Asymptotic normality of randomized periodogram for estimating quadratic variation in mixed Brownian–fractional Brownian model

DOI:10.15559/15-VMSTA24

Asymptotic normality of randomized periodogram for estimating quadratic variation in mixed

Brownian–fractional Brownian model

Ehsan Azmoodeh^a,∗, Tommi Sottinen^b, Lauri Viitasaari^c

aMathematics Research Unit, Luxembourg University, P.O. Box L-1359, Luxembourg

bDepartment of Mathematics and Statistics, University of Vaasa, P.O. Box 700, FIN-65101 Vaasa, Finland

cDepartment 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

ehsan.azmoodeh@uni.lu(E. Azmoodeh),tommi.sottinen@iki.fi(T. Sottinen), lauri.viitasaari@aalto.fi(L. Viitasaari)

Received: 17 November 2014, Revised: 30 March 2015, Accepted: 24 April 2015, Published online: 11 May 2015

Abstract We study asymptotic normality of the randomized periodogram estimator of qua- dratic variation in the mixed Brownian–fractional Brownian model. In the semimartingale case, that is, where the Hurst parameterH of the fractional part satisfiesH ∈(3/4,1), the central limit theorem holds. In the nonsemimartingale case, that is, whereH ∈(1/2,3/4], the conver- gence toward the normal distribution with a nonzero mean still holds ifH =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.

Keywords Central limit theorem, multiple Wiener integrals, Malliavin calculus, fractional Brownian motion, quadratic variation, randomized periodogram

2010 MSC 60G15, 60H07, 62F12

∗Corresponding author.

© 2015 The Author(s). Published by VTeX. Open access article under theCC BYlicense.

www.i-journals.org/vmsta

1 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 [8] 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 [10,11] 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 [13], 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 [6,23].

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 Brown- ian 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 [1,5,7].

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-calledreal- ized quadratic variation. Although the consistency result (see Section2.1) 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 [4,3,16,14,15] and references therein. Another approach, suggested by Dzhaparidze and Spreij [12], is to use the randomized periodogram estimator. In [12], the case of semimartingales was studied. In [2], 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 asymp- totic normality of the randomized periodogram estimator for the mixed Brownian–

fractional Brownian model.

The rest of the paper is organized as follows. In Section2, we briefly introduce the two estimators for the quadratic variation already mentioned. In Section3, we in- troduce the stochastic analysis for Gaussian processes needed for our results. In par- ticular, we introduce the Föllmer pathwise calculus and Malliavin calculus. Section4 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 AppendixA.1and AppendixA.2.

2 Two methods for estimating quadratic variation 2.1 Using discrete observations: realized quadratic variation

It is well known that (see [22, Chapter 6]) for a semimartingaleX, the bracket[X, X] can be identified with

[X, X]t =P- lim

|π|→0

t_k∈π

(X_tk −X_tk−1)²,

whereπ = {tk : 0 = t0 < t1 < · · · < tn = t}is a partition of the interval[0, t],

|π| = max{tk −tk−1 : tk ∈ π}, andP- lim means convergence in probability. Sta- tistically 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 [3] studied precision of the realized quadratic vari- ation 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.

2.2 Using continuous observations: randomized periodogram

Dzhaparidze and Spreij [12] suggested another characterization of the bracket[X, X].

LetF^Xbe the filtration ofX, andτ be a finite stopping time. Forλ ∈ R, define the periodogramIτ(X;λ)ofXatτ by

I_τ(X;λ): = τ

0

e^iλsdXs

²

=2Re _τ

0

_t

0

e^iλ(t−s)dXsdXt+ [X, X]τ (by Itô formula). (1) Letξ be a symmetric random variable independent of the filtrationF^Xwith density gξand real characteristic functionϕξ. For givenL >0, define the randomized peri- odogram by

EξIτ(X;Lξ )=

RIτ(X;Lx)gξ(x)dx. (2) If the characteristic functionϕ_ξis of bounded variation, then Dzhaparidze and Spreij have shown that we have the following characterization of the bracket asL→ ∞:

EξI_τ(X;Lξ )→ [^P X, X]τ. (3) Recently, the convergence (3) is extended in [2] to some class of stochastic pro- cesses which contains nonsemimartingales in general. LetW = {W_t}t∈[0,T] be a standard Brownian motion, andB^H = {B_t^H}t∈[0,T]be a fractional Brownian motion with Hurst parameterH ∈ (¹₂,1), independent of the Brownian motionW. Define the mixed Brownian–fractional Brownian motionX_t by

X_t =W_t+B_t^H, t∈ [0, T].

Remark 1. It is known that(see [9])the processXis an(F^X,P)-semimartingale if H ∈ (³₄,1), and forH ∈ (¹₂,³₄],Xis not a semimartingale with respect to its own filtrationF^X. Moreover, in both cases, we have

[X, X]t =t. (4)

If the partitions in (4) are nested, that is, for eachn, we haveπ⁽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.

Givenλ∈R, define the periodogram ofXatT as(1), that is, I_T(X;λ)=

_T

0

e^iλtdXt

²

=

e^iλTX_T −iλ _T

0

X_te^iλtdt ²

=X²_T +X_{T T}

0

iλ

e^{iλ(T−t )}−e^{−iλ(T−t )}

X_tdt+λ² _T

0

e^iλtX_tdt ². Let(Ω,˜ F˜,P˜)be another probability space. We identify theσ-algebra F with F⊗ {φ,Ω˜}on the product space(Ω× ˜Ω,F⊗ ˜F,P⊗ ˜P). Letξ : ˜Ω→Rbe a real symmetric random variable with densityg_ξand independent of the filtrationF^X. For any positive real numberL, define the randomized periodogramEξI_T(X;Lξ )as in (2) by

EξI_T(X;Lξ ):=

RI_T(X;Lx)g_ξ(x)dx, (5) where the termI_T(X;Lx)is understood as before. Azmoodeh and Valkeila [2] proved the following:

Theorem 1. Assume that X is a mixed Brownian–fractional Brownian motion, EξI_T(X;Lξ )be the randomized periodogram given by(5), and

Eξ²<∞. Then, asL→ ∞, we have

EξI_T(X;Lξ )−→ [^P X, X]T.

3 Stochastic analysis for Gaussian processes 3.1 Pathwise Itô formula

Föllmer [13] obtained a pathwise calculus for continuous functions with finite quadratic variation. The next theorem essentially belongs to Föllmer. For a nice ex- position and its use in finance, see Sondermann [24].

Theorem 2([24]). LetX : [0, T] → Rbe a continuous process with continuous quadratic variation[X, X]t, and letF ∈C²(R). Then for anyt ∈ [0, T], the limit of the Riemann–Stieltjes sums

|πlim|→0

t_it

Fx(Xt_i−1)(Xt_i−Xt_i−1):=

_t

0

Fx(Xs)dXs

exists almost surely. Moreover, we have

F (X_t)=F (X₀)+ _t

0

F_x(X_s)dXs+1 2

_t

0

F_xx(X_s)d[X, X]s. (6) 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. [17,18] for further details. In what follows, we assume that all the random objects are defined on a complete probability space(Ω,F,P).

3.2 Isonormal Gaussian processes derived from covariance functions

LetX= {X_t}t∈[0,T]be a centered continuous Gaussian process on the interval[0, T] withX₀ = 0 and continuous covariance functionR_X(s, t ). We assume that F is generated byX. Denote byE the set of real-valued step functions on[0, T], and let Hbe the Hilbert space defined as the closure ofEwith 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,His not a space of functions, for example, when Xis a fractional Brownian motion with Hurst parameterH ∈ (¹₂,1)(see [21]). The mapping 1_[0,t] −→ X_t can be extended to a linear isometry between H and the Gaussian spaceH1 spanned by a Gaussian processX. We denote this isometry by ϕ −→X(ϕ), and{X(ϕ);ϕ ∈ H}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, ϕ_{2 H}

=

[0,T]²ϕ1(s)ϕ2(t )dRX(s, t ), ϕ1, ϕ2∈E,

where dRX(s, t ) = RX(ds,dt ) stands for the measure induced by the covariance functionRXon[0, T]². LetSbe the space of smooth and cylindrical random vari- ables of the form

F =f

X(ϕ₁), . . . , X(ϕ_n)

, (7)

wheref ∈ C_b^∞(Rⁿ)(f and all its partial derivatives are bounded). For a random variableF of the form (7), we define its Malliavin derivative as theH-valued random variable

DF = n i=1

∂f

∂x_i

X(ϕ₁), . . . , X(ϕ_n) ϕ_i.

By iteration, themth derivativeD^mF ∈L²(Ω;H^⊗m)is defined for everym≥2.

Form≥1,D^m,2denotes the closure ofSwith respect to the norm · m,2, defined by the relation

F²_m,2 = E

|F|² +

m i=1

E

DⁱF²_H⊗i

.

Letδbe the adjoint of the operatorD, also called thedivergence operator. A random elementu∈L²(Ω,H)belongs to the domain ofδ, denoted Dom(δ), if and only if it satisfies

EDF, u H≤cuFL²

for anyF ∈ D^1,2, wherecuis a constant depending only onu. Ifu ∈Dom(δ), then the random variableδ(u)is defined by the duality relationship

E F δ(u)

= EDF, u _H, (8)

which holds for everyF ∈ D^1,2. The divergence operatorδis also called the Sko- rokhod integral because when the Gaussian processX is a Brownian motion, it co- incides with the anticipating stochastic integral introduced by Skorokhod [18]. We denoteδ(u)= ₀^Tu_tδX_t.

For every q ≥ 1, the symbol Hq stands for the qth Wiener chaos of X, de- fined as the closed linear subspace ofL²(Ω)generated by the family{H_q(X(h)) : h∈ H,hH=1}, whereH_qis theqth Hermite polynomial defined as

H_q(x)=(−1)^qe^x

2 2 d^q

dx^q e^−x

2 2

. (9)

We write by conventionH0=R. For anyq ≥1, the mappingI_q^X(h^⊗q)=H_q(X(h)) can be extended to a linear isometry between the symmetric tensor product H^q (equipped with the modified norm√

q! · H⊗q) and theqth Wiener chaosHq. For q=0, we write by conventionI₀^X(c)=c,c∈R. For anyh∈H^q, the random vari- ableI_q^X(h)is called a multiple Wiener–Itô integral of orderq. A crucial fact is that ifH=L²(A,A, ν), whereνis aσ-finite and nonatomic measure on the measurable space(A,A), thenH^q =L²_s(ν^q), whereL²_s(ν^q)stands for the subspace ofL²(ν^q) composed of the symmetric functions. Moreover, for everyh ∈H^q =L²_s(ν^q), the random variableI_q^X(h)coincides with theq-fold multiple Wiener–Itô integral ofh with respect to the centered Gaussian measure (with controlν) generated byX(see [18]). We will also use the following central limit theorem for sequences living in a fixed Wiener chaos (see [20,19]).

Theorem 3. Let{Fn}n≥1be a sequence of random variables in theqth Wiener chaos, q ≥2, such thatlimn→∞E(F_n²)=σ². Then, asn → ∞, the following asymptotic statements are equivalent:

(i) Fnconverges in law toN(0, σ²).

(ii) DFn²_Hconverges inL²toqσ².

To obtain Berry–Esseen-type estimate, we shall use the following result from [17, Corollary 5.2.10].

Theorem 4. Let{F_n}n≥1be a sequence of elements in the second Wiener chaos such thatE(F_n²)→ σ²andVarDF_n²_H→ 0asn→ ∞. Then,F_n ^law→ Z ∼N(0, σ²), and

sup

x∈R

P(F_n< x)−P(Z < x)≤ 2 E(F_n²)

VarDF_n²_H+ 2|E(F_n²)−σ²| max{E(F_n²), σ²}. 3.3 Isonormal Gaussian process associated with two Gaussian processes

In this subsection, we briefly describe how two Gaussian processes can be embed- ded into an isonormal Gaussian process. LetX₁andX₂be two independent centered continuous Gaussian processes withX₁(0)=X₂(0)=0 and continuous covariance functionsR_X1 andR_X2, respectively. Assume thatH₁andH₂denote the associated Hilbert spaces as explained in Section3.2. The appropriate setE˜of elementary func- tions is the set of the functions that can be written asϕ(t, i)=δ1iϕ1(t )+δ2iϕ2(t )for

(t, i)∈ [0, T] × {1,2}, whereϕ1, ϕ2∈E, andδijis the Kronecker’s delta. On the set E, we define the inner product˜

ϕ, ψ _H˜ : =

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 ), (10) where dRXi(s, t )=R_Xi(ds,dt ), i=1,2.

LetH denote the Hilbert space that is the completion ofE˜ with respect to the inner product (10). Notice thatH ∼= H1⊕H2, whereH1⊕H2is the direct sum of the Hilbert spacesH1 andH2, 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)H₁⊕H₂ := h1, g1 H₁ + h2, g2 H₂.

Now, for any ϕ ∈ ˜E, we define X(ϕ) := X₁(ϕ(·,1)) +X₂(ϕ(·,2)). Using the independence ofX₁ andX₂, we infer that E(X(ϕ)X(ψ )) = ϕ₁, ψ _H for all ϕ, ψ∈ ˜E. Hence, the mappingXcan be extended to an isometry onH, and therefore {X(h), h ∈ H}defines an isonormal Gaussian process associated to the Gaussian processesX₁andX₂.

3.4 Malliavin calculus with respect to (mixed Brownian) fractional Brownian motion

The fractional Brownian motionB^H = {B_t^H}t∈Rwith Hurst parameterH ∈(0,1)is a zero-mean Gaussian process with covariance function

E

B_t^HB_s^H

=R_H(s, t )=1 2

|t|^2H + |s|^2H− |t−s|^2H

. (11)

LetHdenote the Hilbert space associated to the covariance functionR_H; see Sec- tion3.2. It is well known that forH = ¹₂, we haveH = L²([0, T]), whereas for H > ¹₂, we haveL²([0, T])⊂L^H1([0, T])⊂ |H| ⊂H, where|H|is defined as the linear space of measurable functionsϕon[0, T]such that

ϕ²_|H|:=α_{H T}

0

_T

0

ϕ(s)ϕ(t )|t−s|^2H−2dsdt <∞, whereα_H =H (2H−1).

Proposition 1([18], Chapter 5). LetHdenote the Hilbert space associated to the covariance functionR_HforH∈(0,1). IfH= ¹₂, that is,B^H is a Brownian motion, then for any ϕ, ψ ∈ H = L²([0, T],dt ), the inner product ofH is given by the well-known Itô isometry

E

B¹²(ϕ)B¹²(ψ )

= ϕ, ψ _H= _T

0

ϕ(t )ψ (t )dt.

IfH > ¹₂, then for anyϕ, ψ∈ |H|, we have E

B^H(ϕ)B^H(ψ )

= ϕ, ψ H=αH

_T

0

_T

0

ϕ(s)ψ (t )|t−s|^2H−2dsdt. (12)

The following proposition establishes the link between pathwise integral and Sko- rokhod integral in Malliavin calculus associated to fractional Brownian motion and will play an important role in our analysis.

Proposition 2 ([18]). Let u = {u_t}t∈[0,T] be a stochastic process in the space D^1,2(|H|)such that almost surely

_T

0

_T

0

|D_su_t||t−s|^2H−2dsdt <∞. Thenuis pathwise integrable, and we have

T 0

utdB_t^H = T

0

utδB_t^H+αH

T 0

T 0

Dsut|t−s|^2H−2dsdt.

For further use, we also need the following ancillary facts related to the isonor- mal Gaussian process derived from the covariance function of the mixed Brownian–

fractional Brownian motion. Assume thatX=W+B^Hstands for a mixed Brownian–

fractional Brownian motion withH > ¹₂. We denote byHthe Hilbert space associated to the covariance function of the processXwith inner product·,· H. Then a direct application of relation (10) and Proposition1 yields the following facts. We recall that in what follows the notationsI₁^XandI₂^X stand for multiple Wiener integrals of orders 1 and 2 with respect to isonormal Gaussian processX; see Section3.2.

Lemma 1. For anyϕ₁, ϕ₂, ψ₁, ψ₂∈L²([0, T]), we have E

I₁^X(ϕ)I₁^X(ψ )

= ϕ, ψ _H

= T

0

ϕ(t )ψ (t )dt+α_H T

0

T 0

ϕ(s)ψ (t )|t−s|^2H−2dsdt.

Moreover, E

I₂^X(ϕ₁⊗ϕ₂)I₂^X(ψ₁⊗ψ₂)

=2ϕ₁⊗ϕ₂, ψ₁⊗ψ_{2 H⊗}2

=

[0,T]²ϕ₁(s₁)ψ₁(s₁)ϕ₂(s₂)ψ₂(s₂)ds1ds2

+α_H

[0,T]³ϕ1(s1)ψ1(s1)ϕ2(s2)ψ2(t2)|t2−s2|^2H−2ds1ds2dt2

+α_H

[0,T]³ϕ₁(s₁)ψ₁(t₁)ϕ₂(s₁)ψ₂(s₁)|t₁−s₁|^2H−2ds1dt1ds1

+α_H²

[0,T]⁴ϕ₁(s₁)ψ₁(t₁)ϕ₂(s₂)ψ₂(t₂)

×|t₁−s₁|^2H−2|t₂−s₂|^2H−2ds1dt1ds2dt2. 4 Main results

Throughout this section, we assume that X = W +B^H is a mixed Brownian–

fractional Brownian motion with H > ¹₂, unless otherwise stated. We denote by Hthe Hilbert space associated to processXwith inner product·,· H.

4.1 Central limit theorem

We start with the following fact, which is one of our key ingredients.

Lemma 2 ([2]). Let Eξ² < ∞. Then the randomized periodogram of the mixed Brownian–fractional Brownian motionXgiven by(5)satisfies

EξI_T(X;L ξ )= [X, X]T +2 _T

0

_t

0

ϕ_ξ

L(t−s)

dXsdXt, (13) whereϕ_ξis the characteristic function ofξ, and the iterated stochastic integral in the right-hand side is understood pathwise, that is, as the limit of the Riemann–Stieltjes sums; see Section3.1.

Our first aim is to transform the pathwise integral in (13) into the Skorokhod integral. This is the topic of the next lemma.

Lemma 3. Letu_t = ₀^tϕ_ξ(L(t−s))dXs, whereϕ_ξdenotes the characteristic function of a symmetric random variableξ. Thenu∈Dom(δ), and

T 0

u_tdXt = T

0

u_tδX_t+α_H T

0

T 0

D^(B_s ^H)u_t|t−s|^2H−2dsdt,

where the stochastic integral in the right-hand side is the Skorokhod integral with respect to mixed Brownian–fractional Brownian motionX, and D^(BH) denotes the Malliavin derivative operator with respect to the fractional Brownian motionB^H. Proof. First, note that

ut =u^W_t +u^B_t ^H = _t

0

ϕξ

L(t−s) dWs+

_t

0

ϕξ

L(t−s) dB_s^H.

Moreover, E( ₀^T u²_tdt ) < ∞, so that ut ∈ D^1,2 for almost all t ∈ [0, T] and E( _[0,T]2(Dsut)²dsdt ) < ∞. Hence,u ∈ Dom(δ)by [18, Proposition 1.3.1]. On the other hand,

_T

0

u_tdXt = _T

0

u_tdWt+ _T

0

u_tdB_t^H

= _T

0

u^W_t dWt+ _T

0

u^B_t ^HdWt+ _T

0

u^W_t dB_t^H + _T

0

u^B_t ^HdB_t^H

= T

0

u^W_t δWt + T

0

u^B_t ^HδWt+ T

0

u^W_t δB_t^H+ T

0

u^B_t^HδB_t^H +α_H

T 0

T 0

D^(B_s ^H)u^B_t ^H|t−s|^2H−2dsdt

= _T

0

u_tδW_t + _T

0

u_tδB_t^H +α_{H T}

0

_T

0

D_s^(BH)u_t|t−s|^2H−2dsdt, where we have used the independence ofW andB^H, 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 ₀^TutδWt + ₀^T utδB_t^H = ₀^T utδXt.

We will also pose the following assumption for characteristic function ϕξ of a symmetric random variableξ.

Assumption 1. The characteristic functionϕ_ξ satisfies _∞

0

ϕ_ξ(x)dx <∞.

Remark 2. Note that Assumption1is satisfied for many distributions. Especially, if the characteristic functionϕξ is positive and the density functiongξ(x)is differen- tiable, then we get by applying Fubini’s theorem and integration by part that

_∞

0

ϕ_ξ(x)dx =2 _∞

0

_∞

0

cos(yx)g_ξ(y)dydx =πg_ξ(0) <∞.

We continue with the following technical lemma, which in fact provides a correct normalization for our central limit theorems.

Lemma 4. Consider the symmetric two-variable functionψ_L(s, t ):=ϕ_ξ(L|t−s|) on[0, T] × [0, T]. Thenψ_L∈H^⊗2, and moreover, asL→ ∞, we have

Llim→∞Lψ_L²_H⊗2 =σ_T²<∞, (14) whereσ_T² :=2T ₀^∞ϕ_ξ²(x)dxis independent of the Hurst parameterH.

Remark 3. We point it out that the variance σ_T² in Lemma 4 is finite. This is a simple consequence of Assumption1and the fact that the characteristic functionϕ_ξ is bounded by one over the real line.

Proof. Throughout the proof,Cdenotes unimportant constant depending onT andH, which may vary from line to line. First, note that clearlyψL ∈ H^⊗2sinceψLis a bounded function. In order to prove (14), we show that, asL→ ∞,

ψ_L²_H⊗2 ∼ 1 L.

Next, by applying Lemma1we obtainψ_L²_H⊗2 =A₁+A₂+A₃, where A₁:=

[0,T]²ϕ²_ξ

L|t−s|

dtds, (15)

A₂:=α_H

[0,T]³ϕ_ξ

L|t−u| ϕ_ξ

L|s−u|

|t−s|^2H−2dtdsdu, (16) A₃:=α_H²

[0,T]⁴ϕ_ξ

L|t−u| ϕ_ξ

L|s−v|

|t−s|^2H−2|v−u|^2H−2dudvdtds.

(17) First, we show thatA₁∼ _L¹. By change of variablesy= ^L_Tsandx= ^L_Ttwe obtain

A1= T² L²

_L

0

_L

0

ϕ_ξ²

T|x−y| dxdy.

Now, by applying L’Hôpital’s rule and some elementary computations we obtain that

Llim→∞L⁻¹ _L

0

_L

0

ϕ_ξ²

T|x−y|

dxdy = lim

L→∞2 _L

0

ϕ_ξ²

T (L−x) dx

= 2 T

_∞

0

ϕ_ξ²(y)dy, which is finite by Assumption1. Consequently, we get

Llim→∞LA1=2T _∞

0

ϕ_ξ²(y)dy,

or, in other words,A1∼L⁻¹. To complete the proof, it is shown in Appendix B that limL→∞L(A2+A3)=0.

We also apply the following proposition. The proof is rather technical and is post- poned to Appendix A.

Proposition 3. Consider the symmetric two-variable function ψ_L(s, t ) :=

ϕ_ξ(L|t−s|)on[0, T] × [0, T]. Denote

ψ˜_L(t, s)= ψL(s, t )

√2ψLH^⊗2

.

Then, for anyH∈(¹₂,1), asL→ ∞, we have I₂^X(ψ˜_L)−→^law N(0,1).

Our main theorem is the following.

Theorem 5. Assume that the characteristic functionϕ_ξof a symmetric random vari- ableξ satisfies Assumption1and letσ_T²be given by(14). Then, asL→ ∞, we have the following asymptotic statements:

1. ifH ∈(³₄,1), then

√L

EξI_T(X;L ξ )− [X, X]T

law

−→N 0, σ_T²

. 2. ifH =³₄, then

√L

EξI_T(X;L ξ )− [X, X]T

law

−→N μ, σ_T²

, whereμ=2αHT ₀^∞ϕ_ξ(x)x^2H−2dx.

3. ifH ∈(¹₂,³₄), then L^2H−1

EI_T(X;L ξ )− [X, X]T

_P

−→μ,

where the real numberμis given in item2. Notice that whenH ∈(¹₂,³₄), we have2H−1< ¹₂.

Proof. First, by applying Lemmas2and3we can write EI_T(X;Lξ )− [X, X]T =I₂^X(ψ_L)+α_H

_T

0

_T

0

ϕ_ξ

L|t−s|

|t−s|^2H−2dsdt.

Consequently, we obtain

√L

EI_T(X;Lξ )− [X, X]T

=√

L I₂^X(ψ_L)+√ L α_H

T 0

T 0

ϕ_ξ

L|t−s|

|t−s|^2H−2dsdt :=A₁+A₂.

Now, thanks to Proposition3, for anyH ∈(¹₂,1), we have A1=√

Lψ_LH^⊗2I₂^X(ψ˜_L)^law→N 0, σ_T²

,

whereσ_H² is given by (14). Hence, it remains to study the termA₂. Using change of variablesy =^L_Tsandx =^L_Tt, we obtain

_T

0

_T

0

ϕ_ξ

L|t−s|

|t−s|^2H−2dsdt

=T^2HL^−2H _L

0

_L

0

ϕ_ξ

T|x−y|

|x−y|^2H−2dxdy, where by L’Hôpital’s rule we obtain

Llim→∞L⁻¹ _L

0

_L

0

ϕξ

T|x−y|

|x−y|^2H−2dxdy=2T^1−2H _∞

0

ϕξ(x)x^2H−2dx.

Note also that the integral in the right-hand side of the last identity is finite by As- sumption1. Consequently, we obtain

Llim→∞L^2H−1α_H T

0

T 0

ϕ_ξ

L|t−s|

|t−s|^2H−2dsdt

=2αHT _∞

0

ϕξ(x)x^2H−2dx=μ. (18)

Therefore,

Llim→∞A₂= lim

L→∞L^32−2Hμ,

which converges to zero forH ∈(³₄,1), and item 1 of the claim is proved. Similarly, forH =³₄, we obtain

Llim→∞A2=μ,

which proves item 2 of the claim. Finally, for item 3, from (18) we infer that, as L→ ∞,

L^2H−1αH

_T

0

_T

0

ϕξ

L|t−s|

|t−s|^2H−2dsdt −→μ.

Furthermore, for the termI₂^X(ψL), we obtain L^2H−1I₂^X(ψL)=L^2H−32 ×√

L I₂^X(ψL)→^P 0 asL→ ∞. This is becauseH < ³₄implies 2H−³₂ <0 and moreover√

L I₂^X(ψ_L)^law→ N(0,1)andL^2H−32 →0.

Corollary 1. WhenX=W is a standard Brownian motion, that is, if the fractional Brownian motion part drops, then with similar arguments as in Theorem5, we obtain

√L

EξI_T(X;L ξ )− [X, X]T

law

−→N 0, σ_T²

, whereσ_T² =2T ₀^∞ϕ_ξ²(x)dx, andϕ_ξis the characteristic function ofξ.

Remark 4. Note that the proof of Theorem5reveals that in the caseH∈(¹₂,³₄), for any > ³₂−2H, we have that, asL→ ∞,

√L

EI_T(X;L ξ )− [X, X]T

_P

−→ ∞, and, moreover,

L¹²⁻

EIT(X;L ξ )− [X, X]T

_P

−→0.

4.2 The Berry–Esseen estimates

As a consequence of the proof of Theorem5, we also obtain the following Berry–

Esseen bound for the semimartingale case.

Proposition 4. Let all the assumptions of Theorem5hold, and letH ∈ (³₄,1). Fur- thermore, letZ ∼ N(0, σ_T²), where the varianceσ_T² is given by (14). Then there exists a constantC(independent ofL)such that for sufficiently largeL, we have

sup

x∈R

P(√ L

Eξ

IT(X;L ξ )− [X, X]T

< x

−P(Z < x)≤Cρ(L),

where

ρ(L)=max

L^32−2H, _∞

L

ϕ_ξ²(T z)dz

. Proof. By proof of Theorem5we have

√L

EI_T(X;Lξ )− [X, X]T

=√

L I₂^X(ψL)+√ L αH

_T

0

_T

0

ϕξ

L|t−s|

|t−s|^2H−2dsdt

=:A₁+A₂, where

A₁=√

2Lψ_LH⊗2I₂^X(ψ˜_L).

Now, we know that the deterministic termA₂converges to zero with rateL^32−2Hand the termA₁ ^law→ N(0, σ_T²). Hence, in order to complete the proof, it is sufficient to show that

sup

x∈R

P(A₁< x)−P(Z < x)≤Cρ(L).

Now, by using the proof of Proposition3in Appendix A we have

VarDF_L²_H≤L⁻¹² ≤L^32−2H. Finally, using the notation of the proof of Lemma4, we have

E F_n²

=Lψ_L²_H⊗2 =L×(A₁+A₂+A₃), whereA₂+A₃≤CL^−2H. Consequently,

L×(A₂+A₃)≤CL^1−2H ≤CL^32−2H. To complete the proof, we have

LA₁= T² L

_L

0

_L

0

ϕ_ξ²

T|x−y|

dydx =T² L

_L

0

_L−x

−x

ϕ_ξ²(T z)dzdx

= T² L

_L

−L

_L−z

−z

ϕ_ξ²(T z)dxdz=T² _L

−L

ϕ_ξ²(T z)dz

=2T² _L

0

ϕ_ξ²(T z)dz.

This gives us

LA₁−σ_T² =2T² _∞

L

ϕ²_ξ(T z)dz.

Now, the claim follows by an application of Theorem4.

Remark 5. In many cases of interest, the leading term inρ(L)is the polynomial termL^32−2H, which reveals that the role of the particular choice ofϕ_ξ affects only to the constant. In particular, ifϕ_ξ admits an exponential decay, that is,|ϕ_ξ(t )| ≤ C₁e^−C2t for some constantsC₁, C₂>0, then _L^∞ϕ_ξ²(T z)dz≤C₃e^−C4L≤CL^32−2H for some constantsC₃, C₄, C > 0. As examples, this is the case if ξ is a standard normal random variable with characteristic functionϕ_ξ(t )=e^−t

2

2 or ifξis a standard Cauchy random variable with characteristic functionϕξ(t )=e^−|t|.

Remark 6. Consider the caseX =W, that is,Xis a standard Brownian motion. In this case, the correction termA2in the proof of Theorem5disappears, and we have

E F_L²

−σ_T²=2T² _∞

L

ϕ_ξ²(T x)dx.

Furthermore, by applying L’Hôpital’s rule twice and some elementary computations it can be shown that

E

DFL²H−EDFL²H

2

≤ϕξ(T L)L⁻¹.