An It\^o's type formula for the fractional Brownian motion in Brownian time

Let $X$ be a (two-sided) fractional Brownian motion of Hurst parameter $H\in (0,1)$ and let $Y$ be a standard Brownian motion independent of $X$. Fractional Brownian motion in Brownian motion time (of index $H$), recently studied in \cite{13}, is by definition the process $Z=X\circ Y$. It is a continuous, non-Gaussian process with stationary increments, which is selfsimilar of index $H/2$. The main result of the present paper is an It\^{o}'s type formula for $f(Z_t)$, when $f:\R\to\R$ is smooth and $H\in [1/6,1)$. When $H>1/6$, the change-of-variable formula we obtain is similar to that of the classical calculus. In the critical case $H=1/6$, our change-of-variable formula is in law and involves the third derivative of $f$ as well as an extra Brownian motion independent of the pair $(X,Y)$. We also discuss briefly the case $H<1/6$.


Introduction
If f : R + → R is C 1 then f (t) = f (0) + t 0 f ′ (s)ds for all t 0 whereas, if W is a standard Brownian motion and if f : R → R is C 2 then, by the Itô's formula, In (1.1) the Itô integral, namely is of forward type. It is well-known that the additional bracket term 1 2 t 0 f ′′ (W s )ds appearing in (1.1) comes from the non-negligibility of the quadratic variation of W in the large limit; more precisely, Based on (1.4), it is then not difficult to prove the following two facts: 1. If H > 1 2 and f : R → R is C 2 (actually, C 1 is enough), then Thus, H = 1 2 appears to be a critical value for the change-of-variable formula involving the forward integral (1.2). This is because it is precisely the value from which the sign of 2H − 1 changes in (1.4). The chain rule being (1.1) in the critical case H = 1 2 , one has a complete picture for the forward integral (1.2).
To go one step further, one may wonder what kind of change-of-variable formula one would obtain after replacing the definition (1.2) by its symmetric counterpart, namely X k2 −n + X (k+1)2 −n (Y (k+1)2 −n − Y k2 −n ) (1.5) (provided the limit exists in some sense). As it turns out, it is arguably a much more difficult problem, which has been solved only recently. In this context, the crucial quantity is now the cubic variation. And this latter is known to satisfy, for any H < 1 2 , With a lot of efforts, one can prove (see [5,6] when H = 1 6 and [16] when H = 1 6 ) the following three facts, which hold for any smooth enough real function f : R → R: Thus, as we see, the critical value for the symmetric integral is now H = 1 6 ; it is exactly the value of H from which the sign of 3H − 1 2 changes in (1.6).
In [1,2] (see also [3]), Burdzy has introduced the so-called iterated Brownian motion. This process, which can be regarded as the realization of a Brownian motion on a random fractal, is defined as where X is a two-sided Brownian motion and Y is a standard (one-sided) Brownian motion independent of X. Note that Z is self-similar of order 1 4 and has stationary increments; hence, in some sense, Z is close to the fractional Brownian motion B 1 4 of index H = 1 4 . As is the case for B 1 4 , Z is neither a Dirichlet process nor a semimartingale or a Markov process in its own filtration. A crucial question is therefore how to define a stochastic calculus with respect to it. This issue has been tackled by Khoshnevisan and Lewis in [10,11], where the authors develop a Stratonovich-type stochastic calculus with respect to Z, by extensively using techniques based on the properties of some special arrays of Brownian stopping times, as well as on excursion-theoretic arguments. See also the paper [14] which may be seen as a follow-up of [10]. The formula obtained in [10,11] reads, unsurprisingly (due to (1.7) and the similarities between Z and B 1 4 ) and losely speaking, as follows: (1.10) The change-of-variable formula (1.10) is of the same kind than (1.7). In view of what has been done so far for the fractional Brownian motion B H , aiming to provide an answer to the following problem is somehow natural: can we also reinterpret (1.10) in a dynamical way, in the spirit of (1.7), (1.8) and (1.9)? To this end, we first need to introduce a family of processes that contains the iterated Brownian motion Z as a particular element. The family consisting in the so-called fractional Brownian motions in Brownian time, studied in [17] by the second-named author, does the job. More specifically, it is the family {Z H } H∈(0,1) defined as follows: where X H is a two-sided fractional Brownian motion of index H and Y is a standard (onesided) Brownian motion independent of X. Roughly speaking, in the present paper we are going to show the following three assertions (see Theorem 2.1 for a precise statement): for any smooth real function f : R → R, 2. If H = 1 6 then, with W a standard Brownian motion independent of the pair (X 1 6 , Y ) and κ 3 ≃ 2.322, The formula (1.11) is related to a recent line of research in which, by means of Malliavin calculus, one aims to exhibit change-of-variable formulas in law with a correction term which is an Itô integral with respect to martingale independent of the underlying Gaussian processes. Papers dealing with this problem and which are prior to our work include [4,7,8,9,12,15,16]; however, it is worthwhile noting that all these mentioned references only deal with Gaussian processes, not with iterated processes (which are arguably more difficult to handle).
A brief outline of the paper is as follows. In Section 2, we introduce the framework in which our study takes place and we provide an exact statement of our result, namely Theorem 2.1. Finally, Section 3 contains the proof of Theorem 2.1, which is divided into several steps.

