Stochastic integral of divergence type with respect to fractional brownian motion with Hurst parameter <span class="mathjax-formula">$H\in (0,\frac{1}{2})$</span>

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Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter H ∈ (0, ¹ ₂ )

Patrick Cheridito

, David Nualart

aORFE, E-Quad, Princeton University, Princeton, NJ 08544, USA

bFacultat de Matemàtiques, Universitat de Barcelona, Gran Via, 585, 08007, Barcelona, Spain Received 25 July 2003; accepted 30 September 2004

Available online 5 February 2005

Abstract

We define a stochastic integral with respect to fractional Brownian motionB^Hwith Hurst parameterH∈(0,¹₂)that extends the divergence integral from Malliavin calculus. For this extended divergence integral we prove a Fubini theorem and establish versions of the formulas of Itô and Tanaka that hold for allH∈(0,¹₂). Then we use the extended divergence integral to show that for everyH ∈(¹₆,¹₂)and allg∈C³(R), the Russo–Vallois symmetric integral_b

ag(B_t^H)d⁰B_t^H exists and is equal to G(B_b^H)−G(B_a^H), whereG=g, while forH∈(0,¹₆],_b

a(B_t^H)²d⁰B_t^Hdoes not exist.

2004 Elsevier SAS. All rights reserved.

Résumé

Nous définissons une intégrale stochastique par rapport au mouvement brownien fractionnaireB^H avec paramètre de Hurst H∈(0,¹₂)qui généralise l’intégrale du type divergence du calcul de Malliavin. Pour cette intégrale de divergence généralisée nous montrons un théorème de Fubini et nous établissons des versions des formules d’Itô et Tanaka pour toutH∈(0,¹₂). Ensuite nous utilisons l’intégrale de divergence généralisée pour démontrer que pourH∈(¹₆,¹₂)etg∈C³(R), l’intégrale symétrique de Russo–Vallois_b

ag(B_t^H)d⁰B_t^H existe et vautG(B_b^H)−G(B_a^H), oùG=g, alors que pourH∈(0,¹₆],_b

a(B_t^H)²d⁰B_t^H n’existe pas.

2004 Elsevier SAS. All rights reserved.

MSC: 60G15; 60H05; 60H07

Keywords: Fractional Brownian motion; Stochastic integration; Malliavin calculus; Symmetric integral

* Corresponding author.

E-mail address: dnualart@ub.edu (D. Nualart).

1 Supported by a research grant of the Generalitat de Catalunya and the Swiss National Science Foundation.

2 Supported by the MCyT grant BFM2000-0598.

0246-0203/$ – see front matter 2004 Elsevier SAS. All rights reserved.

doi:10.1016/j.anihpb.2004.09.004

1. Introduction

A fractional Brownian motion (fBm) B^H = {B_t^H, t ∈R} with Hurst parameter H ∈(0,1) is a continuous Gaussian process with zero mean and covariance function

E[B_t^HB_s^H] =1 2

|t|^2H+ |s|^2H− |t−s|^2H

. (1.1)

IfH=¹₂, then B^H is a two-sided Brownian motion, but forH=¹₂,{B_t^H, t0}is not a semimartingale (for a proof in the caseH∈(¹₂,1)see Example 4.9.2 in Liptser and Shiryaev [18], for a general proof see Maheswaran and Sims [19] or Rogers [31]).

It can easily be seen from (1.1) that E

|B_t^H−B_s^H|²

= |t−s|^2H.

Hence, it follows from Kolmogorov’s continuity criterion (see e.g. Theorem I.2.1 in Revuz and Yor [27]) that on any finite interval, almost all paths ofB^H areβ-Hölder continuous for allβ < H. Therefore, ifuis a stochastic process with Hölder continuous trajectories of orderγ >1−H, then, by Young’s theorem on Stieltjes integrability (see [33]), the path-wise Riemann–Stieltjes integralT

0 u_t(ω)dB_t^H(ω)exists for allT 0. In particular, ifH >¹₂, the path-wise integralT

0 f(B_t^H)dB_t^H exists for allf ∈C²(R), and f (B_T^H)−f (0)=

T 0

f(B_t^H)dB_t^H

(more about path-wise integration with respect to fBm can be found in Lin [17], Mikosch and Norvaiša [21], Zähle [34] or Coutin and Qian [8]).

