60H25 (60H07 60H40 60J55 60J65)We extend the Clark-Ocone formula to a suitable class of generalized Brownian functionals. As an example we derive a representation of Donsker's delta function as (limit of) a stochastic integral

Faria, Margarida de

Oliveira, Maria João

Streit, Ludwig

Random Operators and Stochastic Equations

Repositório Aberto da Universidade Aberta

Random Oper, and Stock. Equ., Vol. 8, No. 2, pp. 163-174 (2000)© VSP 2000A generalized Clark-Ocone formulaMargarida de PARI A1, Maria Jo o OLIVEIRA2 and Ludwig STREIT1'31 COM, Universidade da Madeira, P 9000 Funchal2 Univ. Aberta/Grupo de Fisica Matematica, Av. Prof. Gama Pinto, 2, P 1649-003 Lisboa3BiBoS, Universit t Bielefeld, D 33501 BielefeldReceived for ROSE November 24, 1998Abstract— We extend the Clark-Ocone formula to a suitable class of generalized Brownian function-als. As an example we derive a representation of Donsker's delta function as (limit of) a stochasticintegral.1. INTRODUCTIONFor suitable functionals φ of Brownian motion, expressed in terms of Ito integrals(I)the Clark-Ocone formula [5], [14] provides us with an explicit formula for the integrandra(·), given φ. It has become clear that such an expression should be useful in thedetermination of hedging portfolios, see e.g., [1], [3], [15]. Another application is in thecontext of determining the quadratic variation process of Brownian martingales, seee.g., [7] for a recent example.On the other hand it was pointed out in [2] that the conditions on φ are restrictive. Itseems desirable to extend the validity of (1) and of the Clark-Ocone formula. A possiblesetting is that of generalized functionals of white noise as described, e.g., in [8]-[13],[16]. In particular the generalized function space elaborated in [16], or the larger oneof [8], retain the probabilistic properties that are required for such a generalization. In[3] one finds an announcement of results in terms of the Potthoff-Timpel [4], [16] space(in the meantime elaborated in [1]), for related results in the space D' of Malliavincalculus see [17]. Here we use the space of [8], which we present in the followingsection together with extensions of the Skorohod and Ito integrals, the gradient andsome further auxiliary notions. Section 3 has our generalization of the Clark-Oconeformula, in Section 4 we translate the result into the language, often useful in practicalcalculations, of the 5-transform. A case in point is Donsker's 5-function for which weelaborate the generalized Clark-Ocone formula in Section 5.2. REGULAR GENERALIZED FUNCTIONS OF WHITE NOISE2.1. Regular generalized functionsWe will recall the definition and some properties of the space Q~l of regular generalizedfunctions of White Noise [8], [9].164 A generalized Clark-Ocone formulaWithin the Hubert space Z^(M) = L2(R,Ed), d G N, of vector valued square inte-grable functions we consider the space 5<*(Ε) of vector valued Schwartz test functions.The topology on 5d(K) may be given in terms of a system of increasing Hilbertiannormsd= Σ&Ιρ'£ = &.····&) € Si(RUi 6 S(R),i = l,-,d,P 6 N0.p ^The basic nuclear triple is thusSd(R) COn Srf(R) we fix the canonical Gaussian measure μα which is determined by the char-acteristic function, i €S dThe space L2(S^(l),//d) will be briefly denoted by (L2).We will denote by η the d-tuple (ni,...,n<j), η» € NO, and writen! =t=lThe norm and the inner product in L2(En) will be denote by |·|η and (·, ·)η> respectively.Considering square integrable white noise functionals φ for which the chaos expansionφ(ΰ) == Σ [η J^converges rapidly, i.e.,Ml*nwe define the Hubert space Glq asG\ = (φ 6 (I2) : |M|J < 00} .The space of test functions Ql is defined as the projective limit of the spaces Gj, q G NO,Let GlJ be the dual with respect to (L2) of Gj and G~l the dual space of Ql withrespect to (L2). The corresponding bilinear dual pairing < ·,· > is connected to thesesquilinear inner product on (I/2) byM. Faria, M. J. OJ/veira and L. Stre/t 165(We shall use the same notation < ·, · > for dual pairings in more general settings suchas, e.g., for L2(R) ® i/*1). Since the constant function 1 is in Ql we may extend thedefinition of the expectation E (·) from integrable functions to distributions Φ G Q"1:Ε(Φ)=«Φ,1» .From general duality theory it follows thata-1 = U G-\-,therefore, every distribution is of finite order, i.e., for