Parameter estimation for Gaussian processes with application to the

arXiv:1808.08417v1 [math.PR] 25 Aug 2018

Parameter estimation for Gaussian processes with application to the model with two independent fractional Brownian motions Yuliya Mishura, Kostiantyn Ralchenko and Sergiy Shklyar

Abstract The purpose of the article is twofold. Firstly, we review some recent results on the maximum likelihood estimation in the regression model of the form Xt = θ G(t) + Bt , where B is a Gaussian process, G(t) is a known function, and θ is an unknown drift parameter. The estimation techniques for the cases of discretetime and continuous-time observations are presented. As examples, models with fractional Brownian motion, mixed fractional Brownian motion, and sub-fractional Brownian motion are considered. Secondly, we study in detail the model with two independent fractional Brownian motions and apply the general results mentioned above to this model. Key words: discrete observations, continuous observations, maximum likelihood estimator, strong consistency, fractional Brownian motion, Fredholm integral equation of the first kind

Yuliya Mishura Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 64 Volodymyrska, 01601 Kyiv, Ukraine e-mail: [email protected] Kostiantyn Ralchenko Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 64 Volodymyrska, 01601 Kyiv, Ukraine e-mail: [email protected] Sergiy Shklyar Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 64 Volodymyrska, 01601 Kyiv, Ukraine e-mail: [email protected]

1

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Yuliya Mishura, Kostiantyn Ralchenko and Sergiy Shklyar

1 Introduction Gaussian processes with drift arise in many applied areas, in particular, in telecommunication and on financial markets. An observed process often can be decomposed as the sum of a useful signal and a random noise, where the last one mentioned is usually modeled by a centered Gaussian process, see, e. g., [19, Ch. VII]. The simplest example of such model is the process Yt = θ t + Wt , where W is a Wiener process. In this case the MLE of the drift parameter θ by observations of Y at points 0 = t0 ≤ t1 ≤ . . . ≤ tN = T is given by

θˆ =


YT − Y0 1 N−1 Yti+1 − Yti = , ∑ tN − t0 i=0 T

and depends on the observations at two points, see e. g. [5]. Models of such type are widely used in finance. For example, Samuelson’s model [42] with constant drift parameter µ and known volatility σ has the form   σ2 t + σ Wt , log St = µ − 2 and the MLE of µ equals

µˆ =

log ST − log S0 σ 2 + . T 2

At the same time, the model with Wiener process is not suitable for many processes in natural sciences, computer networks, financial markets, etc., that have long- or short-term dependencies, i. e., the correlations of random noise in these processes decrease slowly with time (long-term dependence) or rapidly with time (short-term dependence). In particular, the models of financial markets demonstrate various kinds of memory (short or long). However, a Wiener process has independent increments, and, therefore, the random noise generated by it is “white”, i. e., uncorrelated. The most simple way to overcome this limitation is to use fractional Brownian motion. In some cases even more complicated models are needed. For example, the noise can be modeled by mixed fractional Brownian motion [9], or by the sum of two fractional Brownian motions [28]. Moreover, recently Gaussian processes with non-stationary increments have become popular such as sub-fractional [6], bifractional [14] and multifractional [2, 36, 40] Brownian motions. In this paper we study rather general model where the noise is represented by a centered Gaussian process B = {Bt ,t ≥ 0} with known covariance function, B0 = 0. We assume that all finite-dimensional distributions of the process {Bt , t > 0} are multivariate normal distributions with nonsingular covariance matrices. We observe the process Xt with a drift θ G(t), that is,

Parameter estimation for Gaussian processes

Xt = θ G(t) + Bt , R

3

(1)

where G(t) = 0t g(s) ds, and g ∈ L1 [0,t] for any t > 0. The paper is devoted to the estimation of the parameter θ by observations of the process X. We consider the MLEs for discrete and continuous schemes of observations. The results presented are based on the recent papers [33, 32]. Note that in [32] the model (1) with G(t) = t was considered, and the driving process B was a process with stationary increments. Then in [33] these results were extended to the case of non-linear drift and more general class of driving processes. In the present paper we apply the theoretical results mentioned above to the models with fractional Brownian motion, mixed fractional Brownian motion and sub-fractional Brownian motion. Similar problems for the model with linear drift driven by fractional Brownian motion were studied in [5, 16, 26, 35]. The mixed Brownian — fractional Brownian model was treated in [7]. In [4, 39] the nonparametric functional estimation of the drift of a Gaussian processes was considered (such estimators for fractional and subfractional Brownian motions were studied in [13] and [43] respectively). In the present paper special attention is given to the model of the form Xt = θ t + BtH1 + BtH2 with two independent fractional Brownian motions BH1 and BH2 . This model was first studied in [28], where a strongly consistent estimator for the unknown drift parameter θ was constructed for 1/2 < H1 < H2 < 1 and H2 − H1 > 1/4 by continuous-time observations of X. Later, in [34], the strong consistency of this estimator was proved for arbitrary 1/2 < H1 < H2 < 1. The details on this approach are given in Remark 2 below. However, the problem of drift parameter estimation by discrete observations in this model was still open. Applying our technique, we obtain the discrete-time estimator of θ and prove its strong consistency for any H1 , H2 ∈ (0, 1). Moreover, we also construct the continuous-time estimator and prove the convergence of the discrete-time estimator to the continuous-time one in the case where H1 ∈ (1/2, 3/4] and H2 ∈ (H1 , 1). It is worth mentioning that the drift parameter estimation is developed for more general models involving fBm. In particular, the fractional Ornstein–Uhlenbeck process is a popular and well-studied model with fBm. The MLE of the drift parameter for this process was constructed in [20] and further investigated in [3, 44, 47]. Several non-standard estimators for the drift parameter of an ergodic fractional Ornstein–Uhlenbeck process were proposed in [15] and studied in [17]. The corresponding non-ergodic case was treated in [1, 10, 45]. In the papers [8, 11, 12, 18, 48, 49] drift parameter estimators were constructed via discrete observations. More general fractional diffusion models were studied in [21, 27] for continuous-time estimators and in [23, 29, 31] for the case of discrete observations. An estimator of the volatility parameter was constructed in [25]. For Hurst index estimators see, e. g., [22] and references cited therein. Mixed diffusion model including fractional Brownian motion and Wiener process was investigated in [21]. We refer to the paper [30] for a survey of the results on parameter estimation in fractional and mixed diffusion models and to the books [24, 38] for a comprehensive study of this topic.