IfH¹₂, the path-wise Riemann–Stieltjes integral_T

0 f(B_t^H)dB_t^H does not exist. ForH=¹₂, the stochastic integral introduced by Itô [16] has proven to be a very fruitful approach and has led to the development of classical stochastic calculus. Gaveau and Trauber [11] and Nualart and Pardoux [23] proved that the Itô stochastic integral coincides with the divergence operator on the Wiener space. Later, several authors have used the divergence op- erator to define stochastic integrals with respect to fBm with arbitraryH∈(0,1). See for instance, Decreusefond and Üstünel [9], Carmona, Coutin and Montseny [6], Alòs, Mazet and Nualart [2,3], Coutin, Nualart and Tudor [7]. In [3] it is shown that ifH∈(¹₄,1), then for all functionsf ∈C²(R)such thatfdoes not grow too fast, the divergence of the process{f(B_t^H), t∈ [0, T]}exists and

f (B_T^H)−f (0)= T 0

f(B_t^H)δB_t^H+H T 0

f(B_t^H)t^2H−1dt. (1.2)

In [7] it is proved that for all H∈(¹₃,1), the process{sign(B_t^H), t∈ [0, T]}is in the domain of the divergence operator, and a fractional version of the Tanaka formula is derived. Privault [26] defined an extended Skorohod integral for a class of processes that satisfy a certain smoothness condition and showed that for this integral, formula (1.2) holds for everyH∈(0,1)and allf ∈C²(R)such thatf, f andf are bounded. However, when using the approach of [26], the integral with respect to fBm withH∈(0,¹₂)cannot be defined directly but must be constructed by approximating fBm with more regular processes. Similarly to [3], Hu [14] defined a stochastic integral with respect to fBm by transforming integrands and integrating them with respect to a standard Brownian motion. Provided they both exist, the integral with respect to fBm defined in [14] coincides with the one in [3].

Duncan, Hu and Pasik-Duncan [10] introduced a stochastic integral for fBm withH∈(¹₂,1)as the limit of finite sums involving the Wick product. It is shown in Section 7 of [3] that again, this integral is the same as the divergence integral if both exist. Leaving the framework of random variables and working in the space of Hida distributions, Hu

and Øksendal [15] as well as Bender [4] developed the integral of [10] further. In [4], for allH∈(0,1), a fractional Tanaka formula is proved, and an extended version of the formula (1.2) is shown to hold under the assumption that f is a tempered distribution that can also depend on t and satisfies some mild regularity conditions. Gradinaru, Russo and Vallois [12] proved a change of variables formula for fBm withH∈ [¹₄,1)that holds for the symmetric integral introduced in Russo and Vallois [28]. If both exist, the Russo–Vallois symmetric integral differs from the divergence integral by a trace term. For more details, see [1] or the introduction of [12].

In this paper we first explore how generally a stochastic integral for fBm can be defined by using the divergence operator from Malliavin calculus, and in particular, whether for the divergence operator, there exist versions of Itô’s and Tanaka’s formula for fBm with anyH∈(0,¹₂). Then, we study Russo–Vallois symmetric integrals of the formb

ag(B_t^H)d⁰B_t^H, for deterministic functionsg:R→R.

It turns out that the standard divergence integral of fBm with respect to itself does not exist ifH ∈(0,¹₄], the reason being that in this case, the paths of fBm are too irregular. However, in the right setup, the standard divergence operator can be extended by a simple change of the order of integration in the duality relationship that defines the divergence operator as the adjoint of the Malliavin derivative. The definition of this extended divergence operator is simpler than the definitions of the stochastic integrals in [26,14,15] and [4]. Moreover, it can be shown that for the extended divergence operator, a Fubini theorem holds as well as versions of the formulas of Itô and Tanaka for fBm with anyH ∈(0,¹₂). By localization, the extended divergence operator can be generalized further, and one can prove that for everyH∈(0,¹₂), formula (1.2) holds for allf ∈C²(R). A similar formula is valid for arbitrary convex functions. Hence, the change of variables formulas that we show for the extended divergence integral in this paper are valid for more general functionsf than the change of variable formulas in [26]. On the other hand, our change of variables formulas for the extended divergence integral are neither more nor less general than the ones in [4]. Whereas in [4]f does not need to be a twice continuously differentiable or convex function, it cannot grow to fast at infinity. Another important difference between the divergence integral in this paper and the stochastic integral of [15] and [4] is that the stochastic integral of divergence type in this paper is always a random variable whereas in [15] and [4], the stochastic integral is defined as a Hida distribution. In the last section we use properties of the extended divergence integral to show that for all real numbersaandbsuch that−∞< a < b <∞and every H∈(¹₆,¹₂), the symmetric integral

