An Itô-type formula for the fractional Brownian motion in Brownian time

El e c t ro nic J

o f

Pr

ob a bi l i t y

Electron. J. Probab.19(2014), no. 99, 1–15.

ISSN:1083-6489 DOI:10.1214/EJP.v19-3184

An Itô-type formula for the fractional Brownian motion in Brownian time

Ivan Nourdin

Raghid Zeineddine

Abstract

LetX be a (two-sided) fractional Brownian motion of Hurst parameterH ∈ (0,1) and letY be a standard Brownian motion independent ofX. Fractional Brownian motion in Brownian motion time (of indexH), recently studied in [17], is by definition the process Z = X ◦Y. It is a continuous, non-Gaussian process with stationary increments, which is selfsimilar of indexH/2. The main result of the present paper is an Itô’s type formula forf(Zt), whenf:R→Ris smooth andH∈[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 caseH = 1/6, our change-of-variable formula is in law and involves the third derivative offas well as an extra Brownian motion independent of the pair(X, Y). We also discuss briefly the caseH <1/6.

Keywords: Fractional Brownian motion in Brownian time; change-of-variable formula in law;

Malliavin calculus.

AMS MSC 2010:60F05; 60H05; 60G15; 60H07.

Submitted to EJP on December 3, 2013, final version accepted on October 6, 2014.

1 Introduction

If f : R⁺ → R ^isC¹ then f(t) = f(0) +Rt

0f⁰(s)ds for allt > 0 whereas, ifW is a standard Brownian motion and iff :R→R^isC²then, by the Itô’s formula,

f(Wt) =f(0) + Z t

0

f⁰(Ws)d⁻Ws+1 2

Z t 0

f⁰⁰(Ws)ds, t>0. (1.1) In (1.1) the Itô integral, namely

Z t 0

Xsd⁻Ys:= lim

n→∞

b2ⁿtc−1

X

k=0

X_k2⁻ⁿ(Y_(k+1)2⁻ⁿ−Y_k2⁻ⁿ), (1.2)

is of forward type. It is well-known that the additional bracket term ¹₂Rt

0f⁰⁰(W_s)ds appearing in (1.1) comes from the non-negligibility of the quadratic variation ofW in

*Support: ANR-10-BLAN-0121 “Malliavin, Stein and Stochastic Equations with Irregular Coefficients”.

†Luxembourg University, Luxembourg. E-mail:[email protected]

‡Université de Nice, France. E-mail:[email protected]

the large limit; more precisely,

b2ⁿtc−1

X

k=0

(W_(k+1)2−n−W_k2−n)² −→^a.s. t asn→ ∞. (1.3)

Introducing a family {B^H}_H∈(0,1) of fractional Brownian motions parametrized by the Hurst parameterH may help to reinterpret (1.1) in a more dynamical way. Let us elaborate this point of view further. Recall thatB¹² is nothing but the standard Brownian motion, whereasB¹is the processB¹_t =tG,t >0,G∼N(0,1). The extension of (1.3) to anyH ∈(0,1)is well-known: one has

2^n(2H−1)

b2ⁿtc−1

X

k=0

(B_(k+1)2^H −n−B_k2^H−n)² −→^a.s. t asn→ ∞. (1.4)

Based on (1.4), it is then not difficult to prove the following two facts:

1. IfH >¹₂ andf :R→R^isC²(actually,C¹is enough), thenRt

0f⁰(B^H_s )d⁻B^H_s exists as a limit in probability and we have

f(B^H_t ) =f(0) + Z t

0

f⁰(B^H_s )d⁻B^H_s , t>0.

2. IfH < ¹₂, then

Z t 0

B^H_s d⁻B_s^H=−∞ a.s.,

meaning that there is no possible change-of-variable formula forf(x) =x². Thus,H = ¹₂ 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 of2H−1changes in (1.4). The chain rule being (1.1) in the critical case H = ¹₂, 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

Z t 0

X_sd^◦Y_s:= lim

n→∞

b2ⁿtc−1

X

k=0

1

2 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 dif- ficult 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 anyH < ¹₂,

2^n(3H−12)

b2ⁿtc−1

X

k=0

(B_(k+1)2^H −n−B_k2^H−n)^{3 law}→ N(0, σ_H²) asn→ ∞. (1.6)

With a lot of efforts, one can prove (see [5, 6] whenH 6= ¹₆ and [16] whenH = ¹₆) the following three facts, which hold for any smooth enough real functionf :R→R^:

1. IfH > ¹₆ thenRt

0f⁰(B_s^H)d^◦B_s^Hexists as a limit in probability and one has f(B_t^H) =f(0) +

Z t 0

f⁰(B_s^H)d^◦B_s^H, t>0. (1.7)

2. IfH = ¹₆ thenRt 0f⁰(B

1

s6)d^◦B

1

s6 exists as a stable limit in law and one has, withW a standard Brownian motion independent ofB¹⁶ andκ3'2.322(see (3.4) for the precise definition ofκ3),

f(B_t¹⁶) =f(0) + Z t

0

f⁰(Bs¹⁶)d^◦Bs¹⁶ −κ₃ 12

Z t 0

f⁰⁰⁰(B¹⁶)dW_s, t>0. (1.8) 3. IfH < ¹₆ then

Z t 0

(B^H_s )²d^◦B_s^H does not exist in law. (1.9) Thus, as we see, the critical value for the symmetric integral is nowH= ¹₆; it is exactly the value ofH from which the sign of3H−¹₂ changes in (1.6).

