THE FBM ITO’S FORMULA THROUGH ANALYTIC CONTINUATION

The Fractional Brownian Motion can be extended to complex values of the parameter (cid:11) for Re (cid:11) > 12 . This is a useful tool. Indeed, the obtained process depends holomorphically on the parameter, so that many formulas, as It^o formula, can be extended by analytic continuation. For large values of Re (cid:11) , the stochastic calculus reduces to a deterministic one, so that formulas are very easy to prove. Hence they hold by analytic continuation for Re (cid:11) (cid:20) 1, containing the classical case (cid:11) = 1.

Note that many of the cited authors prefer to deal with an other kind of FBM, associated to the so-called Hurst parameter H which is real and corresponds to our α through the relation H = Re α − 1 2 . Recall that the LFBM (Liouville Fractional Brownian Motion) is defined by There are three main ideas in the present paper. The first is to deal with the Liouville spaces, J α,p which are the images of L p ([0, T ]) under the Liouville kernel defined by This idea seems to go back to our paper [14] of 1996, in relation with Hölder continuous functions, and Young-Stieltjes integration. Another advantage of the Liouville space, as pointed in [14], is to give the natural isomorphism J α,p (L p (µ)) = L p (µ, J α,p ) where µ is an arbitrary measure (for example the Wiener measure). This gives a nice interpretation of Kolmogorov's lemma, and this gives also some natural Banach spaces of solutions of trace problems.
For a β-Hölder continuous function ϕ, we redefine the FBM-Wiener integral as the natural extension of the linear map for a G ∈ J 1,2 which is the Cameron-Martin space of the Wiener measure, the integral being taken in a little more precise sense than that of Young.
The second idea, after the deep study of [17,22,25], is to use the Itô-Skorohod integral, and to define Here u is a β-Hölder process with values in a Gaussian Sobolev space, and X α T is the Gaussian divergence of a suitable FBM-Wiener integral.
Observe that if u is not adapted, X α is not in general. Hence we get a true stochastic calculus and an Itô-Skorohod formula for anticipative processes.
The third idea is to use complex values of the parameter α, for Re α > 1 2 . The interesting property is that all the preceding objects are holomorphic functions of α. So that our first goal is to show that there exists a differential formula in the ordinary pathwise calculus for Re α > 3 2 . We then extend its validity by analytic continuation on the domain of definition of every term. To this end, we only have to prove that each involved term has a meaning. It appears that doing so, we easily get for α = 1 the so-called Itô-Skorohod stochastic formula, and for Re α > 3 4 the fractional Itô-Skorohod formula we were looking for.
Notice that for Re α < 1, the definition of the remaining term(s) in the Itô formula, needs singular integrals which exist in the sense of Hadamard (Parties finies de Hadamard). Finally for the Itô formula, a natural domain for (α, β) ⊂ C × R is defined by the conditions Note that in [2], the Itô formula for the LFBM is only stated under the stronger condition α + β/2 > 1. Actually for the reason as above (true stochastic calculus), it is not reasonable to consider other values than β < Re α − 1 2 . In conclusion, a natural stochastic calculus can only be obtained for Re α > 3 4 . Observe that the point ( 3 4 , 1) is the most left limit point of the natural domain. As a matter of fact, in all the paper, the only stochastic analysis elements we use, are the Wiener integral and the Sobolev Gaussian space.
For adapted processes, it would be interesting to know that if the domain could be extended by considering simultaneously the method in use in [1][2][3][4] (cutting the Liouville kernel to obtain semi-martingales), and analytic extension of integrals.
Of course, it would be possible to extend this formulas to n-dimensional FBM. There would be no new difficulties, except in writing formulas.
In conclusion, we can say that we have an ordinary pathwise differential calculus for Re α > 3 2 , a "Young stochastic" calculus for Re α > 1, and a "Young-Hadamard stochastic" calculus for Re α > 3 4 .

