A Transfer Principle for Multiple Stochastic Fractional Integrals

Víctor Pérez-Abreu^{∗} Centro de Investigación en Matemáticas, A.C. CIMAT, Apdo. Postal 402, Guanajuato, 36000 Gto. Mexico

Constantin Tudor^{†} University of Bucharest and CIMAT Department of Mathematics 70109 Bucharest, Romania

Abstract

Multiple integrals with respect to a fractional Brownian motion are defined explicitely in terms of multiple Wiener-Itô integrals and fractional integrals and derivatives of deterministic functions of several variables. As applications we define the chaos form of the fractional stochastic integral and we derive the chaos decomposition of the solution of affine fractional equations and the strong law of large numbers for the fractional Brownian motion.

Key Words: Fractional Brownian motion, Multiple fractional integrals, Chaos decomposition AMS Classification: 60H05

∗E-mail address: pabreu@cimat.mx †E-mail address: ctudor@pro.math.unibuc.ro, tudor@cimat.mx

1. Introduction

Fractional Brownian motion (fBm) is playing an increasing role in modeling long-range dependence phenomena in many fields such as telecommunication network, economics, hydrology and biology. From the theoretical point of view a fBm B = {B_{t}^{H}}_{t∈T} is a Gaussian process which is a suitable generalization of the Brownian motion W = {W_{t}}_{t∈T} but exhibiting long-range dependence. Since the pioneering work by Itô [13], multiple Wiener-Itô integrals with respect to a Brownian motion and their corresponding chaos expansions have been studied by many authors, being standard tools for dealing with nonlinear functionals of W as well as with anticipating stochastic calculus for W. It seems then natural to study multiple fractional integrals and chaos expansions for a fractional Brownian motion. This has recently been started by Dasgupta and Kallianpur [4] and Duncan, Hu and Pasik-Duncan [8] for the fBm of Hurst parameter H > \frac{1}{2}, and for general Gaussian process twenty years ago by Huang and Cambanis [12]. The techniques used in these papers involve Wick product and reproducing kernel Hilbert space theory. In this paper our starting point of view is the fact that B_{t}^{H} = ∫ K(t, s)dW_{s} for a nonrandom ”fractional kernel ” kernel K(t, s). Using this representation we are able to define in an explicit manner multiple fractional stochastic integrals in terms of the classical Wiener-Itô or Stratonovich integrals of the Brownian motion and fractional integrals and derivatives for deterministic functions of several variables. By using this ”transfer principle”, we show that our easily defined multiple fractional stochastic integrals coincide with those defined by Dasgupta and Kallianpur [4], [5] and Duncan, Hu and Pasik-Duncan [8] by more complicated manners. Well known properties like multiple product formula, independence of multiple integrals, Hu-Meyer formula, are immediately transferred to multiple fractional integrals in an easy manner. In addition, we define a fractional stochastic integral (divergence operator) and we show that this stochastic integral agrees on relevant classes of integrands with the ones defined by different methods by Alòs, Mazet and Nualart [1] (in the particular case of fractional Brownian motion), Decreusefond and Üstunel [7], Duncan, Hu and Pasik-Duncan [8]. The paper is organized as follows. Section 2 reviews several facts on deterministic fractional integrals and the representation of fractional Brownian motion in terms of a standard Brownian motion. Sections 3 is devoted to multiple fractional integrals and their chaos decomposition. Sections 4 presents the fractional anticipating integral and as an application we derive the chaos decomposition for the solution of affine fractional equations. Finally, Section 5 studies multiple Stratonovich fractional integrals, including the fractional Hu-Meyer formula and an extension of the strong law of large numbers for the fBm (see Dasgupta and Kallianpur [5], Gladyshev [9]

2. Preliminaries

We fix H ∈ (0, 1), 0 < t_{0} < ∞ and 0 < α < 1. We put T = [0, t_{0}] or T = R . For a function f : T^{n} → R we define the Liouville fractional integral by

(I_{t_{0}−}^{α,n}f) (x_{1}, ..., x_{n}) = \frac{1}{[Γ(α)]^{n}} ∫_{x_{1}}^{t_{0}} ... ∫_{x_{n}}^{t_{0}} \frac{f(t_{1}, ..., t_{n})}{(t_{1} − x_{1})^{1−α} ... (t_{n} − x_{n})^{1−α}} dt_{1}...dt_{n}, x_{i} ∈ [0, t_{0}] , (2.1)

(I_{−}^{α,n}f) (x_{1}, ..., x_{n}) = \frac{1}{[Γ(α)]^{n}} ∫_{x_{1}}^{∞} ... ∫_{x_{n}}^{∞} \frac{f(t_{1}, ..., t_{n})}{(t_{1} − x_{1})^{1−α} ... (t_{n} − x_{n})^{1−α}} dt_{1}...dt_{n}, x_{i} ∈ R. (2.2)

The following basic properties of fractional integrals of deterministic functions are proved in Samko, Kilbas and Marichev [21]. Proposition 2.1 (a) The Liouville fractional integral (I_{t_{0}−}^{α,n}f) (x_{1}, ..., x_{n}) (resp. (I_{−}^{α,n}f) (x_{1}, ..., x_{n})) is well defined for almost all x_{1}, ..., x_{n} if f ∈ L^{1} ([0, t_{0}]^{n}) (resp. f ∈ L^{p} (R^{n}) , 1 ≤ p < \frac{1}{α}).

(b) If 1 < p < \frac{1}{α}, q = \frac{p}{1−αp}, then the operators I_{t_{0}−}^{α,n} : L^{p} ([0, t_{0}]^{n}) → L^{q} ([0, t_{0}]^{n}) , I_{−}^{α,n} : L^{p} (R^{n}) → L^{q} (R^{n}) , are bounded and one to one.

(c) I_{t_{0}−}^{α,n} = (I_{t_{0}−}^{α,1})^{⊗n} , I_{−}^{α,n} = (I_{−}^{α,1})^{⊗n} .

(d) If f ∈ I_{T −}^{α,n} (L^{2} ([0, t_{0}]^{n})) (resp. f ∈ I_{−}^{α,n} (L^{2}(R^{n})) then the function φ ∈ L^{2} ([0, t_{0}]^{n}) (resp. φ ∈ L^{2} (R^{n})) such that I_{t_{0}−}^{α,n}φ = f (resp. I_{−}^{α,n}φ = f) is unique and we denote it by D_{t_{0}−}^{α,n}f (resp. D_{−}^{α,n}f ). Define

c_{H} = [\frac{2HΓ(\frac{3}{2} − H)}{Γ(H + \frac{1}{2})Γ(2 − 2H)}]^{\frac{1}{2}} , d_{H} = c_{H}Γ(H + \frac{1}{2}).  (2.3)

Next we fix a normalized fBm {B_{t}^{H}}_{t∈T} with Hurst parameter H, defined on a probability space (Ω, F, P). We assume that F = B (B_{t}^{H} : t ∈ T ) . Recall that B^{H} is a continuous Gaussian process with:

(i) B_{0}^{H} = 0. (ii) For every s, t ∈ T,

E(B_{s}^{H}B_{t}^{H}) = \frac{1}{2} (|t|^{2H} + |s|^{2H} − |t − s|^{2H}) .

Remark 2.2.. It is known that there exists a standard Brownian motion {W_{t}^{H}}_{t∈T} defined on the same probability space (Ω, F, P) such that:

(j) {B_{t}^{H}}_{t∈T} and {W_{t}^{H}}_{t∈T} generate the same filtration F. (jj) (Representation formula). For each t ∈ T,

B_{t}^{H} = ∫_{T} K_{H,T} (t, s)dW_{s}^{H},  (2.4)

K_{H,R}(t, s) = c_{H} [(t − s)_{+}^{H− \frac{1}{2}} − (−s)_{+}^{H− \frac{1}{2}} ],  (2.5)
K_{H,[0,t_{0}]}(t, s) = c_{H} [(t − s)^{H− \frac{1}{2}} +

(\frac{1}{2} − H) ∫_{s}^{t} (u − s)^{H− \frac{3}{2}} (1 − (\frac{s}{u})^{\frac{1}{2} −H}) du] 1_{[0,t)} (s) =

c_{H} (t − s)^{H− \frac{1}{2}} F (H − \frac{1}{2}, \frac{1}{2} − H, H + \frac{1}{2}, 1 − \frac{t}{s})1_{[0,t)} (s) ,  (2.6)

where F is the Gauss hypergeometric function (see Theorems 4.3 and 4.5 of [2] or Theorem 4.8 of [7], or Theorems 3.2 and 5.2 and Remark 5.1 of [17]). If H > \frac{1}{2},

K_{H,[0,t_{0}]}(t, s) = c_{H}Γ (H + \frac{1}{2}) s^{\frac{1}{2} −H}I_{t_{0}−}^{H− \frac{1}{2}} (x^{H− \frac{1}{2}} 1_{[0,t]}) (s) ,  (2.7)

K_{H,R}(t, s) = c_{H}Γ (H + \frac{1}{2}) (I_{−}^{H− \frac{1}{2}} 1_{[0,t)}) (s) ,  (2.8)

and if H < \frac{1}{2},

K_{H,[0,t_{0}]}(t, s) = c_{H}Γ (H + \frac{1}{2}) s^{\frac{1}{2} −H}D_{t_{0}−}^{\frac{1}{2} −H} (x^{H− \frac{1}{2}} 1_{[0,t)}) (s) ,  (2.9)

K_{H,R}(t, s) = c_{H}Γ (H + \frac{1}{2}) (D_{−}^{\frac{1}{2} −H}1_{[0,t)}) (s) .  (2.10)

Remark 2.3 (see [2], [7] and [19]). The Reproducing Kernel Hilbert Space of the fBm is H_{H,T} = K_{H,T} (L^{2}(T)) with the scalar product

⟨K_{H,T} (f), K_{H,T} (g)⟩_{H_{H,T}} = ⟨f, g⟩_{L^{2}(T)} .