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Yuliya Mishura, Kostiantyn Ralchenko and Sergiy Shklyar

The paper is organized as follows. In Section 2 we construct the MLE by discretetime observations and formulate the conditions for its strong consistency. In Section 3 we consider the estimator constructed by continuous-time observations and the relations between discrete-time and continuous-time estimators. In Section 4 these results are applied to various models mentioned above. In particular, the new approach to parameter estimation in the model with two independent fractional Brownian motions is presented in Subsection 4.5. Auxiliary results are proved in the appendices.

2 Construction of drift parameter estimator for discrete-time observations Let the process X be observed at the points 0 < t1 < t2 < . . . < tN . Then the vector of increments ∆ X (N) = (Xt1 , Xt2 − Xt1 , . . . , XtN − XtN−1 )⊤ is a one-to-one function of the observations. We assume in this section that the inequality G(tk ) 6= 0 holds at least for one k. Evidently, vector ∆ X (N) has Gaussian distribution N (θ ∆ G(N) , Γ (N) ), where
⊤ ∆ G(N) = G(t1 ), G(t2 ) − G(t1 ), . . . , G(tN ) − G(tN−1 ) .

Let Γ (N) be the covariance matrix of the vector

∆ B(N) = (Bt1 , Bt2 − Bt1 , . . . , BtN − BtN−1 )⊤ . The density of the distribution of ∆ X (N) w. r. t. the Lebesgue measure is   −1  ⊤   (2π )−N/2 1 (N) (N) (N) x − θ∆G exp − 2 x − θ ∆ G Γ pdf∆ X (N) (x) = √ . (N) det Γ

Then one can take the density of the distribution of the vector ∆ X (N) for a given θ w. r. t. the density for θ = 0 as a likelihood function:   θ2 L(N) (θ ) = exp θ (∆ G(N) )⊤ (Γ (N) )−1 ∆ X (N) − (∆ G(N) )⊤ (Γ (N) )−1 ∆ G(N) . 2 (2) The corresponding MLE equals

θˆ (N) =



−1 ⊤  Γ (N) ∆ X (N) .
⊤
−1 ∆ G(N) Γ (N) ∆ G(N)

∆ G(N)

(3)

Parameter estimation for Gaussian processes

5

Theorem 1 (Properties of the discrete-time MLE [33]). 1. The estimator θˆ (N) is unbiased and normally distributed:   1 (N) ˆ θ − θ ≃ N 0, . (∆ G(N) )⊤ (Γ (N) )−1 ∆ G(N) 2. Assume that

var Bt → 0, G2 (t)

as t → ∞.

(4)

If tN → ∞, as N → ∞, then the discrete-time MLE θˆ (N) converges to θ as N → ∞ almost surely and in L2 (Ω ).

3 Construction of drift parameter estimator for continuous-time observations In this section we suppose that the process Xt is observed on the whole interval [0, T ]. We investigate MLE for the parameter θ based on these observations. R Let h f , gi = 0T f (t)g(t) dt. Assume that the function G and the process B satisfy the following conditions. (A) There exists a linear self-adjoint operator Γ = ΓT : L2 [0, T ] → L2 [0, T ] such that cov(Xs , Xt ) = E Bs Bt =

Z t 0

ΓT 1[0,s] (u) du = hΓT 1[0,s] , 1[0,t] i.

(5)

(B) The R t drift function G is not identically zero, and in its representation G(t) = 0 g(s) ds the function g ∈ L2 [0, T ]. (C) There exists a function hT ∈ L2 [0, T ] such that g = Γ hT . Note that under assumption (A) the covariance between integrals of deterministic functions f ∈ L2 [0, T ] and g ∈ L2 [0, T ] w. r. t. the process B equals E

Z T 0

f (s) dBs

Z T 0

g(t) dBt = hΓT f , gi.

Theorem 2 (Likelihood function and continuous-time MLE [33]). Let T be fixed, assumptions (A)–(C) hold. Then one can choose  Z T  Z θ2 T hT (s) dXs − g(s)hT (s) ds (6) L(θ ) = exp θ 2 0 0 as a likelihood function. The MLE equals RT

hT (s) dXs θˆT = R T0 . 0 g(s)hT (s) ds

(7)

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Yuliya Mishura, Kostiantyn Ralchenko and Sergiy Shklyar

It is unbiased and normally distributed:

θˆT − θ ≃ N

0, R T 0

1 g(s)hT (s) ds

!

.

Theorem 3 (Consistency of the continuous-time MLE [33]). Assume that assumptions (A)–(C) hold for all T > 0. If, additionally, lim inf t→∞

var Bt = 0, G(t)2

(8)

then the estimator θˆT converges to θ as T → ∞ almost surely and in mean square. Theorem 4 (Relations between discrete and continuous MLEs [33]). Let the assumptions of Theorem 2 hold. Construct the estimator θˆ (N) from (3) by observations XT k/N , k = 1, . . . , N. Then 1. the estimator θˆ (N) converges to θˆT in mean square, as N → ∞, n 2. the estimator θˆ (2 ) converges to θˆT almost surely, as n → ∞.

4 Application of estimators to models with various noises 4.1 The model with fractional Brownian motion and linear drift  Definition 1. The fractional Brownian motion BH = BtH ,t ≥ 0 with Hurst index H ∈ (0, 1) is a centered Gaussian process with B0 = 0 and covariance function   2H 1 E BtH BH + s2H − |t − s|2H . s = 2 t Let H ∈ (0, 1) be fixed. Consider the model

Xt = θ t + BtH .

(9)

where X is an observed stochastic process, BH is an unobserved fractional Brownian motion with Hurst index H, and θ is a parameter of interest. Any finite slice of the stochastic process {BtH , t > 0} has a multivariate normal distribution with non
singular covariance matrix. Since var BtH = t 2H , the random process BH satisfies Theorem 1. Hence, we have the following result.

Corollary 1. Under condition tN → +∞ as N → ∞, the estimator θˆ (N) in the model (9) is L2 -consistent and strongly consistent.