Recall on the Liouville space
Throughout the paper, α is a complex parameter such that Re α > 1 2 , β is a real number (the order of Hölder continuity) between 0 and 1, and p is a real number (the Hölder exponent of integrability) strictly between 1 and +∞ when no other precision.
We use the same notations as in [14]. For Re α > 0, the Liouville integral is defined by convolution is a FL p multiplier according to the Marcinkiewicz theorem [21], so that I iγ is a continuous operator of L p ([0, T ]). It then easily follows by the semi-group property that J α,p = J α+iγ,p = J Re α,p .
Note that for Re α > 1 2 , the I α 's are Hilbert-Schmidt. The natural norm of J α,p is given by For β ∈]0, 1[, let C β the space of β-Hölder continuous functions on [0, T ] in the restricted sense, that is those functions such that is a continuous function. This is a separable Banach space under the norm As it was proved in our paper [14], for exponents satisfying the inequalities 1 > β > γ > γ − 1/p > β > 0, the following inclusions hold Now, let B be a complex separable Banach space. Most of the above properties also hold for B-valued functions. For the property concerning the identity J α,p (B) = J Re α,p (B), we need an extra property. We say that a Banach space is a B p -space if it is isomorphic with a closed subspace of an L p space. Hence the required equality holds true for a B p -space.
Note that every separable Hilbert space is a B p -space (even if p = 2), and that a B 2 space is a Hilbert space.
In all of the paper, every involved functional Banach space is separable, and the expression "absolutely convergent integral" of a Banach space valued function means that the function is integrable in the Bochner sense.

Recall on the Wiener space
We denote Ω the standard Wiener space with the Wiener measure µ, that is the space of Rvalued continuous trajectories ω or defined on [0, +∞[ and vanishing at 0. The standard Brownian motion is denoted W t . The µ-expectation is denoted E . The first Wiener chaos, that is the space of µ-measurable linear functions on Ω (cf. . [11], th.22 and [12], th.11) is denoted by H 1 . As for every Gaussian space, the gradient or differential ∇ can be defined. In the particular case of the Wiener space, for every elementary Wiener functional F (ω) = ϕ(W t 1 , . . . , W tn ) Notice that ∇F is linear in the second slot.
The Gaussian Sobolev space D 1,p = D 1,p (Ω, µ) is the completion of elementary functions under the Sobolev norm defined by The divergence operator is defined by transposition. If G is a Wiener-Sobolev functional on Ω × Ω, div G is defined on Ω by Thanks to the theorem of divergence continuity, this definition makes sense for G ∈ D 1,p (Ω × Ω) and p > 1.
In fact, the only interesting values of the divergence are achieved on the functional G which are linear in the second slot . For the particular functions which are of the form G = Φ(ω)X( ) where X is linear (i.e. belongs to the first Wiener chaos), one has where E is the partial expectation w.r. to .

The holomorphic FBM
Let α a complex number such that Re α > 1 2 . We define the complex FBM by the Wiener integral whereẆ is for the white noise on [0, +∞[, that is the "derivative" of the standard Brownian Motion.
This can be justified in the following way: the Cameron-Martin space of the Wiener space is is exactly the µ-measurable linear extension of the bounded linear form (cf. [11], th.38) At this point, one could prouve that the FBM is a Gaussian process with values in the space of holomorphic functions on Re α > 1 2 , with γ-Hölder continuous trajectories for γ < Re α − 1 2 . In fact we shall prove more general results in the next section.
Only observe for the moment that for Re α > 3 2 , W α t has C 1 -trajectories, which is obvious since the Brownian motion has continuous trajectories.

The main lemma
Recall that in [14], we defined the Young-Stieltjes integral T 0 ϕdψ for ϕ ∈ C β and ψ ∈ C γ for β + γ > 1, and the result was Unfortunately, as I α g is only C α − 1 2 , for α < Re α, it seems that we are obliged to assume β + Re α > 3 2 for the existence of the Young integral. Nevertheless we have a more precise result given by the next lemma, which involves an analytic extension, that is in fact a "Partie finie de Hadamard".
Before enouncing this main lemma, it is convenient to introduce some domains of constant use in the sequel.

