proof of the above statement. For precise definitions on type of
Banach spaces and unconditionality we refer to [1]. Since E is a umd space it follows from the
Maurey-Pisier theorem that E has type s with constant T for some s (1,2]. Let (r ) and
s n n 1
(γ ) be a RademacherandstandardGaussiansequence onsome prob∈ability space(Ω,≥F ,P)
n n 1 ′ ′ ′
≥
STOCHASTIC INTEGRABILITY IN INFINITE DIMENSIONS 7
respectively. We claim that for all x E one has (x ) is in ℓs′, where x = e ,x and
∗ ∗ n n 1 n n ∗
where s denotes the conjugate exponen∈t of s. Indeed, let a≥ ℓs, Then h i
′
∈
a x x a e K x r a e
n n ∗ n n ∗ n n n
nX1 ≤k k(cid:13)(cid:13)nX1 (cid:13)(cid:13)≤ k k(cid:13)(cid:13)nX1 (cid:13)(cid:13)L2(Ω′;E)
≥ (cid:13) ≥ (cid:13) (cid:13) ≥ (cid:13)
1/s
TsK x∗ anen s =TsK x∗ (an)n 1 ℓs,
≤ k k(cid:16)nX1k k (cid:17) k kk ≥ k
≥
whereK istheunconditionalityconstantof(en)n 1. Takingthesupremumoverall (an)n 1 ℓs
≥ k ≥ k ≤
1, the claim follows.
With the same notations as in Theorem 2.2, (2.4) and (2.6) with φ,x replaced by φ,x ,
∗
h i h i
remain valid. Indeed, by Jensen’s inequality it suffices to consider 1 < p < 1. By the above
2 s−1
claim one can replace (2.5) with −
Ekhφ,x∗ikpL2(0,1) ≤CpnX0|xn|p|cn|p ≤Cpkxkpℓs′k(cn)n≥1kpℓpr <∞,
≥
where we applied Hölder’s inequality with p + 1 =1.
s′ r
Finally, using Proposition A.2 and the definition of the γ-norm, (2.7) can be replaced by
φ = γ c ξ e supc ξ
γ(L2(0,1),E) n n n n n n
k k (cid:13)(cid:13)nX0 (cid:13)(cid:13)L2(Ω′;E) ≥n≥1
(cid:13) ≥ (cid:13)
and the latter is a.s. infinite by Lemma 2.6.
3. Dudley’s representation theorem
In [14], Dudley obtained a representation theorem for real-valued random variables ξ L0(Ω)
∈
which are σ(W(t) : t 1)-measurable where W is a standard Brownian motion. In [16], this
≤
was further extended to other classes of martingales for more general filtrations which are not
necessarily Brownian. Obviously, Dudley’s result extends to the case of ξ L0(Ω;Rd) using
∈
coordinatewise representations. It seems that an infinite dimensional version is not available yet
and the original proof does not extend to this setting. Moreover, if E = ℓ2 a coordinatewise
representation leads to divergent series in general. Below we provide a representation theorem in
arbitraryBanachspaces. AbriefsurveyofstochasticintegrationtheoryinBanachspacesandthe
definition of γ(L2(a,b),E) can be found in Appendix A.
Theorem 3.1. Let E be a Banach space and 0 a < b . Let ξ : Ω E be strongly
≤ ≤ ∞ →
measurable. Then there exists a strongly progressively measurable process φ:[a,b] Ω E which
× →
is stochastically integrable and satisfies
b
φdW =ξ
Z
a
if and only if ξ is σ(F :t<b)-measurable.
t
Remark 3.2.
(1) If F is left continuous or b= (e.g. Browniancase), then one has F =σ(F :t<b).
b t
∞
(2) Itwillbeclearfromtheproofthatφ L0(Ω;γ(L2(a,b),E)) L0(Ω;L2(a,b;E)). Moreover,
∈ ∩
because of the special structure of the process,no geometricconditions on E are required
for its stochastic integrability.
(3) Note that, as in the scalar case, the representation is not unique (see the text below
t
Theorem2.2). Moreover,in general φdW will not be a martingale,but only a
a
continuous local martingale. (cid:16)R (cid:17)t∈[a,b]
(4) Ifφ ,φ areE-valued,stronglymeasurableandadapted,weaklyL1(Ω;L2(a,b))andsatisfy
1 2
b
φ dW =ξ, i=1,2,
i
Z
a
then φ =φ a.e. Indeed, this follows as in Remark 2.3.