3. Multiple fractional integrals

3.1. Case \frac{1}{2} < H < 1

We denote by L_{H}^{2}([0, t_{0}]^{n}) the class of all measurable functions f : [0, t_{0}]^{n} → R such that

∫_{[0,t_{0}]_{2n}} φ(x_{1}, y_{1})...φ(x_{n}, y_{n}) |f(x_{1}, ..., x_{n})f(y_{1}, ..., y_{n})| dx_{1}...dx_{n}dy_{1}...dy_{n} < ∞,  (3.1)

where

φ : R^{2} → R, φ(s, t) = H(2H − 1) |s − t|^{2H−2} ,

and L_{H}^{2}(R^{n}) the class of all f ∈ L^{2}(R^{n}) such that

∫_{R^{2n}} φ(x_{1}, y_{1})...φ(x_{n}, y_{n}) |f(x_{1}, ..., x_{n})f(y_{1}, ..., y_{n})| dx_{1}...dx_{n}dy_{1}...dy_{n} < ∞,  (3.2)

For f, g ∈ L_{H}^{2}(T^{n}) we define

⟨f, g⟩_{H,T^{n}} = ∫_{T^{2n}} φ(x_{1}, y_{1})...φ(x_{n}, y_{n})f(x_{1}, ..., x_{n})g(y_{1}, ..., y_{n})dx_{1}...dx_{n}dy_{1}...dy_{n},  (3.3)

|f|_{H,T^{n}} = ⟨f, f⟩_{H,T^{n}} .

L_{s,H}^{2}(T^{n}) is the subspace of all f ∈ L_{H}^{2}(T^{n}), f symmetric. Next for a function f : T^{n} → R we denote by sym(f) its symmetrization.

Lemma 3.1 (a) We have the continuous and dense inclusions L^{2}([0, t_{0}]^{n}) ⊂ L_{H}^{2}([0, t_{0}]^{n}) ⊂ L^{1}([0, t_{0}]^{n}). Also we have the inclusions

L^{2}(R^{n}) ∩ L^{1}(R^{n}) ⊂ L_{H}^{2}(R^{n}) ⊂ L_{loc}^{1}(R^{n}),

the first one being dense.

(b) The space (L_{H}^{2}(T^{n}), ⟨., .⟩_{H,T^{n}}) is an incomplete separable pre-Hilbert space and the application

Γ_{H,T}^{(n)} : L_{H}^{2}(T^{n}) → L^{2}(T^{n}) ,
(Γ_{H,T}^{(n)} f) (t_{1}, ..., t_{n}) = d_{H}^{n} (t_{1}...t_{n})^{\frac{1}{2} −H} ×

× (I_{t_{0}−}^{H− \frac{1}{2} ,n}) ((x_{1}...x_{n})^{H− \frac{1}{2}} f(x_{1}, ..., x_{n})) (t_{1}, ..., t_{n}), if T = [0, t_{0}] ,  (3.4)
(Γ_{H,T}^{(n)} f) (t_{1}, ..., t_{n}) = d_{H}^{n} (I_{−}^{H− \frac{1}{2} ,n}f) (t_{1}, ..., t_{n}) if T = R,  (3.5)

is an isometry

(c) If f ∈ L_{H}^{2}(T^{n}) then

|sym(f)|_{H,T^{n}} ≤ |f|_{H,T^{n}} .  (3.6)

and for T = [0, t_{0}] , the mapping

Γ_{H,T}^{(n)} : L^{2}(T^{n}) → L^{2}(T^{n}),

is continuous.

Proof. (a) If f ∈ L^{2}([0, t_{0}]^{n}) then by Schwartz’s inequality

|f|_{H,T^{n}}^{2} ≤ ∫_{[0,t_{0}]_{2n}} |f(t_{1}, ..., t_{n})|^{2} φ(t_{1}, s_{1})...φ(t_{n}, s_{n})dt_{1}..dt_{n}ds_{1}..ds_{n} ≤

C ∫_{[0,t_{0}]_{n}} |f(t_{1}, ..., t_{n})|^{2} [t_{1}^{2H−1} + (t_{0} − t_{1})^{2H−1}] ×
... × [t_{n}^{2H−1} + (t_{0} − t_{n})^{2H−1}] dt_{1}..dt_{n} ≤ C_{1}^{n} ∥f∥_{L^{2}([0,t_{0}]^{n})}^{2} .  (3.7)

If f ∈ L_{H}^{2}([0, t_{0}]^{n}) then

(∫_{[0,t_{0}]_{n}} |f(t_{1}, ..., t_{n})| dt_{1}..dt_{2})^{2} = ∫_{[0,t_{0}]_{2n}} |f(t_{1}, ..., t_{n})| |f(s_{1}, ..., s_{n})| dt_{1}..dt_{2}ds_{1}..ds_{n}

≤ C ∫_{[0,t_{0}]_{2n}} |f(t_{1}, ..., t_{n})| |f(s_{1}, ..., s_{n})| φ(t_{1}, s_{1})...φ(t_{n}, s_{n})dt_{1}..dt_{n}ds_{1}..ds_{n} < ∞.

If f ∈ L_{H}^{2}([0, t_{0}]^{n})(resp. f ∈ L^{1}([0, t_{0}]^{n})) then f_{k} = f1_{{|f|≤k}} ⟶ f in L_{H}^{2}([0, t_{0}]^{n}) (resp. L^{1}([0, t_{0}]^{n})). The inclusion L^{2}(R^{n}) ∩ L^{1}(R^{n}) ⊂ L_{H}^{2}(R^{n}) is proved in [19] for n = 1 (for arbitrary n the proof is similar) and the density is obtained by truncation. Now if f ∈ L_{H}^{2}(R^{n}) and a < b then

(∫_{[a,b]_{2n}} |f(t_{1}, ..., t_{n})| dt_{1}..dt_{2})^{2} = ∫_{[a,b]_{2n}} |f(t_{1}, ..., t_{n})| |f(s_{1}, ..., s_{n})| dt_{1}..dt_{2}ds_{1}..ds_{n} ≤

C ∫_{[a,b]_{2n}} |f(t_{1}, ..., t_{n})| |f(s_{1}, ..., s_{n})| φ(t_{1}, s_{1})...φ(t_{n}, s_{n})dt_{1}..dt_{n}ds_{1}..ds_{n} ≤ C |f|_{R^{n},H}^{2} .

(b) First we show that

∥Γ_{H,T}^{(n)} f∥_{L^{2}(T^{n})}^{2} = |f|_{H,T^{n}}^{2} .  (3.8)

The case T = R is considered in [11]-Lemma 2.1(see also [10]-Proposition 2.2). Assume now that T = [0, t_{0}] . By using the equality (see [25])

(x_{1}x_{2})^{H− \frac{1}{2}} ∫_{0}^{min(x_{1},x_{2})} t^{1−2H} (x_{1} − t)^{H− \frac{3}{2}} (x_{2} − t)^{H− \frac{3}{2}} dt =

|x_{1} − x_{2}|^{2H−2} \frac{Γ(H − \frac{1}{2})Γ(2 − 2H)}{Γ(\frac{3}{2} − H)},  (3.9)

and Fubini’s theorem we obtain

∫_{[0,t_{0}]_{n}} |(Γ_{H,T}^{(n)} f) (t_{1}, ..., t_{n})|^{2} dt_{1}...dt_{n} = d_{H}^{2n} ∫_{[0,t_{0}]_{n}} (t_{1}...t_{n})^{1−2H} ×

× \{∫_{t_{1}}^{t_{0}} ... ∫_{t_{n}}^{t_{0}} ∫_{t_{1}}^{t_{0}} ... ∫_{t_{n}}^{t_{0}} (x_{1}...x_{n})^{H− \frac{1}{2}} (y_{1}...y_{n})^{H− \frac{1}{2}} [(x_{1} − t_{1})(y_{1} − t_{1})]^{H− \frac{3}{2}} ...×
× [(x_{n} − t_{n})(y_{n} − t_{n})]^{H− \frac{3}{2}} f(x_{1}, ..., x_{n})f(y_{1}, ..., y_{n})dx_{1}..dx_{n}dy_{1}..dy_{n}\} dt_{1}..dt_{n} =

d_{H}^{2n} ∫_{[0,t_{0}]_{2n}} ∏_{j=1}^{n} ∫_{0}^{min(x_{j},y_{j})} t^{1−2H} (x_{j} − t)^{H− \frac{3}{2}} (y_{j} − t)^{H− \frac{3}{2}} dt×

× ∏_{j=1}^{n} (x_{j}y_{j})^{H− \frac{1}{2}} f(x_{1}, ..., x_{n})f(y_{1}, ..., y_{n})dx_{1}..dx_{n}dy_{1}..dy_{n} = |f|_{H,T^{n}}^{2} .