Framework and exact statement of our results
For simplicity, throughout the paper we remove the superscript H, that is, we write Z (resp. X) instead of Z H (resp. X H ).
Let Z be a fractional Brownian motion in Brownian time of Hurst parameter H ∈ (0, 1), defined as where X is a two-sided fractional Brownian motion of parameter H and Y is a standard (one-sided) Brownian motion independent of X.
The paths of Z being very irregular (precisely: Hölder continuous of order α if and only if α is strictly less than H/2), we will not be able to define a stochastic integral with respect to it as the limit of Riemann sums with respect to a deterministic partition of the time axis. However, a winning idea borrowed from Khoshnevisan and Lewis [10,11] is to approach deterministic partitions by means of random partitions defined in terms of hitting times of the underlying Brownian motion Y . As such, one can bypass the random "time-deformation" forced by (2.12), and perform asymptotic procedures by separating the roles of X and Y in the overall definition of Z.
Following Khoshnevisan and Lewis [10,11], we start by introducing the so-called intrinsic skeletal structure of Z H . This structure is defined through a sequence of collections of stopping times (with respect to the natural filtration of Y ), noted which are in turn expressed in terms of the subsequent hitting times of a dyadic grid cast on the real axis. More precisely, let D n = {j2 −n/2 : j ∈ Z}, n 1, be the dyadic partition (of R) of order n/2. For every n 1, the stopping times T k,n , appearing in (2.13), are given by the following recursive definition: T 0,n = 0, and Note that the definition of T k,n , and therefore of T n , only involves the one-sided Brownian motion Y , and that, for every n 1, the discrete stochastic process defines a simple random walk over D n . As shown in [10, Lemma 2.2], as n tends to infinity the collection {T k,n : 1 k 2 n t} approximates the common dyadic partition {k2 −n : 1 k 2 n t} of order n of the time interval [0, t]. More precisely, sup 0 s t T ⌊2 n s⌋,n − s → 0 almost surely and in L 2 (Ω). (2.14) Based on this fact, one may introduce the counterpart of (1.5) based on T n , namely, Let C ∞ b denote the class of those functions f : R → R that are C ∞ and bounded together with their derivatives. We then have the following result. Here, 15) as n → ∞ (its existence is part of the conclusion), W is a two-sided Brownian motion independent of the pair (X, Y ) defining Z, and the integral with respect to W is understood in the Wiener-Itô sense.
This means that there is no way to get a change-of-variable formula for f (x) = x 3 .
3 Proof of Theorem 2.1

Elements of Malliavin calculus
In this section, we gather some elements of Malliavin calculus we shall need thoughout the proof of Theorem 2.1. The reader already familiar with this topic may skip this section. We continue to denote by X = (X t ) t∈R a two-sided fractional Brownian motion with Hurst parameter H ∈ (0, 1). That is, X is a zero mean Gaussian process, defined on a complete probability space (Ω, A , P ), with the covariance function We suppose that A is the σ-field generated by X. For all n ∈ N * , we let E n be the set of step functions on [−n, n], and E : Let H be the Hilbert space defined as the closure of E with respect to the inner product The mapping ξ t → X t can be extended to an isometry between H and the Gaussian space H 1 associated with X. We will denote this isometry by ϕ → X(ϕ).
Let F be the set of all smooth cylindrical random variables, i.e. of the form where l ∈ N * , φ : R l → R is C ∞ b and t 1 < ... < t l are some real numbers. The derivative of F with respect to X is the element of L 2 (Ω, H ) defined by In particular D s X t = ξ t (s). For any integer k 1, we denote by D k,2 the closure of the set of smooth random variables with respect to the norm The Malliavin derivative D satisfies the chain rule. If ϕ : R n → R is C 1 b and if F 1 , . . . , F n are in D 1,2 , then ϕ(F 1 , ..., F n ) ∈ D 1,2 and we have We have the following Leibniz formula, whose proof is straightforward by induction on q. Let ϕ, ψ ∈ C q b (q 1), and fix 0 u < v and 0 s < t.
where⊗ stands for the symmetric tensor product. A similar statement holds fo u < v 0 and s < t 0.
If a random element u ∈ L 2 (Ω, H ) belongs to the domain of the divergence operator, that is, if it satisfies For every n 1, let H n be the nth Wiener chaos of X, that is, the closed linear subspace of L 2 (Ω, A , P ) generated by the random variables {H n (B(h)), h ∈ H , h H = 1}, where H n is the nth Hermite polynomial. The mapping I n (h ⊗n ) = H n (B(h)) provides a linear isometry between the symmetric tensor product H ⊙n and H n . For H = 1 2 , I n coincides with the multiple Wiener-Itô integral of order n. The following duality formula holds for any element h ∈ H ⊙n and any random variable F ∈ D n,2 .
Let {e k , k 1} be a complete orthonormal system in H . Given f ∈ H ⊙n and g ∈ H ⊙m , for every r = 0, ..., n ∧ m, the contraction of f and g of order r is the element of H ⊗(n+m−2r) defined by f, e k 1 ⊗ ... ⊗ e kr H ⊗r ⊗ g, e k 1 ⊗ ... ⊗ e kr H ⊗r .
Note that f ⊗ r g is not necessarily symmetric: we denote its symmetrization by f⊗ r g ∈ H ⊙(n+m−2r) . Finally, we recall the following product formula: if f ∈ H ⊙n and g ∈ H ⊙m then I n (f )I m (g) = n∧m r=0 r! n r m r I n+m−2r (f⊗ r g).