1 2
8 MARTINONDREJÁTANDMARKVERAAR
Proof. Step 1: Let Y, τ and h be defined as in Lemma 2.4 with η =1. Then
C 1min t 1/2,1 P(h(τ)>t) Cmin t 1/2,1 , t>0.
− − −
{ }≤ ≤ { }
In particular, for any random variable θ :Ω R which is independent of h(τ), one has
+
→
(3.1) C 1Emin θ1/2t 1/2,1 P(θh(τ)>t) CEmin θ1/2t 1/2,1 , t>0.
− − −
{ }≤ ≤ { }
Note that
1 1
Emin θ1/2t 1/2,1 dt= θ +E θ1/2t 1/2dt
− L0(Ω) −
Z { } k k Z
0 θ 1
∧
θ +2E(θ1/2 1) 3 θ 1/2
≤k kL0(Ω) ∧ ≤ k kL0(Ω)
where we used Jensen’s inequality and x x1/2 for x [0,1]. It follows that (with a different C)
≤ ∈
(3.2) C 1 θ θh(τ) C θ 1/2 .
− k kL0(Ω) ≤k kL0(Ω) ≤ k kL0(Ω)
Step 2: Let [a,b] := [a, ) if b = . Now let ξ : Ω E be as in the theorem. Choose a
∞ ∞ →
sequence (a ) in [a,b] with a b and random variables (ξ ) such that for each n 1,
n n 1 n n n 1
ξ is F -meas≥urable and ξ ξ ↑ < 4 n. Let ξ = 0. Fo≥r each n 1, let Y , τ ≥and
n an k − nkL0(Ω;E) − 0 ≥ n n
h be defined as in step 1 with a = a and b = a . For each n 1, define φ : [a,b] E
n n n+1 n
by φ (t) = (ξ ξ )1 (t)(a t) 1. Then each φ is st≥ochastically integrable→and
n n − n−1 (an,τn) n+1 − − n
ζn(b)=ξn−ξn−1, where ζn = a·φndW. Note that
R
η := (ξ ξ ) h (τ )1/2 = φ = φ .
n n n 1 E n n n γ(L2(a,b),E) n L2(a,b;E)
k − − k k k k k
Since ξ ξ and h (τ ) are independent it follows from (3.2) that
n n 1 n n
− −
η C ξ ξ 1/2
nX1k nkL0(Ω) ≤ nX1k n− n−1kL0(Ω)
≥ ≥
C ξ1 L0(Ω;E)+C (4−n+4−n+1)1/2 =C ξ1 L0(Ω;E)+C51/22−1.
≤ k k k k
nX2
≥
By completeness it follows that η convergesin L0(Ω). Therefore, the series φ:= φ
n 1 n n 1 n
converges(absolutely)in γ(L2(aP,b),≥E) and in L2(a,b;E) almostsurely. Since eachφ isPstr≥ongly
n
progressivelymeasurable it follows that φ is strongly progressivelymeasurable.
Note that by the special form of φ , for all n 1 one has
n
≥
t
ζ (t) = φ dW , t [a,b].
n n
k k (cid:12)Z0 k k (cid:12) ∈
(cid:12) (cid:12)
Let ε (0,1) be arbitrary. Let δn = (2/(cid:12)3)nε. Then by(cid:12) Proposition A.3 with p = 2 and β2,R = 1
∈
and c =1 we obtain
2
t
P ζ >ε =P sup φ dW >ε (4/9)n+P((3/2)nη >ε)
(cid:16)k nkCb([a,b];E) (cid:17) (cid:16)t [a,b](cid:12)Za k nk (cid:12) (cid:17)≤ n
∈ (cid:12) (cid:12)
For n 2, integrating over ε (0,1) yield(cid:12)s (cid:12)
≥ ∈
ζ (4/9)n+ (3/2)nη (4/9)n+(3/2)n η
k nkL0(Ω;Cb([a,b];E)) ≤ k nkL0(Ω) ≤ k nkL0(Ω)
(4/9)n+(3/2)nC ξ ξ 1/2 (4/9)n+C(3/4)n.