Next if |f|_{H,R^{n}}^{2} = 0 = ∥Γ_{H,T}^{(n)} f∥_{L^{2}(R^{n})}^{2} then Γ_{H,T}^{(n)} f = 0 and by [21]-Theorem 6.1 it follows that

f = 0. Now if |f|_{H,[0,t_{0}]^{n}}^{2} = 0 = ∥Γ_{H,[0,t_{0}]}^{(n)}f∥_{L^{2}([0,t_{0}]^{n})}^{2} then

(I_{t_{0}−}^{H− \frac{1}{2} ,n}f) ((x_{1}...x_{n})^{H− \frac{1}{2}} f(x_{1}, ..., x_{n})) (t_{1}, ..., t_{n}) = 0 for a.a. t_{i},

and since I_{t_{0}−}^{H− \frac{1}{2} ,n} is one to one we obtain that f = 0. The other properties of the scalar product are obvious by (3.8). The fact that (L_{H}^{2}(T^{n}), ⟨., .⟩_{H,T^{n}}) is incomplete is shown for n = 1 in [19] if T = R and in [23] if T = [0, t_{0}] (the general case is similar).

(c) We have

|sym(f)|_{H,T^{n}} = ∥Γ_{H,T}^{(n)} (sym(f))∥_{L^{2}(T^{n})} = ∥sym(Γ_{H,T}^{(n)} f)∥_{L^{2}(T^{n})} ≤
∥Γ_{H,T}^{(n)} f∥_{L^{2}(T^{n})} = |f|_{H,T^{n}} .

The continuity of the operator Γ_{H,T}^{(n)} : L^{2}([0, t_{0}]^{n}) → L^{2}([0, t_{0}]^{n}) follows from (a) and (b)(see also [21]-Theorem 10.4). ■

Remark 3.2. From the previous Lemma it follows that the Liouville fractional integral I_{−}^{H− \frac{1}{2} ,n}f is well defined also for f : R^{n} → R which satisfies (3.2) (f not necessarily in L^{2}(R^{n})) and moreover I_{−}^{H− \frac{1}{2} ,n}f ∈ L^{2}(R^{n}).

We shall denote by I_{p}^{\frac{1}{2} ,T} (f) the multiple Wiener-Itô integral of f ∈ L^{2}(T^{p}) with respect to W^{H}. Definition 3.3 If f ∈ L_{H}^{2}(T^{p}) then we define the multiple fractional integral of order p of f with respect to B^{H} by

I_{p}^{H,T} (f) = I_{p}^{\frac{1}{2} ,T} (Γ_{H,T}^{(p)} f).  (3.10)

From the corresponding properties of multiple Wiener-Itô integrals it follows:

Lemma 3.4 (a) I_{p}^{H,T} (f) = I_{p}^{H,T} (sym(f)).

(b) E [I_{p}^{H,T} (f)] = 0.

(c) E [I_{p}^{H,T} (f)I_{n}^{H,T} (g)] = 0 if p ≠ n.

(d) E [|I_{p}^{H,T} (f)|^{2}] = p! |sym(f)|_{H,T^{p}}^{2} .

Proposition 3.5 (a) (The exponential). If f ∈ L_{H}^{2}(T) then

e^{H}(f) := exp \{I_{1}^{H,T} (f) − \frac{1}{2} |f|_{H,T}^{2} \} = ∑_{p=0}^{∞} \frac{1}{p!}I_{p}^{H,T} (f^{⊗p}).  (3.11)

(b) (Product formula). Let f_{m} ∈ L_{s,H}^{2}(T^{m}), g_{n} ∈ L_{s,H}^{2}(T^{n}) be such that for any 1 ≤ r ≤ min(m, n),

f_{m}(t_{1}, ..., t_{m−r}, .), g_{n}(s_{1}, ..., s_{n−r}, .) ∈ L_{H}^{2}(T^{r}) for a.e. t_{1}, ..., t_{m−r}, s_{1}, ..., s_{n−r},
t_{1}, ..., t_{m−r} → |f_{m}(t_{1}, ..., t_{m−r}, .)|_{H,T^{r}} ∈ L_{H}^{2}(T^{m−r}),
s_{1}, ..., s_{n−r} → |g_{n}(s_{1}, ..., s_{n−r}, .)|_{H,T^{r}} ∈ L_{H}^{2}(T^{n−r}).

Then

I_{m}^{H,T} (f_{m})I_{n}^{H,T} (g_{n}) =
∑_{r=0}^{min(m,n)} r!C_{m}^{r}C_{n}^{r}I_{m+n−2r}^{H,T} (⟨f_{m}(t_{1}, ..., t_{m−r}, .), g_{n}(s_{1}, ..., s_{n−r}, .)⟩_{T^{r},H}) .  (3.12)

(c) Let f_{m} ∈ L_{s,H}^{2}(T^{m}), g_{n} ∈ L_{s,H}^{2}(T^{n}) be such that

f_{m}(t_{1}, ..., t_{m−1}, .), g_{n}(s_{1}, ..., s_{n−1}, .) ∈ L_{H}^{2}(T) for a.e. t_{1}, ..., t_{m−1}, s_{1}, ..., s_{n−1},

t_{1}, ..., t_{m−1} → |f_{m}(t_{1}, ..., t_{m−1}, .)|_{H,T} ∈ L_{H}^{2}(T^{m−1}),
s_{1}, ..., s_{n−1} → |g_{n}(s_{1}, ..., s_{n−1}, .)|_{H,T} ∈ L_{H}^{2}(T^{n−1}).

Then the multiple fractional integrals I_{m}^{H,T} (f_{m}), I_{n}^{H,T} (g_{n}) are independent if and only if

⟨f_{m}(t_{1}, ..., t_{m−1}, .), g_{n}(s_{1}, ..., s_{n−1}, .)⟩_{H,T} = 0  (3.13)

for a.a. t_{1}, ..., t_{m−1,}s_{1}, ..., s_{n−1}. Proof (a) It follows from the chaos decomposition of the exponential in the Brownian case.

(b) It is a consequence of the product formula for the Brownian motion (see [18] or [22]),

I_{m}^{\frac{1}{2} ,T} (f′_{m})I_{n}^{\frac{1}{2} ,T} (g′_{n}) = ∑_{r=0}^{min(m,n)} r!C_{m}^{r}C_{n}^{r}I_{m+n−2r}^{H,T} (f′_{m} ⊗_{r} g′_{n})) ,

where

(f′_{m} ⊗_{r} g′_{n})(t_{1}, ..., t_{m−r}, s_{1}, ..., s_{n−r}) =
∫_{T^{r}} f′_{m}(t_{1}, ..., t_{m−r}, u_{1}, ..., u_{r})g′_{n}(s_{1}, ..., s_{n−r}, u_{1}, ..., u_{r})du_{1}...du_{r},

and of the equality

[(Γ_{H,T}^{(m)} f_{m}) ⊗_{r} (Γ_{H,T}^{(n)} g_{n})] (t_{1}, ..., t_{m−r}, s_{1}, ..., s_{n−r}) =
Γ_{H,T}^{(m+n−2r)}(⟨f_{m}(t_{1}, ..., t_{m−r}, .), g_{n}(s_{1}, ..., s_{n−r}, .)⟩_{T^{r},H}).  (3.14)

(c) From [15] or [24] the multiple Wiener-Itô integrals I_{m}^{\frac{1}{2} ,T} (Γ_{H,T}^{(m)} f_{m}), I_{n}^{\frac{1}{2} ,T} (Γ_{H,T}^{(n)} g_{n}) are independent if and only if

(Γ_{H,T}^{(m)} f_{m}) ⊗_{1} (Γ_{H,T}^{(n)} g_{n}) (t_{1}, ..., t_{m−1}, s_{1}, ..., s_{n−1}) = 0

for a.a. t_{1}, ..., t_{m−1}, s_{1}, ..., s_{n−1}. Now we conclude by (3.14) and Proposition 2.1-(b) (one to one part). ■

We denote

L_{H}^{2} (Ω, F, P) = ⊕_{p=0}^{∞}I_{p}^{H,T} (L_{s,H}^{2}(T^{p})),  (3.15)

i.e., F ∈ L_{H}^{2} (Ω, F, P) if F has an orthogonal fractional chaos decomposition of the form

F = E(F) + ∑_{p=1}^{∞} I_{p}^{H,T} (f_{p}),  (3.16)

where f_{p} ∈ L_{s,H}^{2}(T^{p}) and

∑_{p=1}^{∞} p! |f_{p}|_{H,T^{p}}^{2} < ∞.  (3.17)

Proposition 3.6. The chaos decomposition (3.16) is unique and the subspace L_{H}^{2} (Ω, F, P) is total in L^{2} (Ω, F, P) . Moreover, for every p ≥ 1, the fractional Wiener chaos of order p, I_{p}^{H,T} (L_{s,H}^{2}(T^{p})) is not closed. In particular L_{H}^{2} (Ω, F, P) is not closed and it is strictly included in L^{2} (Ω, F, P) .