Remark 1. Bertin et al. [5] considered the MLE in the model (9) in the discrete scheme of observations, where the trajectory of X was observed at the points tk = Nk , k = 1, 2, . . . , N α , α > 1. Hu et al. [16] investigated the MLE by discrete observations

Parameter estimation for Gaussian processes

7

at the points tk = kh, k = 1, 2, . . . , N. They considered even more general model of the form Xt = θ t + σ BtH with unknown σ . In both papers L2 -consistency and strongly consistency of the MLEs were proved. Note that in Corollary 1 both these schemes of observations are allowed, since the only condition tN → ∞ is required. Now we consider the case of continuous-time observations and apply the results of Section 3 to the model (9). Let H ∈ ( 12 , 1). Denote by ΓH the corresponding operator Γ for the model (9). Then (ΓH f )(t) = H(2H − 1)

Z T 0

f (s) |t − s|2−2H

ds.

(10)

For the function hT (s) = CH s1/2−H (T − s)1/2−H ,

−1 , we have that CH = H(2H − 1)B H − 12 , 23 − H

ΓH hT = 1[0,T] ,

(11)

see [35]. The MLE is given by

θˆT =

T 2H−2 B(3/2 − H, 3/2 − H)

Z T 0

s1/2−H (T − s)1/2−H dXs .

This estimator was studied in [26, 35], see also [32, Example 3.11]. Corollary 2. Let H ∈ ( 12 , 1). The conditions of Theorems 2, 3 and 4, are satisfied. The estimator θˆT is L2 -consistent and strongly consistent. For fixed T , it can be approximated by discrete-sample estimator in mean-square sense.

4.2 Model with fractional Brownian motion and power drift Now we generalize the model (9) for the case of the non-linear drift function G(t) = t α +1 . Let 0 < H < 1 and α > −1. Consider the process Xt = θ t α +1 + BtH

(12)

This is a particular case of model (1), with g(t) = (α + 1)t α . Now verify the conditions of the theorems. The condition (4) holds true if and only if α > H − 1. Corollary 3. If α > H − 1, the model (12) satisfies the conditions of Theorem 1. The estimator θˆ (N) in the model (12) is L2 -consistent and strongly consistent (provided that limN→∞ tN = +∞). The condition (B): g ∈ L2 [0, T ] holds true if and only if α > − 12 . The integral equation Γ h = g is rewritten as

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Yuliya Mishura, Kostiantyn Ralchenko and Sergiy Shklyar

Z T 0

α +1 α h(s) ds = t . 2H−2 |t − s| (2H − 1)H

If α > 2H − 2, then the solution is h(t) = const ·

Tα 1

1

t H− 2 (T − t)H− 2

− αt

α +1−2H

W

T t ,

α, H −

1 2

!


,

(13)


R T −1 1 where W Tt , α , H − 12 = 0t (v + 1)α −1 v 2 −H dv. The asymptotic behaviour of
the function W Tt , α , H − 12 as t → 0+ is  3
if α < H − 12 , B 2 − H, H − 12 − α   
if α = H − 12 , W Tt , α , H − 12 ∼ ln(T /t) 1  α −H+ 2  2 T  if α > H − 12 . 2α +1−2H α −H+ 1 t

2

Therefore, the function h(t) defined in (13) is square integrable if α + 1 − 2H −
max 0, α − H + 12 > − 12 , which holds if α > 2H − 23 . Note that if α > 2H − 32 , then the following inequalities hold true: α > 2H − 2 (whence h defined in (13) is indeed a solution to the integral equation Γ h = g), α > H − 1 (whence conditions (4) and so (8) are satisfied), and α > − 21 (whence condition (B) is satisfied). Corollary 4. If α > 2H − 32 , the conditions of Theorems 2, 3 and 4, are satisfied. The estimator θˆT is L2 -consistent and strongly consistent. For fixed T , it can be approximated by discrete-sample estimator in mean-square sense.

4.3 The model with Brownian and fractional Brownian motion Consider the following model: Xt = θ t + Wt + BtH ,

(14)

where W is a standard Wiener process, BH is a fractional Brownian motion with Hurst index H, and random processes W and BH are independent. The corresponding operator Γ is Γ = I + ΓH , where ΓH is defined by (10). The operator ΓH is self-adjoint and positive semi-definite. Hence, the operator Γ is invertible. Thus Assumption (C) holds true. In other words, the problem is reduced to the solving of the following Fredholm integral equation of the second kind hT (u) + H2(2H2 − 1)

Z T 0

hT (s) |s − u|2H2 −2 ds = 1,

u ∈ [0, T ].

(15)

Parameter estimation for Gaussian processes

9

This approach to the drift parameter estimation in the model with mixed fractional Brownian motion was first developed in [7]. Note also that the function hT = ΓT−1 1[0,T] can be evaluated iteratively 1 2

∞

hT =

∑ k=0

H


Γ I − Γ H k 1[0,T] T T .


k+1 1 + 21 ΓTH

(16)

4.4 Model with subfractional Brownian motion n o Definition 2. The subfractional Brownian motion BeH = BetH ,t ≥ 0 with Hurst pa-

rameter H ∈ (0, 1) is a centered Gaussian random process with covariance function  2 |t|2H + 2 |s|2H − |t − s|2H − |t + s|2H  H e = cov BeH , B . s t 2

(17)

We refer [6, 46] for properties of this process. Obviously, neither BeH, nor its increments are stationary. If {BtH , t ∈ R} is a fractional Brownian motion, then the ranBH +BH

dom process t √2 −t is a subfractional Brownian motion. Evidently, mixed derivative of the covariance function (17) equals   eH ∂ 2 cov BeH
s , Bt KH (s,t) := = H (2H − 1) |t − s|2H−2 − |t + s|2H−2 . (18) ∂t ∂s

If H ∈ ( 12 , 1), then the operator Γ = ΓeH that satisfies (5) for BeH equals

ΓeH f (t) =

Z T 0

KH (s,t) f (s) ds.

Consider the model (1) for G(t) = t and B = BeH : Xt = θ t + BetH .

(19)

(20)

Let us construct the estimators θˆ (N) and θˆT from (3) and (7) respectively and establish their properties. In particular, Proposition 1 allows to define finite-sample estimator θˆ (N) . Proposition 1. The linear equation ΓeH f = 0 has only trivial solution in L2 [0, T ]. As  H a consequence, the finite slice Bet , . . . , BetH with 0 < t1 < . . . < tN has a multivari1

N

ate normal distribution with nonsingular covariance matrix.  
Since var BetH = 2 − 22H−1 t 2H , the random process BeH satisfies Theorem 1. Hence, we have the following result.