In [1, 2] (see also [3]), Burdzy has introduced the so-called iterated Brownian mo- tion. This process, which can be regarded as the realization of a Brownian motion on a random fractal, is defined as

Zt=X(Yt), t>0,

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 ¹₄ and has stationary increments; hence, in some sense,Z is close to the fractional Brownian motionB¹⁴ of indexH = ¹₄. As is the case forB¹⁴,Zis 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 betweenZ andB¹⁴) and losely speaking, as follows:

f(Z_t) =f(0) + Z t

0

f⁰(Z_s)d^◦Z_s, t>0. (1.10) The change-of-variable formula (1.10) is of the same kind as (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:

Z_t^H=X^H(Y_t), t>0,

whereX^H is a two-sided fractional Brownian motion of index H and Y is a standard (one-sided) Brownian motion independent ofX. 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 functionf :R→R^,

1. IfH > ¹₆ then

f(Z_t^H) =f(0) + Z t

0

f⁰(Z_s^H)d^◦Z_s^H, t>0.

2. If H = ¹₆ then, with W a standard Brownian motion independent of the pair (X¹⁶, Y)andκ₃'2.322(see (3.4) for the precise definition ofκ₃),

f(Z

1 6

t) =f(0) + Z t

0

f⁰(Z

1

s6)d^◦Z

1

s6 −κ3

12 Z t

0

f⁰⁰⁰(Zs)d^◦3Zs, t>0. (1.11)

3. IfH < ¹₆, then

Z t 0

(Z_s^H)²d^◦Z_s^Hdoes not exist.

The formula (1.11) is related to a recent line of research in which, by means of Malli- avin 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.

2 Framework and exact statement of our results

For simplicity, throughout the paper we remove the superscriptH, that is, we write Z(resp. X) instead ofZ^H (resp.X^H).

Let Z be a fractional Brownian motion in Brownian time of Hurst parameter H ∈ (0,1), defined as

Z_t=X(Y_t), t>0, (2.1)

whereX is a two-sided fractional Brownian motion of parameterH andY is a standard (one-sided) Brownian motion independent ofX.

The paths ofZ being very irregular (precisely: Hölder continuous of orderαif and only ifαis strictly less thanH/2), we will not be able to define a stochastic integral with respect to it as the limit of Riemann sums with respect to adeterministicpartition 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 motionY. As such, one can bypass the random

“time-deformation” forced by (2.1), and perform asymptotic procedures by separating the roles ofX andY in the overall definition ofZ.

Following Khoshnevisan and Lewis [10, 11], we start by introducing the so-called intrinsic skeletal structure ofZ. This structure is defined through a sequence of collec- tions of stopping times (with respect to the natural filtration ofY), noted

Tn={Tk,n:k>0}, n>1, (2.2) which are in turn expressed in terms of the subsequent hitting times of a dyadic grid cast on the real axis. More precisely, letDn ={j2^−n/2 : j ∈ Z},n >1, be the dyadic partition (of R^{) of order}n/2. For every n > 1, the stopping times Tk,n, appearing in (2.2), are given by the following recursive definition:T_0,n= 0, and

Tk,n= inf

s > Tk−1,n: Y(s)∈Dn\ {Y(Tk−1,n)} , k>1.

Note that the definition ofT_k,n, and therefore ofTn, only involves the one-sided Brow- nian motionY, and that, for everyn>1, the discrete stochastic process

Yn={Y(T_k,n) :k>0}

defines a simple random walk overDn. As shown in [10, Lemma 2.2], as n tends to infinity the collection{Tk,n: 16 k6 2ⁿt}approximates the common dyadic partition {k2⁻ⁿ : 16k62ⁿt}of ordernof the time interval[0, t]. More precisely,

sup

06s6t

T_b2nsc,n−s

→0 almost surely and inL²(Ω). (2.3) Based on this fact, one may introduce the counterpart of (1.5) based onTn, namely,

V_n(f, t) =

b2ⁿtc−1

X

k=0

1

2 f(Z_Tk,n) +f(Z_Tk+1,n)

(Z_Tk+1,n−Z_Tk,n). (2.4) LetC_b^∞denote the class of those functionsf :R→R^{that are}C^∞and bounded together with their derivatives. We then have the following result.

Theorem 2.1.Letf ∈C_b^∞andt >0. 1. IfH > ¹₆ then

f(Zt)−f(0) = Z t

0

f⁰(Zs)d^◦Zs, (2.5)

whereRt

0f⁰(Zs)d^◦Zsis the limit in probability ofVn(f⁰, t)defined in (2.4) asn→ ∞. 2. IfH =¹₆ then, withκ3'2.322(see (3.4) for the precise definition ofκ3),

f(Z_t)−f(0) +κ₃ 12

Z t 0

f⁰⁰⁰(Z_s)d^◦3Z_s ^(law)= Z t

0

f⁰(Z_s)d^◦Z_s. (2.6) Here,Rt

0f⁰(Zs)d^◦Zsdenotes the limit in law ofVn(f⁰, t)defined in (2.4) asn→ ∞ (its existence is part of the conclusion.) Moreover, we have, for allt>0,

Z t 0

f⁰⁰⁰(Zs)d^◦3Zs:=

Z Y_t 0

f⁰⁰⁰(Xs)dWs,

whereW is a two-sided Brownian motion independent of the pair(X, Y)defining Z, and where the integral with respect toW is understood in the Wiener-Itô sense.