Proof. The uniqueness of the orthogonal chaos decomposition is a consequence of the Lemma 3.1-(b) and the norm property of |.|_{H,T^{p}} . By Theorem 3.1 of [8] the linear span of the exponentials is dense in L^{2} (Ω, F, P). Then we apply Proposition 3.5-(a). The Wiener chaos I_{p}^{H,T} (L_{s,H}^{2}(T^{p})) is not closed since L_{s,H}^{2}(T^{p}) is not complete (Lemma 3.1). Next if f ∈ L^{2}(T) and f ∉ Γ_{H,T}^{(1)} (L_{H}^{2}(T)) then I_{1}^{H,T} (f) has no fractional chaos decomposition.■

Theorem 3.7. Let Ĩ_{p}^{H,T} (f) be the multiple fractional integral as defined in [4] and [8]. If f ∈ L_{s,H}^{2}(T^{p}) then

I_{p}^{H,T} (f) = Ĩ_{p}^{H,T} (f).  (3.18)

Proof. Enough to prove (3.18) for f = g^{⊗p}, g ∈ L_{H}^{2}(T). First note that we have

I_{1}^{H,T} (g) = Ĩ_{1}^{H,T} (g), g ∈ L_{H}^{2}(T),  (3.19)

since

I_{1}^{H,T} (1_{[0,t]}) = Ĩ_{1}^{H,T} (1_{[0,t]}) = B_{t}^{H},

and the elementary functions are dense in L_{H}^{2}(T) (see [19], [23]). Assume that T = R. From [8]-Theorem 6.7 we have that

Ĩ_{p}^{H,R}(f) = p! |g|_{H,R}^{p} H_{p} (\frac{I_{1}^{H,R}(g)}{|g|_{H,R}}) ,  (3.20)

where H_{p} is the Hermite polynomial of degree p. Also, it is well known that

Ip^{\frac{1}{2}} (Γ_{H,T}^{(p)} (f)) = p! ∥Γ_{H,T}^{(1)} (g)∥_{L^{2}(R)}^{p} H_{p} (\frac{I_{1}^{\frac{1}{2} ,T} (Γ_{H,T}^{(1)} (g))}{∥Γ_{H,T}^{(1)} (g)∥_{L^{2}(R)}} ) ,

and we conclude by (3.19) and Lemma 3.1. Assume now that T = [0, t_{0}] . We prove the assertion by induction over p. If the assertion is true up to p − 1 then for p it follows from the product formula (3.12) and Lemma 6.2 of [5].■

Remark 3.8. Instead of L_{s,H}^{2}(T^{p}) we could consider the following larger classes of integrands.

We denote by Λ_{H}^{2}([0, t_{0}]^{n}) the class of all measurable functions f : [0, t_{0}]^{n} → R such that

I_{t_{0}−}^{H− \frac{1}{2} ,n} ((x_{1}....x_{n})^{H− \frac{1}{2}} |f(x_{1}, ..., x_{n})|) < ∞ for a.e. x_{i} ∈ [0, t_{0}] ,

and

(Γ_{H,[0,t_{0}]}^{(n)}f) (t_{1}, ..., t_{n}) = d_{H}^{n} (t_{1}...t_{n})^{\frac{1}{2} −H} ×
× (I_{t_{0}−}^{H− \frac{1}{2} ,n}) ((x_{1}...x_{n})^{H− \frac{1}{2}} f(x_{1}, ..., x_{n})) (t_{1}, ..., t_{n}) ∈ L^{2}([0, t_{0}]^{n}),  (3.21)

with the norm

∥f∥_{H,[0,t_{0}]^{n}} = ∥Γ_{H,[0,t_{0}]}^{(n)}f∥_{L^{2}([0,t_{0}]^{n})} .  (3.22)

Also we denote by Λ_{H}^{2}(R^{n}) the class of all measurable functions f : R^{n} → R such that

I_{−}^{H− \frac{1}{2} ,n} (|f|) (x_{1}, ..., x_{n}) < ∞ for a.e. x_{i} ∈ R,

and

(Γ_{H,R}^{(n)}f) (t_{1}, ..., t_{n}) = d_{H}^{n} (I_{−}^{H− \frac{1}{2} ,n}f) (t_{1}, ..., t_{n}) ∈ L^{2}(R^{n}),  (3.23)

with the norm

∥f∥_{H,R^{n}} = ∥Γ_{H,R}^{(n)}f∥_{L^{2}(R^{n})} .  (3.24)

Again (Λ_{H}^{2}(T^{n}), ∥.∥_{H,T^{n}}) is an incomplete pre-Hilbert space (see [19] and [23]). 3.2. Case 0 < H < \frac{1}{2} We define

L_{H}^{2}([0, t_{0}]^{n}) = I_{t_{0}−}^{\frac{1}{2} −H,n}(L^{2}([0, t_{0}]^{n})),  (3.25)
(Γ_{H,[0,t_{0}]}^{(n)}f) (t_{1}, ..., t_{n}) = d_{H}^{n} (t_{1}...t_{n})^{\frac{1}{2} −H} ×
× (D_{t_{0}−}^{\frac{1}{2} −H,n}) ((x_{1}...x_{n})^{H− \frac{1}{2}} f(x_{1}, ..., x_{n})) (t_{1}, ..., t_{n}), f ∈ L_{H}^{2}([0, t_{0}]^{n}),  (3.26)
L_{H}^{2}(R^{n}) = I_{−}^{\frac{1}{2} −H,n}(L^{2}(R^{n})),  (3.27)
(Γ_{H,R}^{(n)}f) (t_{1}, ..., t_{n}) = d_{H}^{n} (D_{−}^{\frac{1}{2} −H,n}f) (t_{1}, ..., t_{n}), f ∈ L_{H}^{2}(R^{n}).  (3.28)

Proposition 3.9 (a) The space L_{H}^{2}(T^{n}) is a Hilbert space with respect to the inner product

⟨f, g⟩_{H,T^{n}} = ⟨Γ_{H,T}^{(n)} f, Γ_{H,T}^{(n)} g⟩_{L^{2}(T^{n})} .  (3.29)

(b) The map Γ_{H,T}^{(n)} : L_{H}^{2}(T^{n}) → L^{2}(T^{n}) is one to one and onto.

Proof. (a) The proof follows along the same arguments as in [19] , [23].

(b) If f ∈ L^{2}([0, t_{0}]^{n}) (resp. f ∈ L^{2}(R^{n})) then the function

φ(t_{1}, ..., t_{n}) = d_{H}^{n} (t_{1}...t_{n})^{\frac{1}{2} −H} (I_{t_{0}−}^{\frac{1}{2} −H,n}) ((x_{1}...x_{n})^{H− \frac{1}{2}} f) (t_{1}, ..., t_{n}) ∈ L_{H}^{2}(T^{n}),

resp.

φ = d_{H}^{n} (I_{−}^{\frac{1}{2} −H,n}) (f) ∈ L_{H}^{2}(R^{n}),

and is the unique solution of (Γ_{H,T}^{(n)} ) (φ) = f. ■

If f ∈ L_{H}^{2}(T^{p}) then we define the multiple fractional integral of order p of f with respect to B^{H} as in (3.10).

Remark 3.10. The assertions of Proposition 3.5 still holds in the case 0 < H < \frac{1}{2}. We define L_{H}^{2} (Ω, F, P) as in (3.15). Then we have the following result (which is different from the case \frac{1}{2} < H < 1 ).

Proposition 3.11. For every p ≥ 1, the fractional Wiener chaos of order p, I_{p}^{H,T} (L_{s,H}^{2}(T^{p}))

is closed and agrees with the Wiener chaos I_{p}^{\frac{1}{2} ,T} (L_{s}^{2}(T^{p})) . We have the equality

L_{H}^{2} (Ω, F, P) = L^{2} (Ω, F, P) .  (3.30)

Proof. It follows from the onto property of Γ_{H,T}^{(n)} and the classical Itô chaos decomposition. ■

4. Chaos form of the fractional anticipating integral

Throughout this section we assume that \frac{1}{2} < H < 1. The case H < \frac{1}{2} will be considered elsewhere. Definition 4.1. We define D_{H}^{ch} as the class of all measurable processes {u (t)}_{t∈T} such that:

(1) u ∈ L^{2} (Ω, F, P, L_{H}^{2}(T)) ,

(2) The process u has the chaos decomposition in L^{2} (Ω, F, P, L_{H}^{2}(T)) ,

u(t) = ∑_{p=0}^{∞} I_{p}^{H,T} (f_{p}(., t)), f_{p}(., t) ∈ L_{s,H}^{2}(T^{p}), f_{p} ∈ L_{H}^{2}(T^{p+1}), ∀p ≥ 1,  (4.1)

and the series

δ_{H}^{ch}(u) := ∑_{p=0}^{∞} I_{p+1}^{H,T} (sym (f_{p})),  (4.2)

converges in L^{2} (Ω, F, P). Note that from (3.6) sym(f_{p}) ∈ L_{s,H}^{2}(T^{p+1}). Here sym(f_{p}) is the symmetrization of f_{p}(t_{1}, ..., t_{p}, t) with respect to all its p + 1-variables, i.e.,

sym (f_{p}) (t_{1}, ..., t_{p}, t) = \frac{1}{p + 1} \{f_{p}(t_{1}, ..., t_{p}, t) + ∑_{j=1}^{p} f_{p}(t_{1}, .., t_{j−1}, t, t_{j+1}, .., t_{p}, t_{j})\} .

We say that the measurable process u ∈ L^{2} (Ω, F, P, L_{H}^{2}(T)) is integrable with respect to B^{H}(u ∈Dom(δ_{H}^{ch}) for short) if there exists u_{n} ∈ D_{H}^{ch}, such that

E (|u_{n} − u|_{H,T}^{2} ) → 0, E (|δ_{H}^{ch}(u_{n}) − v|^{2}) → 0.

We define in this case δ_{H}^{ch}(u) = v and we call δ_{H}^{ch}(u) the fractional stochastic integral of u with respect to B^{H} and we denote it sometimes also by ∫_{T} u(t)dB_{t}^{H}. If u1_{[a,b]} ∈Dom(δ_{H}^{ch}) then we define

∫_{a}^{b} u(t)dB_{t}^{H} = δ_{H}^{ch}(u1_{[a,b]}).