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Yuliya Mishura, Kostiantyn Ralchenko and Sergiy Shklyar

Corollary 5. Under condition tN → +∞ as N → ∞, the estimator θˆ (N) in the model (20) is L2 -consistent and strongly consistent. In order to define the continuous-time MLE (7), we have to solve an integral equation. The following statement guarantees the existence of the solution. Proposition 2. If

1 2

Z T 0

< H < 43 , then the integral equation ΓeH h = 1[0,T ] , that is KH (s,t)h(s) ds = 1

for almost all t ∈ (0, T )

(21)

has a unique solution h ∈ L2 [0, T ]. Corollary 6. If 12 < H < 34 , then the random process BeH satisfies Theorems 2, 3, and 4. As the result, L(θ ) defined in (6) is the likelihood function in the model (20), and θˆT defined in (7) is the MLE. The estimator is L2 -consistent and strongly consistent. For fixed T , it can be approximated by discrete-sample estimator in mean-square sense.

4.5 The model with two independent fractional Brownian motions Consider the following model: Xt = θ t + BtH1 + BtH2 ,

(22)

where BH1 and BH2 are two independent fractional Brownian motion with Hurst indices H1 , H2 ∈ ( 12 , 1). Obviously, the condition (4) is satisfied:   var BtH1 + BtH2 t 2H1 + t 2H2 = → 0, t → ∞. 2 t t2 Theorem 5. Under condition tN → +∞ as N → ∞, the estimator θˆ (N) in the model (22) is L2 -consistent and strongly consistent. Evidently, the corresponding operator Γ for the model (22) equals ΓH1 + ΓH2 , where ΓH is defined by (10). Therefore, in order to verify the assumptions of Theorem 2 we need to show that  there exists a function hT such that (ΓH1 + ΓH2 ) hT =

ΓH2 hT = ΓH−1 1[0,T ] , since the operator ΓH1 is 1[0,T ] . This is equivalent to I + ΓH−1 1 1 injective and its range contains 1[0,T ] , see (11) and Theorem 7. Hence, it suffices ΓH2 is invertible. This is done in Theorem 8 in to prove that the operator I + ΓH−1 1 Appendix 8 for H1 ∈ (1/2, 3/4], H2 ∈ (H1 , 1). Thus, in this case the assumptions of Theorem 2 hold with  −1 Γ ΓH−1 1[0,T ] . hT = I + ΓH−1 H 2 1 1

Parameter estimation for Gaussian processes

11

Therefore, we have the following result for the estimator R

T hT (s) dXs . θˆT = R0 T 0 hT (s) ds

Theorem 6. If H1 ∈ (1/2, 3/4] and H2 ∈ (H1 , 1), then the random process BH1 + BH2 satisfies Theorems 2, 3, and 4. As the result, L(θ ) defined in (6) is the likelihood function in the model (22), and θˆT is the maximum likelihood estimator. The estimator is L2 -consistent and strongly consistent. For fixed T , it can be approximated by discrete-sample estimator in mean-square sense. Remark 2. Another approach to the drift parameter estimation in the model with two fractional Brownian motions was proposed in [28] and developed in [34]. It is based on the solving of the following Fredholm integral equation of the second kind (2 − 2H1)h˜ T (u)u1−2H1 +

Z T 0

h˜ T (s)k(s, u) ds = (2 − 2H1)u1−2H1 ,

u ∈ (0, T ], (23)

where k(s, u) =

Z s∧u 0

∂s KH1 ,H2 (s, v)∂u KH1 ,H2 (u, v) dv,

1/2−H2

KH1 ,H2 (t, s) = cH1 βH2 s cH1 =

Z t

(t − u)1/2−H1 uH2 −H1 (u − s)H2 −3/2 du,

s

Γ (3 − 2H1) 3 2H1Γ ( 2 − H1)3Γ (H1 + 12 )

!1 2

, βH2 =

2H2 H2 − 12


2

Γ ( 32 − H2 ) Γ (H2 + 12 )Γ (2 − 2H2)

! 12

.

Then for 1/2 ≤ H1 < H2 < 1 the estimator is defined as

θˆ (T ) = where δH1 = cH1 B gale,

3 2

N(T ) , δH1 hNi(T )


− H1, 23 − H1 , N(t) is a square integrable Gaussian martinN(T ) =

Z T 0

h˜ T (t) dX(t),

h˜ T (t) is a unique solution to (23) and hNi(T ) = (2 − 2H1)

Z T 0

h˜ T (t)t 1−2H1 dt.

This estimator is also unbiased, normal and strongly consistent. The details of this method can be found also in [24, Sec. 5.5].

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Yuliya Mishura, Kostiantyn Ralchenko and Sergiy Shklyar

5 Integral equation with power kernel Theorem 7. Let 0 < p < 1 and b > 0. 1. If y ∈ L1 [0, b] is a solution to integral equation Z b y(s) ds 0

|t − s| p

= f (t)

for almost all t ∈ (0, b),

(24)

then y(x) satisfies    Γ (p) cos π2p (1−p)/2 1−p (1−p)/2 f (x) y(x) = D x D0+ π x(1−p)/2 b− x(1−p)/2

(25)

α and D α are the Riemann–Liouville almost everywhere on [0, b], where Da+ b− fractional derivatives, that is Z x  d f (t) 1 α Da+ dt , f (x) = Γ (1 − α ) dx a (x − t)α Z b  d f (t) −1 α Db− f (x) = dt . Γ (1 − α ) dx x (t − x)α

2. If y1 ∈ L1 [0, b] and y2 ∈ L1 [0, b] are two solutions to integral equation (24), then y1 (x) = y2 (x) almost everywhere on [0, b]. 3. If y ∈ L1 [0, b] satisfies (25) almost everywhere on [0, b] and the fractional derivatives are solutions to respective Abel integral equations, that is

Γ



1 1−p 2



Z t (1−p)/2 D0+ ( f (x)x(p−1)/2 ) 0

(t − x)(p+1)/2

dx =

f (t) , t (1−p)/2

for almost all t ∈ (0, b) and

Γ



1 1−p 2



Z b π y(s)s(1−p)/2 x

Γ (p) cos π2p

ds (1−p)/2 = x1−p D0+ (s − x)(p+1)/2



f (x) x(1−p)/2

(26)