3. IfH < ¹₆ then

Vn(g, t)does not converge, even stably in law, (2.7) where g(x) = x². This means that there is no way to get a change-of-variable formula forf(x) =x³.

3 Proof of Theorem 2.1

3.1 Elements of Malliavin calculus

In this section, we gather some elements of Malliavin calculus we shall need though- out the proof of Theorem 2.1. The reader already familiar with this topic may skip this section.

We continue to denote byX = (Xt)_t∈Ra two-sided fractional Brownian motion with Hurst parameterH ∈(0,1).That is,X is a zero mean Gaussian process, defined on a complete probability space(Ω,A, P), with covariance function,

CH(t, s) =E(XtXs) = 1

2(|s|^2H+|t|^2H− |t−s|^2H), s, t∈R.

We suppose thatA ^{is the}σ-field generated byX. For alln∈N^{∗, we let}En be the set of step functions on[−n, n], andE :=∪_nEn. Set ξ_t = 1_[0,t] (resp. 1_[t,0]) ift > 0 (resp.

t <0). LetH be the Hilbert space defined as the closure ofE with respect to the inner product

hξt, ξsi_H =CH(t, s), s, t∈R.

The mappingξt 7→ Xt can be extended to an isometry between H and the Gaussian spaceH¹associated withX. We will denote this isometry byϕ7→X(ϕ).

LetF be the set of all smooth cylindrical random variables, i.e. of the form F =φ(Xt1, ..., Xt_l),

wherel∈N^∗,φ:R^l→R^isC_b^∞andt₁< ... < t_lare some real numbers. The derivative ofF with respect toX is the element ofL²(Ω,H)defined by

DsF =

l

X

i=1

∂φ

∂x_i(Xt₁, ..., Xt_l)ξt_i(s), s∈R.

In particularD_sX_t=ξ_t(s). For any integerk>1, we denote byD^k,2the closure of the set of smooth random variables with respect to the norm

kFk²_k,2=E(F²) +

k

X

j=1

E[kD^jFk²_H⊗j].

The Malliavin derivativeDsatisfies the chain rule. Ifϕ:Rⁿ →R^isC_b¹and ifF1, . . . , Fn

are inD^{1,2, then}ϕ(F1, ..., Fn)∈D^1,2and we have Dϕ(F1, ..., Fn) =

n

X

i=1

∂ϕ

∂xi

(F1, ..., Fn)DFi.

We have the following Leibniz formula, whose proof is straightforward by induction onq. Letϕ, ψ∈C_b^q(q>1), and fix06u < vand06s < t.Thenϕ(X_t−X_s)ψ(X_v−X_u)∈D^q,2 and

D^q ϕ(X_t−X_s)ψ(X_v−X_u)

=

q

X

a=0

q a

ϕ^(a)(X_t−X_s)ψ^(q−a)(X_v−X_u)1^⊗a_[s,t]⊗˜1^⊗(q−a)_[u,v] ,

(3.1) where⊗˜ stands for the symmetric tensor product. A similar statement holds fou < v6 0ands < t60.

If a random elementu∈L²(Ω,H)belongs to the domain of the divergence operator, that is, if it satisfies

|EhDF, ui_H|6cu

pE(F²) for anyF ∈F, thenI(u)is defined by the duality relationship

E F I(u)

=E hDF, ui_H ,

for everyF ∈D^1,2.

For every n > 1, let Hn be thenth Wiener chaos of X, that is, the closed linear subspace ofL²(Ω,A, P)generated by the random variables{Hn(B(h)), h∈H,khk_H = 1},whereHnis thenth Hermite polynomial. The mappingIn(h^⊗n) =Hn(B(h))provides a linear isometry between the symmetric tensor productH^{n and}H^{n. For}H = ¹₂,In

coincides with the multiple Wiener-Itô integral of ordern. The following duality formula holds

E F In(h)

=E hDⁿF, hi_H^⊗n

, (3.2)

for any elementh∈Hⁿand any random variableF ∈D^n,2.

Let {e_k, k >1}be a complete orthonormal system inH. Givenf ∈ H^{n and} g ∈ H^m,for everyr= 0, ..., n∧m,the contraction off andg of orderris the element of H^{⊗(n+m−2r)defined by}

f⊗rg=

∞

X

k₁,...,k_r=1

hf, ek₁⊗...⊗ek_ri_H⊗r⊗ hg, ek₁⊗...⊗ek_ri_H⊗r.

Note thatf ⊗rgis not necessarily symmetric: we denote its symmetrization byf⊗˜rg∈ H^(n+m−2r).Finally, we recall the following product formula: iff ∈H^nandg∈H^m then

I_n(f)I_m(g) =

n∧m

X

r=0

r!

n r

m r

I_n+m−2r(f⊗˜_rg). (3.3)

3.2 Notation and reduction of the problem

Throughout all the proof, we shall use the following notation. For all k, n ∈ N ^we write

ξ_k2−n/2 = 1_[0,k2−n/2], ξ_k2⁻_−n/2 =1_[−k2−n/2,0],

δ_k2−n/2 = 1_[(k−1)2−n/2,k2^−n/2], δ⁻_k2−n/2 =1_[−k2−n/2,(−k+1)2^−n/2].

Also,h·,·i(k·k, respectively) will always stand for inner product (the norm, respectively) in an appropriate tensor productH^⊗s.

On the other hand, for allj ∈ N^letG_j =X_j−X_j−1. The family{Gj}is Gaussian, stationary, centered, with variance 1; moreover its covarianceρis given by

ρ(j−j⁰) =E[G_jG_j⁰] = 1

2 |j−j⁰+ 1|^2H+|j−j⁰−1|^2H−2|j−j⁰|^2H ,

so thatP

|ρ(a)|<∞ifH 6 ¹2. Then, for allr∈N^∗, we define κ_2r−1:=

s

(2r−1)!X

a∈Z

ρ(a)^2r−1. (3.4)

Note thatP

a∈Z|ρ(a)|^2r−1<∞if and only ifH <1−1/(2(2r−1)), which is satisfied for allr>1if we suppose thatH61/2(the caseH = 1/2may be treated separately).

In the sequel, we only consider the caseH < ¹₂. The proof of (2.5) in the caseH > ¹₂ is easier and left to the reader, whereas the proof whenH = ¹₂ was already done in [10, 11] by Khoshnevisan and Lewis.

That said, we now divide the proof of Theorem 2.1 in several steps.

3.3 Step 1: A key algebraic lemma

For each integern>1,k∈Zand real numbert>0, letUj,n(t)(resp. Dj,n(t)) denote the number ofupcrossings (resp. downcrossings) of the interval [j2^−n/2,(j+ 1)2^−n/2]

within the firstb2ⁿtcsteps of the random walk{Y(T_k,n)}k>1, that is, Uj,n(t) =]

k= 0, . . . ,b2ⁿtc −1 :

Y(Tk,n) = j2^−n/2andY(Tk+1,n) = (j+ 1)2^−n/2 ; D_j,n(t) =]

k= 0, . . . ,b2ⁿtc −1 :

Y(Tk,n) = (j+ 1)2^−n/2andY(Tk+1,n) =j2^−n/2 . 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 variationVn^(2r−1)(f, t)of oddorder2r−1, defined as:

V_n^(2r−1)(f, t) =

b2ⁿtc−1

X

k=0

1

2 f(ZT_k,n) +f(ZT_k+1,n)

(ZT_k+1,n−ZT_k,n)^2r−1, t>0.

Its main feature is to separateXfromY, thus providing a representation ofVn^(2r−1)(f, t) which is amenable to analysis.

Lemma 3.1.Fixf ∈C_b^∞,t>0andr∈N^{∗. Then} V_n^(2r−1)(f, t)

= X

j∈Z

1 2

f(X_j2−n

2) +f(X_(j+1)2−n 2)

X_(j+1)2−n

2 −X_j2−n 2

2r−1

U_j,n(t)−D_j,n(t) .

Observe thatVn⁽¹⁾(f, t) =V_n(f, t), see (2.4).

3.4 Step 2: Transforming the weighted power variations of odd order By [10, Lemma 2.5], one has

Uj,n(t)−Dj,n(t) =







1_{06j<j^∗_(n,t)} ifj^∗(n, t)>0

0 ifj^∗= 0

−1_{j^∗(n,t)6j<0} ifj^∗(n, t)<0 ,

wherej^∗(n, t) = 2^n/2YT_{b2n tc,n}. As a consequence,Vn^(2r−1)(f, t)is equal to







2^{−nH(r−12)}Pj^∗(n,t)−1 j=0

1

2 f(X_j2⁺_−n/2) +f(X_(j+1)2⁺ _−n/2)

X_j+1^n,+−X_j^n,+2r−1

ifj^∗(n, t)>0

0 ifj^∗= 0

2^{−nH(r−12)}P|j^∗(n,t)|−1 j=0

1

2 f(X_j2⁻_−n/2) +f(X_(j+1)2⁻ _−n/2)

X_j+1^n,−−X_j^n,−2r−1

ifj^∗(n, t)<0 ,

whereX_t⁺ := X_t fort > 0, X_−t⁻ := X_t fort < 0, X_t^n,+ := 2^nH/2X₂⁺_−n/2t fort > 0 and X_−t^n,− := 2^nH/2X₂⁻_−n/2

(−t)fort <0.

Let us now introduce the following sequence of processes W_±,n^(2r−1), in which Hp

stands for thepth Hermite polynomial:

W_±,n^(2r−1)(f, t) =

b2^n/2tc−1

X

j=0

1 2 f(X^±

j2⁻ⁿ2) +f(X^±

(j+1)2⁻ⁿ2)

H_2r−1(X_j+1^n,±−X_j^n,±), t>0

W_n^(2r−1)(f, t) =

( W_+,n^(2r−1)(f, t) ift>0 W_−,n^(2r−1)(f,−t) ift <0 . We then have, using the decompositionx^2r−1=Pr

l=1ar,lH_2l−1(x)(withar,r = 1, which is the only explicit value ofal,rwe will need in the sequel),

V_n^(2r−1)(f, t) = 2^{−nH(r−12)}

r

X

l=1

ar,lW_n^(2l−1)(f, YT_{b2n tc,n}). (3.5)

3.5 Step 3: Known results for fractional Brownian motion

We recall the following result taken¹ from [13] . Ifm>2and H ∈ _4m−2¹ ,¹₂ then, for anyf ∈C_b^∞and asn→ ∞,

X_t,2^−n/4W_±,n^(2m−1)(f, t)

t>0

−→fdd

X_t, κ_2m−1 Z t

0

f(X_s^±)dW_s^±

t>0

, (3.6)

whereκ_2m−1is defined in (3.4),W_t⁺=W_tift >0andW_t⁻=W_−tift <0, withW a two- sided Brownian motion independent ofX, and whereRt

0f(X_s^±)dW_s^±must be understood in the Wiener-Itô sense.

Note that in the boundary case m = 2and H = ¹₆, (3.6) continues to hold, as was shown in [16, Theorem 3.1].

In the casem= 1, it was shown in [13, Theorem 4] (caseH > ¹₆) and [16, Theorem 2.13] (case H = ¹₆) that, for any fixed t > 0, the sequence W_±,n⁽¹⁾(f, t) converges in probability (when H > ¹₆) or only in law (when H = ¹₆) to a non degenerate limit as n→ ∞.

3.6 Step 4: Moment bounds forWn^(2r−1)(f,·)

Fix an integerr >1 as well as a functionf ∈C_b^∞. We claim the existence ofc > 0 such that, for all real numberss < tand alln∈N^,

E

W_n^(2r−1)(f, t)−W_n^(2r−1)(f, s)²

6 cmax(|s|^2H,|t|^2H) |t−s|2^n/2+ 1 . (3.7) In order to prove (3.7), we will need the following lemma.

Lemma 3.2.Ifs, t, u >0or ifs, t, u <0then

|E Xu(Xt−Xs)

|6|t−s|^2H. (3.8)

Proof. Whens, t, u >0we have E X_u(X_t−X_s)

= 1

2 t^2H−s^2H +1

2 |s−u|^2H− |t−u|^2H .

Since|b^2H−a^2H|6|b−a|^2H for anya, b∈R⁺, we immediately deduce (3.8). The proof whens, t, u <0is similar.

We are now ready to show (3.7). We distinguish two cases according to the signs of s, t∈R(and reducing the problem by symmetry):

(1) if06s < t(the cases < t60being similar), then

E[(W_n^(2r−1)(f, t)−W_n^(2r−1)(f, s))²] =E[(W_+,n^(2r−1)(f, t)−W_+,n^(2r−1)(f, s))²]

= 1

4

b2^n/2tc−1

X

j,j⁰=b2^n/2sc

E f(X⁺

j2⁻ⁿ2) +f(X⁺

(j+1)2⁻ⁿ2)

× f(X⁺

j⁰2⁻ⁿ2) +f(X⁺

(j⁰+1)2⁻ⁿ2)

H_2r−1(X_j+1^n,+−X_j^n,+)H_2r−1(X_j^n,+0+1−X_j^n,+0 )

= 1

42^nH(2r−1)

b2^n/2tc−1

X

j,j⁰=b2^n/2sc

E

Θⁿ_jf(X⁺)Θⁿ_j0f(X⁺)I_2r−1(δ^⊗(2r−1)_(j+1)2−n/2)I_2r−1(δ_(j^⊗(2r−1)_0+1)2−n/2) ,

1More precisely: a careful inspection shows that there is no additional difficulty to prove (3.6) 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_±,n^(r) is of symmetric type, whereas all the quantities of interest studied in [13] are of forward type.

with obvious notation. Thanks to the product formula (3.3), we deduce that this latter quantity is less than or equal to

1

42^nH(2r−1)

b2^n/2tc−1

X

j,j⁰=b2^n/2sc 2r−1

X

l=0

l!

2r−1 l

²

hδ_(j+1)2−n/2;δ_(j0+1)2^−n/2i

l

×

E

Θⁿ_jf(X⁺)Θⁿ_j0f(X⁺)I_4r−2−2l(δ^{⊗(2r−1−l)}_(j+1)2−n/2⊗δ˜ ^{⊗(2r−1−l)}_(j0+1)2^−n/2)

=: 1 4

2r−1

X

l=0

l!

2r−1 l

2

Q^(+,r,l)_n (s, t). (3.9)

By the duality formula (3.2) and the Leibniz rule (3.1), one has that

d^(+,r,l)_n (j, j⁰) :=E

Θⁿ_jf(X⁺)Θⁿ_j0f(X⁺)I_4r−2−2l(δ_(j+1)2^{⊗(2r−1−l)}_−n/2⊗δ˜ _(j^{⊗(2r−1−l)}_0+1)2−n/2)

= E

D^4r−2−2l(Θⁿ_jf(X⁺)Θⁿ_j0f(X⁺)) ; δ^{⊗(2r−1−l)}_(j+1)2−n/2⊗δ˜ ^{⊗(2r−1−l)}_(j0+1)2^−n/2

=

4r−2−2l

X

a=0

4r−2−2l a

E

f^(a)(X_j2⁺−n/2)ξ_j2^⊗a−n/2+f^(a)(X_(j+1)2⁺ −n/2)ξ_(j+1)2^⊗a −n/2

⊗˜ f(4r−2−2l−a)(X_j⁺_02−n/2)ξ⊗(4r−2−2l−a)

j⁰2^−n/2 +f(4r−2−2l−a)(X_(j⁺_0+1)2−n/2)ξ⊗(4r−2−2l−a) (j⁰+1)2^−n/2

;

δ_(j+1)2^{⊗(2r−1−l)}_−n/2⊗δ˜ _(j^{⊗(2r−1−l)}0+1)2^−n/2

.

Let nowcdenote a generic constant that may differ from one line to another and recall thatf ∈C_b^∞. We then have the following estimates.

• Casel= 2r−1

Q^(+,r,2r−1)_n (s, t)

6 c2^nH(2r−1)

b2^n/2tc−1

X

j,j⁰=b2^n/2sc

hδ_(j+1)2−n/2;δ_(j0+1)2^−n/2i

2r−1

= c

b2^n/2tc−1

X

j,j⁰=b2^n/2sc

1

2(|j−j⁰+ 1|^2H+|j−j⁰−1|^2H−2|j−j⁰|^2H)

2r−1

= c

b2^n/2tc−1

X

j=b2^n/2sc

j−b2^n/2sc

X

q=j−b2^n/2tc+1

ρ(q)

2r−1,

withρ(q) :=¹₂(|q+ 1|^2H+|q−1|^2H−2|q|^2H). By a Fubini argument, it follows that Q^(+,r,2r−1)_n (s, t)

6 c

b2^n/2tc−b2^n/2sc−1

X

q=b2^n/2sc−b2^n/2tc+1

|ρ(q)|^2r−1

(q+b2^n/2tc)∧ b2^n/2tc −(q+b2^n/2sc)∨ b2^n/2sc

6 c

b2^n/2tc−b2^n/2sc−1

X

q=b2^n/2sc−b2^n/2tc+1

|ρ(q)|^2r−1 b2^n/2tc − b2^n/2sc

6 cX

q∈Z

|ρ(q)|^2r−1

b2^n/2tc − b2^n/2sc =c

b2^n/2tc − b2^n/2sc

6 c

b2^n/2tc −2^n/2t

+ 2^n/2 t−s

+

b2^n/2sc −2^n/2s

6 c(1 + 2^n/2|t−s|). (3.10)

Note thatP

q∈Z|ρ(q)|^2r−1<∞sinceH < ¹₂ 61−_4r−2¹ .

• Preparation to the cases where06l62r−2

In order to handle the termsQ^(+,r,l)n (s, t)whenever06l62r−2, we will make use of the following decomposition:

|d^(+,r,l)_n (j, j⁰)|6

1

X

u,v=0

Ω^(u,v,r,l)_n (j, j⁰), (3.11)

where

Ω^(u,v,r,l)_n (j, j⁰) =

4r−2−2l

X

a=0

4r−2−2l a

E[f^(a)(X_(j+u)2⁺ −n/2)f(4r−2−2l−a)(X_(j⁺0+v)2^−n/2)]

×

ξ^⊗a_(j+u)2−n/2⊗ξ˜ ⊗(4r−2−2l−a)

(j⁰+v)2^−n/2 ;δ_(j+1)2^{⊗(2r−1−l)}_−n/2⊗δ˜ _(j^{⊗(2r−1−l)}_0+1)2−n/2 .

• Case16l62r−2(only whenr>2)

Sincef belongs toC_b^∞ and since, by (3.8), we have|hξt;δ_(j+1)2−n/2i| 62^−nH for all t>0and allj ∈N, we deduce that

|d^(+,r,l)_n (j, j⁰)|6c2−nH(4r−2−2l).

As a consequence, and relying to the same arguments that have been used previously in the casel= 2r−1, we get

Q^(+,r,l)_n (s, t) 6 c2−nH(4r−2−2l)2^nH(2r−1)

b2^n/2tc−1

X

j,j⁰=b2^n/2sc

hδ_(j+1)2−n/2;δ_(j0+1)2^−n/2i

l

6 c2−nH(4r−2−2l)2^nH(2r−1)2^−nHlX

q∈Z

|ρ(q)|^l(1 + 2^n/2|t−s|)

= c2−nH(2r−1−l)(1 + 2^n/2|t−s|)6c(1 + 2^n/2|t−s|). (3.12)

• Casel= 0

Relying to the decomposition (3.11), we get

Q^(+,r,0)_n (s, t) 6 2^nH(2r−1)

b2^n/2tc−1

X

j,j⁰=b2^n/2sc 1

X

u,v=0

Ω^(u,v,r,0)_n (j, j⁰). (3.13)

We will study only the term corresponding toΩ^(0,1,r,0)n (j, j⁰)in (3.13), which is repre- sentative of the difficulty. It is given by

2^nH(2r−1)

b2^n/2tc−1

X

j,j⁰=b2^n/2sc 4r−2

X

a=0

4r−2 a

E[f^(a)(X_j2⁺_−n/2)f^(4r−2−a)(X_(j⁺_0+1)2−n/2)]

×

ξ^⊗a_j2−n/2⊗ξ˜ _(j^{⊗(4r−2−a)}_0+1)2−n/2;δ_(j+1)2^⊗(2r−1)_−n/2⊗δ˜ _(j^⊗(2r−1)_0+1)2−n/2

6 c2^nH(2r−1)

b2^n/2tc−1

X

j,j⁰=b2^n/2sc 4r−2

X

a=0

ξ^⊗a_j2−n/2⊗ξ˜ _(j^{⊗(4r−2−a)}_0+1)2−n/2;δ_(j+1)2^⊗(2r−1)_−n/2⊗δ˜ _(j^⊗(2r−1)_0+1)2−n/2 .

We defineEn^(a,r)(j, j⁰) :=

ξ^⊗a_j2−n/2⊗ξ˜ _(j^{⊗(4r−2−a)}0+1)2^−n/2;δ_(j+1)2^⊗(2r−1)_−n/2⊗δ˜ _(j^⊗(2r−1)0+1)2^−n/2

.By (3.8), recall that |hξt;δ_(j+1)2−n/2i| 6 2^−nH for all t > 0 and allj ∈ N. We thus get, with ˜ca some combinatorial constants,

E_n^(a,r)(j, j⁰) 6 ˜c_a2^{−nH(4r−3)} |hξ_j2−n/2;δ_(j+1)2−n/2i|+|hξ_j2−n/2;δ_(j0+1)2^−n/2i|

+|hξ_(j0+1)2^−n/2;δ_(j+1)2−n/2i|+|hξ_(j0+1)2^−n/2;δ_(j0+1)2^−n/2i|

. For instance, we can write

b2^n/2tc−1

X

j,j⁰=b2^n/2sc

|hξ_(j0+1)2^−n/2;δ_(j+1)2−n/2i|

= 2^−nH−1

b2^n/2tc−1

X

j,j⁰=b2^n/2sc

(j+ 1)^2H−j^2H+|j⁰−j+ 1|^2H− |j⁰−j|^2H

6 2^−nH−1

b2^n/2tc−1

X

j,j⁰=b2^n/2sc

(j+ 1)^2H−j^2H

+2^−nH−1 X

b2^n/2sc6j6j⁰6b2^n/2tc−1

(j⁰−j+ 1)^2H−(j⁰−j)^2H

+2^−nH−1 X

b2^n/2sc6j⁰<j6b2^n/2tc−1

(j−j⁰)^2H−(j−j⁰−1)^2H

6 3

22^−nH b2^n/2tc − b2^n/2sc

b2^n/2tc^2H6 3t^2H

2 1 + 2^n/2|t−s|

.

Similarly,

b2^n/2tc−1

X

j,j⁰=b2^n/2sc

|hξ_j2−n/2;δ_(j+1)2−n/2i| 6 3t^2H

2 1 + 2^n/2|t−s|

;

b2^n/2tc−1

X

j,j⁰=b2^n/2sc

|hξ_j2−n/2;δ_(j0+1)2^−n/2i| 6 3t^2H

2 1 + 2^n/2|t−s|

;

b2^n/2tc−1

X

j,j⁰=b2^n/2sc

|hξ_(j0+1)2^−n/2;δ_(j0+1)2^−n/2i| 6 3t^2H

2 1 + 2^n/2|t−s|

.

As a consequence, we deduce

Q^(+,r,0)_n (s, t)6c2^{−nH(2r−2)}t^2H 2^n/2|t−s|+ 1)6c t^2H 2^n/2|t−s|+ 1). (3.14) Combining (3.9), (3.10), (3.12) and (3.14) finally shows our claim (3.7).

(2) ifs <06t, then

E[(W_n^(2r−1)(f, t)−W_n^(2r−1)(f, s))²] =E[(W_+,n^(2r−1)(f, t)−W_−,n^(2r−1)(f,−s))²] 6 2E[(W_+,n^(2r−1)(f, t))²] + 2E[(W_−,n^(2r−1)(f,−s))²].

By (1) withs= 0, one can write

E[(W_+,n^(2r−1)(f, t))²]6c t^2H(t2^n/2+ 1).

Similarly

E[(W_−,n^(2r−1)(f,−s))²]6c(−s)^2H (−s)2^n/2+ 1 We deduce that

E[(W_n^(2r−1)(f, t)−W_n^(2r−1)(f, s))²]6cmax(t^2H,(−s)^2H) (t−s)2^n/2+ 1 .

That is, (3.7) also holds true in this case.

3.7 Step 5: Limits of the weighted power variations of odd order Fixf ∈C_b^∞andt>0. We claim that, ifH ∈₁

6,¹₂

andr>3then, asn→ ∞,

V_n^(2r−1)(f, t)^prob−→0. (3.15)

Moreover, ifH ∈ ¹₆,¹₂

then, asn→ ∞,

V_n⁽³⁾(f, t)^prob−→0, (3.16)

whereas, ifH = ¹₆ then, asn→ ∞,

Xt, Yt, V_n⁽³⁾(f, t)

t>0

fdd→ Xt, Yt, κ3

Z Y_t 0

f(Xs)dWs

!

t>0

, (3.17)

withW = (Wt)t∈Ra two-sided Brownian motion independent of the pair(X, Y).

Indeed, using the decomposition (3.5), the conclusion of Step 4 (to pass fromYT_{b2n tc,n}

toY_t) and since by [10, Lemma 2.3], we haveY_{Tb2n tc,n}−→^L2 Y_tasn→ ∞, we deduce that the limit ofVn^(2r−1)(f, t)is the same as that of

2^{−nH(r−12)}

r

X

l=1

a_r,lW_n^(2l−1)(f, Y_t).

Thus, the proofs of (3.15), (3.16) and (3.17) then follow directly from the results recalled in Step 3, as well as the fact thatX andY are independent.

3.8 Step 6: Proving (2.5) and (2.6)

We assumeH ∈[¹₆,¹₂). We will make use of the following Taylor’s type formula. Fix f ∈C_b^∞. For anya, b∈Rand for some constantsc_rwhose explicit values are immaterial here,

f(b)−f(a) = 1

2 f⁰(a) +f⁰(b)

(b−a)− 1

24 f⁰⁰⁰(a) +f⁰⁰⁰(b) (b−a)³

+

7

X

r=3

c_r f^(2r−1)(a) +f^(2r−1)(b)

(b−a)^2r−1+O(|b−a|¹⁴),

where|O(|b−a|¹⁴)| 6C_f|b−a|¹⁴, C_f being a constant depending only on f. One can thus write

f(ZT_{b2n tc,n})−f(0) =

b2ⁿtc−1

X

k=0

f(ZT_k+1,n)−f(ZT_k,n)

=

b2ⁿtc−1

X

k=0

1

2 f(ZT_k,n) +f(ZT_k+1,n)

(ZT_k+1,n−ZT_k,n) (3.18)

−1

12V_n⁽³⁾(f, t) +

7

X

r=3

2crV_n^(2r−1)(f, t) +

b2ⁿtc−1

X

k=0

O((ZT_k+1,n−ZT_k,n)¹⁴).

As far as the bigOin (3.18) is concerned, we have, withG∼N(0,1),

E

b2ⁿtc−1

X

k=0

O((ZT_k+1,n−ZT_k,n)¹⁴)

6Cf b2ⁿtc−1

X

k=0

E

(ZT_k+1,n−ZT_k,n)¹⁴

= Cf b2ⁿtc−1

X

k=0

2^−7nHE[G¹⁴]6CfE[G¹⁴]t2^n(1−7H)→n→∞0 sinceH> ¹6. (3.19) On the other hand, by continuity off◦Zand due to (2.3), one has, almost surely and asn→ ∞,

f(Z_{Tb2n tc,n})−f(0)→f(Z_t)−f(0). (3.20) Finally, whenH > ¹₆ the desired conclusion (2.5) follows from (3.19), (3.20), (3.15) and (3.16) plugged into (3.18). The proof of (2.6) when H = ¹₆ is similar, the only difference being that one has (3.17) instead of (3.16), thus leading to the bracket term

κ3

12

RYt

0 f⁰⁰⁰(Xs)dWs=:^κ₁₂³ Rt

0f⁰⁰⁰(Zs)d^◦3Zsin (2.6).

3.9 Step 7: Proving (2.7)

Using b³−a³ = ³₂(a²+b²)(b−a)− ¹₂(b−a)³, one can write, with 1 denoting the function constantly equal to 1,

Vn(g, t)−1

3Z_t³ = 1

6V_n⁽³⁾(1, t) +1 3

b2ⁿtc−1

X

k=0

(Z_T³_k+1,n−Z_T³_k,n)−1 3Z_t³

= 1

6V_n⁽³⁾(1, t) +1

3(Z_T³_{b2n tc,n}−Z_t³).

As a result, and thank to (2.3), one deduces that ifV_n(g, t)converges stably in law, then Vn⁽³⁾(1, t)must converge as well. But it is shown in [17, Corollary 1.2] that2^{−n(1−6H)/4}Vn⁽³⁾(1, t) converges in law to a non degenerate limit. This being clearly in contradiction with the convergence ofVn⁽³⁾(1, t), we deduce that (2.7) holds.

References

[1] K. Burdzy (1993): Some path properties of iterated Brownian motion. In Seminar on Stochastic Processes(E. Cinlar, K. L. Chung and M. J. Sharpe, eds.) pp. 67-87. Birkhaüser, Boston. MR-1278077

[2] K. Burdzy (1994): Variation of iterated Brownian motion. InWorkshop and Conference on Measure-Valued Processes, Stochastic Partial Differential Equations and Interacting Particle Systems(D. A. Dawson, ed.), pp. 35-53. Amer. Math. Soc., Providence, RI. MR-1278281 [3] K. Burdzy and D. Khoshnevisan (1998): Brownian motion in a Brownian crack.Ann. Appl.

Probab.8, pp. 708-748. MR-1627764