VMSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA24 10.15559/15-VMSTA24 Research Article Asymptotic normality of randomized periodogram for estimating quadratic variation in mixed Brownian–fractional Brownian model AzmoodehEhsanehsan.azmoodeh@uni.lua SottinenTommitommi.sottinen@iki.fib ViitasaariLaurilauri.viitasaari@aalto.fic Mathematics Research Unit, Luxembourg University, P.O. Box L-1359, Luxembourg Department of Mathematics and Statistics, University of Vaasa, P.O. Box 700, FIN-65101 Vaasa, Finland Department of Mathematics and System Analysis, Aalto University School of Science, Helsinki, P.O. Box 11100, FIN-00076 Aalto, Finland Department of Mathematics, Saarland University, Post-fach 151150, D-66041 Saarbrücken, Germany Corresponding author. 2015 1152015212949 17112014 3032015 2442015 © 2015 The Author(s). Published by VTeX2015 Open access article under the CC BY license.

We study asymptotic normality of the randomized periodogram estimator of quadratic variation in the mixed Brownian–fractional Brownian model. In the semimartingale case, that is, where the Hurst parameter H of the fractional part satisfies H(3/4,1) , the central limit theorem holds. In the nonsemimartingale case, that is, where H(1/2,3/4] , the convergence toward the normal distribution with a nonzero mean still holds if H=3/4 , whereas for the other values, that is, H(1/2,3/4) , the central convergence does not take place. We also provide Berry–Esseen estimates for the estimator.

Central limit theorem multiple Wiener integrals Malliavin calculus fractional Brownian motion quadratic variation randomized periodogram 60G15 60H07 62F12
Introduction and motivation

The quadratic variation, or the pathwise volatility, of stochastic processes is of paramount importance in mathematical finance. Indeed, it was the major discovery of the celebrated article by Black and Scholes  that the prices of financial derivatives depend only on the volatility of the underlying asset. In the Black–Scholes model of geometric Brownian motion, the volatility simply means the variance. Later the Brownian model was extended to more general semimartingale models. Delbaen and Schachermayer  gave the final word on the pricing of financial derivatives with semimartingales. In all these models, the volatility simply meant the variance or the semimartingale quadratic variance. Now, due to the important article by Föllmer , it is clear that the variance is not the volatility. Instead, one should consider the pathwise quadratic variation. This revelation and its implications to mathematical finance has been studied, for example, in .

An important class of pricing models is the mixed Brownian–fractional Brownian model. This is a model where the quadratic variation is determined by the Brownian part and the correlation structure is determined by the fractional Brownian part. Thus, this is a pricing model that captures the long-range dependence while leaving the Black–Scholes pricing formulas intact. The mixed Brownian–fractional Brownian model has been studied in the pricing context, for example, in .

By the hedging paradigm the prices and hedges of financial derivative depend only on the pathwise quadratic variation of the underlying process. Consequently, the statistical estimation of the quadratic variation is an important problem. One way to estimate the quadratic variation is to use directly its definition by the so-called realized quadratic variation. Although the consistency result (see Section 2.1) does not depend on a specific choice of the sampling scheme, the asymptotic distribution does. There are numerous articles that study the asymptotic behavior of realized quadratic variation; see  and references therein. Another approach, suggested by Dzhaparidze and Spreij , is to use the randomized periodogram estimator. In , the case of semimartingales was studied. In , the randomized periodogram estimator was studied for the mixed Brownian–fractional Brownian model, and the weak consistency of the estimator was proved. This article investigates the asymptotic normality of the randomized periodogram estimator for the mixed Brownian–fractional Brownian model.

The rest of the paper is organized as follows. In Section 2, we briefly introduce the two estimators for the quadratic variation already mentioned. In Section 3, we introduce the stochastic analysis for Gaussian processes needed for our results. In particular, we introduce the Föllmer pathwise calculus and Malliavin calculus. Section 4 contains our main results: the central limit theorem for the randomized periodogram estimator and an associated Berry–Esseen bound. Finally, some technical calculations are deferred into Appendix A.1 and Appendix A.2.

Two methods for estimating quadratic variation Using discrete observations: realized quadratic variation

It is well known that (see [22, Chapter 6]) for a semimartingale X, the bracket [X,X] can be identified with [X,X]t=P-lim|π|0tkπ(XtkXtk1)2, where π={tk:0=t0<t1<<tn=t} is a partition of the interval [0,t] , |π|=max{tktk1:tkπ} , and P-lim means convergence in probability. Statistically speaking, the sums of squared increments (realized quadratic variation) is a consistent estimator for the bracket as the volume of observations tends to infinity. Barndorff-Nielsen and Shephard  studied precision of the realized quadratic variation estimator for a special class of continuous semimartingales. They showed that sometimes the realized quadratic variation estimator can be a rather noisy estimator. So one should seek for new estimators of the quadratic variation.

Using continuous observations: randomized periodogram

Dzhaparidze and Spreij  suggested another characterization of the bracket [X,X] . Let FX be the filtration of X, and τ be a finite stopping time. For λR , define the periodogram Iτ(X;λ) of X at τ by Iτ(X;λ):=|0τeiλsdXs|2=2Re0τ0teiλ(ts)dXsdXt+[X,X]τ(by Itô formula). Let ξ be a symmetric random variable independent of the filtration FX with density gξ and real characteristic function φξ . For given L>0 0$]]>, define the randomized periodogram by EξIτ(X;Lξ)=RIτ(X;Lx)gξ(x)dx. If the characteristic function φξ is of bounded variation, then Dzhaparidze and Spreij have shown that we have the following characterization of the bracket as L : EξIτ(X;Lξ)P[X,X]τ. Recently, the convergence (3) is extended in  to some class of stochastic processes which contains nonsemimartingales in general. Let W={Wt}t[0,T] be a standard Brownian motion, and BH={BtH}t[0,T] be a fractional Brownian motion with Hurst parameter H(12,1) , independent of the Brownian motion W. Define the mixed Brownian–fractional Brownian motion Xt by Xt=Wt+BtH,t[0,T]. It is known that (see ) the process X is an (FX,P) -semimartingale if H(34,1) , and for H(12,34] , X is not a semimartingale with respect to its own filtration FX . Moreover, in both cases, we have [X,X]t=t. If the partitions in (4) are nested, that is, for each n, we have π(n)π(n+1) , then the convergence can be strengthened to almost sure convergence. Hereafter, we always assume that the sequences of partitions are nested. Given λR , define the periodogram of X at T as (1), that is, IT(X;λ)=|0TeiλtdXt|2=|eiλTXTiλ0TXteiλtdt|2=XT2+XT0Tiλ(eiλ(Tt)eiλ(Tt))Xtdt+λ2|0TeiλtXtdt|2. Let (Ω˜,F˜,P˜) be another probability space. We identify the σ-algebra F with F{ϕ,Ω˜} on the product space (Ω×Ω˜,FF˜,PP˜) . Let ξ:Ω˜R be a real symmetric random variable with density gξ and independent of the filtration FX . For any positive real number L, define the randomized periodogram EξIT(X;Lξ) as in (2) by EξIT(X;Lξ):=RIT(X;Lx)gξ(x)dx, where the term IT(X;Lx) is understood as before. Azmoodeh and Valkeila  proved the following: Assume that X is a mixed Brownian–fractional Brownian motion, EξIT(X;Lξ) be the randomized periodogram given by (5), and Eξ2<. Then, as L , we have EξIT(X;Lξ)P[X,X]T. Stochastic analysis for Gaussian processes Pathwise Itô formula Föllmer  obtained a pathwise calculus for continuous functions with finite quadratic variation. The next theorem essentially belongs to Föllmer. For a nice exposition and its use in finance, see Sondermann . ([<xref ref-type="bibr" rid="j_vmsta24_ref_024">24</xref>]). Let X:[0,T]R be a continuous process with continuous quadratic variation [X,X]t , and let FC2(R) . Then for any t[0,T] , the limit of the Riemann–Stieltjes sums lim|π|0titFx(Xti1)(XtiXti1):=0tFx(Xs)dXs exists almost surely. Moreover, we have F(Xt)=F(X0)+0tFx(Xs)dXs+120tFxx(Xs)d[X,X]s. The rest of the section contains the essential elements of Gaussian analysis and Malliavin calculus that are used in this paper. See, for instance, Refs.  for further details. In what follows, we assume that all the random objects are defined on a complete probability space (Ω,F,P) . Isonormal Gaussian processes derived from covariance functions Let X={Xt}t[0,T] be a centered continuous Gaussian process on the interval [0,T] with X0=0 and continuous covariance function RX(s,t) . We assume that F is generated by X. Denote by E the set of real-valued step functions on [0,T] , and let H be the Hilbert space defined as the closure of E with respect to the scalar product 1[0,t],1[0,s]H=RX(t,s),s,t[0,T]. For example, when X is a Brownian motion, H reduces to the Hilbert space L2([0,T],dt) . However, in general, H is not a space of functions, for example, when X is a fractional Brownian motion with Hurst parameter H(12,1) (see ). The mapping 1[0,t]Xt can be extended to a linear isometry between H and the Gaussian space H1 spanned by a Gaussian process X. We denote this isometry by φX(φ) , and {X(φ);φH} is an isonormal Gaussian process in the sense of [18, Definition 1.1.1], that is, it is a Gaussian family with covariance function E(X(φ1)X(φ2))=φ1,φ2H=[0,T]2φ1(s)φ2(t)dRX(s,t),φ1,φ2E, where dRX(s,t)=RX(ds,dt) stands for the measure induced by the covariance function RX on [0,T]2 . Let S be the space of smooth and cylindrical random variables of the form F=f(X(φ1),,X(φn)), where fCb(Rn) (f and all its partial derivatives are bounded). For a random variable F of the form (7), we define its Malliavin derivative as the H -valued random variable DF= i=1nfxi(X(φ1),,X(φn))φi. By iteration, the mth derivative DmFL2(Ω;Hm) is defined for every m2 . For m1 , Dm,2 denotes the closure of S with respect to the norm ·m,2 , defined by the relation Fm,22=E[|F|2]+ i=1mE(DiFHi2). Let δ be the adjoint of the operator D, also called the divergence operator. A random element uL2(Ω,H) belongs to the domain of δ, denoted Dom(δ) , if and only if it satisfies |EDF,uH|cuFL2 for any FD1,2 , where cu is a constant depending only on u. If uDom(δ) , then the random variable δ(u) is defined by the duality relationship E(Fδ(u))=EDF,uH, which holds for every FD1,2 . The divergence operator δ is also called the Skorokhod integral because when the Gaussian process X is a Brownian motion, it coincides with the anticipating stochastic integral introduced by Skorokhod . We denote δ(u)=0TutδXt . For every q1 , the symbol Hq stands for the qth Wiener chaos of X, defined as the closed linear subspace of L2(Ω) generated by the family {Hq(X(h)):hH,hH=1} , where Hq is the qth Hermite polynomial defined as Hq(x)=(1)qex22dqdxq(ex22). We write by convention H0=R . For any q1 , the mapping IqX(hq)=Hq(X(h)) can be extended to a linear isometry between the symmetric tensor product Hq (equipped with the modified norm q!·Hq ) and the qth Wiener chaos Hq . For q=0 , we write by convention I0X(c)=c , cR . For any hHq , the random variable IqX(h) is called a multiple Wiener–Itô integral of order q. A crucial fact is that if H=L2(A,A,ν) , where ν is a σ-finite and nonatomic measure on the measurable space (A,A) , then Hq=Ls2(νq) , where Ls2(νq) stands for the subspace of L2(νq) composed of the symmetric functions. Moreover, for every hHq=Ls2(νq) , the random variable IqX(h) coincides with the q-fold multiple Wiener–Itô integral of h with respect to the centered Gaussian measure (with control ν) generated by X (see ). We will also use the following central limit theorem for sequences living in a fixed Wiener chaos (see ). Let {Fn}n1 be a sequence of random variables in the qth Wiener chaos, q2 , such that limnE(Fn2)=σ2 . Then, as n , the following asymptotic statements are equivalent: Fn converges in law to N(0,σ2) . DFnH2 converges in L2 to qσ2 . To obtain Berry–Esseen-type estimate, we shall use the following result from [17, Corollary 5.2.10]. Let {Fn}n1 be a sequence of elements in the second Wiener chaos such that E(Fn2)σ2 and VarDFnH20 as n . Then, FnlawZN(0,σ2) , and supxR|P(Fn<x)P(Z<x)|2E(Fn2)VarDFnH2+2|E(Fn2)σ2|max{E(Fn2),σ2}. Isonormal Gaussian process associated with two Gaussian processes In this subsection, we briefly describe how two Gaussian processes can be embedded into an isonormal Gaussian process. Let X1 and X2 be two independent centered continuous Gaussian processes with X1(0)=X2(0)=0 and continuous covariance functions RX1 and RX2 , respectively. Assume that H1 and H2 denote the associated Hilbert spaces as explained in Section 3.2. The appropriate set E˜ of elementary functions is the set of the functions that can be written as φ(t,i)=δ1iφ1(t)+δ2iφ2(t) for (t,i)[0,T]×{1,2} , where φ1,φ2E , and δij is the Kronecker’s delta. On the set E˜ , we define the inner product φ,ψH˜:=φ(·,1),ψ(·,1)H1+φ(·,2),ψ(·,2)H2=[0,T]2φ(s,1)ψ(t,1)dRX1(s,t)+[0,T]2φ(s,2)ψ(t,2)dRX2(s,t), where dRXi(s,t)=RXi(ds,dt),i=1,2 . Let H denote the Hilbert space that is the completion of E˜ with respect to the inner product (10). Notice that HH1H2 , where H1H2 is the direct sum of the Hilbert spaces H1 and H2 , that is, it is a Hilbert space consisting of elements of the form of ordered pairs (h1,h2)H1×H2 equipped with the inner product (h1,h2),(g1,g2)H1H2:=h1,g1H1+h2,g2H2 . Now, for any φE˜ , we define X(φ):=X1(φ(·,1))+X2(φ(·,2)) . Using the independence of X1 and X2 , we infer that E(X(φ)X(ψ))=φ1,ψH for all φ,ψE˜ . Hence, the mapping X can be extended to an isometry on H , and therefore {X(h),hH} defines an isonormal Gaussian process associated to the Gaussian processes X1 and X2 . Malliavin calculus with respect to (mixed Brownian) fractional Brownian motion The fractional Brownian motion BH={BtH}tR with Hurst parameter H(0,1) is a zero-mean Gaussian process with covariance function E(BtHBsH)=RH(s,t)=12(|t|2H+|s|2H|ts|2H). Let H denote the Hilbert space associated to the covariance function RH ; see Section 3.2. It is well known that for H=12 , we have H=L2([0,T]) , whereas for H>12 \frac{1}{2}$]]>, we have L2([0,T])L1H([0,T])|H|H , where |H| is defined as the linear space of measurable functions φ on [0,T] such that φ|H|2:=αH0T0T|φ(s)||φ(t)||ts|2H2dsdt<, where αH=H(2H1) .

([<xref ref-type="bibr" rid="j_vmsta24_ref_018">18</xref>], Chapter 5).

Let H denote the Hilbert space associated to the covariance function RH for H(0,1) . If H=12 , that is, BH is a Brownian motion, then for any φ,ψH=L2([0,T],dt) , the inner product of H is given by the well-known Itô isometry E(B12(φ)B12(ψ))=φ,ψH=0Tφ(t)ψ(t)dt. If H>12 \frac{1}{2}$]]>, then for any φ,ψ|H| , we have E(BH(φ)BH(ψ))=φ,ψH=αH0T0Tφ(s)ψ(t)|ts|2H2dsdt. The following proposition establishes the link between pathwise integral and Skorokhod integral in Malliavin calculus associated to fractional Brownian motion and will play an important role in our analysis. ([<xref ref-type="bibr" rid="j_vmsta24_ref_018">18</xref>]). Let u={ut}t[0,T] be a stochastic process in the space D1,2(|H|) such that almost surely 0T0T|Dsut||ts|2H2dsdt<. Then u is pathwise integrable, and we have 0TutdBtH=0TutδBtH+αH0T0TDsut|ts|2H2dsdt. For further use, we also need the following ancillary facts related to the isonormal Gaussian process derived from the covariance function of the mixed Brownian–fractional Brownian motion. Assume that X=W+BH stands for a mixed Brownian–fractional Brownian motion with H>12 \frac{1}{2}$]]>. We denote by H the Hilbert space associated to the covariance function of the process X with inner product ·,·H . Then a direct application of relation (10) and Proposition 1 yields the following facts. We recall that in what follows the notations I1X and I2X stand for multiple Wiener integrals of orders 1 and 2 with respect to isonormal Gaussian process X; see Section 3.2.

For any φ1,φ2,ψ1,ψ2L2([0,T]) , we have E(I1X(φ)I1X(ψ))=φ,ψH=0Tφ(t)ψ(t)dt+αH0T0Tφ(s)ψ(t)|ts|2H2dsdt. Moreover, E(I2X(φ1φ2)I2X(ψ1ψ2))=2φ1φ2,ψ1ψ2H2=[0,T]2φ1(s1)ψ1(s1)φ2(s2)ψ2(s2)ds1ds2+αH[0,T]3φ1(s1)ψ1(s1)φ2(s2)ψ2(t2)|t2s2|2H2ds1ds2dt2+αH[0,T]3φ1(s1)ψ1(t1)φ2(s1)ψ2(s1)|t1s1|2H2ds1dt1ds1+αH2[0,T]4φ1(s1)ψ1(t1)φ2(s2)ψ2(t2)×|t1s1|2H2|t2s2|2H2ds1dt1ds2dt2.

Main results

Throughout this section, we assume that X=W+BH is a mixed Brownian–fractional Brownian motion with H>12 \frac{1}{2}$]]>, unless otherwise stated. We denote by H the Hilbert space associated to process X with inner product ·,·H . Central limit theorem We start with the following fact, which is one of our key ingredients. ([<xref ref-type="bibr" rid="j_vmsta24_ref_002">2</xref>]). Let Eξ2< . Then the randomized periodogram of the mixed Brownian–fractional Brownian motion X given by (5) satisfies EξIT(X;Lξ)=[X,X]T+20T0tφξ(L(ts))dXsdXt, where φξ is the characteristic function of ξ, and the iterated stochastic integral in the right-hand side is understood pathwise, that is, as the limit of the Riemann–Stieltjes sums; see Section 3.1. Our first aim is to transform the pathwise integral in (13) into the Skorokhod integral. This is the topic of the next lemma. Let ut=0tφξ(L(ts))dXs , where φξ denotes the characteristic function of a symmetric random variable ξ. Then uDom(δ) , and 0TutdXt=0TutδXt+αH0T0TDs(BH)ut|ts|2H2dsdt, where the stochastic integral in the right-hand side is the Skorokhod integral with respect to mixed Brownian–fractional Brownian motion X, and D(BH) denotes the Malliavin derivative operator with respect to the fractional Brownian motion BH . First, note that ut=utW+utBH=0tφξ(L(ts))dWs+0tφξ(L(ts))dBsH. Moreover, E(0Tut2dt)< , so that utD1,2 for almost all t[0,T] and E([0,T]2(Dsut)2dsdt)< . Hence, uDom(δ) by [18, Proposition 1.3.1]. On the other hand, 0TutdXt=0TutdWt+0TutdBtH=0TutWdWt+0TutBHdWt+0TutWdBtH+0TutBHdBtH=0TutWδWt+0TutBHδWt+0TutWδBtH+0TutBHδBtH+αH0T0TDs(BH)utBH|ts|2H2dsdt=0TutδWt+0TutδBtH+αH0T0TDs(BH)ut|ts|2H2dsdt, where we have used the independence of W and BH , Proposition 2, and the fact that for adapted integrands, the Skorokhod integral coincides with the Itô integral. To finish the proof, we use the very definition of Skorokhod integral and relation (8) to obtain that 0TutδWt+0TutδBtH=0TutδXt . □ We will also pose the following assumption for characteristic function φξ of a symmetric random variable ξ. The characteristic function φξ satisfies 0|φξ(x)|dx<. Note that Assumption 1 is satisfied for many distributions. Especially, if the characteristic function φξ is positive and the density function gξ(x) is differentiable, then we get by applying Fubini’s theorem and integration by part that 0φξ(x)dx=200cos(yx)gξ(y)dydx=πgξ(0)<. We continue with the following technical lemma, which in fact provides a correct normalization for our central limit theorems. Consider the symmetric two-variable function ψL(s,t):=φξ(L|ts|) on [0,T]×[0,T] . Then ψLH2 , and moreover, as L , we have limLLψLH22=σT2<, where σT2:=2T0φξ2(x)dx is independent of the Hurst parameter H. We point it out that the variance σT2 in Lemma 4 is finite. This is a simple consequence of Assumption 1 and the fact that the characteristic function φξ is bounded by one over the real line. Throughout the proof, C denotes unimportant constant depending on T and H, which may vary from line to line. First, note that clearly ψLH2 since ψL is a bounded function. In order to prove (14), we show that, as L , ψLH221L. Next, by applying Lemma 1 we obtain ψLH22=A1+A2+A3 , where A1:=[0,T]2φξ2(L|ts|)dtds, A2:=αH[0,T]3φξ(L|tu|)φξ(L|su|)|ts|2H2dtdsdu, A3:=αH2[0,T]4φξ(L|tu|)φξ(L|sv|)|ts|2H2|vu|2H2dudvdtds. First, we show that A11L . By change of variables y=LTs and x=LTt we obtain A1=T2L20L0Lφξ2(T|xy|)dxdy. Now, by applying L’Hôpital’s rule and some elementary computations we obtain that limLL10L0Lφξ2(T|xy|)dxdy=limL20Lφξ2(T(Lx))dx=2T0φξ2(y)dy, which is finite by Assumption 1. Consequently, we get limLLA1=2T0φξ2(y)dy, or, in other words, A1L1 . To complete the proof, it is shown in Appendix B that limLL(A2+A3)=0 . □ We also apply the following proposition. The proof is rather technical and is postponed to Appendix A. Consider the symmetric two-variable function ψL(s,t):=φξ(L|ts|) on [0,T]×[0,T] . Denote ψ˜L(t,s)=ψL(s,t)2ψLH2. Then, for any H(12,1) , as L , we have I2X(ψ˜L)lawN(0,1). Our main theorem is the following. Assume that the characteristic function φξ of a symmetric random variable ξ satisfies Assumption 1 and let σT2 be given by (14). Then, as L , we have the following asymptotic statements: if H(34,1) , then L(EξIT(X;Lξ)[X,X]T)lawN(0,σT2). if H=34 , then L(EξIT(X;Lξ)[X,X]T)lawN(μ,σT2), where μ=2αHT0φξ(x)x2H2dx . if H(12,34) , then L2H1(EIT(X;Lξ)[X,X]T)Pμ, where the real number μ is given in item 2. Notice that when H(12,34) , we have 2H1<12 . First, by applying Lemmas 2 and 3 we can write EIT(X;Lξ)[X,X]T=I2X(ψL)+αH0T0Tφξ(L|ts|)|ts|2H2dsdt. Consequently, we obtain L(EIT(X;Lξ)[X,X]T)=LI2X(ψL)+LαH0T0Tφξ(L|ts|)|ts|2H2dsdt:=A1+A2. Now, thanks to Proposition 3, for any H(12,1) , we have A1=LψLH2I2X(ψ˜L)lawN(0,σT2), where σH2 is given by (14). Hence, it remains to study the term A2 . Using change of variables y=LTs and x=LTt , we obtain 0T0Tφξ(L|ts|)|ts|2H2dsdt=T2HL2H0L0Lφξ(T|xy|)|xy|2H2dxdy, where by L’Hôpital’s rule we obtain limLL10L0Lφξ(T|xy|)|xy|2H2dxdy=2T12H0φξ(x)x2H2dx. Note also that the integral in the right-hand side of the last identity is finite by Assumption 1. Consequently, we obtain limLL2H1αH0T0Tφξ(L|ts|)|ts|2H2dsdt=2αHT0φξ(x)x2H2dx=μ. Therefore, limLA2=limLL322Hμ, which converges to zero for H(34,1) , and item 1 of the claim is proved. Similarly, for H=34 , we obtain limLA2=μ, which proves item 2 of the claim. Finally, for item 3, from (18) we infer that, as L , L2H1αH0T0Tφξ(L|ts|)|ts|2H2dsdtμ. Furthermore, for the term I2X(ψL) , we obtain L2H1I2X(ψL)=L2H32×LI2X(ψL)P0 as L . This is because H<34 implies 2H32<0 and moreover LI2X(ψL)lawN(0,1) and L2H320 . □ When X=W is a standard Brownian motion, that is, if the fractional Brownian motion part drops, then with similar arguments as in Theorem 5, we obtain L(EξIT(X;Lξ)[X,X]T)lawN(0,σT2), where σT2=2T0φξ2(x)dx , and φξ is the characteristic function of ξ. Note that the proof of Theorem 5 reveals that in the case H(12,34) , for any ϵ>322H \frac{3}{2}-2H$]]>, we have that, as L , L(EIT(X;Lξ)[X,X]T)P, and, moreover, L12ϵ(EIT(X;Lξ)[X,X]T)P0.

The Berry–Esseen estimates

As a consequence of the proof of Theorem 5, we also obtain the following Berry–Esseen bound for the semimartingale case.

Let all the assumptions of Theorem 5 hold, and let H(34,1) . Furthermore, let ZN(0,σT2) , where the variance σT2 is given by (14). Then there exists a constant C (independent of L) such that for sufficiently large L, we have supxR|P(L(Eξ(IT(X;Lξ)[X,X]T)<x)P(Z<x)|Cρ(L), where ρ(L)=max{L322H,Lφξ2(Tz)dz}.

By proof of Theorem 5 we have L(EIT(X;Lξ)[X,X]T)=LI2X(ψL)+LαH0T0Tφξ(L|ts|)|ts|2H2dsdt=:A1+A2, where A1=2LψLH2I2X(ψ˜L). Now, we know that the deterministic term A2 converges to zero with rate L322H and the term A1lawN(0,σT2) . Hence, in order to complete the proof, it is sufficient to show that supxR|P(A1<x)P(Z<x)|Cρ(L).

Now, by using the proof of Proposition 3 in Appendix A we have VarDFLH2L12L322H. Finally, using the notation of the proof of Lemma 4, we have E(Fn2)=LψLH22=L×(A1+A2+A3), where A2+A3CL2H . Consequently, L×(A2+A3)CL12HCL322H. To complete the proof, we have LA1=T2L0L0Lφξ2(T|xy|)dydx=T2L0LxLxφξ2(Tz)dzdx=T2LLLzLzφξ2(Tz)dxdz=T2LLφξ2(Tz)dz=2T20Lφξ2(Tz)dz. This gives us LA1σT2=2T2Lφξ2(Tz)dz. Now, the claim follows by an application of Theorem 4.  □

In many cases of interest, the leading term in ρ(L) is the polynomial term L322H , which reveals that the role of the particular choice of φξ affects only to the constant. In particular, if φξ admits an exponential decay, that is, |φξ(t)|C1eC2t for some constants C1,C2>0 0$]]>, then Lφξ2(Tz)dzC3eC4LCL322H for some constants C3,C4,C>0 0$]]>. As examples, this is the case if ξ is a standard normal random variable with characteristic function φξ(t)=et22 or if ξ is a standard Cauchy random variable with characteristic function φξ(t)=e|t| .

Consider the case X=W , that is, X is a standard Brownian motion. In this case, the correction term A2 in the proof of Theorem 5 disappears, and we have E(FL2)σT2=2T2Lφξ2(Tx)dx. Furthermore, by applying L’Hôpital’s rule twice and some elementary computations it can be shown that E[DFLH2EDFLH2]2|φξ(TL)|L1. Consequently, in this case, we obtain the Berry–Esseen bound supxR|P(L(Eξ(IT(X;Lξ)[X,X]T)<x)P(Z<x)|Cρ(L), where ρ(L)=max{|φξ(TL)|L1,Lφξ2(Tz)dz}, which is in fact better in many cases of interest. For example, if φξ admits an exponential decay, then we obtain ρ(L)ecL for some constant c.

Acknowledgments

Azmoodeh is supported by research project F1R-MTH-PUL-12PAMP from University of Luxembourg, and Lauri Viitasaari was partially funded by Emil Aaltonen Foundation. The authors are grateful to Christian Bender for useful discussions.

Appendix section Proof of Proposition <xref rid="j_vmsta24_stat_017">3</xref>

Denote FL=I2X(ψ˜L) and note that by the definition of ψ˜L we have E(FL2)=1 . Hence, it is sufficient to prove that, as L , E[DFLH2EDFLH2]20. Now, using the definition of the Malliavin derivative, we get DsFL=2I1X(ψ˜L(s,·))=2ψLH2I1X(φξ(L|s·|)). For the rest of the proof, C denotes unimportant constants, which may vary from line to line. Furthermore, we also use the short notation K(ds,dt)=δ0(ts)dsdt+αH|ts|2H2dsdt, where δ0 denotes the Kronecker delta function, to denote the measure associated to the Hilbert space H generated by the mixed Brownian–fractional Brownian motion X. Furthermore, without loss of generality, we assume that φξ0 . Indeed, otherwise we simply approximate the integral by taking absolute values inside the integral, which is consistent with Assumption 1. Now we have DsFLH2=CψLH220T0TI1X(φξ(L|u·|))I1X(φξ(L|v·|))K(du,dv). Next, using the multiplication formula for multiple Wiener integrals, we see that I1X(φξ(L|u·|))I1X(φξ(L|v·|))=φξ(L|u·|),φξ(L|v·|)H+I2X(φξ(L|u·|)˜φξ(L|v·|))=:J1(u,v)+J2(u,v), where the term J1 is deterministic, and J2 has expectation zero. Hence, we need to show that E[1ψLH220T0TJ2(u,v)K(du,dv)]20. Therefore, by applying Fubini’s theorem it suffices to show that, as L , 1ψLH24[0,T]4E[J2(u1,v1)J2(u2,v2)]K(du1,dv1)K(du2,dv2)0. First, using isometry (iii) [18, p. 9] , we get that E[J2(u1,v1)J2(u2,v2)]=2[0,T]4(φξ(L|u1·|)˜φξ(L|v1·|))(x1,y1)×(φξ(L|u2·|)˜φξ(L|v2·|))(x2,y2)K(dx1,dx2)K(dy1,dy2). By plugging into (20) we obtain that it suffices to have 1ψLH24[0,T]8(φξ(L|u1·|)˜φξ(L|v1·|))(x1,y1)×(φξ(L|u2·|)˜φξ(L|v2·|))(x2,y2)×K(dx1,dx2)K(dy1,dy2)K(du1,dv1)K(du2,dv2)0. The rest of the proof is based on similar arguments as the proof of Lemma 4. Indeed, again by the symmetric property of measures K(dx,dy) and functions φξ(L|u1·|)˜φξ(L|v1·|) we obtain five different terms, denoted by Ak , k=1,2,3,4,5 , of the forms A1=[0,T]4φξ(L|ux|)φξ(L|uy|)φξ(L|vx|)φξ(L|yv|)dxdydvdu,A2=αH[0,T]5φξ(L|ux1|)φξ(L|uy|)φξ(L|vx2|)φξ(L|yv|)×|x1x2|2H2dx1dx2dydvdu,A3=αH2[0,T]6φξ(L|ux1|)φξ(L|uy1|)φξ(L|vx2|)φξ(L|y2v|)×|x1x2|2H2|y1y2|2H2dx1dx2dy1dy2dvdu,A4=αH3[0,T]7φξ(L|u1x1|)φξ(L|v1y1|)φξ(L|vx2|)φξ(L|y2v|)×|x1x2|2H2|y1y2|2H2|u1v1|2H2dx1dx2dy1dy2dv1du1dv,A5=αH4[0,T]8φξ(L|u1x1|)φξ(L|v1y1|)φξ(L|u2x2|)×φξ(L|y2v2|)|x1x2|2H2|y1y2|2H2|u1v1|2H2×|u2v2|2H2dx1dx2dy1dy2dv1du1dv2du2. Next, we prove that A3CL3 . First, by change of variables we obtain A3=CL4H2[0,L]6φξ(T|ux1|)φξ(T|uy1|)φξ(T|vx2|)φξ(T|y2v|)×|x1x2|2H2|y1y2|2H2dx1dx2dy1dy2dvdu. Note that Assumption 1 implies that 0Lφξ(T|xy|)dxC , where the constant C does not depend on L and y. Similarly, we have 0L|xy|2H2dxCL2H1, where again the constant C is independent of L and y. Moreover, we have φξ(T|uv|)1 for any u,vR . Hence, we can estimate A3CL4H2[0,L]6φξ(T|ux1|)φξ(T|uy1|)φξ(T|vx2|)×φξ(T|y2v|)|x1x2|2H2|y1y2|2H2dx1dx2dy1dy2dvduCL4H2[0,L]61×φξ(T|uy1|)φξ(T|vx2|)φξ(T|y2v|)×|x1x2|2H2|y1y2|2H2dx1dx2dy1dy2dvdu=CL4H2[0,L]4φξ(T|vx2|)φξ(T|y2v|)|y1y2|2H2×([0,L]2φξ(T|uy1|)|x1x2|2H2dudx1)dx2dy1dy2dvCL4H2×L2H1[0,L]4φξ(T|vx2|)φξ(T|y2v|)×|y1y2|2H2dx2dy1dy2dv=CL2H3[0,L]3φξ(T|y2v|)|y1y2|2H2×(0Lφξ(T|vx2|)dx2)dy1dy2dvCL2H3[0,L]2|y1y2|2H2(0Lφξ(T|y2v|)dv)dy1dy2CL2H3[0,L]2|y1y2|2H2dy1dy2=CL3. To conclude, treating A1 , A2 , A4 , and A5 similarly, we deduce that k=15|Ak|CL3. Hence, by applying ψLH22L1 we obtain (21), which completes the proof.

