# Quasi-Likelihood Analysis of Fractional Brownian Motion with Constant Drift under High-Frequency Observations

Consider an estimation of the Hurst parameter H∈(0,1) and the volatility parameter σ>0 for a fractional Brownian motion with a drift term under high-frequency observations with a finite time interval. In the present paper, we propose a consistent estimator of the parameter θ=(H,σ) combining the ideas of a quasi-likelihood function based on a local Gaussian approximation of a high-frequently observed time series and its frequency-domain approximation. Moreover, we prove an asymptotic normality property of the proposed estimator for all H∈(0,1) when the drift process is constant.

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## 1 Introduction

Let

be a complete probability space. Consider the stochastic process

defined on of the form

 dXθt=μtdt+σdBHt,  Xθ0=ξ0,  θ=(H,σ)∈(0,1)×(0,∞), (1.1)

where is a fractional Brownian motion (fBm) with Hurst parameter , is a continuous stochastic process and

is a random variable. The stochastic process in (

1.1) is, for example, used for the log-volatility process and such volatility models recently attract much attention from researchers in mathematical finance and financial econometrics and practitioners in the financial industry, e.g. see [11], [2] and [9] for details. The aim of the present paper is to investigate an asymptotic distribution of a quasi-likelihood-type estimator of the parameter based on the high-frequency data with when the sample size goes to infinity.

First, note that is the only parameter to be estimated in (1.1) because we can not consistently estimate drift parameters even if a sample path of is continuously observed. Under high-frequency asymptotics, i.e.  as , we can show

 Xθjδn−Xθ(j−1)δn=σ(BHjδn−BH(j−1)δn)+OP(δn)  as n→∞ (1.2)

and, thanks to the self-similarity property of the fBm, we have

 σ(BHjδn−BH(j−1)δn)L=σδHn(BHj−BHj−1),  j=1,⋯,n, (1.3)

where means that the equality holds in law. If we know the Hurst parameter , then the volatility is the only parameter to be estimated and it is well-known that the QMLE (Quasi-Maximum Likelihood Estimator) of based on the local Gaussian approximation (1.2), which is same as the quadratic variation of with equidistant sampling since the fBm has the independent increments property when , is consistent and asymptotically normal as under some mild technical assumptions of , e.g. see [6]. Then the drift term can be seen as a nuisance parameter because the QMLE of can be computed without identifying the drift term and its asymptotic distribution does not depend on the drift term.

Thanks to (1.2) and (1.3), even if is unknown, it would be possible to consistently estimate the parameter without identifying the drift term under high-frequency asymptotics using a quasi-likelihood-type estimator based on the local Gaussian approximation (1.2). On the other hand, it is unclear whether an asymptotic distribution of the quasi-likelihood-type estimator does not depend on the drift term because (1.2) and (1.3) imply that it becomes more difficult to distinguish between the noise and drift terms of from high-frequency data when approaches .

Recently, [10] proposed an estimator of the parameter using the change-of-frequency method and proved its asymptotic normality property for all with under the technical condition for some satisfying if . We remark that the condition assumed in the case is not standard because it implies the length of the observation period converges to zero. Therefore, it is not trivial whether their proposed estimator enjoys the same asymptotic normality property when under the condition . Moreover, their proposed estimator is, of course, not optimal because the covariance structure of the noise is not used in their estimation procedure.

In the present paper, we propose an estimator of the parameter combining the following two ideas in the similar way to [9]: (1) the local Gaussian approximation (1.2) and (2) the frequency domain approximation, the so-called Whittle approximation, of the quasi-likelihood function, see Section 3.1 for details. Then we can easily prove the consistency of the proposed estimator in the similar way to Theorem 2.8 of [9]. Our contribution in the present paper is to prove that  the proposed estimator enjoys the asymptotic normality property for all even when as , which implies the technical condition of assumed in [10] is not essential to derive asymptotic distributions of estimators of , and  an asymptotic distribution of the proposed estimator does not depend on the drift term for all , at least, when is . constant.

The present paper is organized as follows. We summarize preliminary results and notation in Section 2. In Section 3, our proposed estimator is defined and a main theorem in the present paper is given. The main theorem is proven in Section 4 and a preliminary lemma used in the proof of the main theorem is proven in Section 5.

## 2 Preliminary Results

### 2.1 Fractional Brownian Motion

A centered continuous Gaussian process , defined on a complete probability space , is called a fractional Brownian motion (fBm) with Hurst parameter if -a.s. and it satisfies the following scaling property:

 E[|BHt−BHs|2]=|t−s|2H  for any s,t∈R. (2.1)

From (2.1), it is obvious that the fBm has the stationary increments and self-similarity properties. Moreover, it is well-known that the spectral density function of the stationary increments process is given by

 fH(λ):=CH{2(1−cosλ)}∑j∈Z1|λ+2πj|1+2H,  λ∈[−π,π], (2.2)

with , e.g. see [14].