b a

g(B_t^H)d⁰B_t^H (1.3)

in the Russo–Vallois sense exists for allg∈C³(R)and is equal toG(B_b^H)−G(B_a^H), whereG=g, while on the other hand, forH∈(0,¹₆], the symmetric integral_b

a(B_t^H)²d⁰B_t^H does not exist.

ThatH=¹₆ is a barrier for the existence of integrals of the form (1.3) was simultaneously and independently discovered in the paper [13] by Gradinaru, Nourdin, Russo and Vallois. Their method of proof is different from ours and to show that the integral (1.3) exists for allH >¹₆, they need thatg∈C⁵(R). On the other hand, their result holds for more general symmetric stochastic integrals than the one considered in this paper.

The structure of the paper is as follows. In Section 2, we collect some facts from the theory of fractional calculus and discuss the first chaos of fBm with Hurst parameterH∈(0,¹₂). In Section 3, we show that ifH∈(0,¹₄], then for−∞< a < b <∞, the processB_t^H1_(a,b](t )is not in the domain of the standard divergence operator. We then introduce an extended divergence operator and prove a Fubini theorem. Section 4 contains versions of the formulas of Itô and Tanaka for fBm with Hurst parameterH∈(0,¹₂). In Section 5, we show that for−∞< a < b <∞, the Russo–Vallois symmetric integral_b

a g(B_t^H)d⁰B_t^H exists for allg∈C³(R)if and only ifH > ¹₆, in which case it is equal toG(B_b^H)−G(B_a^H), whereG=g.

2. The first chaos of fBm withH∈(0,¹₂)

Let{B_t^H, t∈R}be a fBm with Hurst parameterH∈(0,¹₂)on a probability space(Ω,F,P)such that F=σ{B_t^H, t∈R}.

ByEH we denote the linear space of step functions _n

j=1

a_j1(t_j,t_j+1]: n1, −∞< t₁< t₂<· · ·< t_n+1<∞, a_j∈R , equipped with the inner product

_n

j=1

a_j1_(tj,tj+1], m k=1

b_k1_(sk,sk+1]

EH

:=E _n

j=1

a_j(B_t^H

j+1−B_t^H

j) m k=1

b_k(B_s^H

k+1−B_s^H

k)

. Obviously, the linear map

n j=1

a_j1_(tj,tj+1]→ n j=1

a_j(B_t^H

j+1−B_t^H

j) (2.1)

is an isometry between the inner product spacesEH and span{B_t^H, t∈R} ⊂L²(Ω),

where span denotes the linear span.

There exists a Hilbert space of functions which containsEHas a dense subspace. To describe this Hilbert space, we need the following notions of fractional calculus. We refer the reader to Samko, Kilbas and Marichev [32] for a complete presentation of this theory.

Letα=¹₂−H. The fractional integralsI₊^αϕandI₋^αϕof a functionϕon the whole real axis are given by I₊^αϕ(t ):= 1

(α) t

−∞

(t−s)^α−1ϕ(s)ds, t∈R, and

I₋^αϕ(t ):= 1 (α)

∞ t

(s−t )^α−1ϕ(s)ds, t∈R,

respectively (see page 94 of [32]). The Marchaud fractional derivatives D^α₊ϕand D^α₋ϕof a functionϕon the whole real line are defined by

D^α_±ϕ(t ):=lim

ε 0D^α_±,εϕ(t ), t∈R, where

D^α_±,εϕ(t ):= α (1−α)

∞ ε

ϕ(t )−ϕ(t∓s)

s^1+α ds, t∈R

(compare page 111 of [32]). It follows from Theorem 5.3 of [32] thatI₊^αandI₋^αare bounded linear operators from L²(R)toL^1/H(R). Theorem 6.1 of [32] implies that for allϕ∈L²(R),

D^α₊I₊^αϕ=ϕ and D^α₋I₋^αϕ=ϕ. (2.2)

In Corollary 1 to Theorem 11.4 of [32] it is shown that I^α

L²(R) :=I₊^α

L²(R)

=I₋^α L²(R)

. Let

c_H =

H+1 2

^∞

0

(1+s)^H−1/2−s^H−1/22

ds+ 1 2H

₋1/2

.