every Φ G G~l there exists q 6 NOsuch that Φ G GlJ. It turns out from the definition that the Hubert space GlJ can bedescribed as follows:SGiven ξ*€ 5^(1), let us consider the Wick exponential:exp<w,£>: = exp ( < ΰ,ξ > -i E/i\ i=lSince the sumnnH2converges if and only if 2q \ξ\ 2 < 1, the Wick exponentials are not test functions ini/1, but they are in those Gj for which 2* 1(1 < 1. Thus it is still possible to defineI lLj(M.)an 5-transform in the space Q"1 because every distribution is of finite order. GivenΦ G (Γ1, there exists q G N0 such that Φ G GlJ. For all ξ G 5d(R) with 2q 1(1 < 1we define the 5-transform of Φ as=<Φ,: expThis definition extends to complex vectors η G S^cW such that 2^ |=« Φ,: exp < -,77< 1,(2)Therefore, for Φ G GlJ, (2) defines the 5-transform for every η from the open neigh-borhood of zero, Uq = {ff G 5d,c(l) : < 1}, q G N0.166 A generalized Clark-Ocone formula2.2. Generalization of the gradient operatorWe begin with the observation that the Hida derivative dt fails to be pointwise definedon the spaces G±J. However we still may consider the gradient (<9V)i<i<d a·8 an operatorΘ L2 RGiven ψ a test function from Ql with kernel functions ψχ, η G NO, we define the operatorgradient of ψ, νψ = (di .il>)\<i<d, where, for each 1 < i < d, d\^ is the functional fromΖ/2 (R) 0 G J, ς € NO, characterized by the sequence;^ ^ η = (ni,...,n<i).In fact,mή,η>1which proves that θ\ψ G L2 (E.) <g) Glq for every non negative integer number q and,moreover, the continuity of the linear operators 9* : G^-^L2 (E) 0 G* :for every q 6 NQ. Hence,which proves that the linear operator V : Gj->Rd Θ L2 (R) <8> G^ is continuous.We extend the operator gradient from test functions on Q1 (introduced above) todistributions on Q~l .Given a regular generalized function Φ from C/"1, Φ € GlJ for some g 6 NO, charac-terized by the sequence (Φ^), η G NO, Φη € L2(Rn), consider the functional character-ized by the sequenceN0.Using the inequality 2~kk2 < 2 for k = (p - q)n > 0, we have2Φη + <£ (·,*,·)Φtn + o i-^η+1)2"(ρ-<7^η+1)(ηί 4-1) Φn + i n+l (3)Φn + Ji n+l• Σ 2-^|Φη|2,η,η>1M. Faria, M. J. Oliveira and L. Streit 167where the above sum is convergent because Φ G G_J. Hence, the sequence (Φ^),n G NO, defines a functional from L2(R) <g> Gl*. Keeping the terminology and thenotation introduced above, we will denote this functional by 91Φ and the operator(οιΦ)ι<»<<* will be called the gradient of Φ, denoted by νΦ. Using this notation, itfollows from (3) thatfor every pair ρ > q > 0, which proves that the gradient is a bounded linear operatorfrom GlJ into Rd ® L2(R) 0 GlJ if ρ > 9.2.3. An extension of the Skorohod and Ito integralsIn [10] the Skorohod integral was discussed in a white noise setting. An extension tocertain generalized white noise integrands can be found in [6].Considering an element Φ from L2 (R) <8> GlJ, for some q € NO, characterized by thesequence Φ^(·; ·) 6 L2 (R)0L2(Rn), n G NO, let us consider the functional characterizedby the sequenceΦΐ? = i^where Φ denotes the symmetrization of Φη - di η -Since, for each 1 < i < d,in the variablesΣη,η>1Φη - n,n>lΦη —(4)L2H.the sequence (Φ^), η € NO, defines a distribution from G_J. We denote it by /ί(Φ).For every test function ·φ from Q1 with kernel functions1 < t < d,), η € N0, we have for each= /Rdiit « Φ(ί; Ο,^Υ »=« Φ,οψ »,„_!and /ί(Φ) is the unique functional from "1 for which the above equality holds for everytest function ψ £ Q1; if /'(Φ) is another functional in such conditions, it turns out that<£ 7j(f) - /'(Φ), · » is identically equal to zero on Q1.168 A generalized Clark-Ocone formulaWe may now formulate theDefinition. Given Φ an element from Rd <8> L2 (R) <g> GlJ, for some q G N0, we callgeneralized Skorohod integral of Φ the distribution on Q~l, /(Φ), defined by the sumd/(Φ) = ^Λ·(ΦΟ,i=lwhere, for each i = 1, ...,d, /»(Φ;) is the unique regular generalized function from Q~lfor which the following equalityholds for every test function ψ from Q1.This definition generalizes the notion of Skorohod integral. In fact, in the particularsituation Φ e Rd <8> L2 (R) <S> D,00the generalized Skorohod integral /(Φ) coincides with the Skorohod integral. In view ofthe relation between the Skorohod and It integral we may add the following remark.Remark 1. For t G R, let T t denote the σ-algebra generated by the random vari-ables {B(s), s < t}, where (#t)t€ is