(27)

for almost all x ∈ (0, b), then y(s) is a solution to integral equation (24). Proof. Firstly, transform the left-hand side of (24). By [35, Lemma 2.2(i)], for 0 < s 0, t > 0, s 6= t

Parameter estimation for Gaussian processes

Z min(s,t) 0

13





B p, 1−p 2 dτ = (1−p)/2 (1−p)/2 . 1−p (p+1)/2 (p+1)/2 (t − τ ) (s − τ ) τ s t |t − s| p

Hence Z b y(s) ds

|t − s| p

0

=

=

Z b (1−p)/2 (1−p)/2 Z s t y(s) min(s,t)



0

B p,

1−p 2



0

dτ ds. (t − τ )(p+1)/2(s − τ )(p+1)/2τ 1−p

Change the order of integration, noting that {(s, τ ) : 0 < s < b, 0 < τ < min(s,t)} = {(s, τ ) : 0 < τ < t, τ < s < b} for 0 < t < b: Z b y(s) ds 0

|t − s| p

=

t (1−p)/2   B p, 1−p 2

Z t 0

1 (t − τ )(p+1)/2τ 1−p

Z b (1−p)/2 s y(s) ds τ

(s − τ )(p+1)/2

dτ .

(28)

The right-hand side of (28) can be rewritten with fractional integration:  2   Γ 1−p 2 1 (1−p)/2 (1−p)/2 (1−p)/2 (1−p)/2   = I (x y(x)) x I 0+ |x − s| p B p, 1−p x1−p b− 2

Z b y(s) ds 0

α and I α are fractional integrals for 0 < x < b, where Ia+ b− α Ia+ f (x) =

1 Γ (α )

Z x

f (t) dt (x − t)(1−α )

α f (x) = Ib−

1 Γ (α )

Z b

f (t) dt. (t − x)(1−α )

and

a

x

The constant coefficient can be simplified:

  2    1−p p+1 Γ 1−p Γ Γ 2 2 2 π =  = 1−p Γ (p) Γ (p) cos B p, 2

πp
. 2

Thus integral equation (24) can be rewritten with use of fractional integrals:   π 1 (1−p)/2 (1−p)/2 (1−p)/2 (1−p)/2
x I0+ I (x y(x)) = f (x) (29) x1−p b− Γ (p) cos π2p

for almost all x ∈ (0, b). Whenever y ∈ L1 [0, b], the function x(1−p)/2y(x) is obviously integrable on [0, b]. (1−p)/2 (1−p)/2 Now prove that the function x p−1 Ib− (x y(x)) is also integrable on [0, b].

14

Yuliya Mishura, Kostiantyn Ralchenko and Sergiy Shklyar

Indeed, (1−p)/2 (1−p)/2 (x y(x)) ≤ Ib−





Z b (1−p)/2 (1−p)/2 (x y(x)) Ib− 0



x1−p

=

Γ



1

1−p 2



Γ

Z b 0



1 1−p 2

1  dx ≤  1−p Γ 2

t (1−p)/2 |y(t)|

Z t



Z b (1−p)/2 t |y(t)|

(t − x)(p+1)/2

x

Z b 0

1 x1−p

dt,

Z b (1−p)/2 t |y(t)| dt x

dx

0 x1−p (t − x)(p+1)/2

(t − x)(p+1)/2

dx

dt

  Z b B p, 1−p 2   = |y(t)| dt < ∞, 0 Γ 1−p 2

and the integrability is proved. α φ (x), x ∈ (a, b), Due to [41, Theorem 2.1], the Abel integral equation f (x) = Ia+ may have not more that one solution φ (x) within L1 [a, b]. If the equation has such α f (x). Similarly, the Abel integral a solution, then the solution φ (x) is equal to Da+ α equation f (x) = Ib− φ (x) may have not more that one solution φ (x) ∈ L1 [a, b], and α f. if it exists, φ = Db− Therefore, if y ∈ L1 [0, b] is a solution to integral equation, then is also satisfies (29), so !! f (x) 1 (1−p)/2 π (1−p)/2 (1−p)/2
x y(x) = (1−p)/2 , I0+ I (30) x1−p b− Γ (p) cos π2p x !   1 (1−p)/2 π x(1−p)/2y(x) f (x) (1−p)/2
= D0+ I , x1−p b− Γ (p) cos π2p x(1−p)/2    π x(1−p)/2y(x) f (x) (1−p)/2 1−p (1−p)/2
x D = D 0+ b− Γ (p) cos π2p x(1−p)/2

for almost all x ∈ (0, b). Thus y(x) satisfies (25). Statement 1 of Theorem 7 is proved, and statement 2 follows from statement 1. From equations (26) and (27), which can be rewritten with fractional integration operator, f (x) ( f (x)x(1−p)/2 ) = (1−p)/2 , t !   (1−p)/2 f (x) π y(x)x 1−p (1−p)/2 = x D 0+ Γ (p) cos π2p x(1−p)/2

(1−p)/2

I0+ (1−p)/2

Ib−

(1−p)/2

D0+

(30) follows, and (30) is equivalent to (24). Thus statement 3 of Theorem 7 holds true. ⊓ ⊔

Parameter estimation for Gaussian processes

15

Remark 3. The integral equation (24) was solved explicitly in [26, Lemma 3] under the assumption f ∈ C([0, b]). Here we solve this equation in L1 [0, b] and prove the uniqueness of a solution in this space. Note also that the formula for solution in the handbook [37, formula 3.1.30] is incorrect (it is derived from the incorrect formula 3.1.32 of the same book, where an operator of differentiation is missing; this error comes from the book [50]).

6 Boundedness and invertibility of operators This appendix is devoted to the proof of the following result, which plays the key role in the proof of the strong consistency of the MLE for the model with two independent fractional Brownian motions.  Theorem 8. Let H1 ∈ 12 , 34 , H2 ∈ (H1 , 1), and ΓH be the operator defined by (10). Then ΓH−1 ΓH2 : L2 [0, T ] → L2 [0, T ] is a compact linear operator defined on the entire 1 space L2 [0, T ], and the operator I + ΓH−1 ΓH2 is invertible. 1 The proof consists of several steps.