It follows from (2.2) that the spaceI^α(L²(R))equipped with the inner product ϕ, ψΛ_H :=c²_HD^α₋ϕ, D₋^αψL²(R),

is a Hilbert space. We denote it byΛ_H. It is shown in Pipiras and Taqqu [24] that for allϕ, ψ∈EH, ϕ, ψΛ_H = ϕ, ψ_EH

and thatEHis dense inΛ_H. Therefore, the isometry (2.1) can be extended to an isometry betweenΛ_Hand the first chaos of{B_t^H, t∈R},

span^L2(Ω){B_t^H, t∈R}. We will denote this isometry by

ϕ→B^H(ϕ).

Remark 2.1. Let−∞< a < b <∞, and set Λ^(a,b_H ^]:=

ϕ∈ΛH: ϕ=ϕ1(a,b](·) .

Letϕ∈ΛH\Λ^(a,b_H ^]. SinceI₋^α is a bounded linear operator fromL²(R)toL^1/H(R), there exists a constantc >0 such that for allψ∈Λ^(a,b_H ^],

cϕ−ψΛ_H ϕ−ψL^1/H(R)

(−∞,a]∪(b,∞)

ϕ(t )^1/Hdt H

>0.

This shows thatΛ^(a,b_H ^]is a closed subspace ofΛ_H. On the other hand, let E_H^(a,b]:=

ϕ∈EH: ϕ=ϕ1(a,b](·) ,

and denote byE_H^(a,b]the closure ofE_H^(a,b]inΛ_H. Ifϕ∈E_H^(a,b], there exists a sequence{ϕ_n}^∞_n=1of functions inE_H^(a,b] such thatϕ_n→ϕinΛ_H and therefore also inL^1/H(Ω). It follows thatϕ∈Λ^(a,b_H ^]. This shows thatE_H^(a,b]⊂Λ^(a,b_H ^].

The right-sided fractional integralI_b^α₋ϕof a functionϕon the interval(a, b]is given by

I_b^α₋ϕ(t ):= 1 (α)

b t

(s−t )^α−1ϕ(s)ds, t∈(a, b]

(see Definition 2.1 of [32]). The right-sided Riemann–Liouville fractional derivativeD^α_b−ϕ of a functionϕon the interval(a, b]is given by

Db^α−ϕ(t ):= − 1 (1−α)

d dt

b t

(s−t )^−αϕ(s)ds, t∈(a, b]

(see Definition 2.2 in [32]). It is shown in Theorem 2.6 of [32] thatI_b^α₋is a bounded linear operator fromL¹(a, b] toL¹(a, b]. It follows from Theorem 2.4 and formula (2.19) of [32] that for allϕ∈L¹(a, b],

D^αb−I_b^α₋ϕ=ϕ.

Clearly, the linear maps

M^−α:L²(a, b] →L¹(a, b], f (t )→(t−a)^−αf (t ) and

M^α:L¹(a, b] →L¹(a, b], f (t )→(t−a)^αf (t ) are bounded and injective. It follows that the map

J:=M^α◦I_b^α₋◦M^−α:L²(a, b] →L¹(a, b]

is bounded and injective. Therefore,λ^(a,b_H ^]=J (L²(a, b])with the inner product ϕ, ψ_λ^(a,b]

H := π(2H−1)H

(2−2H )sin(π(H−1/2))J⁻¹ϕ, J⁻¹ψ_L²_(a,b]

is a Hilbert space. In [25], Pipiras and Taqqu have shown thatE_H^(a,b]is dense inλ^(a,b_H ^]. Let{ϕn}^∞_n=1be a Cauchy- sequence inE_H^(a,b]. Then, there exist functionsϕ∈E_H^(a,b]andψ∈λ^(a,b_H ^]such that