Notation and reduction of the problem
Throughout all the proof, we shall use the following notation. For all k, n ∈ N we write . Also, ·, · ( · , respectively) will always stand for inner product (the norm, respectively) in an appropriate tensor product H ⊗s .
In the sequel, we only consider the case H < 1 2 . The proof of (2.16) in the case H > 1 2 is easier and left to the reader, whereas the proof when H = 1 2 was already done in [10,11] by Khoshnevisan and Lewis.
That said, we now divide the proof of Theorem 2.1 in several steps.
While easy, the following lemma taken from [10, Lemma 2.4] is going to be the key when studying the asymptotic behavior of the weighted power variation V (2r−1) n (f, t) of odd order 2r − 1, defined as: Its main feature is to separate X from Y , thus providing a representation of V (2r−1) n (f, t) which is amenable to analysis.
Observe that V

Step 2: Transforming the weighted power variations of odd order
By [10, Lemma 2.5], one has where j * (n, t) = 2 n/2 Y T ⌊2 n t⌋,n . As a consequence, V where X + t := X t for t 0, X − −t := X t for t < 0, X n,+ (t) := 2 nH/2 X + 2 −n/2 t for t 0 and X n,− (−t) := 2 nH/2 X − 2 −n/2 (−t) for t < 0. Let us now introduce the following sequence of processes W (2r−1) ±,n , in which H p stands for the pth Hermite polynomial: We then have, using the decomposition x 2r−1 = r l=1 a r,l H 2l−1 (x) (with a r,r = 1, which is the only explicit value of a l,r we will need in the sequel), (3.22)

Step 3: Known results for fractional Brownian motion
We recall the following result taken * from [13] . If m 2 and H ∈ 1 4m−2 , 1 2 then, for any f ∈ C ∞ b and as n → ∞, (3.23) * More precisely: a careful inspection would show that there is no additional difficulty to prove (3.23) by following the same route than the one used to show [13,Theorem 1,(1.15)]. The only difference is that the definition of W (r) ±,n is of symmetric type, whereas all the quantities of interest studied in [13] are of forward type. ±,n (f, t) converges in probability (when H > 1 6 ) or only in law (when H = 1 6 ) to a non degenerate limit as n → ∞.

3.6
Step 4: Moment bounds for W Fix an integer r 1 as well as a function f ∈ C ∞ b . We claim the existence of c > 0 such that, for all real numbers s < t and all n ∈ N, In order to prove (3.24), we will need the following lemma.
Proof. When s, t, u > 0 we have Since |b 2H − a 2H | |b − a| 2H for any a, b ∈ R + , we immediately deduce (3.25). The proof when s, t, u < 0 is similar.
We are now ready to show (3.24). We distinguish two cases according to the signs of s, t ∈ R (and reducing the problem by symmetry): (1) if 0 s < t (the case s < t 0 being similar), then with obvious notation. Relying to the product formula (3.21), we deduce that this latter quantity is less than or equal to By the duality formula (3.20) and the Leibniz rule (3.19), one has that Let now c denote a generic constant that may differ from one line to another and recall that f ∈ C ∞ b . We then have the following estimates.

Step 5: Limits of the weighted power variations of odd order
Fix f ∈ C ∞ b and t 0. We claim that, if H ∈ 1 6 , 1 2 and r 3 then, as n → ∞, Moreover, if H ∈ 1 6 , 1 2 then, as n → ∞, whereas, if H = 1 6 then, as n → ∞, with W = (W t ) t∈R a two-sided Brownian motion independent of the pair (X, Y ). Indeed, using the decomposition (3.22), the conclusion of Step 4 (to pass from Y T ⌊2 n t⌋,n to Y t ) and the convergence (2.14), we deduce that the limit of V (2r−1) n (f, t) is the same as that of Thus, the proofs of (3.32), (3.33) and (3.34) then follow directly from the results recalled in Step 3, as well as the fact that X and Y are independent.

3.8
Step 6: Proving (2.16) and (2.17) We assume H ∈ [ 1 6 , 1 2 ). We will make use of the following Taylor's type formula. Fix f ∈ C ∞ b . For any a, b ∈ R and for some constants c r whose explicit values are immaterial here,