6.1 Convolution operator If φ ∈ L1 [−T, T ], then the following convolution operator L f (x) =

Z T 0

φ (t − s) f (s) ds

(31)

is a linear continuous operator L2 [0, T ] → L2 [0, T ], and kLk ≤

Z T

−T

|φ (t)| dt.

(32)

Moreover, L is a compact operator. The adjoint operator of the operator (31) is L∗ f (x) =

Z T 0

φ (s − t) f (s) ds.

If the function φ is even, then the linear operator L is self-adjoint. Let us consider the following convolution operators. Definition 3. For α > 0, the Riemann–Liouville operators of fractional integration are defined as

16

Yuliya Mishura, Kostiantyn Ralchenko and Sergiy Shklyar

Z

t f (s) ds 1 , Γ (α ) 0 (t − s)1−α Z T 1 f (s) ds ITα− f (t) = . Γ (α ) t (t − s)1−α

α f (t) = I0+

α and I α are mutually adjoint. Their norm can be bounded as The operators I0+ T− follows Z T 1 ds Tα α k≤ . (33) = kITα− k = kI0+ Γ (α ) 0 s1−α Γ (α + 1)

Let

1 2

< H < 1 and ΓH be the operator defined by (10). Then
2H−1 2H−1 ΓH = H Γ (2H) I0+ + IT− .

α , I α for α > 0, and Γ for The linear operators I0+ H T−

1 2

(34)

< H < 1 are injective.

6.2 Semigroup property of the operator of fractional integration Theorem 9. For α > 0 and β > 0 the following equalities hold β

α +β

β

α +β

α I0+ = I0+ , I0+

ITα− IT − = IT − . This theorem is a particular case of [41, Theorem 2.5]. Proposition 3. For 0 < α ≤

1 2

and f ∈ L2 [0, T ], α hI0+ f , ITα− f i ≥ 0.

Equality is achieved if and only if • f = 0 almost everywhere on [0, T ] for 0 < α < 21 ; R • 0T f (t) dt = 0 for α = 12 .

α and I α are mutually adjoint, by semigroup property, Proof. Since the operators I0+ T− we have that α α α 2α hI0+ f , ITα− f i = hI0+ I0+ f , f i = hI0+ f , f i,

α f , ITα− f i = h f , ITα− ITα− f i = h f , IT2α− f i. hI0+

Adding these equalities, we obtain 1 2α α f , ITα− f i = hI0+ f + IT2α− f , f i. hI0+ 2 If 0 < α < 12 , then

(35)

Parameter estimation for Gaussian processes

17

1 hΓH f , f i = 2H Γ (2H) Z T 2 1 = E f (t) dBtH ≥ 0. 2H Γ (2H) 0

α hI0+ f , ITα− f i =

where H = α + 12 , 21 < H < 1, and BtH is a fractional Brownian motion. Let us consider the case α = 21 . Since 1 I0+ f (t) + IT1 − f (t) = 1 1 I0+ f + IT− f=



1 I0+ f

+ IT1 − f ,

f =

we see from (35) that D



Z t

Z T

f (s) ds +

0

Z T 0

Z T 0

1/2 1/2 I0+ f , IT − f

f (s) ds =

t

Z T

f (s) ds,

0

f (s) ds 1[0,T ] , f (s) ds h1[0,T ] , f i =

E

1 = 2

Z

T

0

Z

T 0

2 f (s) ds ,

2 f (s) ds ≥ 0.

(36)

α f , I α f i = 0 can be easily found by analyzing Conditions for the equality hI0+ T− 1 the proof. Indeed, if 0 < α < 2 and H = α + 21 , then ΓH is a self-adjoint positive α f , Iα f i = compact operator whose eigenvalues are all positive. Then 2H Γ (2H)hI0+ T− 1/2

α f , I α f i = 0 holds true if and hΓH f , f i = kΓH f k2 . In this case, the equality hI0+ T− only if f = 0 almost everywhere on [0, T ]. If α = 21 , then the condition for the equality follows from (36). ⊓ ⊔

Proposition 4. For 0 < α ≤

1 2

and f ∈ L2 [0, T ],

α α kI0+ f k + kIT− fk ≤

Consequently, for

1 2

K 2 k f k2 . (39) Put

Parameter estimation for Gaussian processes

19 N

g=

xk

∑ λ k ek .

k=1

Then kgk2 = B−1 g =

N

x2

N

∑ λk2 = sN ,

k=1

k

√ sk ,

g ∈ B(L2 [0, T ]),

∗ −1 

 A B g, f = B−1 g, A f =

xk

∑ λ 2 ek ,

k=1

kgk =

k

N

xk

∑ λ 2 xk = sN .

k=1

k

By the Cauchy–Schwarz inequality, ∗ −1

 A B g, f ≤ A∗ B−1 g k f k ≤ kA∗ B−1 k kgk k f k = K √sN k f k.

Hence,

√ sN ≤ K sN k f k.

(40)

The inequalities (39) and (40) contradict each other. Thus, the operator B−1 A is defined on the entire space L2 [0, T ]. Now let us prove boundedness of the operator B−1 A and the inequality

−1

B A ≤ K. Suppose that this is not so. Then there exists an element f of the space L2 [0, T ] such that

−1

B A f > Kk f k. (41) We use the same decomposition of the vector A f into the eigenvectors of B as above, see (38). Then ∞ x2

−1 2 xk k

e , B A f = , k ∑ ∑ λk 2 λ k=1 k k=1

2 lim sn = B−1 A f > K 2 k f k2 ,

B−1 A f =

∞

n→∞

by (41). Therefore, there exists N ∈ N such that the inequality (39) holds. Arguing as above, we get a contradiction. Hence, kB−1 Ak ≤ K. It remains to prove the opposite inequality kB−1Ak ≥ K. The operator A∗ B−1 is defined on the set B(L2 [0, T ]). For all f ∈ B(L2 [0, T ]) from the domain of the operator A∗ B−1 , we have kA∗ B−1 f k2 = hB−1 AA∗ B−1 f , f i ≤

≤ kB−1 AA∗ B−1 f k k f k ≤ kB−1 Ak kA∗B−1 f k k f k,

whence kA∗ B−1 f k ≤ kB−1 Ak k f k.