An example. As consequence of the product formula (3.12) it follows the equality

∫_{a}^{b} B_{t}^{H}dB_{t}^{H} = \frac{(B_{b}^{H} − B_{a}^{H})^{2} − (b − a)^{2H}}{2} .  (4.3)

Theorem 4.2. Let u be a measurable process, u ∈ L^{2} (Ω, F, P, L_{H}^{2}(T)) . Assume that Γ_{H,T}^{(1)} u ∈

L1,2_{\frac{1}{2}},i.e.,

Γ_{H,T}^{(1)} u_{t} = ∑_{p} Ip^{\frac{1}{2}} (f_{p}(., t)), f_{p}(., t) ∈ L_{s}^{2} ([0, t_{0}]^{p}) , f_{p} ∈ L^{2} ([0, t_{0}]^{p+1}) ,  (4.4)

∑_{p} (p + 1)! ∥f_{p}∥_{L^{2}([0,t_{0}]^{p+1})}^{2} < ∞.  (4.5)

Then

δ_{H}^{ch}(u) = δ_{\frac{1}{2}} (Γ_{H,T}^{(1)} u).  (4.6)

In particular if u ∈ L1,2_{\frac{1}{2}} then u ∈Dom(δ_{H}^{ch}) and (4.6) holds. Proof. Let Δ_{i} = [s_{i}, s_{i+1}) , i = 0, ..., k, be a partition of [0, t_{0}] . Define

f_{p}^{Δ}(t_{1}, ..., t_{p_{+1}}) = ∑_{i_{1},...,i_{p+1}=1}^{k} [\frac{1}{|Δ_{i_{1}}| ... |Δ_{i_{p+1}} |} ∫_{Δ_{i_{1}}} .. ∫_{Δ_{i_{p+1}}} f_{p}(u_{1}, .., u_{p+1})du_{1}..du_{p+1}] ×

×1_{Δ_{i_{1}}}(t_{1})...1_{Δ_{i_{p+1}}}(t_{p+1}), t_{i}, t ∈ [0, t_{0}] .

We have

∥f_{p}^{Δ}∥_{L^{2}(T^{p+1})} ≤ ∥f_{p}∥_{L^{2}(T^{p+1})} , f_{p}^{Δ} ⟶_{∥Δ∥→0}^{L^{2}(T^{p+1})} f_{p}.  (4.7)

From [21]-page 33 there is φ_{i} ∈ L_{H}^{2} ([0, t_{0}]) such that Γ_{H,T}^{(1)} φ_{i} = 1_{Δ_{i}}. Denote

f̄_{p}^{Δ}(t_{1}, ..., t_{p+1}) = ∑_{i_{1},...,i_{p+1}=1}^{k} [\frac{1}{|Δ_{i_{1}}| ... |Δ_{i_{p+1}} |} ∫_{Δ_{i_{1}}} .. ∫_{Δ_{i_{p+1}}} f_{p}(u_{1}, .., u_{p+1})du_{1}..du_{p+1}] ×

×φ_{i_{1}}(t_{1})...φ_{i_{p+1}}(t_{p+1}), t_{i}, t ∈ [0, t_{0}] ,  (4.8)
u_{t}^{Δ} = ∑_{p} I_{p}^{H,T} (f̄_{p}^{Δ}(., t)), t ∈ [0, t_{0}] .  (4.9)

Clearly we have

Γ_{H,T}^{(p+1)}f̄_{p}^{Δ} = f_{p}^{Δ}.  (4.10)

By a density argument it follows the equality

Γ_{H,T}^{(1)} (I_{p}^{H,T} (f̄_{p}^{Δ}(., t))) = Γ_{H,T}^{(1)} (I_{p}^{\frac{1}{2} ,T} (Γ_{H,T}^{(p)} f̄_{p}^{Δ}(., t))) =  (4.11)
I_{p}^{\frac{1}{2} ,T} (Γ_{H,T}^{(p+1)} \binom{}{f̄_{p}^{Δ}}) = I_{p}^{\frac{1}{2} ,T} (f_{p}^{Δ}) in L^{2} (T × Ω)  (4.12)

and this implies

Γ_{H,T}^{(1)} u_{t}^{Δ} = ∑_{p} I_{p}^{\frac{1}{2} ,T} (f_{p}^{Δ}(., t)), in L^{2} (T × Ω) .  (4.13)

From (4.7), (4.13) it follows that

E (|u^{Δ} − u|_{H,T}^{2} ) = E (∥Γ_{H,T}^{(1)} u^{Δ} − Γ_{H,T}^{(1)} u∥_{L^{2}(T)}^{2}) =

∑_{p} p! ∫_{0}^{t_{0}} ∥f_{p}^{Δ}(., t) − f_{p}(., t)∥_{L^{2}(T^{p})}^{2} dt = ∑_{p} p! ∥f_{p}^{Δ} − f_{p}∥_{L^{2}(T^{p+1})}^{2} ⟶_{∥Δ∥→0} 0.

Next we have by (4.7)

∑_{p} (p + 1)! ∥sym (f_{p}^{Δ})∥_{L^{2}(T^{p+1})}^{2} ≤

∑_{p} (p + 1)! ∥f_{p}^{Δ}∥_{L^{2}(T^{p+1})}^{2} ≤ ∑_{p} (p + 1)! ∥f_{p}∥_{L^{2}(T^{p+1})}^{2} < ∞,

and hence u^{Δ} ∈ D_{H}^{ch} and by (4.10),(4.13)

δ_{H}^{ch}(u^{Δ}) = ∑_{p} I_{p+1}^{H,T} (sym(f̄_{p}^{Δ})) =  (4.14)
∑_{p} I_{p+1}^{\frac{1}{2} ,T} (sym (Γ_{H,T}^{(p+1)}f̄_{p}^{Δ})) = ∑_{p} I_{p+1}^{\frac{1}{2} ,T} (sym (f_{p}^{Δ})) = δ_{\frac{1}{2}} (Γ_{H,T}^{(1)} u^{Δ}) .  (4.15)

Finally by (4.5) and (4.7),

E (∥δ_{H}^{ch}(u^{Δ}) − δ_{\frac{1}{2}} (Γ_{H,T}^{(1)} u)∥^{2}) =

∑_{p} (p + 1)! ∥sym (f_{p}^{Δ} − f_{p})∥_{L^{2}(T^{p+1})}^{2} ≤

∑_{p} (p + 1)! ∥f_{p}^{Δ} − f_{p})∥_{L^{2}(T^{p+1})}^{2} ⟶_{∥Δ∥→0} 0,

and hence u ∈Dom (δ_{H}^{ch}) and δ_{H}^{ch}(u) = δ_{\frac{1}{2}} (Γ_{H,T}^{(1)} u) . Next if u ∈ L1,2_{\frac{1}{2}} then it is easily seen that Γ_{H,T}^{(1)} u ∈ L1,2_{\frac{1}{2}} and (4.6) holds. ■

Remark 4.3. The stochastic integral from the right-hand side of (4.6) is introduced in [1] for Gaussian processes of the form

X_{t} = ∫_{0}^{t} K(t, s)dW_{s},

where W is a Brownian motion and K a square integrable kernel. Taking K = K_{H} one obtains the particular case of the fractional Brownian motion.

Theorem 4.4. Let δ_{H}^{1} be the fractional stochastic integrals introduced in [6], [7] for the fractional Wiener space. If u is as in Theorem 4.2 then u ∈Dom(δ_{H}^{1}) and

δ_{H}^{ch}(u) = δ_{H}^{1}(u) = δ_{\frac{1}{2}} (Γ_{H,T}^{(1)} u) .  (4.16)

We need the following lemma. Lemma 4.5. Let ∇ be the Gross-Sobolev derivative as defined in [7] for the fractional Wiener space. If f_{p} ∈ L^{2} (T^{p}) , T = [0, t_{0}] , p ≥ 1, then

∇I_{p}^{H,T} (f_{p}) = p (K_{H,T} ∘ Γ_{H,T}^{(1)} ) (I_{p−1}^{H,T} (f_{p}(t_{1}, ..., t_{p−1}, .))) .  (4.17)

Proof. We prove first (4.17) for f_{p} = f^{⊗p}, f ∈ L^{2} (T), by induction over p. If p = 1 and f_{1} = 1_{[0,t]} then

I_{1}^{H,T} (1_{[0,t]}) = B_{t}^{H}, ∇B_{t}^{H} = R(t, .),

and by Lemma 3.1 of [7],

(K_{H,T} ∘ Γ_{H,T}^{(1)} ) (1_{[0,t]}) = K_{H,T} (K_{H,T} (t, .)) = R(t, .).