It converges absolutely for Re α > 3 2 and is of class C 1 in t. Moreover it admits a unique holomorphic extension in the domain D 1 (β). This extension is absolutely continuous for
where K T (α, β, γ) is locally bounded (and even continuous) on the admissible domain Proof: The assertion concerning the case Re α > 3 2 is obvious. Now one has where we introduced the continuous function (cf. section 2) First observe that y α t is exactly I α (ϕg)(t), hence it has a holomorphic extension until Re α > 1 2 , which belongs to J α,2 . Then for Re α > 1, it is absolutely continuous. Now, the double integral defining z α t converges absolutely and is holomorphic until Re α > 1− β. It remains to prove inequality (2). Put a = Re α. First assume that a ≤ 1. We get which belongs to L 2 loc (dt, B). Hence z α t ∈ J 1,2 (B) ⊂ J α,2 (B) for a ≤ 1. Moreover z α t is absolutely continuous.
From these different inclusions, the following inequality follows where K T (α, β) is locally bounded on D 1 . Inequality (2) follows from the inclusions (1). Now assume that 1 ≤ a ≤ 3 2 . As y α belongs to J α,2 (B), we get by the same inclusions Hence, replacing τ with s, we get the required The proof is complete.
2 Remarks : a) Observe that z α t vanishes for α = 1, so that x α t reduces to the obvious formula 3 2 , it would be more correct to write 3 2 , and but in general, we shall omit these tedious notations. The context shall recall the meaning of the singular integrals.

The FBM Wiener integral
The following definition is equivalent to the one given in [14], page 12.
In the case ϕ = 1, we recover W α t (take β > 1 2 ). In the case α = 1, we recover the ordinary Wiener integral. Now, it follows from the formulas of the main lemma with the same notations, that we have From this inequality we infer that X α t makes sense even if ϕ is H-valued for a separable Hilbert space H.
Proof: According to inequality (2) of lemma 1, we have for the restriction of X α t to the Cameron- As the first Wiener chaos H 1 is naturally isometrically isomorphic to the dual space of the Cameron-Martin space (cf. [11,12]) we get the norm in As X α t is Gaussian, we get for every p ≥ 2 Integrating over a compact set L ⊂ D 1 (β), w.r. to the Lebesgue measure σ on C , we get The right member is finite. As the topology of H(D 1 (β), H) is induced by L p loc (D 1 (β), σ, H), the Kolmogorov lemma gives all the results.

Remarks : a) The coefficient
√ p − 1 follows from an easy extension of the Nelson inequalities (cf. [16], remarque 9) to Gaussian vectors. b) This applies to the case X α t = W α t (take ϕ = 1 and β > 1 2 so that α ∈ D 0 ). This is an improvement of a result of [5], where it is proved an analogous result for W α t , but only for C ∞ -functions of real α. c) As in our article [14], we could extend some of these considerations to the fractional Brownian sheet, and get the separately Hölder continuity for the sheet with values in H-valued holomorphic functions.
Proof: This is true for . This extends to X α t by the definition of X α t . 7 Theorem : Assume that (α, β) ∈ D 1 , β > 1/p, Re α + β > 1 + 1/p with p ≥ 2, and that ϕ belongs to C β (B) where B is a B p -space. Then the conclusions of the previous theorem still hold.

The FBM-Itô-Skorohod integral of u is defined by
As the FBM-Wiener integral X α t belongs C γ (D 1,2 ) for every γ < Re α− 1 2 , not only the divergence is well defined but also the result is a process which belongs to C γ (L 2 (µ)). 9 Theorem : Let (α, β) ∈ D 1 , and that u ∈ p C β (D 1,p ). Then X α t belongs to p L p (µ, C γ ) for every γ such that 0 < γ < Re α − 1 2 . Moreover, for a fixed γ, X belongs to p L p (µ, C γ (H)) where H is the space of holomorphic functions on Re α > 1 2 . Proof: Applying theorem 7 for p > 1/β and the continuity of the divergence yields Proof: Compute the divergence which is worth 12 Theorem : Let Re α > 3 4 , and let F be a polynomial, one has Proof: Note that for α > 3 2 formula (5) is nothing but formula (4). The second step consists to remark that formula (5) makes sense for every complex α in the domain {Re α > 3 4 } in view of theorem 9, since F (W α s ) and F (W α s ) are β-Hölder for every 0 < β < Re α − 1 2 . Indeed one has (α, β) ∈ D 1 for such a β. Hence the equality holds true by analytic continuation for Re α > has an analytic extension all over D 0 = {Re α > 1 2 } for every polynomial G. This remark is not so trivial if we deal with the n-dimensional Brownian motion. Some analogous properties will come below. b) If F is not a polynomial, formula (5) extends by routine arguments, for real α > 3 4 , to a suitable subspace of C 2 -functions F . 9 The FBM Itô-Skorohod differential 14 Proposition : Let (α, β) ∈ D 1 , and let u a process belonging to C β ([0, T ], D 1,2 ). Put If X α vanishes (for every t), then u = 0.
Proof: According to the definition, we have for every t ≤ T . As ψ runs through a total set in L 2 ([0, T ]), f runs through a total set in H 1 , and e f runs through a total set in L 2 (Ω, µ). Hence we have u t = 0.