### 2.2 Notation

Consider the parameter space , where is a compact set of . Denote by the true value of the parameter . Let be the sample size and be the length of sampling intervals. Denote by and

 ΔXθn:=(Xθδn−Xθ0,Xθ2δn−Xθδn,⋯,Xθnδn−Xθ(n−1)δn), ΔBθn:=(σBHδn,σ(BH2δn−BHδn),⋯,σ(BHnδn−BH(n−1)δn)).

Moreover, set and .

Let . For an integrable function , the -th Fourier coefficient of is defined by

 ˆf(τ):=∫π−πe√−1τxf(x)dx.

We denote by the -Toeplitz matrix whose -element is given by for each . Thanks to the self-similarity property of the fBm, we can write with . Define by

 b(H):=exp(12π∫π−πlogfH(λ)dλ),  gH(λ):=b(H)−1fH(λ), hH(λ):=1/(4π2gH(λ)),  ν2n(H):=1n⟨ΔXθ0n,Tn(hH)ΔXθ0n⟩Rn.

Set and for . Finally, denotes the convergence in law under the probability measure .

## 3 Main Result

### 3.1 Quasi-Whittle Likelihood Estimator (QWLE)

In the present paper, we propose the estimator defined by

 ˆHn:=arg minH∈ΘHν2n(H),  ˆσn:=δ−ˆHnnb(ˆHn)−12√ν2n(ˆHn). (3.1)

We call the QWLE (Quasi-Whittle Likelihood Estimator) in the following. In the rest of this subsection, we make several remarks on the proposed estimator in order.

###### Remark 3.1 (Local Gaussian Approximation and Whittle Approximation).

Thanks to (1.2) and (1.3), it would be possible that

is approximated by the Gaussian vector

in some suitable sense under high-frequency asymptotics. Therefore, we use the likelihood function of as a quasi-likelihood function of . Actually, we utilize an approximate likelihood function of in the frequency domain, the so-called Whittle likelihood function, as a quasi-likelihood function of because the quasi-maximum likelihood estimator is computationally infeasible when the sample size is quite large due to the high computational cost of the inverse and determinant of .

In the rest of this remark, we explain why the estimator is defined by (3.1). Set and . First note that is the spectral density function of the stationary Gaussian sequence with respect to the reparameterized parameter . Since we have

 12π∫π−πloggH(λ)dλ=0 (3.2)

for all , the quasi-Whittle likelihood function of with respect to the reparameterized parameter is defined by

 L(0)n(H,νn):=12(logν2n+1ν2nν2n(H)).

Since is a minimizer of the quasi-Whittle likelihood function on for each , the estimator can be defined as (3.1) using the estimator and the relation .

###### Remark 3.2 (Reparameterization).

Under high-frequency observations, the effects of and fuse in the limit and the asymptotic Fisher information matrix when a “diagonal” rate-matrix is used becomes singular due to the self-similarity property of the fractional Gaussian noise. As a result, it is necessary to reparametrize the parameter in order to obtain a limit theorem of an estimator. See [3] and [7] for more details.

In the rest of this remark, we briefly explain of our strategy to prove asymptotic properties of the QWLE . First note that is also a minimizer of the function

 ˜σ2n(H):=δ−2H0nb(H0)−1ν2n(H)

with respect to on so that the random variable with is a minimizer of the function

 Ln(θ):=12(logσ2+1σ2˜σ2n(H))=12(logσ2+1σ2n⟨Δ˜Xn,Tn(hH)Δ˜Xn⟩Rn) (3.3)

with respect to on . Note that is not an estimation function since the true value is used in its definition. It plays, however, a similar role to the usual estimation function in proofs of asymptotic properties of the QWLE because is the quasi-Whittle likelihood function of the suitably rescaled random vector and the quasi-spectral density function which appears in no longer depends on the asymptotic parameter . Therefore, it would be possible that the random variable converges to in some suitable sense as and asymptotic properties of the estimator can be proven using the convergence of . See Section 5.2 of [7], Section 4 of [9] and Section 4.2 for more details.

###### Remark 3.3 (Implementation).

In this remark, we briefly explain how to efficiently implement the QWLE. First note that we can write

 ν2n(H)=12π∫π−πIn(λ,ΔXθ0n)gH(λ)dλ, (3.4)

where is the periodogram defined by

 In(λ,x):=12πn∣∣ ∣∣n∑j=1xje√−1jλ∣∣ ∣∣,  λ∈[−π,π],  x=(x1,⋯,xn)∈Rn.