a d-dimensional Brownian motion. If F GRd <8> L2(R)<8>(L2) and it is adapted to the filtration (^t)t€]j, then the generalizedSkorohod integral I(F) is equal to the It integral of F. Without the first condition wespeak of a generalized It integral.Before ending this subsection, we return to the sequence of inequalities (4) whichimplyi.e., / is a bounded linear operator from Rd<8> L2(R)<g)Glq into Glq, q G N0.2.4. Some notations and definitionsFor the next sections it is useful to introduce some notations and recall some definitions.We shall denote by Qt the Heaviside functionn f * ί I if s < tQ^ = \Vifs>tand also the linear operator given byΘ, :/(·)->©, (-) /( · ) ·The functional derivatives jm\> for suitable G, are defined as«Ao ε J 'uw Sf(t)M. Faria, M. J. Oliveira and L. Streit 169In particular for cylinder functions G(f) = g (fh(t)f(t)dt) where g is a differentiablefunction, then(r). (5)For what follows it is also helpful to define the notion of second quantization of Qt,t G R, defined on the space G~°°. Recall:For bounded linear operators A on L2 (R) the linear map which transforms eachsequence (φ.), n G NO, φα € L2 (Rn), to the sequence (ΑΘη</?^), n G NO, is called thesecond quantization of A. It is denoted by Γ(Α).LEMMA 2.1 For bounded linear operators A on L2 (R), T(A) is a continuous operatoronQ~l.Proof. Given a bounded linear operator A on I/2(R), for every element Φ € G~l(belonging to GlJ, for some ς), we have,2~pnfor every non negative integer number p. If || A\\2 <2P q for some ρ € NO the above sumis majorized by | |Φ| |*ι and the second quantization of A is a bounded linear operatorfrom GlJ into Gl*. In particular, if \\A\\ < l, T (A) is a bounded linear operator fromG~l into itself. The lemma is proved.In particular, for A = Qt, for some t € R, it follows thatfor every g G NQ. Hence Γ(Θ^) is a bounded linear operator from the space of regulargeneralized functions G~l into itself.Remark 2. Consider the cr-algebra T t generated by the random variables {B(s), $ < t}.Γ(Θ*)Φ coincides with the conditional expectation for elements Φ from Q~l with respectto Ft, as introduced in [8].3. THE GENERALIZED CLARK-OCONE FORMULANow we are prepared to present the main result of this note. It generalizes the wellknown Clark-Ocone formula to regular generalized functions of white noise, i.e., to thespace G~l.THEOREM 3.1 (Generalized Clark-Ocone Formula) Let Φ be a regular generalizedfunction, Φ € Q~l. Then it can be written as a generalized ltd integralΦ = Ε(Φ)+/(τη)withmi(t) =170 A generalized Clark-Ocone formulaProof. We begin by noting that the integrand m is non-anticipating. Let Φ be anarbitrary element from C/"1, i.e., Φ € GlJ for some q, characterized by the sequence(Φ^), η G N0. Hence, for every test function ψ € Gl with kernel functions given by(Ψη)> η € NO, we have« Φ, </;»= Ε (Φ)Ε (ψ) +η,η>1where, for each η-tuple η = (ni, . . . ,η^) such that η > l,Taking each variable s*·, j = l , . . . ,n i , i = l , . . . ,d, in term, in the range s} >sup 4, the above integral can be written asΣΣ L dsi> Γ dn"li=l j=l J^ J-<*>which is equal tod ί ΓΣη* dr Ι ^η~1i=:1 JK J—ooby the symmetry of Φ^ and the kernel function ψχ with respect to each n^-tuple ofvariables (s\ , . . . , s*ni ), i = 1, . . . , d. This means,Σ η\Σή,η>1 =lt=l n,ni>ld ( \= Σ /RdrEn! θ?»^ + 1)Φ _ (-,τ,·),^ + 1)^ _ . (·,τ,-)<=1 Λ V nTJi n + (5i / n= Σ <Γ(θ.)ο.*Φ,ο.ν».i=lTherefore,<ί« φ, ψ »= Ε(Φ)Ε(ι/;) + ^  « ^ (ΓίθΟοίΦ)^ »,ί=1for every ψ belongs to C/1, which implies the result. Theorem 3.1 is proved.4. S-TRANSFORMIn this section, we find an expression to the 5-transform of a regular generalized functionΦ € Q~l which corresponds to the Clark-Ocone formula established above.M. Far/a, M. J. OJiveira and L. Streit 171THEOREM 4.1 Given α regular generalized function Φ from Q~l and g G NO such thatΦ G Gli, its S-transform is equal tofor every ή= (ηι,...,ηα) G Uq.Proof. Taking Φ G "1 characterized by the sequence (Φη), η G NO, such that Φ G GlJfor some q G NO, for every test function ή = (τ/ι,...,^) G Sd,cW with 29 IT?]^/]^ < 1we haved3Φ(η) = Ε (Φ) + « /i(r(0.) ^),: exp < ·,τ/>:>> . (6)Here,£ « /ί(Γ(θ.)β.'