By a density argument, the closedness of ∇ and the continuity of K_{H,T} and Γ_{H,T}^{(1)} we obtain the equality for f_{1} ∈ L^{2} (T) . Assume now that (4.17) is satisfied up to p − 1 and prove it for p. From the product formula (3.12) we have

I_{p}^{H,T} (f^{⊗p}) = I_{p−1}^{H,T} (f^{⊗(p−1)})I_{1}^{H,T} (f) − (p − 1) |f|_{H,T}^{2} I_{p−2}^{H,T} (f^{⊗(p−2)}),

and then

∇I_{p}^{H,T} (f^{⊗p}) = I_{1}^{H,T} (f)∇I_{p−1}^{H,T} (f^{⊗(p−1)}) + I_{p−1}^{H,T} (f^{⊗(p−1)})∇I_{1}^{H,T} (f)−

(p − 1) |f|_{H,T}^{2} ∇I_{p−2}^{H,T} (f^{⊗(p−2)}) =
(p − 1) I_{1}^{H,T} (f) (K_{H,T} ∘ Γ_{H,T}^{(1)} ) (f)I_{p−2}^{H,T} (f^{⊗(p−2)})+
I_{p−1}^{H,T} (f^{⊗(p−1)}) (K_{H,T} ∘ Γ_{H,T}^{(1)} ) (f)−
(p − 1) (p − 2) |f|_{H,T}^{2} (K_{H,T} ∘ Γ_{H,T}^{(1)} ) (f)I_{p−3}^{H,T} (f^{⊗(p−3)}) =
(p − 1) (K_{H,T} ∘ Γ_{H,T}^{(1)} ) (f) [I_{1}^{H,T} (f)I_{p−2}^{H,T} (f^{⊗(p−2)}) − (p − 2) |f|_{H,T}^{2} I_{p−3}^{H,T} (f^{⊗(p−3)})] +
I_{p−1}^{H,T} (f^{⊗(p−1)}) (K_{H,T} ∘ Γ_{H,T}^{(1)} ) (f) = (p − 1) (K_{H,T} ∘ Γ_{H,T}^{(1)} ) (f)I_{p−1}^{H,T} (f^{⊗(p−1)})+
I_{p−1}^{H,T} (f^{⊗(p−1)}) (K_{H,T} ∘ Γ_{H,T}^{(1)} ) (f) = p (K_{H,T} ∘ Γ_{H,T}^{(1)} ) (f)I_{p−1}^{H,T} (f^{⊗(p−1)}) =
p (K_{H,T} ∘ Γ_{H,T}^{(1)} ) (I_{p−1}^{H,T} (f^{⊗p}(t_{1}, ..., t_{p−1}, .)). ■

Now by polarization we obtain (4.17) for f_{p} ∈ H_{1} =Sp{f_{1} ⊗ .. ⊗ f_{p}, f_{i} ∈ L^{2} (T)} . Next let f_{p} ∈ L^{2} (T^{p}) and let f_{p}^{k} ∈ H_{1} be such that f_{p}^{k} ⟶_{k→∞}^{L^{2}(T^{p})} f_{p}. In particular

f_{p}^{k} ⟶_{k→∞}^{L_{H}^{2}(T^{p})} f_{p} and thus I_{p}^{H,T} (f_{p}^{k}) ⟶_{k→∞}^{L^{2}(Ω)} I_{p}^{H,T} (f_{p}). On the other hand Remark 2.3 and Lemma 3.1 yield

∥∇I_{p}^{H,T} (f_{p}^{k}) − p (K_{H,T} ∘ Γ_{H,T}^{(1)} ) (I_{p−1}^{H,T} (f_{p}(t_{1}, ..., t_{p−1}, .)))∥_{L^{2}(Ω×H_{H,T})}^{2} =
p^{2} ∥(K_{H,T} ∘ Γ_{H,T}^{(1)} ) (I_{p−1}^{H,T} ((f_{p}^{k} − f_{p}) (t_{1}, ..., t_{p−1}, .)))∥_{L^{2}(Ω×H_{H,T})}^{2} =
p^{2}E [∥Γ_{H,T}^{(1)} (I_{p−1}^{H,T} ((f_{p}^{k} − f_{p}) (t_{1}, ..., t_{p−1}, .)))∥_{L^{2}(T)}^{2}] =
p^{2}E [|I_{p−1}^{\frac{1}{2} ,T}(Γ_{H,T}^{(p+1)} (f_{p}^{k} − f_{p}) (t_{1}, ..., t_{p−1}, .))|_{L^{2}(T)}^{2}] ≤
pp! |f_{p}^{k} − f_{p}|_{L^{2}(T)}^{2} → 0.

Finally from the closedness of ∇ we get (4.17) for f_{p}. ■

Proof of Theorem 4.4. Assume that T = [0, t_{0}] and Γ_{H,T}^{(1)} u ∈ L1,2_{\frac{1}{2}}. By using an approximating argument and Lemma 3.1-(a), we need to check the equality for

u(t) = (I_{p}^{H,T} (f_{p})) g(t), f_{p} ∈ L_{s}^{2}(T^{p}), g ∈ L^{2}(T).

Let δ_{H} be the adjoint of ∇. Then, by using Lemma 3.1, Lemma 4.5 and the product formula,

δ_{1}^{H}(u) := δ_{H} ((K_{H,T} ∘ Γ_{H,T}^{(1)} ) (u)) = δ_{H} ((I_{p}^{H,T} f_{p}) (K_{H,T} ∘ Γ_{H,T}^{(1)} )(g)) =
(I_{p}^{H,T} f_{p}) δ_{H} ((K_{H,T} ∘ Γ_{H,T}^{(1)} )(g)) − ⟨∇ (I_{p}^{H,T} f_{p}) , (K_{H,T} ∘ Γ_{H,T}^{(1)} )(g)⟩_{H_{H,T}} =
(I_{p}^{H,T} f_{p}) δ_{H} ((K_{H,T} ∘ Γ_{H,T}^{(1)} )(g)) −
p ⟨(K_{H,T} ∘ Γ_{H,T}^{(1)} ) (I_{p−1}^{H,T} f_{p}(t_{1}, ..., t_{p−1}, .)) , (K_{H,T} ∘ Γ_{H,T}^{(1)} )(g)⟩_{H_{H,T}} =
I_{p}^{H,T} (f_{p}) I_{1}^{H,T} (g) − p ⟨(Γ_{H,T}^{(1)} ) (I_{p−1}^{H,T} f_{p}(t_{1}, ..., t_{p−1}, .)) , (Γ_{H,T}^{(1)} )(g)⟩_{L^{2}(T)} =
I_{p}^{H,T} (f_{p}) I_{1}^{H,T} (g) − p ⟨I_{p−1}^{H,T} f_{p}(t_{1}, ..., t_{p−1}, .), g⟩_{H,T} = I_{p}^{H,T} (f_{p}) I_{1}^{H,T} (g) −
pI_{p−1}^{H,T} (⟨f_{p}(t_{1}, ..., t_{p−1}, .), g⟩_{H,T} ) = I_{p+1}^{H,T} (sym(f_{p} ⊗ g)) = δ_{H}^{ch}(u).

■

For a random variable F ∈ L^{2} (Ω, F, P) we denote by D_{s}^{H}F the H-derivative as defined in [8]. Recall that a process u is H-differentiable on [0, t_{0}] if for each t ∈ [0, t_{0}] , u_{t} has H-derivative D_{s}^{H}u_{t} which is jointly measurable. We denote by L (0, t_{0}) the class of all processes {u_{t}}_{t∈[0,t_{0}]} such that:

(i_{1}) u is H-differentiable on [0, t_{0}] and E (|u|_{H,[0,t_{0}]}^{2}) < ∞.

(i_{2}) The trace {D_{s}^{H}u_{t}}_{0≤s,t≤t_{0}} exists and E (∫_{0}^{t_{0}} |D_{s}^{H}u_{s}|^{2} ds) < ∞. (i_{3}) For each sequence of partitions π : 0 = s_{0} < s_{1} < ... < s_{n} = t_{0} with |π| → 0 we have

E (|u^{π} − u|2_{H,[0,t_{0}]}) → 0,

∑_{i=0}^{n−1} E (|∫_{t_{i}}^{t_{i+1}} |D_{s}^{H}u_{s_{i}}^{π} − D_{s}^{H}u_{s}| ds|^{2}) → 0,

where u_{s}^{π} = u_{s_{i}}^{π} if s ∈ [s_{i}, s_{i+1}) . Remark 4.6. In [8] an anticipating stochastic integral ∫_{0}^{t_{0}} u_{t}δ_{2}B_{t}^{H} is defined for u ∈ L (0, t_{0}) as the limit of Riemann sums defined in terms of Wick product. Under rather restrictive conditions, it can be proved that the stochastic integral ∫_{0}^{t_{0}} u_{t}δ_{2}B_{t}^{H} is related with δ_{H}^{ch}. In the next proposition we show this in a simple case.

Proposition 4.7. Let {u_{t}}_{t∈R} = {I_{p}^{H,R}(f^{⊗p})g(t)}_{t∈R} , f, g ∈ L_{H}^{2}(R), be a process such that

|g^{π} − g|_{H,[0,t_{0}]}^{2} → 0,  (4.18)

∫_{0}^{t_{0}} |∫_{R} f(v)φ(v, s)dv|^{2} ds < ∞,  (4.19)

∫_{0}^{t_{0}} |∫_{R} f(v)φ(v, s)dv|^{2} |g(s)|^{2} ds < ∞.  (4.20)

Then

∫_{0}^{t_{0}} u_{t}dB_{t}^{H} = ∫_{0}^{t_{0}} u_{t}δ_{2}B_{t}^{H}.  (4.21)

Proof. First we show that

D_{s}^{H} (I_{p}^{H,R}(f^{⊗p})) = p [∫_{R} f(v)φ(v, s)dv] I_{p−1}^{H,R}(f^{⊗(p−1)}).  (4.22)

It is easy to check that

D_{s}^{H} (I_{1}^{H,R}(f)) = ∫_{R} f(v)φ(v, s)dv.  (4.23)

From (3.18),(3.20) and the chain rule we obtain

D_{s}^{H} (I_{p}^{H,R}(f^{⊗p})) = p! |f|_{H,R}^{p} D_{s}^{H} (H_{p} (\frac{I_{1}^{H,R}(f)}{|f|_{H,R}})) =

p! |f|_{H,R}^{p} H′_{p} (\frac{I_{1}^{H,R}(f)}{|f|_{H,R}}) |f|_{H,R}^{−1} D_{s}^{H} (I_{1}^{H,R}(f) =

p! |f|_{R,H}^{p−1} H_{p−1} (\frac{I_{1}^{H,R}(f)}{|f|_{R,H}}) [∫_{R} f(v)φ(v, s)dv] =

pI_{p−1}^{H,R}(f^{⊗(p−1)}) [∫_{R} f(v)φ(v, s)dv] .