≤ k n− n−1kL0(Ω;E) ≤
Hence ζ < and therefore ζ converges in L0(Ω;C ([a,b];E)).
n 2k nkL0(Ω;Cb([a,b];E)) ∞ n 1 n b
SincPe ea≥ch φn is stochastically integrable and bothP ≥
(a) ∞n=1φn =φ in L0(Ω;L2(a,b;E)),
(b) P0· Nn=1φndW = Nn=1ζn convergesin L0(Ω;Cb([a,b];E)) as N →∞,
it foRllowPs that φ is stocPhastically integrable, and moreover,
b
φdW = ζ (b)= ξ ξ =ξ.
n n n 1
Za nX1 nX1 − −
≥ ≥
(cid:3)
STOCHASTIC INTEGRABILITY IN INFINITE DIMENSIONS 9
Remark 3.3. If additionally to the assumptions of Theorem 3.1 one knows that ξ Lp(Ω;E)
∈
with p (0,1). Then the process φ can be chosen in Lp(Ω;γ(L2(a,b),E)) Lp(Ω;L2(a,b;E)).
Indeed,∈choose(ξ ) suchthatξ isF -measurableand ξ ξ p <∩4 n. Fromthisone
n n≥1 n an k − nkLp(Ω;E) −
can deduce that φ converges φ in Lp(Ω;γ(L2(a,b),E)) Lp(Ω;L2(a,b;E)) and λ P-a.s.
n 1 n ∩ ⊗
Moreover, P ≥
·φdW = ·φ dW =ζ with convergence in Lp(Ω;C([a,b];E)).
n
Z Z
a nX1 a
≥
This yields an infinite dimensional version of Garling’s result [18]. To prove the above facts it
suffices to note that from (3.1) with θ replaced by θ2 one obtains
(3.3) C 1Emin θt 1/p,1 P(θh(τ)1/2 >t1/p) CEmin θt 1/p,1 , t>0.
− − −
{ }≤ ≤ { }
It is easy to check that
∞Emin θt 1/p,1 dt= 1 θ p .
Z { − } 1 pk kLp(Ω)
0 −
Therefore,integrating(3.3)showsthatC 1 θ θh(τ)1/2 C θ foradifferent
− Lp(Ω) Lp(Ω) Lp(Ω)
k k ≤k k ≤ k k
constant C. Now one can proceed as in the proof of Theorem 3.1 with Lp(Ω) instead of L0(Ω)
with translation invariant metric p and using Proposition A.2 instead of Proposition A.3
k·kLp(Ω)
for E =R and p (0,1).
∈
Thenextresultisneweveninthescalarsetting. ItcanbeseenasauniversalversionofDudley’s
representation theorem. Namely, we construct a locally stochastically integrable process and
consideritsindefinitestochasticintegralasacontinuouscurveinL0. Thenthesetofaccumulation
points of this curve in L0 coincides with L0.
Theorem 3.4. Let E be a separable Banach space. Assume F is P-countably generated. Then
there exists a strongly predictable process φ:R Ω E which∞is locally stochastically integrable
+
× →
and such that the following holds
foreveryξ L0(Ω,F ;E)thereexistsanincreasingsequence(n ) ofnaturalnumbers
k k 0
• ∈ ∞ ≥
such that
lim ζ(n )=ξ in L0(Ω;E),
k
k
→∞
t
where ζ(t)= φdW.
0
R
Clearly, φ is only locally stochastically integrable, because lim ζ(t) does not exist.
t
→∞
Proof. Step 1: Let ξ L0(Ω,F ;E). We claim that there exist an F-predictable process η
such that lim η(t)∈= ξ almos∞t surely. By Proposition B.1, there exists a left-continuous F-
t
adapted proce→ss∞θ : R Ω E satisfying θ (t) = E ξ F a.s. for every t > 0 and
lim θ (t)= ξ′ al+mo×st su→rely. Define η :R′ Ω (cid:0)E1+bkyξk(cid:12) t−(cid:1)
t→∞ ′ 1+kξk +× → (cid:12)
θ (t)
(3.4) η(t)=1kθ′(t)k<11−k′θ′(t)k.
Then η is also predictable, and by the previous observation lim η(t)=ξ almost surely.
t
Step 2: Since F is P-countably generated and E is separ→ab∞le, the space L0(Ω,F ;E) is
separable as well. L∞et (ξ ) be a dense sequence in L0(Ω,F ;E). For each n 0∞, step 1
n n 0
≥ ∞ ≥
implies that there exists a predictable η such that lim η (t)=ξ almost surely.
n t n n
Let α:N N be such that for each j 0, →∞
→ ≥
(3.5) # k N:α(k)=j = .