Theorem : Let u t and v t belonging to
Then we have where Finally we have the following computational rule Proof: First observe that the last term in the right member of formula (6) is non-ambiguous thanks to proposition 14, and this is a FBM-Wiener integral. Secondly the regularity conditions are satisfied for (α, β) ∈ D 1 , and all the quantities are holomorphic w.r. to α.
Hence it suffices to prove formulas (6) and (6 ) for real large enough values of α.
In this case formula (6) reads and this is also formula (6 ).

The main FBM Itô-Skorohod formula
Recall that the domains D 1 and D 1 (β) were defined by We deal with a process and a polynomial F . We introduce the following domains It is well known that in the Itô formula are involved many terms. So, before claiming the formula, we need to analyze the existence of the two following terms. The first one is According to theorem 9, X α t belongs to p C γ (L p (µ)) for every 0 < γ < Re α − 1 2 , so that, for the existence of this term we need to assume that (α, More generally we have is holomorphically extendable for α ∈ D 1 (β).
Proof: Denote ∆ T 0 the simplex defined by the condition where the triple integral absolutely converges. We then have to apply the following lemma 17 Lemma : Let ϕ, ψ ∈ C β (B) where B is a Banach space. Let b a bilinear map with values in another Banach space B 1 , that we denote b(ϕ, ψ) = ϕψ. Then converges absolutely for Re α > 1 and admits a holomorphic extension for α ∈ D 1 (β).
As for J α 2 , one finds that J α 321 holomorphically extends for α + β > 1. Finally we have so that we are done.
18 Remarks : For α = 1 every integral vanishes except J α 322 , and we recover Now we can claim the Itô formula.

Then we have the FBM Itô-Skorohod formula
This formula can also be written Proof: By the preceding considerations, we know that in formulas (8) and (8 ), every term but maybe the last makes sense and is holomorphic with respect to α.
First we prove formulas (8) and (8 ) for Re α large enough (for example Re α > 5). In this case every computation can be made pathwise (as for the little Itô formula). We then get On the other hand, we havė so that the sum of the two first terms of the right hand side is worth We then obtain It remains to compute We have so that J(ω) splits into two terms J 1 (ω) and J 2 (ω) Now we compute J 1 (ω). First we have Indeed, for every test functional G(ω) ∈ D 1,2 , we have Using the Wiener representation of ∇u s w.r. to that is Hence we find Now we prove formula (8 ). We return to As above we get This yields and formula (8 ) is proved.
So, formulas are proved for Re α large enough.
Every term but the last admits an analytic continuation for α ∈ D 2 (β), as we have seen above. Hence the last term (in the formulas (8) and (8 )) has also an analytic continuation. This establishes the formulas, and the following corollary.
20 Corollary : Let F be a polynomial. Then the integral admits an analytic continuation for α ∈ D 2 (β).
Proof: it suffices to notice that every polynomial is the second derivative of another polynomial, and to apply formula (8).

Recovering the case α = 1
Note that the last integral in formula (8) is singular, even in the case α = 1. Nevertheless, the previous corollary proves that the symbolic writing In this section we prove that such a formula can be justified under an additional trace hypothesis.
First we prove a lemma 21 Lemma : Let B be a Banach space and q > 1. Consider a function Φ(r, t) which belongs to the space L q ([0, T ], dr, C β (B)). Then the following integral makes sense and is holomorphic w.r. to α for Re α + β > 1/q. Moreover, its value for α = 1 is where Φ(t, t) belongs to L q ([0, T ], dr).
23 Corollary (The classical Itô-Skorohod formula, cf. [22]): With the additional trace hypothesis, we have 24 Proposition : If u is an adapted process, and for α = 1, the last term vanishes, so that we recover the classical Itô formula.
In the right hand member, the Skorohod integral extends for α ∈ D 2 (β), the following term extends for α ∈ D 2 (β) thanks to lemma 17. The last term extends by the extra trace property for v, thanks to theorem 22. The proof is complete.
Proof: Put Y α t = F (X α t ), so that Y α t satisfies the hypotheses of the last theorem, and applies the analytic continuation from the case Re α > 3 2 .
Notes Added After Proof: After the acceptance of this paper, we have learned of the following relevant and interesting preprint: