TheAnnalsofProbability 2008,Vol.36,No.6,2159–2175 DOI:10.1214/07-AOP385 (cid:13)c InstituteofMathematicalStatistics,2008 ASYMPTOTIC BEHAVIOR OF WEIGHTED QUADRATIC AND CUBIC VARIATIONS OF FRACTIONAL BROWNIAN MOTION 9 0 By Ivan Nourdin 0 2 Universit´e Paris VI n a Thepresentarticleisdevotedtoafinestudyoftheconvergenceof J renormalized weighted quadraticand cubic variations of afractional 9 Brownian motion B with Hurst index H. In the quadratic (resp. 1 cubic) case, when H <1/4 (resp. H <1/6), we show by means of MalliavincalculusthattheconvergenceholdsinL2towardanexplicit ] limit which only depends on B. This result is somewhat surprising R when compared with thecelebrated Breuer and Major theorem. P . h 1. Introduction and main result. The study of single path behavior of t a stochastic processes is often based on the study of their power variations m and there exists a very extensive literature on the subject. Recall that, a [ real κ>1 being given, the κ-power variation of a process X, with respect 4 to a subdivision π = 0=t <t < <t =1 of [0,1], is defined to n n,0 n,1 n,n v { ··· } be the sum 0 7 n−1 5 X X κ. 0 | tn,k+1 − tn,k| k=0 . X 5 For simplicity, consider from now on the case where t = k/n, for n 70 N∗ and k 0,...,n . In the present paper, we wish tno,kpoint out som∈e ∈ { } 0 interesting phenomena when X =B is a fractional Brownian motion and : v when the value of κ is 2 or 3. In fact, we will also drop the absolute value i (when κ=3) and we will introduce some weights. More precisely, we will X consider r a n−1 (1.1) h(B )(∆B )κ, κ 2,3 , k/n k/n ∈{ } k=0 X Received May 2007; revised August 2007. AMS 2000 subject classifications. 60F05, 60G15, 60H07. Key words and phrases. Fractional Brownian motion, Breuer and Major theorem, weighted quadratic variation, weighted cubic variation, exact rate of convergence, Malli- avin calculus. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Probability, 2008,Vol. 36, No. 6, 2159–2175. This reprint differs from the original in pagination and typographic detail. 1 2 I.NOURDIN where the function h:R R is assumed to be smooth enough and where → ∆B denotes the increment B B . k/n (k+1)/n k/n − The analysis of the asymptotic behavior of quantities of type (1.1) is mo- tivated, for instance, by the study of the exact rates of convergence of some approximation schemes of scalar stochastic differential equations driven by B (see [5, 10] and [11]), besides, of course, the traditional applications of quadratic variations to parameter estimation problems. Now, let us recall some known results concerning the κ-power variations (for κ=2,3,4,...),whicharetoday moreorless classical. First,assumethat the Hurst index H of B is 1/2, that is, B is the standard Brownian motion. Let µ denote the κ-moment of a standard Gaussian random variable G κ N (0,1). By the scaling property of the Brownian motion and using th∼e central limit theorem, it is immediate that, as n : →∞ n−1 1 (1.2) [nκ/2(∆B )κ µ ] Law N (0,µ µ2). √n k/n − κ −→ 2κ− κ k=0 X When weights are introduced, an interesting phenomenon appears: instead of Gaussian random variables, we rather obtain mixing random variables as limit in (1.2). Indeed, when κ is even, it is a very particular case of a more general result by Jacod [7] (see also [13]) that we have, as n : →∞ 1 n−1 1 (1.3) h(B )[nκ/2(∆B )κ µ ] Law µ µ2 h(B )dW . √n k/n k/n − κ −→ 2κ− κ s s kX=0 q Z0 Here, W denotes another standard Brownian motion, independent of B. When κ is odd, we have this time, as n : →∞ n−1 1 h(B )[nκ/2(∆B )κ] k/n k/n √n k=0 (1.4) X 1 Law h(B )( µ µ2 dW +µ dB ); −→ s 2κ− κ+1 s κ+1 s Z0 q see [13]. Second,assumethat H =1/2, thatis,thecase wherethefractionalBrow- 6 nianmotionB hasnoindependentincrementsanymore.Then(1.2)hasbeen extended by Breuer and Major [1], Dobrushin and Major [3], Giraitis and Surgailis [4] or Taqqu [16]. Precisely, four cases are considered according to the evenness of κ and the value of H: If κ is even and if H (0,3/4), as n , • ∈ →∞ n−1 1 (1.5) [nκH(∆B )κ µ ] Law N (0,σ2 ). √n k/n − κ −→ H,κ k=0 X WEIGHTED QUADRATICAND CUBIC VARIATIONSOFFBM 3 If κ is even and if H (3/4,1), as n , • ∈ →∞ n−1 n1−2H [nκH(∆B )κ µ ] Law “Rosenblatt r.v.” k/n κ − −→ k=0 X If κ is odd and if H (0,1/2], as n , • ∈ →∞ n−1 1 (1.6) nκH(∆B )κ Law N (0,σ2 ). √n k/n −→ H,κ k=0 X If κ is odd and if H (1/2,1), as n , • ∈ →∞ n−1 n−H nκH(∆B )κ Law N (0,σ2 ). k/n −→ H,κ k=0 X Here, σ >0 denotes a constant depending only on H and κ, which may H,κ be different from one formula to another one, and which can be computed explicitly. The term “Rosenblatt r.v.” denotes a random variable whose dis- tribution is the same as that of Z at time one, for Z the Rosenblatt process defined in [16]. Now, let us proceed with the results concerning the weighted power vari- ations in the case where H = 1/2. In what follows, h denotes a regular 6 enough function such that h together with its derivatives has subexponen- tial growth. If κ is even and H (1/2,3/4), then by Theorem 2 in Leo´n and ∈ Luden˜a [9] (see also Corcuera, Nualart and Woerner [2] for related results on the asymptotic behavior of the p-variation of stochastic integrals with respect to B) we have, as n : →∞ 1 n−1 1 (1.7) h(B )[nκH(∆B )κ µ ] Law σ h(B )dW , k/n k/n κ H,κ s s √n − −→ k=0 Z0 X where, once again, W denotes a standard Brownian motion independent of B. Thus, (1.7) shows for (1.1) a similar behavior to that observed in the standard Brownian case; compare with (1.3). In contradistinction, the asymptotic behavior of (1.1) can be completely different from (1.3) or (1.7) for other values of H. The first result in this direction has been observed by Gradinaru, Russo and Vallois [6] and continued in [5]. Namely, if κ is odd and H (0,1/2), we have, as n : ∈ →∞ (1.8) nH−1n−1h(B )[nκH(∆B )κ] L2 µκ+1 1h′(B )ds. k/n k/n −→− 2 s k=0 Z0 X Before giving the main result of this paper, let us make three comments. First, we stress that the limit obtained in (1.8) does not involve an indepen- dent standard Brownian motion anymore, as was the case for (1.3) or (1.7). 4 I.NOURDIN Second, notice that (1.8) agrees with (1.6) because, when H (0,1/2), we ∈ have (κ+1)H 1<κH 1/2. Thus, (1.8) with h 1 is actually a corol- − − ≡ lary of (1.6). Third, observe that the same type of convergence as (1.8) with H =1/4 had already been performed in [8], Theorem 4.1, when in (1.8) the fractional Brownian motion B with Hurst index 1/4 is replaced by an iter- ated Brownian motion Z. It is not very surprising, since this latter process is also centered, self-similar of index 1/4 and has stationary increments. Fi- nally, let us mention that Swanson announced in [15] that, in a joint work with Burdzy, they will prove that the same also holds for the solution of the stochastic heat equation driven by a space–time white noise. Now, let us go back to our problem. In the sequel, we will make use of the following hypothesis on real function h: (Hm) The function h belongs to Cm and, for any p (0, ) and any 0 ∈ ∞ ≤ i m, we have sup E h(i)(B )p < . ≤ t∈[0,1] {| t | } ∞ The aim of the present work is to prove the following result: Theorem 1.1. Let B be a fractional Brownian motion with Hurst index H. Then: 1. If h:R R verifies (H4) and if H (0,1/4), we have, as n : → ∈ →∞ n−1 1 (1.9) n2H−1 h(B )[n2H(∆B )2 1] L2 1 h′′(B )du. k/n k/n − −→ 4 u k=0 Z0 X 2. If h:R R verifies (H6) and if H (0,1/6), we have, as n : → ∈ →∞ n−1 n3H−1 [h(B )n3H(∆B )3+ 3h′(B )n−H] k/n k/n 2 k/n k=0 (1.10) X 1 L2 1 h′′′(B )du. −→−8 u Z0 BeforegivingtheproofofTheorem1.1,letusroughlyexplainwhy(1.9)is onlyavailablewhenH <1/4[ofcourse,thesametypeofargumentcouldalso be applied to understand why (1.10) is only available when H <1/6]. For this purpose,let usfirstconsider thecase whereB is thestandardBrownian motion (i.e., when H =1/2). By using the independence of increments, we easily compute n−1 E h(B )[n2H(∆B )2 1] =0 k/n k/n ( − ) k=0 X WEIGHTED QUADRATICAND CUBIC VARIATIONSOFFBM 5 and n−1 2 n−1 E h(B )[n2H(∆B )2 1] =2E h2(B ) k/n k/n k/n ( − ! ) ( ) k=0 k=0 X X 1 2nE h2(B )du . u ≈ (cid:26)Z0 (cid:27) Although these two facts are of course not sufficient to guarantee that (1.3) holdswhenκ=2,theyhoweverroughlyexplainwhyitistrue.Now,letusgo back to the general case, that is, the case where B is a fractional Brownian motion of index H (0,1/2). In the sequel, we will show (see Lemmas 2.2 ∈ and 2.3 for precise statements) that n−1 n−1 E h(B )[n2H(∆B )2 1] 1n−2H E[h′′(B )], ( k/n k/n − )≈ 4 k/n k=0 k=0 X X and, when H (0,1/4): ∈ n−1 2 E h(B )[n2H(∆B )2 1] k/n k/n ( − ! ) k=0 X n−1 1 n−4H E[h′′(B )h′′(B )] ≈ 16 k/n ℓ/n k,ℓ=0 X 1 n2−4HE h′′(B )h′′(B )dudv . ≈ 16 (cid:26)Z Z[0,1]2 u v (cid:27) This explains the convergence (1.9). At the opposite, when H (1/4,1/2), ∈ one can prove that n−1 2 1 E h(B )[n2H(∆B )2 1] σ2 nE h2(B )du . ( k=0 k/n k/n − ! )≈ H (cid:26)Z0 u (cid:27) X Thus, when H (1/4,1/2), the quantity n−1h(B )[n2H(∆B )2 1] ∈ k=0 k/n k/n − behaves as in the standard Brownian motion case, at least for the first- P and second-order moments. In particular, one can expect that the following convergence holds when H (1/4,1/2): as n , ∈ →∞ 1 n−1 1 (1.11) h(B )[n2H(∆B )2 1] Law σ h(B )dW , √n k/n k/n − −→ H s s k=0 Z0 X withW astandardBrownianmotionindependentofB.Infact,inthesequel of the present paper, which is a joint work with Nualart and Tudor [12], we show that (1.11) is true and we also investigate the case where H 3/4. ≥ 6 I.NOURDIN Finally, let us remark that, of course, convergence (1.9) agrees with con- vergence (1.5). Indeed, we have 2H 1< 1/2 if and only if H <1/4 (it is − − another fact explaining the condition H <1/4 in the first point of Theorem 1.1). Thus, (1.9) with h 1 is actually a corollary of (1.5). Similarly, (1.10) ≡ agrees with (1.5), since we have 3H 1< 1/2 if and only if H <1/6 (it − − explains the condition H <1/6 in the second point of Theorem 1.1). Now, the rest of our article is devoted to the proof of Theorem 1.1. In- stead of the pedestrian technique performed in [6] (as their authors called it themselves), we stress the fact that we chose here a more elegant way via Malliavin calculus. It can be viewed as another novelty of this paper. 2. Proof of the main result. 2.1. Notation and preliminaries. We begin by briefly recalling some ba- sic facts about stochastic calculus with respect to a fractional Brownian motion. One may refer to [14] for further details. Let B =(Bt)t∈[0,1] be a fractional Brownian motion with Hurst parameter H (0,1/2) defined on a probability space (Ω,A,P).We mean that B is a cent∈ered Gaussian process with the covariance function E(B B )=R (s,t), where s t H (2.1) R (s,t)= 1(t2H +s2H t s 2H). H 2 −| − | We denote by E the set of step R-valued functions on [0,1]. Let H be the Hilbert space defined as the closure of E with respect to the scalar product 1 ,1 =R (t,s). [0,t] [0,s] H H h i We denote by the associate norm. The mapping 1 B can be ex- H [0,t] t |·| 7→ tended to an isometry between H and the Gaussian space (B) associated 1 H with B. We denote this isometry by ϕ B(ϕ). Let S be the set of all smooth cylin7→drical random variables, that is, of the form F =f(B(φ ),...,B(φ )) 1 n where n 1, f:Rn R is a smooth function with compact support and ≥ → φ H. The Malliavin derivative of F with respect to B is the element of i ∈ L2(Ω,H) defined by n ∂f D F = (B(φ ),...,B(φ ))φ (s), s [0,1]. s 1 n i ∂x ∈ i i=1 X In particular D B =1 (s). As usual, D1,2 denotes the closure of the set s t [0,t] of smooth random variables with respect to the norm kFk21,2=E[F2]+E[|D·F|2H]. WEIGHTED QUADRATICAND CUBIC VARIATIONSOFFBM 7 The Malliavin derivative D verifies the following chain rule: if ϕ:Rn R is → continuously differentiable with a bounded derivative, and if (F ) is a i i=1,...,n sequence of elements of D1,2, then ϕ(F ,...,F ) D1,2 and we have, for any 1 n ∈ s [0,1]: ∈ n ∂ϕ D ϕ(F ,...,F )= (F ,...,F )D F . s 1 n 1 n s i ∂x i=1 i X The divergence operator I is the adjoint of the derivative operator D. If a random variable u L2(Ω,H) belongs to the domain of the divergence ∈ operator, that is, if it verifies E DF,u c F for any F S, H u L2 | h i |≤ k k ∈ then I(u) is defined by the duality relationship E(FI(u))=E DF,u , H h i for every F D1,2. ∈ 2.2. Proof of (1.9). In this section, we assume that H (0,1/4). For ∈ simplicity, we note δ =1 and ε =1 . k/n [k/n,(k+1)/n] k/n [0,k/n] Also C will denote a generic constant that can be different from line to line. We first need three lemmas. The proof of the first one follows directly from a convexity argument: Lemma 2.1. For any x 0, we have 0 (x+1)2H x2H 1. ≥ ≤ − ≤ Lemma 2.2. For h,g:R R verifying (H2), we have → n−1 E h(B )g(B )[n2H(∆B )2 1] k/n ℓ/n k/n { − } k,ℓ=0 X (2.2) n−1 = 1n−2H E h′′(B )g(B ) +o(n2−2H). 4 { k/n ℓ/n } k,ℓ=0 X Proof. For 0 ℓ,k n 1, we can write ≤ ≤ − E h(B )g(B )n2H(∆B )2 k/n ℓ/n k/n { } =E h(B )g(B )n2H∆B I(δ ) k/n ℓ/n k/n k/n { } =E h′(B )g(B )n2H∆B ε ,δ k/n ℓ/n k/n k/n k/n H { }h i +E h(B )g′(B )n2H∆B ε ,δ +E h(B )g(B ) . k/n ℓ/n k/n ℓ/n k/n H k/n ℓ/n { }h i { } 8 I.NOURDIN Thus, n−2HE h(B )g(B )[n2H(∆B )2 1] k/n ℓ/n k/n { − } ′ =E h(B )g(B )I(δ ) ε ,δ k/n ℓ/n k/n k/n k/n H { }h i ′ +E h(B )g (B )I(δ ) ε ,δ k/n ℓ/n k/n ℓ/n k/n H (2.3) { }h i =E h′′(B )g(B ) ε ,δ 2 k/n ℓ/n k/n k/n H { }h i ′ ′ +2E h(B )g (B ) ε ,δ ε ,δ k/n ℓ/n k/n k/n H ℓ/n k/n H { }h i h i +E h(B )g′′(B ) ε ,δ 2. k/n ℓ/n ℓ/n k/n H { }h i But ε ,δ = 1n−2H((k+1)2H k2H 1), (2.4) h k/n k/niH 2 − − ε ,δ = 1n−2H((k+1)2H k2H ℓ k 12H + ℓ k 2H). h ℓ/n k/niH 2 − −| − − | | − | In particular, ε ,δ 2 1n−4H |h k/n k/niH− 4 | (2.5) = 1n−4H(((k+1)2H k2H)2 2((k+1)2H k2H)) |4 − − − | 3n−4H((k+1)2H k2H) by Lemma 2.1. ≤ 4 − Consequently, under (H2): n−1 n2H E h′′(B )g(B ) ( ε ,δ 2 1n−4H) | { k/n ℓ/n } h k/n k/niH− 4 | k,ℓ=0 X n−1 Cn1−2H ((k+1)2H k2H)=Cn. ≤ − k=0 X Similarly, using again Lemma 2.1, we deduce ε ,δ ε ,δ + ε ,δ 2 k/n k/n H ℓ/n k/n H ℓ/n k/n H |h i h i | |h i | Cn−4H((k+1)2H k2H + ℓ k 2H ℓ k 12H ). ≤ | − | || − | −| − − | | Since, obviously n−1 (2.6) ℓ k 2H ℓ k 12H Cn2H+1, || − | −| − − | |≤ k,ℓ=0 X we obtain, again under (H2): n−1 n2H ( 2E h′(B )g′(B ) ε ,δ ε ,δ k/n ℓ/n k/n k/n H ℓ/n k/n H | { }h i h i | k,ℓ=0 X + E h(B )g′′(B ) ε ,δ 2 ) Cn. k/n ℓ/n ℓ/n k/n H | { }h i | ≤ WEIGHTED QUADRATICAND CUBIC VARIATIONSOFFBM 9 Finally,recallingthatH <1/4<1/2,equality(2.2)followssincen=o(n2−2H). (cid:3) Lemma 2.3. For h,g:R R verifying (H4), we have → n−1 E h(B )g(B )[n2H(∆B )2 1][n2H(∆B )2 1] k/n ℓ/n k/n ℓ/n { − − } k,ℓ=0 X (2.7) n−1 = 1 n−4H E h′′(B )g′′(B ) +o(n2−4H). 16 { k/n ℓ/n } k,ℓ=0 X Proof. For 0 ℓ,k n 1, we can write ≤ ≤ − E h(B )g(B )[n2H(∆B )2 1]n2H(∆B )2 k/n ℓ/n k/n ℓ/n { − } =E h(B )g(B )[n2H(∆B )2 1]n2H∆B I(δ ) k/n ℓ/n k/n ℓ/n ℓ/n { − } =E h′(B )g(B )[n2H(∆B )2 1]n2H∆B ε ,δ k/n ℓ/n k/n ℓ/n k/n ℓ/n H { − }h i +E h(B )g′(B )[n2H(∆B )2 1]n2H∆B ε ,δ k/n ℓ/n k/n ℓ/n ℓ/n ℓ/n H { − }h i +2E h(B )g(B )n4H∆B ∆B δ ,δ k/n ℓ/n k/n ℓ/n k/n ℓ/n H { }h i +E h(B )g(B )[n2H(∆B )2 1] . k/n ℓ/n k/n { − } Thus, E h(B )g(B )[n2H(∆B )2 1][n2H(∆B )2 1] k/n ℓ/n k/n ℓ/n { − − } =E h′(B )g(B )[n2H(∆B )2 1]n2HI(δ ) ε ,δ k/n ℓ/n k/n ℓ/n k/n ℓ/n H { − }h i +E h(B )g′(B )[n2H(∆B )2 1]n2HI(δ ) ε ,δ k/n ℓ/n k/n ℓ/n ℓ/n ℓ/n H { − }h i +2E h(B )g(B )n4H∆B I(δ ) δ ,δ k/n ℓ/n k/n ℓ/n k/n ℓ/n H { }h i =n2HE h′′(B )g(B )[n2H(∆B )2 1] ε ,δ 2 k/n ℓ/n k/n k/n ℓ/n H { − }h i +2n2HE h′(B )g′(B )[n2H(∆B )2 1] k/n ℓ/n k/n { − } ε ,δ ε ,δ k/n ℓ/n H ℓ/n ℓ/n H ×h i h i +4n4HE h′(B )g(B )∆B ε ,δ δ ,δ k/n ℓ/n k/n k/n ℓ/n H k/n ℓ/n H { }h i h i +4n4HE h(B )g′(B )∆B ε ,δ δ ,δ k/n ℓ/n k/n ℓ/n ℓ/n H k/n ℓ/n H { }h i h i +2n4HE h(B )g(B ) δ ,δ 2 k/n ℓ/n k/n ℓ/n H { }h i +n2HE h(B )g′′(B )[n2H(∆B )2 1] ε ,δ 2 k/n ℓ/n k/n ℓ/n ℓ/n H { − }h i 10 I.NOURDIN 6 , Ai . k,ℓ,n i=1 X We claim that, for 1 i 5, we have n−1 Ai =o(n2−4H). Let us first ≤ ≤ k,ℓ=0| k,ℓ,n| consider the case where i=1. By using Lemma 2.1, we deduce that P n2H ε ,δ = 1((ℓ+1)2H ℓ2H ℓ k+12H + ℓ k 2H) h k/n ℓ/niH 2 − −| − | | − | is bounded. Consequently, under (H4): A1 C ε ,δ . | k,ℓ,n|≤ |h k/n ℓ/niH| AsintheproofofLemma2.2[seemoreprecisely inequality (2.6)],thisyields n−1 A1 Cn.SinceH <1/4,wefinallyobtain n−1 A1 =o(n2−4H). k,ℓ=0| k,ℓ,n|≤ k,ℓ=0| k,ℓ,n| Similarly, by using the fact that n2H ε ,δ is bounded, we prove that P ℓ/n ℓ/n H P h i n−1 A2 =o(n2−4H). k,ℓ=0| k,ℓ,n| Now, let us consider the case of Ai for i=5, the cases where i=3,4 P k,ℓ,n being similar. Again by Lemma 2.1, we have that n2H δ ,δ = 1(k ℓ+12H + k ℓ 12H 2k ℓ 2H) h k/n ℓ/niH 2 | − | | − − | − | − | is bounded. Consequently, under (H4): A5 C(k ℓ+12H + k ℓ 12H 2k ℓ 2H). | k,ℓ,n|≤ | − | | − − | − | − | But, since H <1/2, we have n−1 (k ℓ+12H + k ℓ 12H 2k ℓ 2H) | − | | − − | − | − | k,ℓ=0 X +∞ = (p+12H + p 12H 2p 2H) | | | − | − | | p=−∞ X [(n 1) (n 1 p) 0 ( p)] × − ∧ − − − ∨ − Cn. ≤ This yields n−1 A5 Cn=o(n2−4H), again since H <1/4. k,ℓ=0| k,ℓ,n|≤ It remains to consider the term with A6 . By replacing g by g′′ in P k,ℓ,n identity (2.3)andbyusingargumentssimilartothoseintheproofofLemma 2.2, we can write, under (H4): n−1 n−1 A6 =n4H E h′′(B )g′′(B ) ε ,δ 2 ε ,δ 2 k,ℓ,n { k/n ℓ/n }h k/n k/niHh ℓ/n ℓ/niH k,ℓ=0 k,ℓ=0 X X +o(n2−4H).