Φ), : exp < ·,η >:»=ι=1 η,η>1Using the symmetry of Φ/ϊ in each n^tuple of variables (s j , . . . ,«^.) , i = l,. . . ,d, itfollows thati(^i(s\) · · · r?i(4,-i) Πk=l,k«(r) fn« Γ d^'^ « / — ood= Σί=1Thus dΣ ίΦ-* fi®^} =V n > ' /ηη,η>1whereV^ Γ n~ln^^J-oo S n(But, for each r,μι(τ) =172 A generalized Clark-Ocone formulaHence, (6) can be written as5Φ(ττ) = Ε (Φ) +i=iObserving that, for each r,ftfc(T)there follows the required equality_n n -f <5ί5(Φ)(ΘΓ7/),Theorem 4.1 is proved.5. AN EXAMPLEAs an application of the above let us consider Φ equal to the Donsker delta functionwhich we may consider defined as a Fourier (Bochner) integral [10]_ a) = -L /with S-transform(SS(B(t) - «It is well known that δ(Β(ί) — α) is in Q~l . From Theorems 3.1 and 4.1 it follows that6(B(t) - o) = Ε (i(B(t) - a)) 4- / άΒ(τ)τη(τ)withThe functional derivative of (7) is calculated straightforwardly using (5)(here, l[0,t] denotes the indicator function of the interval [0,£]), so that, projecting the/ with θ τ , we obtainM. Far/a, M. J. Oliveira and L. Streit 173Note that the rhs depends only onX = f ( S ) e ( s ) d swhere e = ^l[o,r] is a unit vector in L2 (E). Consequently, m depends only on thenormal random variableχ=<ω,β>= -j=B(r)vrandwith 5-transform- - . «Ρ -To obtain m itself we must thus calculate the inverse Laplace transform ofwhich gives3/2Ifn m^) / t \h(x) = — — l l (y/rx — a)expSubstitutingwe finally obtainOne notes that m(r) is an adapted random variable in (L2) as long as r < t, and itpermits conventional It integration. It is thus not hard to show that, as a limit in Q~l,S(B(t) - a) = - r r e - + lim dB(r)m(r).ν/2πί v ; v JAcknowledgmentsThis work has had partial support from PRAXIS XXI and FEDER. M. J. 0. is gratefulfor an encouraging discussion with Prof. B. 0ksendal, and for hospitality at CCM;L. S. would like to express his gratitude for the generous hospitality of the Grupo deFisica Matem tica da Universidade de Lisboa, under the auspices of a "Marie Curie"fellowship (ERBFMBICT 971949). Prof. Yu. Kondratiev improved the manuscript bymany helpful comments.174 A generalized Clark-Ocone formulaREFERENCES1. K. Aase, B. 0ksendal, J. Ub0e. White noise generalizations of the Clark-Oconetheorem with application to mathematical finance. MaPhySto Research Report n°30, University of Aarhus (1998).2. K. Aase, B. 0ksendal, J. Ub0e. Using the Donsker delta function to computehedging strategies. Preprint University of Oslo n° 6 (1998).3. F. E. Benth. An Addendum to 'An Introduction to Malliavin Calculus with Appli-cations to Economies'. MaPhySto Miscellanea n° 1, University of Aarhus (1998).4. F. E. Benth, J. Potthoff. On the martingale property for generalized stochasticprocesses, Stochastics Stochastics Rep. 58 (3-4), 349-367 (1996).5. J. M. C. Clark. The representation of functional of Brownian motion by stochasticintegrals, Ann. Math. Stat. 41 1282-1295 (1970) and 42 , 1778 (1971).6. Th. Deck, J. Potthoff, G. Vage. A review of white noise analysis from a probabilisticstandpoint. Ada Appl Math. 48, 91-112 (1997).7. M. de Faria, C. Drumond, L. Streit. The renormalization of self intersection localtimes. I: The chaos expansion. Preprint IFM 1/98, Grupo de Fisica Matemδticada Universidade de Lisboa (1998).8. M. Grothaus, Yu. G. Kondratiev, L. Streit. Regular generalized functions in Gaus-sian analysis. Preprint 23/97, Universidade da Madeira, 1997. To appear in InfiniteDim. Anal. Quantum Prob. 2 (1)(1999).9. M. Grothaus, Yu. G. Kondratiev, G. F. Us. Wick calculus for regular generalizedstochastic functionals, Random Oper. Stochastic Equations 7, 3, 263-290, (1999).10. T. Hida, H. H. Kuo, J. Potthoff, L. Streit. White Noise. An Infinite DimensionalCalculus. Kluwer, Dordrecht (1993).11. Yu. G. Kondratiev, P. Leukert, L. Streit. Wick calculus in Gaussian analysis. ActaAppl. Math. 44, 269-294 (1996).12. H. H. Kuo. White Noise Distribution Theory. CRC Press, Boca Raton (1996).13. N. Obata. White Noise Calculus and Fock Space. Lecture Notes in Math. 1577,Springer-Verlag (1994).14. D. Ocone. Malliavin's calculus and stochastic integral representations of functionalsof diffusion processes, Stochastics 12, 161-185 (1984).15. B. 0ksendal. An introduction to Malliavin calculus with applications to economics.Working Paper n° 3/96, Norwegian School of Economics and Business Adminis-tration (1996).16. J. Potthoff, M. Timpel. On a dual pair of spaces of smooth and generalized randomvariables, Potential Analysis 4, 637-654 (1995).17. A. S. Ustόnel. Representation of the distributions on Wiener space and stochasticcalculus of variations, JFA 70, 126-139 (1987).