ϕ_n→ϕ inΛ_H and therefore also inL^1/H(a, b] and

ϕ_n→ψ inλ^(a,b_H ^]and therefore also inL¹(a, b]. It follows thatϕ=ψ. This shows that

λ^(a,b_H ^]=E_H^(a,b]⊂Λ^(a,b_H ^]. (2.3)

3. Extension of the divergence operator

In this section we define an extended divergence operator with respect to{B_t^H, t∈R}forH∈(0,¹₂). We briefly recall the basic notions of the stochastic calculus of variations, also called Malliavin calculus. For more details we refer to the books by Nualart [22] and Malliavin [20]. The set of smooth and cylindrical random variablesSconsists of all random variables of the form

F =f

B^H(ϕ₁), . . . , B^H(ϕn)

, (3.1)

where n1, f ∈Cp^∞(Rⁿ) (f and all its partial derivatives have polynomial growth), and ϕj ∈ΛH. Since F=σ{B_t^H, t∈R},S is dense inL^p(Ω)for allp1. The derivative of a smooth and cylindrical random vari- ableF of the form (3.1) is defined as theΛH-valued random variable

DF= n j=1

∂f

∂x_j

B^H(ϕ₁), . . . , B^H(ϕ_n) ϕ_j.

For allp1,F→DFis a closable unbounded linear operator fromL^p(Ω)toL^p(Ω, Λ_H). We denote the closed operator byDand its domain inL^p(Ω)byD^1,p.

The divergence operatorδis defined as the adjoint of the derivative operator. Forp >1, letp˜=_p^p₋₁>1. By δ_pwe denote the adjoint ofDviewed as operator fromL^p˜(Ω)toL^p˜(Ω, Λ_H), that is, the domain ofδ_p, Domδ_p, is the space of processesu∈L^p(Ω, Λ_H)such that

F→Eu, DFΛ_H

is a bounded linear functional on(S, · p_˜), and foru∈Domδ_p,δ_p(u)is the unique element inL^p(Ω)such that Eu, DFΛ_H =E

δ_p(u)F

, (3.2)

for allF ∈S. Obviously, ifu∈Domδ_p∩Domδ_q, for differentp, q >1, then δ_p(u)=δ_q(u). Hence, one can define

Domδ:=

p>1

Domδ_p, and foru∈Domδ,

δ(u):=δp(u), (3.3)

for somep >1 such thatu∈Domδp.

Remark 3.1. Let−∞< a < b <∞and consider the process{B_t^H, a < tb}on(Ω,F^(a,b],P), where F^(a,b]=σ{B_t^H, a < tb} =σ{B_t^H, atb}.

Letδ^(a,b]be the corresponding divergence operator defined analogously to the divergence operatorδin (3.3). By (2.3), a processu∈

p>1L^p(Ω, λ^(a,b_H ^])can be viewed as a process in

p>1L^p(Ω, Λ_H). It can easily be checked that ifu∈

p>1L^p(Ω, λ^(a,b_H ^])∩Domδ, thenu∈Domδ^(a,b]as well, andδ(u)=δ^(a,b](u).

Proposition 3.2. Let−∞< a < b <∞, and set u_t=B_t^H1_(a,b](t ), t∈R.

Then

P[u∈Λ_H] =1, forH∈ 1

4,1 2

, and

P[u∈Λ_H] =0, forH∈

0,1 4

.

Proof. First, letH∈(¹₄,¹₂). It follows from Kolmogorov’s continuity criterion (compare e.g. Theorem I.2.1 in Revuz and Yor [27]) that there exists a measurable setΩ⊂Ωwith P[Ω] =1 such that for allω∈Ω, there exists a constantC(ω) such that

sup

t∈(a,b]

B_t^H(ω)C(ω) and

sup

t,s∈(a,b];t=s

|B_t^H(ω)−B_s^H(ω)|

|t−s|^1/4 C(ω).

We fix anω∈Ωand set

ϕ(t ):=u_t(ω)=B_t^H(ω)1_(a,b](t ), t∈R,

and

C:= α

(1−α)C(ω), whereα=1 2−H.