Then the Riemann approximation of the integral (3.4) gives

 ν2n(H)≈1nn∑j=1In(λnj,ΔXθ0n)gH(λnj),  λnj:=2πjn, (3.5)

and the sum in (3.5

) can be effectively computed using the fast Fourier transform algorithm. Note that the series appears in the function

can be accurately and efficiently computed using the approximation method proposed by [13]. See also [7] and its supplementary article [8] for more details.

### 3.2 Asymptotic Normality Property of QWLE

First, we introduce a class of sequences of non-diagonal rate matrices which plays a key role to prove an asymptotic normality property of QWLE with a non-degenerate asymptotic variance-covariance matrix.

###### Assumption 3.4.

Assume a sequence of matrices and a matrix of the forms

 φn(θ):=1√n(φ11n(θ)φ12n(θ)φ21n(θ)φ22n(θ)),  ¯¯¯¯φ(θ):=(¯¯¯¯φ11(θ)¯¯¯¯φ12(θ)¯¯¯¯φ21(θ)¯¯¯¯φ22(θ))

satisfy the following properties for each :

1. as ,

2. as ,

3. as ,

4. as ,

5. for each ,

6. .

Then we can prove a main theorem in the present paper as follows.

###### Theorem 3.5.

Consider a sequence of rate matrices satisfying Assumption 3.4. Assume is an interior point of . Then we obtain the following result:

1. The sequence of the QWLEs is (weakly) consistent as .

2. If is identically equal to a -measurable random variable , then the sequence of the QWLEs satisfies the following asymptotic normality property:

 φn(θ0)−1(ˆθn−θ0)L→N(0,I(θ0)−1)  as n→∞, (3.6)

where is the positive definite matrix defined by

 I(θ):=¯¯¯¯φ(θ)∗F(θ)¯¯¯¯φ(θ),  F(θ):=14π∫π−π(∂∂θlogfθ(λ))(∂∂θlogfθ(λ))∗dλ.

Several examples of satisfying Assumption 3.4 can be found in [7]. Particular choices of imply that the convergence rates of and are and respectively. See [7] for details.

###### Remark 3.6.

In the case , i.e. , Theorem 3 of [7] proved that the Whittle estimator, defined in the same way as (3.1), has the same asymptotic distribution as under high-frequency asymptotics. Therefore, Theorem 3.5 (2) implies that the asymptotic distribution of the QWLE does not depend on the drift term, at least, when is constant. We will investigate asymptotic properties of the QWLE when is not constant in the future work.

## 4 Proof of Theorem 3.5

### 4.1 Preliminary Lemma

Before proving Theorem 3.5, we prepare the following lemma.

###### Lemma 4.1.

For any , and , we have

 (4.1) ⟨1n,Tn(h(j)H)Tn(gH)Tn(h(j)H)1n⟩Rn=o(n2(1−H)+ϵ)  as n→∞. (4.2)

The proof of Lemma 4.1 is left to Section 5.

### 4.2 Proof of Theorem 3.5

First, note that the consistency of the QWLE can be proven in the similar way to the proof of Theorem 2.8 of [9]. In the following, we prove only the asymptotic normality property of the QWLE. In the similar way to the proof of Theorem 2.12 of [9] and the proof of Theorem 3 of [7], the asymptotic normality property of the QWLE follows once we have proven

 √n∇Ln(θ0)L→N(0,diag(G(H0),2σ−20))  as n→∞, (4.3)

where the function is defined by (3.3) and

 G(H):=14π∫π−π∣∣∣∂∂HloggH(λ)∣∣∣2dλ.

Now we introduce notation used in the proof. Define by

for . Set and for . By a straight-forward calculation, we can write

 √n∇Ln(θ)=(Y1n(θ),−σ−1Y2n(θ)).

Moreover, in the similar way to the proof of Theorem 2 of [5], we can prove

 (Z1n(θ0),−σ−10Z2n(θ0))L→N(0,diag(G(H0),2σ−20))  as n→∞.

Therefore, in order to prove (4.3), it suffices to prove

 ⟨Δ˜Xn,Tn(h(j)H0)Δ˜Xn⟩Rn=⟨Δ˜Bn,Tn(h(j)H0)Δ˜Bn⟩Rn+oP(nϵ)  as n→∞ (4.4)

for any and . We prove (4.4) in the rest of the proof. Since and , we can write

 ⟨Δ˜Xn,Tn(h(j)H0)Δ˜Xn⟩Rn−⟨Δ˜Bn,Tn(h(j)H0)Δ˜Bn⟩Rn =2μb(H0)−1nH0−1⟨1n,Tn(h(j)H0)Δ˜Bn⟩Rn+μ2b(H0)−2n2(H0−1)⟨1n,Tn(h(j)H0)1n⟩Rn. (4.5)

Moreover, we have

 L{⟨1n,Tn(h(j)H0)Δ˜Bn⟩Rn∣∣P}∼N(0,⟨1n,Tn(h(j)H0)Tn(gH0)Tn(h(j)H0)1n⟩Rn). (4.6)

Therefore (4.4) follows from (4.5), (4.6) and Lemma 4.1. This completes the proof.

## 5 Proof of Lemma 4.1

### 5.1 Notation

Suppose is a real-valued -matrix. Define the operator norm of by

 ∥A∥op:=supx∈Rn∥Ax∥Rn∥x∥Rn

and the Frobenius norm of by

 ∥A∥F:=(Tr[AA∗])12.

In the present paper, we use the following well-known properties:

1. .

2. and .

### 5.2 Preliminary Lemma

Before proving Lemma 4.1, we prove the following preliminary lemma.

###### Lemma 5.1.

For any and ,

 ∥∥∥In−Tn(gH)12Tn(hH)Tn(gH)12∥∥∥F=o(nϵ)  as n→∞.
###### Proof.

First note that we can write

 ∥∥∥In−Tn(gH)12Tn(hH)Tn(gH)12∥∥∥2F =n−2Tr[Tn(gH)Tn(hH)]+Tr[Tn(gH)Tn(hH)Tn(gH)Tn(hH)]

Then the conclusion follows from Theorem 3.1 of [15]. This completes the proof. ∎

### 5.3 Proof of (???) in the case j=0

Thanks to Theorems 4.1 and 5.2 of [1], it suffices to prove

 ⟨1n,Tn(hH)1n⟩Rn=⟨1n,Tn(gH)−11n⟩Rn+o(n2(1−H)+ϵ)  as n→∞

for any . First we can show

 =∣∣⟨1n,(Tn(gH)−1−Tn(hH))1n⟩Rn∣∣ =∣∣∣⟨Tn(gH)−121n,(In−Tn(gH)12Tn(hH)Tn(gH)12)Tn(gH)−121n⟩Rn∣∣∣ ≤∥∥∥Tn(gH)−121n∥∥∥Rn∥∥∥(In−Tn(gH)12Tn(hH)Tn(gH)12)Tn(gH)−121n∥∥∥Rn ≤∥∥∥Tn(gH)−121n∥∥∥2Rn∥∥∥In−Tn(gH)12Tn(hH)Tn(gH)12∥∥∥F.

Then the conclusion follows from Lemma 5.1 and Theorems 4.1 and 5.2 of [1]. This completes the proof.

### 5.4 Proof of (???) in the case j=1

First we can show

 ∣∣⟨1n,Tn(h(1)H)1n⟩Rn∣∣ ≤⟨1n,Tn(|h(1)H|)1n⟩Rn =⟨Tn(hH)121n,(Tn(hH)−12Tn(|h(1)H|)Tn(hH)−12)Tn(hH)121n⟩Rn ≤∥∥∥Tn(hH)121n∥∥∥2Rn∥∥∥Tn(|h(1)H|)12Tn(hH)−12∥∥∥2op.

Then the conclusion follows from in the case and Lemma 2 in the full version of [12]. This completes the proof.

### 5.5 Proof of (???)

First we introduce notation used in the proof. For , set ,

 C(j)n(H):=Tn(gH)−1Tn(g(j)H)Tn(gH)−1, D(j)n(H):=Tn(gH)12(Tn(h(j)H)−C(j)n(H))Tn(gH)12, ˜C(j)n(H):=Tn(gH)12C(j)n(H)Tn(gH)12, ˜Fn(H):=Tr[{(Tn(gH)Tn(h(1)H)Tn(gH)−Tn(g(1)H))Tn(hH)}2].

Note that follows once we have proven that

 ⟨1n,Tn(h(j)H)Tn(gH)Tn(h(j)H)1n⟩Rn =⟨1n,C(j)n(H)Tn(gH)C(j)n(H)1n⟩Rn+o(n2(1−H)+ϵ)  as n→∞ (5.1)

holds for any and each because we can show

 ⟨1n,C(j)n(H)Tn(gH)C(j)n(H)1n⟩Rn=o(n2(1−H)+ϵ)  as n→∞

for any and using Theorem 5.2 of [1] and Lemma 2 in the full version of [12] in the similar way to the proof of Lemma 5.4 (d) of [4]. In the rest of the proof, we will prove (5.1). First we can write

 ⟨1n,Tn(h(j)H)Tn(gH)Tn(h(j)H)1n⟩Rn−⟨1n,C(j)n(H)Tn(gH)C(j)n(H)1n⟩Rn =⟨1n,(Tn(h(j)H)−C(j)n(H))Tn(gH)Tn(h(j)H)1n⟩Rn +⟨1n,C(j)n(H)Tn(gH)(Tn(h(j)H)−C(j)n(H))1n⟩Rn