Therefore K = kA∗ B−1 k ≤ kB−1 Ak.

⊓ ⊔

20

Yuliya Mishura, Kostiantyn Ralchenko and Sergiy Shklyar

6.4 The proof of boundedness and compactness Proposition 5. Let 12 < H1 < H2 < 1 and H1 ≤ 34 . Then ΓH−1 ΓH2 is a compact linear 1 operator defined on the entire space L2 [0, T ]. Proof. The operator ΓH1 is an injective self-adjoint compact operator L2 [0, T ] → is densely defined on L2 [0, T ]. By Proposition 4, L2 [0, T ]. The inverse operator ΓH−1 1

2H1 −1 −1 the operators I0+ ΓH1 and IT2H−1 −1ΓH−1 are bounded. Therefore, by Lemma 1, the 1

I 2H1 −1 and ΓH−1 I 2H1 −1 are also bounded and defined on the entire operators ΓH−1 1 T− 1 0+ space L2 [0, T ]. By (34) and the semigroup property (Theorem 9),   −1 2H2 −1 −1 2H2 −1 ΓH−1 Γ Γ (2H) Γ Γ = H I + I = H2 H1 0+ H1 T − 1   2H1 −1 2(H2 −H1 ) −1 2H1 −1 2(H2 −H1 ) = H Γ (2H) ΓH−1 Γ I I + I I . H T − T − 0+ 0+ 1 1 2(H −H1 )

Since I0+ 2 compact.

2(H −H1 )

and IT − 2

ΓH2 is also are compact operators, the operator ΓH−1 1 ⊓ ⊔

6.5 The proof of invertibility ΓH2 . Indeed, if Now prove that −1 is not an eigenvalue of the linear operator ΓH−1 1 −1 ΓH1 ΓH2 f = − f for some function f ∈ L2 [0, T ], then ΓH2 f + ΓH1 f = 0. Since ΓH2 and ΓH1 are positive definite self-adjoint (and injective) operators, ΓH2 + ΓH1 is also a positive definite self-adjoint and injective operator. Hence f = 0 almost everywhere on [0, T ]. ΓH2 , −1 is a Because −1 is not an eigenvalue of the compact linear operator ΓH−1 1 −1 −1 regular point, i.e., −1 6∈ σ (ΓH1 ΓH2 ), and the linear operator ΓH1 ΓH2 + I is invertible. Acknowledgements The research of Yu. Mishura was funded (partially) by the Australian Government through the Australian Research Council (project number DP150102758). Yu. Mishura and K. Ralchenko acknowledge that the present research is carried through within the frame and support of the ToppForsk project nr. 274410 of the Research Council of Norway with title STORM: Stochastics for Time-Space Risk Models.

References 1. Belfadli, R., Es-Sebaiy, K., Ouknine, Y.: Parameter estimation for fractional Ornstein– Uhlenbeck processes: non-ergodic case. Frontiers in Science and Engineering 1(1), 1–16 (2011) 2. Benassi, A., Cohen, S., Istas, J.: Identifying the multifractional function of a Gaussian process. Stat. Probab. Lett. 39(4), 337–345 (1998)

Parameter estimation for Gaussian processes

21

3. Bercu, B., Coutin, L., Savy, N.: Sharp large deviations for the fractional Ornstein–Uhlenbeck process. Teor. Veroyatn. Primen. 55(4), 732–771 (2010) 4. Berger, J., Wolpert, R.: Estimating the mean function of a Gaussian process and the Stein effect. J. Multivariate Anal. 13(3), 401–424 (1983) 5. Bertin, K., Torres, S., Tudor, C.A.: Maximum-likelihood estimators and random walks in long memory models. Statistics 45(4), 361–374 (2011) 6. Bojdecki, T., Gorostiza, L.G., Talarczyk, A.: Sub-fractional Brownian motion and its relation to occupation times. Stat. Probab. Lett. 69(4), 405–419 (2004) 7. Cai, C., Chigansky, P., Kleptsyna, M.: Mixed Gaussian processes: A filtering approach. Ann. Probab. 44(4), 3032–3075 (2016) 8. C´enac, P., Es-Sebaiy, K.: Almost sure central limit theorems for random ratios and applications to LSE for fractional Ornstein–Uhlenbeck processes. Probab. Math. Statist. 35(2), 285–300 (2015) 9. Cheridito, P.: Mixed fractional Brownian motion. Bernoulli 7(6), 913–934 (2001) 10. El Machkouri, M., Es-Sebaiy, K., Ouknine, Y.: Least squares estimator for non-ergodic Ornstein–Uhlenbeck processes driven by Gaussian processes. J. Korean Statist. Soc. 45(3), 329–341 (2016) 11. Es-Sebaiy, K.: Berry-Ess´een bounds for the least squares estimator for discretely observed fractional Ornstein–Uhlenbeck processes. Stat. Probab. Lett. 83(10), 2372–2385 (2013) 12. Es-sebaiy, K., Ndiaye, D.: On drift estimation for non-ergodic fractional Ornstein–Uhlenbeck process with discrete observations. Afr. Stat. 9(1), 615–625 (2014) 13. Es-Sebaiy, K., Ouassou, I., Ouknine, Y.: Estimation of the drift of fractional Brownian motion. Statist. Probab. Lett. 79(14), 1647–1653 (2009) 14. Houdr´e, C., Villa, J.: An example of infinite dimensional quasi-helix. In: Stochastic models. Seventh symposium on probability and stochastic processes, June 23–28, 2002, Mexico City, Mexico. Selected papers, pp. 195–201. Providence, RI: American Mathematical Society (AMS) (2003) 15. Hu, Y., Nualart, D.: Parameter estimation for fractional Ornstein–Uhlenbeck processes. Statistics and Probability Letters 80(11-12), 1030–1038 (2010) 16. Hu, Y., Nualart, D., Xiao, W., Zhang, W.: Exact maximum likelihood estimator for drift fractional Brownian motion at discrete observation. Acta Math. Sci. Ser. B Engl. Ed. 31(5), 1851– 1859 (2011) 17. Hu, Y., Nualart, D., Zhou, H.: Parameter estimation for fractional Ornstein–Uhlenbeck processes of general Hurst parameter. arXiv preprint arXiv:1703.09372 (2017) 18. Hu, Y., Song, J.: Parameter estimation for fractional Ornstein-Uhlenbeck processes with discrete observations. In: Malliavin calculus and stochastic analysis. A Festschrift in honor of David Nualart, pp. 427–442. New York, NY: Springer (2013) 19. Ibragimov, I.A., Rozanov, Y.A.: Gaussian random processes, Applications of Mathematics, vol. 9. Springer-Verlag, New York-Berlin (1978) 20. Kleptsyna, M.L., Le Breton, A.: Statistical analysis of the fractional Ornstein-Uhlenbeck type process. Statist. Inference Stoch. Process. 5, 229–248 (2002) 21. Kozachenko, Y., Melnikov, A., Mishura, Y.: On drift parameter estimation in models with fractional Brownian motion. Statistics 49(1), 35–62 (2015) 22. Kubilius, K., Mishura, Y.: The rate of convergence of Hurst index estimate for the stochastic differential equation. Stochastic Processes Appl. 122(11), 3718–3739 (2012) 23. Kubilius, K., Mishura, Y., Ralchenko, K., Seleznjev, O.: Consistency of the drift parameter estimator for the discretized fractional Ornstein–Uhlenbeck process with Hurst index H ∈ (0, 21 ). Electron. J. Stat. 9(2), 1799–1825 (2015) 24. Kubilius, K.e., Mishura, Y., Ralchenko, K.: Parameter estimation in fractional diffusion models, Bocconi & Springer Series, vol. 8. Bocconi University Press, [Milan]; Springer, Cham (2017) 25. Kukush, A., Mishura, Y., Valkeila, E.: Statistical inference with fractional Brownian motion. Stat. Inference Stoch. Process. 8(1), 71–93 (2005) 26. Le Breton, A.: Filtering and parameter estimation in a simple linear system driven by a fractional Brownian motion. Statist. Probab. Lett. 38(3), 263–274 (1998)

22

Yuliya Mishura, Kostiantyn Ralchenko and Sergiy Shklyar

27. Mishura, Y.: Stochastic calculus for fractional Brownian motion and related processes, vol. 1929. Springer Science & Business Media (2008) 28. Mishura, Y.: Maximum likelihood drift estimation for the mixing of two fractional Brownian motions. In: Stochastic and Infinite Dimensional Analysis, pp. 263–280. Springer (2016) 29. Mishura, Y., Ralchenko, K.: On drift parameter estimation in models with fractional Brownian motion by discrete observations. Austrian Journal of Statistics 43(3), 218–228 (2014) 30. Mishura, Y., Ralchenko, K.: Drift parameter estimation in the models involving fractional brownian motion. In: V. Panov (ed.) Modern Problems of Stochastic Analysis and Statistics: Selected Contributions In Honor of Valentin Konakov, pp. 237–268. Springer International Publishing, Cham (2017) 31. Mishura, Y., Ralchenko, K., Seleznev, O., Shevchenko, G.: Asymptotic properties of drift parameter estimator based on discrete observations of stochastic differential equation driven by fractional Brownian motion. In: Modern stochastics and applications, Springer Optim. Appl., vol. 90, pp. 303–318. Springer, Cham (2014) 32. Mishura, Y., Ralchenko, K., Shklyar, S.: Maximum likelihood drift estimation for Gaussian process with stationary increments. Austrian J. Statist. 46(3-4), 67–78 (2017) 33. Mishura, Y., Ralchenko, K., Shklyar, S.: Maximum likelihood drift estimation for Gaussian process with stationary increments. Nonlinear Anal. Model. Control 23(1), 120–140 (2018) 34. Mishura, Y., Voronov, I.: Construction of maximum likelihood estimator in the mixed fractional–fractional Brownian motion model with double long-range dependence. Mod. Stoch. Theory Appl. 2(2), 147–164 (2015) 35. Norros, I., Valkeila, E., Virtamo, J.: An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli 5(4), 571–587 (1999) 36. Peltier, R.F., L´evy V´ehel, J.: Multifractional Brownian motion: definition and preliminary results. INRIA research report, vol. 2645 (1995) 37. Polyanin, A., Manzhirov, A.: Handbook of integral equations, second edn. Chapman & Hall/CRC, Boca Raton, FL (2008) 38. Prakasa Rao, B.L.S.: Statistical inference for fractional diffusion processes. Chichester: John Wiley & Sons (2010) 39. Privault, N., R´eveillac, A.: Stein estimation for the drift of Gaussian processes using the Malliavin calculus. Ann. Statist. 36(5), 2531–2550 (2008) 40. Ralchenko, K.V., Shevchenko, G.M.: Paths properties of multifractal Brownian motion. Theory Probab. Math. Statist. 80, 119–130 (2010) 41. Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives. Taylor & Francis (1993) 42. Samuelson, P.A.: Rational theory of warrant pricing. Industrial Management Review 6(2), 13–32 (1965) 43. Shen, G., Yan, L.: Estimators for the drift of subfractional Brownian motion. Comm. Statist. Theory Methods 43(8), 1601–1612 (2014) 44. Tanaka, K.: Distributions of the maximum likelihood and minimum contrast estimators associated with the fractional Ornstein-Uhlenbeck process. Stat. Inference Stoch. Process 16, 173–192 (2013) 45. Tanaka, K.: Maximum likelihood estimation for the non-ergodic fractional Ornstein– Uhlenbeck process. Stat. Inference Stoch. Process. 18(3), 315–332 (2015) 46. Tudor, C.: Some properties of the sub-fractional Brownian motion. Stochastics 79(5), 431–448 (2007) 47. Tudor, C.A., Viens, F.G.: Statistical aspects of the fractional stochastic calculus. The Annals of Statistics 35(3), 1183–1212 (2007) 48. Xiao, W., Zhang, W., Xu, W.: Parameter estimation for fractional OrnsteinUhlenbeck processes at discrete observation. Applied Mathematical Modelling 35, 4196–4207 (2011) 49. Xiao, W.L., Zhang, W.G., Zhang, X.L.: Maximum-likelihood estimators in the mixed fractional Brownian motion. Statistics 45, 73–85 (2011) 50. Zabreyko, P.P., Koshelev, A.I., Krasnosel’skii, M.A., Mikhlin, S.G., Rakovshchik, L.S., Stet’senko, V.Y.: Integral equations: A reference text. Noordhoff, Leyden (1975)