Letε >0. Fort∈(−∞, a], D^α_+,εϕ(t )=0.

Fort∈(a, b],

D^α_+,εϕ(t ) α (1−α)

1_{t−a>ε}

t−a

ε

ϕ(t )−ϕ(t−s) s^1+α

ds+ϕ(t ) ^∞

(t−a)∨ε

s^−1−αds

C

1_{t−a>ε}

t−a

ε

s^−3/4−αds+ ∞ t−a

s^−1−αds

C

1

1/4−α(t−a)^1/4−α+1

α(t−a)^−α

. Fort∈(b,∞),

D^α_+,εϕ(t ) α (1−α)

t−a

t−b

|ϕ(t−s)| s^1+α dsC

t−a

t−b

s^−1−αds=C1 α

(t−b)^−α−(t−a)^−α . Hence, for allε >0, for allt∈R,|D^α_+,εϕ(t )|ψ (t ), where

ψ (t )=





0, ift∈(−∞, a],

C[(t−a)^1/4−α+(t−a)^−α], ift∈(a, b], C[(t−b)^−α−(t−a)^−α], ift∈(b,∞) and

C=C 1

1/4−α∨ 1 α

.

It can easily be checked thatψ∈L²(R). It follows thatϕsatisfies condition (1) of Theorem 6.2 of [32]. Condition (2) is trivially satisfied. Therefore, Theorem 6.2 of [32] implies thatϕ∈ΛH, which proves the first part of the proposition.

Now, let us assume thatH∈(0,¹₄]. The process B_t^H:=B_t^H_+a−B_a^H, t∈R,

is also a fBm with Hurst parameterH. Since it isH-selfsimilar, for allt∈(0, b−a), the random variable t^−2H

b−a−t 0

(B_s^H_+t−B_s^H)²ds

has the same distribution as

b−a−t 0

(B_s/t^H₊₁−B_s/t^H )²ds=t

(b−a)/t−1 0

(B_x^H₊₁−B_x^H)²dx

=(b−a−t ) 1 (b−a)/t−1

b−a/t−1 0

(B_x^H₊₁−B_x^H)²dx. (3.4)

The process(B_x^H₊₁−B_x^H)_x0is stationary and mixing. Therefore, it follows from the ergodic theorem that (3.4) converges to

(b−a)E(B₁^H)²

>0, inL¹ast→0.

Hence, t^−2H

b−a−t 0

(B_s^H_+t−B_s^H)²ds ^L

−→1 (b−a)E(B₁^H)²

, ast→0,

as well. It follows that there exists a measurable setΩ⊂Ω with P[Ω] =1 and a sequence of positive numbers {t_k}^∞_k=1that converges to 0 such that for allω∈Ωandk1,

R

u_s+tk(ω)−u_s(ω)2

ds

b−t_k a

B_s^H_+t

k(ω)−B_s^H(ω)2

ds=

b−a−t_k 0

B_s^H_+t

k(ω)−B_s^H(ω)2

ds

b−a

2 E(B₁^H)²

t_k^2H. (3.5)

Now, assume that there exists anω∈Ωsuch thatu(ω)∈Λ_H. By (6.40) of [32], the functionu(ω)has the property

R

us+t(ω)−us(ω)2

ds=o(t^2α) ast→0. (3.6)

Butu(ω)can only satisfy (3.5) and (3.6) at the same time ifH > α=¹₂−H, which contradictsH¹₄. Therefore, u(ω) /∈Λ_H for allω∈Ω, and the proposition is proved. 2

Since Domδ⊂

p>1L^p(Ω, Λ_H), Proposition 3.2 implies that processes of the form B_t^H1_(a,b](t ),

cannot be in Domδ if H ¹₄. Note that it follows from (2.3) that for H ¹₄, almost surely, no path of {B_t^H, a < tb}is inλ^(a,b_H ^]either, and therefore,{B_t^H, a < tb}∈/Domδ^(a,b]. In the following definition we extend the divergenceδto an operator whose domain also contains processes with paths that are not inΛ_H.

We set

Λ^∗_H:=I₋^α(EH).