Analysis of the variance

We have A2=αH[0,T]3φξ(L|tu|)φξ(L|su|)|ts|2H2dtdsdu, A3=αH2[0,T]4φξ(L|tu|)φξ(L|sv|)|ts|2H2|vu|2H2dudvdtds, which, by change of variable, leads to A2=αHT2H+1L2H1[0,L]3φξ(T|tu|)φξ(T|su|)|ts|2H2dtdsdu,A3=αH2T4HL4H×[0,L]4φξ(T|tu|)φξ(T|sv|)|ts|2H2|vu|2H2dudvdtds. We begin with the term A2 . Denote A˜2(L)=[0,L]3φξ(T|tu|)φξ(T|su|)|ts|2H2dtdsdu. By differentiating we get dA˜2dL(L)=2[0,L]2φξ(T|Lu|)φξ(T|uv|)|Lv|2H2dvdu+[0,L]2φξ(T|Lu|)φξ(T|Lv|)|uv|2H2dudv=:J1+J2. First, we analyze the term J1 . Similarly to Appendix A, we assume that φξ0 . Hence, we have 12J1=[0,L]2φξ(T|Lu|)φξ(T|uv|)|Lv|2H2dvdu=[0,L]2φξ(Tu)φξ(T|uv|)v2H2dvdu=0L1Lφξ(Tu)φξ(T|uv|)v2H2dvdu+0L01φξ(Tu)φξ(T|uv|)v2H2dvdu0L1Lφξ(Tu)φξ(T|uv|)dvdu+0L01φξ(Tu)v2H2dvduC. For the term J2 , we write J2=[0,L]2φξ(T|Lu|)φξ(T|Lv|)|uv|2H2dudv=[0,L]2φξ(Tu)φξ(Tv)|uv|2H2dudv=20L0tφξ(Tu)φξ(Tv)(vu)2H2dudv=2(010t+1L0t1+1Lt1t)φξ(Tu)φξ(Tv)(vu)2H2dudv=:J2,1+J2,2+J2,3. Now, it is straightforward to show that J2,1+J2,2C . Consequently, as L , we obtain A2L2H1A˜2L2H(J1+J2,1+J2,2+J2,3), where L2H(J1+J2,1+J2,2)L2H. For the term J2,3 , we write J2,3(L)=1Lt1tφξ(Tu)φξ(Tv)(vu)2H2dudv, so that dJ2,3dL(L)=L1Lφξ(TL)φξ(Tv)(Lv)2H2dvCφξ(TL). Hence, by L’Hôpital’s rule we have L2HJ2,3L12Hφξ(TL) . On the other hand, we have φξ(TL)=o(L2H2) since φ is integrable by Assumption 1. Hence, L2HJ2,3=o(L1) , which shows that limLLA2=0 . Consequently, A2 does not affect the variance. The term A3 is easier and can be treated with similar elementary computations together with L’Hôpital’s rule. As a consequence, we obtain A3L2H , so that limLLA3=0 . Hence, A3 does not affect the variance either, which justifies (14).

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