A generalized clark-ocone formula

The Repositório Aberto

Random Oper, and Stock. Equ., Vol. 8, No. 2, pp. 163-174 (2000)
© VSP 2000
A generalized Clark-Ocone formula
Margarida de PARI A1, Maria Jo o OLIVEIRA2 and Ludwig STREIT1'3
1
 COM, Universidade da Madeira, P 9000 Funchal
2
 Univ. Aberta/Grupo de Fisica Matematica, Av. Prof. Gama Pinto, 2, P 1649-003 Lisboa
3BiBoS, Universit t Bielefeld, D 33501 Bielefeld
Received for ROSE November 24, 1998
Abstract— We extend the Clark-Ocone formula to a suitable class of generalized Brownian function-
als. As an example we derive a representation of Donsker's delta function as (limit of) a stochastic
integral.
1. INTRODUCTION
For suitable functionals φ of Brownian motion, expressed in terms of Ito integrals
(I)
the Clark-Ocone formula [5], [14] provides us with an explicit formula for the integrand
ra(·), given φ. It has become clear that such an expression should be useful in the
determination of hedging portfolios, see e.g., [1], [3], [15]. Another application is in the
context of determining the quadratic variation process of Brownian martingales, see
e.g., [7] for a recent example.
On the other hand it was pointed out in [2] that the conditions on φ are restrictive. It
seems desirable to extend the validity of (1) and of the Clark-Ocone formula. A possible
setting is that of generalized functionals of white noise as described, e.g., in [8]-[13],
[16]. In particular the generalized function space elaborated in [16], or the larger one
of [8], retain the probabilistic properties that are required for such a generalization. In
[3] one finds an announcement of results in terms of the Potthoff-Timpel [4], [16] space
(in the meantime elaborated in [1]), for related results in the space D' of Malliavin
calculus see [17]. Here we use the space of [8], which we present in the following
section together with extensions of the Skorohod and Ito integrals, the gradient and
some further auxiliary notions. Section 3 has our generalization of the Clark-Ocone
formula, in Section 4 we translate the result into the language, often useful in practical
calculations, of the 5-transform. A case in point is Donsker's 5-function for which we
elaborate the generalized Clark-Ocone formula in Section 5.
2. REGULAR GENERALIZED FUNCTIONS OF WHITE NOISE
2.1. Regular generalized functions
We will recall the definition and some properties of the space Q~l of regular generalized
functions of White Noise [8], [9].
164 A generalized Clark-Ocone formula
Within the Hubert space Z^(M) = L2(R,Ed), d G N, of vector valued square inte-
grable functions we consider the space 5<*(Ε) of vector valued Schwartz test functions.
The topology on 5d(K) may be given in terms of a system of increasing Hilbertian
norms
d
= Σ&Ιρ'£ = &.····&) € Si(RUi 6 S(R),i = l,-,d,P 6 N0.
p
 ^The basic nuclear triple is thus
Sd(R) C
On Srf(R) we fix the canonical Gaussian measure μα which is determined by the char-
acteristic function
, i €S d
The space L2(S^(l),//d) will be briefly denoted by (L2).
We will denote by η the d-tuple (ni,...,n<j), η» € NO, and write
n! =
t=l
The norm and the inner product in L2(En) will be denote by |·|η and (·, ·)η> respectively.
Considering square integrable white noise functionals φ for which the chaos expansion
φ(ΰ) =
= Σ [
η J^
converges rapidly, i.e.,
Ml*
n
we define the Hubert space Glq as
G\ = (φ 6 (I2) : |M|J < 00} .