Now, using (4.18)-(4.20) and (4.22) it is easy to see that (i_{1})-(i_{3}) are satisfied for u. Therefore u ∈ L(0, t_{0}) and we have in L^{2} (Ω) ,

∫_{0}^{t_{0}} u_{t}δ_{2}B_{t}^{H} = lim_{|π|→0} ∑_{i} u_{t_{i}} ⋄ (B_{t_{i+1}}^{H} − B_{t_{i}}^{H}) =

lim_{|π|→0} ∑_{i} {u_{t_{i}} (B_{t_{i+1}}^{H} − B_{t_{i}}^{H}) − ∫_{t_{i}}^{t_{i+1}} D_{s}^{H}u_{s}ds} ,

(here ⋄ is the Wick product). By using the product formula and (4.22) we deduce that

∑_{i} {u_{t_{i}} (B_{t_{i+1}}^{H} − B_{t_{i}}^{H}) − ∫_{t_{i}}^{t_{i+1}} D_{s}^{H}u_{s}ds} = I_{p+1}^{H,R} (sym((f^{⊗p} ⊗ g^{π}1_{[0,t_{0}]}))) .

Finally we have

E (|I_{p+1}^{H,R} (sym((f^{⊗p} ⊗ g^{π}1_{[0,t_{0}]}) − sym((f^{⊗p} ⊗ g1_{[0,t_{0}]}))|^{2}) =

(p + 1)! |sym (f^{⊗p} ⊗ g^{π}1_{[0,t_{0}]}) − sym (f^{⊗p} ⊗ g1_{[0,t_{0}]})|_{H,R^{p+1}}^{2} ≤
(p + 1)! |f^{⊗p} ⊗ g^{π}1_{[0,t_{0}]} − (f^{⊗p} ⊗ g1_{[0,t_{0}]})|_{H,R^{p+1}}^{2} ≤
(p + 1)! |f|_{H,R^{p}}^{p} |g − g^{π}|_{H,[0,t_{0}]}^{2} → 0. ■

Application. We now apply the multiple fractional integrals to obtain the explicit solution of an affine fractional equations. We consider the fractional affine equation

X_{t} = η + ∫_{t_{0}}^{t} [a_{0}(s) + a(s)X_{s}] ds + ∫_{t_{0}}^{t} [b_{0}(s) + b(s)X_{s}] dB_{s}^{H}, t ∈ T,  (4.24)

where T = [t_{0}, t_{1}] ⊂ R, η ∈ L^{2} (Ω, F, P) and a_{0}, b_{0}, a, b : T → R are measurable functions such that a, a_{0} ∈ L^{1} (T) , b, b_{0} ∈ L_{H}^{2} (T) .

Definition 4.8 A process {X_{t}}_{t∈T} is a strong solution of (4.24) if

(i) a(.)X_{.} ∈ L^{1} (T) and 1_{[0,t]}(.)b(.)X_{.} ∈Dom(δ_{H}^{ch}) for a.a. t ∈ T. (ii) For a.a. t ∈ T, the equation (4.24) is satisfied P −a.s.. We introduce the following notation: Given t_{1}, ..., t_{p} and a symmetric function of p−j variables (j < p), we denote by f_{p−j}(t̂_{i_{1}}, .., t̂_{i_{j}}) the function f_{p−j} evaluated in t’s other that t_{i_{1}}, .., t_{i_{j}}. Proposition 4.9 Assume that

η = ∑_{p=0}^{∞} I_{p}^{H,T} (η_{p}), ∑_{p=1}^{∞} p! |η_{p}|_{H,T^{p}}^{2} < ∞.  (4.25)

Define

U(t, s) = exp {∫_{s}^{t} a(r)dr}
Φ(t, s) = exp\{∫_{s}^{t} a(u)du + ∫_{s}^{t} b(u)dB_{u}^{H} − \frac{1}{2}|b1_{[s,t]}|_{H,T}^{2} \}.

Then (4.24) has a unique strong solution X ∈ D_{H}^{ch} with the chaos decomposition

X_{t} = ∑_{p=0}^{∞} I_{p}^{H,T} (f_{p}^{t}),  (4.26)

f_{0}^{t} = U(t, t_{0})η_{0} + ∫_{t_{0}}^{t} U(t, s)a_{0}(s)ds,  (4.27)
f_{1}^{t}(t_{1}) = U(t, t_{0})η_{1}(t_{1}) + 1_{[t_{0},t]}(t_{1})U(t, t_{1})b_{0}(t_{1})+
1_{[t_{0},t]}(t_{1})U(t, t_{0})b(t_{1})η_{0} + 1_{[t_{0},t]}(t_{1})b(t_{1}) ∫_{t_{0}}^{t_{1}} U(t, s)a_{0}(s)ds,  (4.28)

and for p ≥ 2,

f_{p}^{t}(t_{1}, ..., t_{p}) = U(t, t_{0})η_{p}(t_{1}, ..., t_{p}) + \frac{1}{p} ∑_{j=1}^{p} 1_{[t_{0},t]}(t_{j})U(t, t_{j})b(t_{j})f_{p−1}^{t_{j}}(t̂_{j}),  (4.29)

or in an explicit form

f_{p}^{t}(t_{1}, ..., t_{p}) = U(t, t_{0})η_{p}(t_{1}, ..., t_{p})+

U(t, t_{0}) ∑_{j=1}^{p} \frac{1}{j!}sym \{1_{[t_{0},t]}^{j}(t_{1}, ..., t_{j})b(t_{1})...b(t_{j})η_{p−j}(t̂_{1}, .., t̂_{j}\} +

sym {1_{t_{0}<t_{1}<...<t_{p}<t}U(t, t_{1})b_{0}(t_{1})b(t_{2})...b(t_{p})} +
\frac{1}{p!}1_{[t_{0},t]}^{p}(t_{1}, ..., t_{p})b(t_{1})...b(t_{p}) ∫_{t_{0}}^{t_{1}∧...∧t_{p}} U(t, s)a_{0}(s)ds.  (4.30)

In particular if η = η_{0} ∈ R the solution is

X_{t} = Φ(t, t_{0})η_{0} + ∫_{t_{0}}^{t} Φ(t, s)a_{0}(s)ds + ∫_{t_{0}}^{t} Φ(t, s)b_{0}(s)dB_{s}^{H}  (4.31)

Proof. By using Fubini’s theorem for multiple Wiener-Itô integrals (see [16]), it is easy to see that X is a strong solution if and only if it is an evolution solution, i.e.,

X_{t} = U(t, t_{0})η + ∫_{t_{0}}^{t} U(t, s)a_{0}(s)ds+

∫_{t_{0}}^{t} U(t, s) [b_{0}(s) + b(s)X_{s}] dB_{s}^{H}.  (4.32)

Now,by replacing (4.26) in (4.32) and then identifying the kernels we get (4.27)-(4.29). ■ Remark 4.10. (i) If a_{0} = b_{0} = 0 then the solution of (4.24) can be expressed in terms of Wick product in the form

X_{t} = Z_{t} ⋄ η,

where Z_{t} is the solution of (4.24) with the initial condition η = 1(see Buckdahn and Nualart [3] for the case of Brownian motion). (ii) In Proposition 4.9 we can assume η is analytic.

5. Multiple Stratonovich Fractional Integrals

Definition 5.1. If f_{p} ∈ L_{s,H}^{2}(T^{p}) then we define the multiple Stratonovich fractional integral of f_{p} with respect to the fBm by

J_{p}^{H,T} (f_{p}) = J_{p}^{\frac{1}{2} ,T} (Γ_{H,T}^{(p)} f_{p}) ,  (5.1)

where J_{p}^{\frac{1}{2} ,T} (.) is the usual multiple Stratonovich integral corresponding to the Brownian motion {W_{t}^{T} }_{t∈T}.

Following Johnson and Kallianpur [14] we recall the concept of trace. Definition 5.2. Let f_{p} ∈ L_{s}^{2}(T^{p}) and let 0 ≤ k ≤ [\frac{p}{2}], where [\frac{p}{2}] denotes the greatest integer in \frac{p}{2}. We define Tr0_{\frac{1}{2}} f_{p} = f_{p} and for 1 ≤ k ≤ [\frac{p}{2}] we say that the trace of order k of f_{p} exists (in notation Trk_{\frac{1}{2}} f_{p}) and equals h ∈ L_{s}^{2}(T^{p−2k}) if for every complete orthonormal sequence (e_{j}) of L^{2}(T^{k}),

∑_{j=1}^{∞} ∫_{T^{2k}} f_{p}(s_{1}, ..., s_{k}; s_{k+1}, .., s_{2k}, .)e_{j}(s_{1}, .., s_{k})e_{j}(s_{k+1}, .., s_{2k})ds_{1.}..ds_{2k} = h(.),

where the series on the left-hand side converges to h in L^{2}(T^{p−2k}). Definition 5.3. Let f_{p} ∈ L_{s,H}^{2}( T^{p}). Define Tr_{H}^{0}f_{p} = Γ_{H,T}^{(p)} f_{p} and for 1 ≤ k ≤ [\frac{p}{2}] we say

that Tr_{H}^{k}f_{p} exists and equals h ∈ L_{s,H}^{2}(T^{p−2k}) if Trk_{\frac{1}{2}} (Γ_{H,T}^{(p)} f_{p}) exists and

Trk_{\frac{1}{2}} (Γ_{H,T}^{(p)} f_{p}) = Γ_{H,T}^{(p−2k)}h.