SinceEH is dense inL²(R),Λ^∗_His dense inΛ_H. Furthermore, it can easily be checked that for−∞< a < b <∞, I₊^α

(t−a)^H₊^−1/2−(t−b)^H₊^−1/2

= (H+1/2)1_(a,b](t ), t∈R. It follows from (2.2) that

D^α₊1_(a,b](t )= 1 (H+1/2)

(t−a)^H₊^−1/2−(t−b)^H₊^−1/2

, t∈R, which shows that

D^α₊D^α₋(Λ^∗_H)=D^α₊(EH)⊂L^p(R), (3.7)

for all p∈

1

3/2−H, 1 1/2−H

, in particular, forp=2.

Corollary 2 to Theorem 6.2 of [32] implies that for allϕ∈Λ_H andψ∈EH, the following integration by parts formula holds:

R

ϕ(x)D^α₊ψ (x)dx=

R

D^α₋ϕ(x)ψ (x)dx. (3.8)

ByH_nwe denote then-th over-normalized Hermite polynomial, that is, H₀(x):=1, and H_n(x):=(−1)ⁿ

n! e^x2/2 dⁿ

dxⁿ(e^−x2/2), n1.

Furthermore, we setH₋₁(x):=0. It can be shown as in Theorem 1.1.1 of Nualart [22] that for allp1, span

H_n B^H(ϕ)

: n∈N, ϕ∈Λ^∗_H,ϕΛH=1 is dense inL^p(Ω).

Definition 3.3. Letu= {u_t, t∈R}be a measurable process. We say thatu∈Dom^∗δif and only if there exists a δ(u)∈

p>1L^p(Ω)such that for alln∈Nandϕ∈Λ^∗_H withϕΛH=1, the following conditions are satisfied:

(i) for almost allt∈R:u_tH_n−1(B^H(ϕ))∈L¹(Ω), (ii) E[u_·H_n−1(B^H(ϕ))]D^α₊D^α₋ϕ(·)∈L¹(R), and (iii) c²_H

RE[utHn−1(B^H(ϕ))]D^α₊D^α₋ϕ(t )dt=E[δ(u)Hn(B^H(ϕ))].

Note that ifu∈Dom^∗δ, thenδ(u)is uniquely defined, and the mappingδ: Dom^∗δ→

p>1L^p(Ω)is linear.

Remark 3.4.

1. Letn∈Nandϕ∈Λ^∗_H. By (3.7), the process Hn−1

B^H(ϕ)

D^α₊D^α₋ϕ(t ) is inL^p(Ω, L^q(R))for all

p∈ [1,∞) and q∈ 1

3/2−H, 1 1/2−H

. By twice applying Hölder’s inequality, it follows that if

u∈L^p

Ω, L^q(R) for some

p∈(1,∞] and q∈ 1

1/2+H, ∞

, then

utHn−1

B^H(ϕ)

D^α₊D^α₋ϕ(t )∈L¹

Ω, L¹(R)

=L¹(Ω×R), (3.9)

which implies thatusatisfies conditions (i) and (ii) of Definition 3.3.

2. The extended divergence operatorδis closed in the following sense:

Let

p∈(1,∞] and q∈ 1

1/2+H, ∞

.

Let{u^k}^∞_k=1be a sequence in Dom^∗δ∩L^p(Ω, L^q(R))andu∈L^p(Ω, L^q(R))such that lim

k→∞u^k=u inL^p

Ω, L^q(R) . It follows that for alln∈Nandϕ∈Λ^∗_H,

lim

k→∞u^k_tH_n−1 B^H(ϕ)

D^α₊D^α₋ϕ(t )=u_tH_n−1 B^H(ϕ)

D^α₊D^α₋ϕ(t ) inL¹(Ω×R). If there exists apˆ∈(1,∞]and anX∈L^pˆ(Ω)such that

lim

k→∞δ(u^k)=X inL^pˆ(Ω), thenu∈Dom^∗δ, andδ(u)=X.

Proposition 3.5.

Dom^∗δ∩

p>1

L^p(Ω, Λ_H)=Domδ,

and the extended divergence operatorδrestricted to Domδcoincides with the standard divergence operator defined by (3.3).