The space of test functions Ql is defined as the projective limit of the spaces Gj, q G NO,
Let GlJ be the dual with respect to (L2) of Gj and G~l the dual space of Ql with
respect to (L2). The corresponding bilinear dual pairing < ·,· > is connected to the
sesquilinear inner product on (I/2) by
M. Faria, M. J. OJ/veira and L. Stre/t 165
(We shall use the same notation < ·, · > for dual pairings in more general settings such
as, e.g., for L2(R) ® i/*1). Since the constant function 1 is in Ql we may extend the
definition of the expectation E (·) from integrable functions to distributions Φ G Q"1:
Ε(Φ)=«Φ,1» .
From general duality theory it follows that
a-1 = U G-\-,
therefore, every distribution is of finite order, i.e., for every Φ G G~l there exists q 6 NO
such that Φ G GlJ. It turns out from the definition that the Hubert space GlJ can be
described as follows:
S
Given ξ*€ 5^(1), let us consider the Wick exponential
:exp<w,£>: = exp ( < ΰ,ξ > -i E/i
\ i=l
Since the sum
n
n
H2
converges if and only if 2q \ξ\
 2 < 1, the Wick exponentials are not test functions in
i/1, but they are in those Gj for which 2* 1(1 < 1. Thus it is still possible to define
I lLj(M.)
an 5-transform in the space Q"1 because every distribution is of finite order. Given
Φ G (Γ1, there exists q G N0 such that Φ G GlJ. For all ξ G 5d(R) with 2q 1(1 < 1
we define the 5-transform of Φ as
=<Φ,: exp
This definition extends to complex vectors η G S^cW such that 2^ |
=« Φ,: exp < -,77
< 1,
(2)
Therefore, for Φ G GlJ, (2) defines the 5-transform for every η from the open neigh-
borhood of zero, Uq = {ff G 5d,c(l) : < 1}, q G N0.
166 A generalized Clark-Ocone formula
2.2. Generalization of the gradient operator
We begin with the observation that the Hida derivative dt fails to be pointwise defined
on the spaces G±J. However we still may consider the gradient (<9V)i<i<d a·8 an operator
Θ L2 R
Given ψ a test function from Ql with kernel functions ψχ, η G NO, we define the operator
gradient of ψ, νψ = (di .il>)\<i<d, where, for each 1 < i < d, d\^ is the functional from
Ζ/2 (R) 0 G J, ς € NO, characterized by the sequence
;^ ^ η = (ni,...,n<i).
In fact,
m
ή,η>1
which proves that θ\ψ G L2 (E.) <g) Glq for every non negative integer number q and,
moreover, the continuity of the linear operators 9* : G^-^L2 (E) 0 G* :
for every q 6 NQ. Hence,
which proves that the linear operator V : Gj->Rd Θ L2 (R) <8> G^ is continuous.
We extend the operator gradient from test functions on Q1 (introduced above) to
distributions on Q~l .
Given a regular generalized function Φ from C/"1, Φ € GlJ for some g 6 NO, charac-
terized by the sequence (Φ^), η G NO, Φη € L2(Rn), consider the functional character-
ized by the sequence
N0.
Using the inequality 2~kk2 < 2 for k = (p - q)n > 0, we have
2
Φ
η + <£ (·,*,·)
Φt
n + o i
-^
η+1)2"(ρ-<7^η+1)(ηί 4-1) Φ
n + i n+l (3)
Φ
n + Ji n+l
• Σ 2-^|Φη|2,
η,η>1
M. Faria, M. J. Oliveira and L. Streit 167
where the above sum is convergent because Φ G G_J. Hence, the sequence (Φ^),
n G NO, defines a functional from L2(R) <g> Gl*. Keeping the terminology and the
notation introduced above, we will denote this functional by 91Φ and the operator
(οιΦ)ι<»<<* will be called the gradient of Φ, denoted by νΦ. Using this notation, it
follows from (3) that
for every pair ρ > q > 0, which proves that the gradient is a bounded linear operator
from GlJ into Rd ® L2(R) 0 GlJ if ρ > 9.
2.3. An extension of the Skorohod and Ito integrals
In [10] the Skorohod integral was discussed in a white noise setting. An extension to
certain generalized white noise integrands can be found in [6].
Considering an element Φ from L2 (R) <8> GlJ, for some q € NO, characterized by the
sequence Φ^(·; ·) 6 L2 (R)0L2(Rn), n G NO, let us consider the functional characterized
by the sequence
Φΐ? = i^
where Φ denotes the symmetrization of Φ
η - di η -
Since, for each 1 < i < d,
in the variables
Σ
η,η>1
Φ
η - n,n>l
Φ
η —
(4)
L2
H.
the sequence (Φ^), η € NO, defines a distribution from G_J. We denote it by /ί(Φ).