Remark 5.4 (1) If f_{p} = g^{⊗p}, g ∈ L_{H}^{2}(T) then

Tr_{H}^{k}f_{p}(t_{1}, ..., t_{p−2k}) = |g|_{H,T}^{2k} g^{⊗(p−2k)}.  (5.2)

(2) In [5]) it is shown that a sufficient condition (in the case of T = [0, t_{0}] and H > \frac{1}{2}) for the existence of \{Tr_{H}^{k}f_{p}\}_{1≤k≤[\frac{p}{2}]} is f_{p} ∈ L^{2}(T^{p}, μ_{p}), where

μ_{p}(dx_{1}, ..., dx_{p}) = \begin{matrix}[ & \frac{p}{2} & ] \\ ∑ \\ k=0 \end{matrix} ν_{p−2k}(dx_{1}, ..., dx_{p}),  (5.3)

ν_{p−2k}(dx_{1}, ..., dx_{p}) = [H(2H − 1)]^{p−k} [\frac{2t_{0}^{2α+1}}{2α(2α + 1)}]^{k} c_{p,k}^{2}(p − 2k)! (\frac{1}{2α})^{p−2k} ×

×φ(x_{1}, x_{2})...φ(x_{2k−1}, x_{2k}) [x_{2k+1}^{2α} + (t_{0} − x_{2k+1})^{2α}] .. [x_{p}^{2α} + (t_{0} − x_{p})^{2α}] dx_{1}..dx_{p},  (5.4)

and

α = H − \frac{1}{2}, c_{p,k} = \frac{p!}{2^{k}k! (p − 2k)!}.  (5.5)

We have in this case the equality

Tr_{H}^{k}f_{p}(t_{1}, ..., t_{p−2k}) = ∫_{T^{2k}} f_{p}(s_{1}, ..., s_{2k}, t_{1}, ..., t_{p−2k})×

×φ(s_{1}, s_{2})....φ(s_{2k−1}, s_{2k})ds_{1}...ds_{2k}.  (5.6)

Proposition 5.5. (a) If f_{p} =sym(g_{1} ⊗ ... ⊗ g_{p}), g_{i} ∈ L_{H}^{2}(T), then

J_{p}^{H,T} (f_{p}) = J_{1}^{H,T} (g_{1})...J_{p}^{H,T} (g_{p}).  (5.7)

(b) (The fractional Hu-Meyer formula). Let f_{p} ∈ L_{s,H}^{2}( T^{p}) be such that (Tr_{H}^{k}f_{p})_{1≤k≤[\frac{p}{2}]} exist. Then

J_{p}^{H,T} (f_{p}) = \begin{matrix}[ & \frac{p}{2} & ] \\ ∑ \\ k=0 \end{matrix} c_{p,k}I_{p−2k}^{H,T}(Tr_{H}^{k}f_{p}),  (5.8)

Corollary 5.6. If f_{p} ∈ L_{s,H}^{2}( T^{p}) and (Tr_{H}^{k}f_{p})_{1≤k≤[\frac{p}{2}]} exist, then

E [J_{p}^{H,T} (f_{p})] = \{ \begin{matrix}0, & p odd \\ c_{p,[\frac{p}{2}]}TrH^{\frac{p}{2}}f_{p}, & p even, \end{matrix}  (5.9)

E [|J_{p}^{H,T} (f_{p})|^{2}] = \begin{matrix}[ & \frac{p}{2} & ] \\ ∑ \\ k=0 \end{matrix} (c_{p,k})^{2} (p − 2k)! |Tr_{H}^{k}f_{p}|_{H,T^{p−2k}}^{2} .  (5.10)

In particular if f_{p} ∈ L^{2}(T^{p}, μ_{p}) and H > \frac{1}{2},

E [J_{p}^{H,T} (f_{p})] =

\{ \begin{matrix}0, & p odd \\ c_{p,[\frac{p}{2}]} ∫_{T} f_{p}(s_{1}, t_{1}, .., s_{\frac{p}{2}} , t_{\frac{p}{2}} )φ(s_{1}, t_{1})..φ(s_{\frac{p}{2}} , t_{\frac{p}{2}} )ds_{1}dt_{1}..ds_{\frac{p}{2}} dt_{\frac{p}{2}} , & p even, \end{matrix}  (5.11)

E [|J_{p}^{H,T} (f_{p})|^{2}] =

\begin{matrix}[ & \frac{p}{2} & ] \\ ∑ \\ k=0 \end{matrix} (c_{p,k})^{2} (p − 2k)! ∫_{T^{2(p−2k)}} φ(u_{2k+1}, v_{2k+1})...φ(u_{p}, v_{p})du_{2k+1}dv_{2k+1}..du_{p}dv_{p}×

∫_{T^{2k}} f_{p}(s_{1}, t_{1}, ..., s_{k}, t_{k}, u_{2k+1}, ..., u_{p})φ(s_{1}, t_{1})...φ(s_{k}, t_{k})ds_{1}dt_{1}...ds_{k}dt_{k}×

∫_{T^{2k}} f_{p}(s_{1}, t_{1}, ..., s_{k}, t_{k}, v_{2k+1}, ..., v_{p})φ(s_{1}, t_{1})...φ(s_{k}, t_{k})ds_{1}dt_{1}...ds_{k}dt_{k}.  (5.12)

Proof. It follows from the corresponding equalities for multiple Stratonovich integrals and Lemma 3.1. ■ Application. As an application of multiple Stratonovich fractional integrals we obtain the following result (see [5] and [9]). Proposition 5.7. (strong ley of large numbers for fBm). Let H > \frac{1}{2}, [a, b] ⊂ R and let us consider the partition a < a + \frac{b−a}{2^{n}} < .... < a + \frac{(2^{n}−1)(b−a)}{2^{n}} < b of [a, b] and denote Δ_{j} =

[a + \frac{(j−1)(b−a)}{2^{n}}, a + \frac{j(b−a)}{2^{n}}) . Let p be a positive integer and let f : [a, b] → R be a measurable function such that 0 < C_{1} ≤ f ≤ C_{2}. Then

S_{n,p} := \frac{b − a}{2^{n}} ∑_{j=1}^{2^{n}} \frac{1}{|f1_{Δ_{j}} |_{H,R}^{2p}} [I_{1}^{H,R} (f1_{Δ_{j}})]^{2p} ⟶_{n→∞}^{a.s.,L^{2}} \frac{(2p)!}{2^{p}p!}.  (5.13)

In particular

\frac{1}{2^{n(1−2pH)}} ∑_{j=1}^{2^{n}} [B^{H}(\frac{j}{2^{n}}) − B^{H}(\frac{j − 1}{2^{n}})]^{2p} ⟶_{n→∞}^{a.s.,L^{2}} \frac{(2p)!}{2^{p}p!}.  (5.14)

Proof. For simplicity we take [a, b] = [0, 1] . If we define

g_{n,2p} = \frac{1}{2^{n}} ∑_{j=1}^{2^{n}} \frac{1}{|f1_{Δ_{j}} |_{H,R}^{2p}} (f1_{Δ_{j}})^{⊗2p} ,

then we have

J_{2p}^{H,R}(g_{n,2p}) = S_{n,p}, E(S_{n,p}) = \frac{(2p)!}{2^{p}p!}.

It is enough to prove that ∑_{n=1}^{∞} D^{2} (S_{n,p}) < ∞. From (5.2) it follows that

(Tr_{H}^{j}g_{n,2p}) (t_{1}, ..., t_{2p−2j}) =
∫_{R^{2j}} g_{n,2p}(s_{1}, .., s_{2j}, t_{1}, ..., t_{2p−2j})φ(s_{1}, s_{2})...φ(s_{2j−1}, s_{2j})ds_{1}...ds_{2j} = g_{n},_{2p−2j} ,

and then

D^{2} (S_{n,p}) = ∑_{j=0}^{p−1} c_{2p,j}^{2}(2p − 2j)! |g_{n,2p−2j}|_{H,R^{2p−2j}}^{2} =

∑_{j=0}^{p−1} c_{2p,j}^{2}(2p − 2j)! ∥Γ^{(2p−2j)}g_{n,2p−2j}∥_{L^{2}(R^{2p−2j})}^{2} .

Therefore all we need to show is that

∑_{n=1}^{∞} ∥Γ_{H,T}^{(2p−2j)}g_{n,2p−2j}∥_{L^{2(p−j)}(R^{2(p−j)})}^{2} < ∞, ∀0 ≤ j ≤ p − 1.

Since, by Proposition 2.1-(b) the operator Γ_{H,T}^{(2r)} : L^{H^{−1}}(R^{2r}) → L^{2}(R^{2r}) is continuous, we have for r ≥ 1,

∑_{n=1}^{∞} ∥Γ_{H,T}^{(2r)}g_{n,r}∥_{L^{2}(R^{2r})}^{2} ≤ C ∑_{n=1}^{∞} ∥g_{n,r}∥_{L^{H^{−1}}(R^{2r})}^{2} =

C ∑_{n=1}^{∞} \frac{1}{2^{\frac{n}{H}}} ∑_{j=1}^{2^{n}} \frac{1}{|f1_{Δ_{j}} |H,R^{\frac{2r}{H}}} (∫_{Δ_{j}} |f(t)|^{\frac{1}{H}} dt)^{2r} ≤ C_{3}(r, H) ∑_{n=1}^{∞} \frac{1}{2^{n(H^{−1}−1)}} < ∞. ■

Remark 5.8. The result (5.14) still holds if H < \frac{1}{2}( see for example [20]).

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