Proof. Letu∈Domδ=

p>1Domδp. Then, there exists ap >1 such thatu∈Domδp, andδp(u)∈L^p(Ω). In particular,u∈L^p(Ω, Λ_H). Hence, it follows from Theorem 5.3 of [32] thatu∈L^p(Ω, L^1/H(R)). Therefore, by Remark 3.4.1,usatisfies conditions (i) and (ii) of Definition 3.3.

Now, letn∈Nandϕ∈Λ^∗_H withϕΛH =1. The duality relation (3.2), the expression DHn

B^H(ϕ)

=Hn−1

B^H(ϕ) ϕ

and the fractional integration by parts formula (3.8), yield E

δ_p(u)H_n

B^H(ϕ)

=E

u, DH_n

B^H(ϕ)

Λ_H =E H_n−1

B^H(ϕ)

u, ϕΛH

=c²_HE Hn−1

B^H(ϕ)

D^α₋u,D^α₋ϕL²(R)

=c²_HE

H_n−1 B^H(ϕ)

R

u_tD^α₊D^α₋ϕ(t )dt

. (3.10)

Since (3.9) is valid, Fubini’s theorem implies that (3.10) is equal to c_H²

R

E

u_tH_n−1

B^H(ϕ)

D^α₊D^α₋ϕ(t )dt,

which shows thatualso fulfills condition (iii) of Definition 3.3. Hence, Domδ⊂Dom^∗δ∩

p>1

L^p(Ω, Λ_H).

and the operatorδfrom Definition 3.3 is an extension of the one defined by (3.3).

Ifu∈Dom^∗δ∩

p>1L^p(Ω, Λ_H), then there exists ap >1 such that u∈L^p(Ω, Λ_H)andδ(u)∈L^p(Ω).

Letn∈Nandϕ∈Λ^∗_H. Theorem 5.3 of [32] implies thatu∈L^p(Ω, L^1/H(R)), and it follows that (3.9) holds.

Therefore, Fubini’s theorem applies, and we get

E

u, DH_n

B^H(ϕ)

Λ_H =E

c²_H

R

D^α₋u_tH_n−1 B^H(ϕ)

D^α₋ϕ(t )dt

=E

c²_H

R

u_tD^α₊D^α₋ϕ(t )dt H_n−1

B^H(ϕ)

=c²_H

R

E utHn−1

B^H(ϕ)

D^α₊D^α₋ϕ(t )dt

=E

δ(u)H_n

B^H(ϕ) .

It can be deduced from this by an approximation argument that Eu, DFΛ_H =E

δ(u)F

for allF ∈S, which shows thatu∈Domδ, and therefore, Dom^∗δ∩

p>1

L^p(Ω, Λ_H)⊂Domδ. 2

Proposition 3.6. Letu∈Dom^∗δsuch thatE[u_.] ∈L²(R). ThenE[u_.] ∈Λ_H.

Proof. By Definition 3.3,δ(u)∈L^p(Ω)for somep >1. Letϕ∈Λ^∗_H withϕΛH =1. Forn=1, condition (iii) of Definition 3.3 yields

c²_H

R

E[ut]D^α₊D^α₋ϕ(t )dt =E

δ(u)B^H(ϕ) δ(u)

L^p(Ω) B^H(ϕ)

L^p˜(Ω), wherep˜=_p^p₋₁. Since there exists a constantγ_p˜such that for allϕ∈ΛH,

B^H(ϕ)

L^p˜(Ω)=γ_p˜ B^H(ϕ)

L²(Ω)=γ_p˜ϕΛ_H, the mapping

ϕ→c²_H

R

E[u_t]D^α₊D^α₋ϕ(t )dt

is a continuous linear functional onΛ^∗_H=I₋^α(EH)⊂Λ_H, which can be extended to a continuous linear functional onΛ_H. Therefore, there exists aψ∈Λ_H such that for allϕ∈Λ^∗_H,

c²_H

R

E[u_t]D^α₊D^α₋ϕ(t )dt= ψ, ϕΛH=c²_H

R

D^α₋ψ (t )D^α₋ϕ(t )dt. (3.11)

It follows from the integration by parts formula (3.8) that (3.11) is equal to c²_H

R

ψ (t )D^α₊D^α₋ϕ(t )dt.

Hence,

R

ψ (t )D^α₊D^α₋ϕ(t )dt

E[u_.]

L²(R)D^α₊D^α₋ϕL²(R)