For every test function ·φ from Q1 with kernel functions
1 < t < d,
), η € N0, we have for each
= /Rdi
it « Φ(ί; Ο,^Υ »=« Φ,οψ »,
„_!
and /ί(Φ) is the unique functional from "1 for which the above equality holds for every
test function ψ £ Q1; if /'(Φ) is another functional in such conditions, it turns out that
<£ 7j(f) - /'(Φ), · » is identically equal to zero on Q1.
168 A generalized Clark-Ocone formula
We may now formulate the
Definition. Given Φ an element from Rd <8> L2 (R) <g> GlJ, for some q G N0, we call
generalized Skorohod integral of Φ the distribution on Q~l, /(Φ), defined by the sum
d
/(Φ) = ^Λ·(ΦΟ,
i=l
where, for each i = 1, ...,d, /»(Φ;) is the unique regular generalized function from Q~l
for which the following equality
holds for every test function ψ from Q1.
This definition generalizes the notion of Skorohod integral. In fact, in the particular
situation Φ e Rd <8> L2 (R) <S> D,
00
the generalized Skorohod integral /(Φ) coincides with the Skorohod integral. In view of
the relation between the Skorohod and It integral we may add the following remark.
Remark 1. For t G R, let T t denote the σ-algebra generated by the random vari-
ables {B(s), s < t}, where (#t)t€ is a d-dimensional Brownian motion. If F G
Rd <8> L2(R)<8>(L2) and it is adapted to the filtration (^t)t€]j, then the generalized
Skorohod integral I(F) is equal to the It integral of F. Without the first condition we
speak of a generalized It integral.
Before ending this subsection, we return to the sequence of inequalities (4) which
imply
i.e., / is a bounded linear operator from Rd<8> L2(R)<g)Glq into Glq, q G N0.
2.4. Some notations and definitions
For the next sections it is useful to introduce some notations and recall some definitions.
We shall denote by Qt the Heaviside function
n f * ί I if s < tQ
^ = \Vifs>t
and also the linear operator given by
Θ, :/(·)->©, (-) /( · ) ·
The functional derivatives jm\> for suitable G, are defined as
«Ao ε J 'uw Sf(t)
M. Faria, M. J. Oliveira and L. Streit 169
In particular for cylinder functions G(f) = g (fh(t)f(t)dt) where g is a differentiable
function, then
(r). (5)
For what follows it is also helpful to define the notion of second quantization of Qt,
t G R, defined on the space G~°°. Recall:
For bounded linear operators A on L2 (R) the linear map which transforms each
sequence (φ.), n G NO, φα € L2 (Rn), to the sequence (ΑΘη</?^), n G NO, is called the
second quantization of A. It is denoted by Γ(Α).
LEMMA 2.1 For bounded linear operators A on L2 (R), T(A) is a continuous operator
onQ~l.
Proof. Given a bounded linear operator A on I/2(R), for every element Φ € G~l
(belonging to GlJ, for some ς), we have,
2~pn
for every non negative integer number p. If || A\\2 <2P q for some ρ € NO the above sum
is majorized by | |Φ| |*ι and the second quantization of A is a bounded linear operator
from GlJ into Gl*. In particular, if \\A\\ < l, T (A) is a bounded linear operator from
G~l into itself. The lemma is proved.
In particular, for A = Qt, for some t € R, it follows that
for every g G NQ. Hence Γ(Θ^) is a bounded linear operator from the space of regular
generalized functions G~l into itself.
Remark 2. Consider the cr-algebra T t generated by the random variables {B(s), $ < t}.
Γ(Θ*)Φ coincides with the conditional expectation for elements Φ from Q~l with respect
to Ft, as introduced in [8].
3. THE GENERALIZED CLARK-OCONE FORMULA
Now we are prepared to present the main result of this note. It generalizes the well
known Clark-Ocone formula to regular generalized functions of white noise, i.e., to the
space G~l.