{ ∈ } ∞
Letρ =η (n) forn 0. Since η ispredictable,for eachn 0,ρ isstronglyFn-measurable.
n α(n) n
≥ ≥
Moreover,the set of accumulation points of (ρ ) is equal to L0(Ω,F ;E). Indeed, it suffices
n n 0
≥ ∞
to show that each ξ is an accumulation point of (ρ ) . This is obvious from (3.5) and step 1.
j n n 0
≥
10 MARTINONDREJÁTANDMARKVERAAR
Step 3: By Theorem 3.1, for each n 2 we can find a stochastic integrable process φ :
n
[n,n+1] E such that n+1φ dW =ρ≥ ρ . Define φ:R E by
→ n n n−1− n−2 + →
R
φ= 1 φ .
[n,n+1) n
nX2
≥
Thenφispredictableandlocallystochasticallyintegrable. Letζ beitsstochasticintegralprocess.
For each integer k 2 one has
≥
k n+1 k k
ζ(k)= φ dW = (ρ ρ )= (ρ ρ )=ρ .
n n n 1 n n 1 k
nX=2Zn nX=2 − − nX=1 − −
Now the theorem follows from Step 2. (cid:3)
Remark 3.5. Using the result of Remark 3.3 one can modify the proof of Theorem 3.4 to ob-
tain a universal representation theorem for Lp(Ω;E) with p (0,1) in the sense that for every
∈
ξ Lp(Ω;E) one can find a sequence of natural numbers (n ) such that lim ζ(n ) = ξ
k k 1 k k
in∈Lp(Ω;E). To proof this fact one can follow the arguments ≥of the above proo→f∞with θ(t) =
E ξξ1−p Ft (convention 00 =0) and (3.4) replaced by η(t)=θ′(t)kθ′(t)k1−pp.
(cid:0)k k (cid:12) (cid:1)
(cid:12)
4. Weak characterizations of stochastic integrability
Inthissectionweobtaintwonewweakcharacterizationsofstochasticintegrabilityforprocesses
Φ : [0,T] Ω L(H,E), where E is a umd Banach space and H is a separable Hilbert space.
× →
For details on stochastic integration theory of vector-valuedprocesses we refer to Appendix A.
Thefollowingstrengtheningof[38,Theorem5.9]holds,wherethecasewithζ L0(Ω;C ([0,T];E))
b
∈
was considered.
Theorem 4.1. Let E be a umd Banach space. Let Φ : [0,T] Ω L(H,E) be an H-strongly
× →
measurable and adapted process which is weakly in L0(Ω;L2(0,T;H)). Let ζ : [0,T] Ω E be
× →
a process with almost surely bounded paths. If for all x E almost surely, one has
∗ ∗
∈
t
Φ∗x∗dWH = ζ(t),x∗ , t [0,T],
Z h i ∈
0
then Φ represents an element in L0 (Ω;γ(L2(0,T;H),E)), and almost surely one has
F
t
(4.1) ΦdW =ζ(t), t [0,T].
H
Z ∈
0
Moreover, ζ is a local martingale with a.s. continuous paths.
Before we turn to the proof, we mention the following consequence of the above result. It
extends a part of [38, Theorem 3.6] to the case p=1.
Corollary 4.2. Let E be a umd Banach space. Let Φ : [0,T] Ω L(H,E) be an H-strongly
× →
measurable and adapted process which is weakly in L1(Ω;L2(0,T;H)). Let ξ L1(Ω;E). If for
∈
all x E one has almost surely
∗ ∗
∈
T
Φ x dW = ξ,x ,
∗ ∗ H ∗
Z h i
0
then Φ represents an element in L0 (Ω;γ(L2(0,T;H),E)), and
F
T
ΦdW =ξ.
H
Z
0
Moreover, for all q (0,1) one has Φ Lq(Ω;γ(L2(0,T;H),E)).
∈ ∈
Remark 4.3.
(1) Note that Corollary4.2 is false if one only assumesthat Φ is weaklyinLp(Ω;L2(0,T;H))
for all p [0,1) and ξ Lq(Ω;E) for some q [0, ] (see Theorem 2.2).
∈ ∈ ∈ ∞