THEOREM 3.1 (Generalized Clark-Ocone Formula) Let Φ be a regular generalized
function, Φ € Q~l. Then it can be written as a generalized ltd integral
Φ = Ε(Φ)+/(τη)
with
mi(t) =
170 A generalized Clark-Ocone formula
Proof. We begin by noting that the integrand m is non-anticipating. Let Φ be an
arbitrary element from C/"1, i.e., Φ € GlJ for some q, characterized by the sequence
(Φ^), η G N0. Hence, for every test function ψ € Gl with kernel functions given by
(Ψη)> η € NO, we have
« Φ, </;»= Ε (Φ)Ε (ψ) +
η,η>1
where, for each η-tuple η = (ni, . . . ,η^) such that η > l,
Taking each variable s*·, j = l , . . . ,n i , i = l , . . . ,d, in term, in the range s} >
sup 4, the above integral can be written as
ΣΣ L dsi> Γ dn"l
i=l j=l J^ J-<*>
which is equal to
d ί ΓΣη* dr Ι ^η~1
i=:1 JK J—oo
by the symmetry of Φ^ and the kernel function ψχ with respect to each n^-tuple of
variables (s\ , . . . , s*ni ), i = 1, . . . , d. This means,
Σ η\Σ
ή,η>1 =l
t=l n,ni>l
d ( \
= Σ /RdrEn! θ?»^ + 1)Φ _ (-,τ,·),^ + 1)^ _ . (·,τ,-)
<=1
 Λ V nTJi n + (5i / n
= Σ <Γ(θ.)ο.*Φ,ο.ν».
i=l
Therefore,
<ί
« φ, ψ »= Ε(Φ)Ε(ι/;) + ^  « ^ (ΓίθΟοίΦ)^ »,
ί=1
for every ψ belongs to C/1, which implies the result. Theorem 3.1 is proved.
4. S-TRANSFORM
In this section, we find an expression to the 5-transform of a regular generalized function
Φ € Q~l which corresponds to the Clark-Ocone formula established above.
M. Far/a, M. J. OJiveira and L. Streit 171
THEOREM 4.1 Given α regular generalized function Φ from Q~l and g G NO such that
Φ G Gli, its S-transform is equal to
for every ή= (ηι,...,ηα) G Uq.
Proof. Taking Φ G "1 characterized by the sequence (Φη), η G NO, such that Φ G GlJ
for some q G NO, for every test function ή = (τ/ι,...,^) G Sd,cW with 29 IT?]^/]^ < 1
we have
d
3Φ(η) = Ε (Φ) + « /i(r(0.) ^),: exp < ·,τ/>:>> . (6)
Here,
£ « /ί(Γ(θ.)β.'Φ), : exp < ·,η >:»=
ι=1 η,η>1
Using the symmetry of Φ/ϊ in each n^tuple of variables (s j , . . . ,«^.) , i = l,. . . ,d, it
follows that
i(^i(s\) · · · r?i(4,-i) Π
k=l,k
«(r) fn« Γ d^'
^ « / — oo
d
= Σ
ί=1
Thus d
Σ ίΦ-* fi®^} =V n > ' /η
η,η>1
where
V^ Γ n~l
n^^J-oo S n(
But, for each r,
μι(τ) =
172 A generalized Clark-Ocone formula
Hence, (6) can be written as
5Φ(ττ) = Ε (Φ) +
i=i
Observing that, for each r,
ftfc(T)
there follows the required equality
_
n n -f <5ί
5(Φ)(ΘΓ7/),
Theorem 4.1 is proved.
5. AN EXAMPLE
As an application of the above let us consider Φ equal to the Donsker delta function
which we may consider defined as a Fourier (Bochner) integral [10]
_ a) = -L /
with S-transform
(SS(B(t) - «
It is well known that δ(Β(ί) — α) is in Q~l . From Theorems 3.1 and 4.1 it follows that
6(B(t) - o) = Ε (i(B(t) - a)) 4- / άΒ(τ)τη(τ)
with
The functional derivative of (7) is calculated straightforwardly using (5)
(here, l[0,t] denotes the indicator function of the interval [0,£]), so that, projecting the
/ with θ τ , we obtain
M. Far/a, M. J. Oliveira and L. Streit 173
Note that the rhs depends only on
X = f ( S ) e ( s ) d s
where e = ^l[o,r] is a unit vector in L2 (E). Consequently, m depends only on the
normal random variable
χ=<ω,β>= -j=B(r)
vr
and
with 5-transform
- - . «Ρ -
To obtain m itself we must thus calculate the inverse Laplace transform of
which gives
3/2Ifn m^) / t \h(x) = — — l l (y/rx — a)exp
Substituting
we finally obtain
One notes that m(r) is an adapted random variable in (L2) as long as r < t, and it
permits conventional It integration. It is thus not hard to show that, as a limit in Q~l,
S(B(t) - a) = - r r e - + lim dB(r)m(r).
ν/2πί v ; v J
Acknowledgments
This work has had partial support from PRAXIS XXI and FEDER. M. J. 0. is grateful
for an encouraging discussion with Prof. B. 0ksendal, and for hospitality at CCM;
L. S. would like to express his gratitude for the generous hospitality of the Grupo de
Fisica Matem tica da Universidade de Lisboa, under the auspices of a "Marie Curie"
fellowship (ERBFMBICT 971949). Prof. Yu. Kondratiev improved the manuscript by
many helpful comments.
174 A generalized Clark-Ocone formula
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