Robust model selection for a semimartingale continuous time regression from discrete data

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Robust model selection for a semimartingale continuous time regression from discrete data

Victor Konev, Serguei Pergamenchtchikov

To cite this version:

Victor Konev, Serguei Pergamenchtchikov. Robust model selection for a semimartingale continuous time regression from discrete data. Stochastic Processes and their Applications, Elsevier, 2015, 125, pp.294 - 326. �10.1016/j.spa.2014.08.003�. �hal-02334868�

Robust model selection for a semimartingale continuous time regression from discrete data

Konev Victor ^†and Pergamenchtchikov Serguei^‡

Abstract

The paper considers the problem of estimating a periodic func- tion in a continuous time regression model observed under a general semimartingale noise with an unknown distribution in the case when continuous observation cannot be provided and only discrete time measurements are available. Two specific types of noises are stud- ied in detail: a non - Gaussian Ornstein - Uhlenbeck process and a time - varying linear combination of a Brownian motion and com- pound Poisson process. We develop new analytical tools to treat the adaptive estimation problems from discrete data. A lower bound for the frequency sampling, needed for the efficiency of the procedure constructed by discrete observations, has been found. Sharp non- asymptotic oracle inequalities for the robust quadratic risk have been derived. New convergence rates for the efficient procedures have been obtained. An example of the regression with a martingale noise ex- hibits that the minimax robust convergence rate may be both higher or lower as compared with the minimax rate for the ”white noise”

model. The results of Monte-Carlo simulations are given.

Keywords: Semimartingale regression; Estimation from discrete data; Ro- bust risk estimation; Model selection; Sharp oracle inequalities.

AMS 2000 Subject Classifications: Primary: 62G08; Secondary: 62G05

∗This study was partially supported by Russian Science Fondation (research project No. 14-49-00079) and by the National Research Tomsk State University.

†Department of Applied Mathematics and Cybernetics, Tomsk State University and Department of High Mathematics and Mathematical Physics Tomsk Polytechnical Uni- versity, Tomsk, Russia, e-mail: [email protected]

‡Laboratoire de Math´ematiques Rapha¨el Salem, UMR 6085 CNRS-Universit´e de Rouen, France and National Research University - Higher School of Economics, Laboratory of Quantitative Finance, Moscow, Russia, email: [email protected].

1 Introduction

Consider a regression model in continuous time

dy_t =S(t)dt+ dξ_t, 0≤t≤n , (1.1) where S is an unknown function which belongs to the linear space V₁ of 1 - periodic R→Rcadlag functions;ξ = (ξ_t)_t≥0 is an unobservable semimartin- gale noise with the values in the Skorokhod space D[0, n] such that for each function f from L₂[0, n] the stochastic integral

I_n(f) = Z n

0

f(s)dξ_s (1.2)

is well defined and has the properties:

E_QI_n(f) = 0 and E_QI_n²(f)≤κQ

Z n 0

f²(s)ds , (1.3) where κQ >0 is some positive constant depending on the noise distribution Q on the space D[0, n]; E_Q denotes the expectation under the distribution Q. The noise distribution Q is unknown and assumed to belong to some distribution family Q_n specified below. All necessary tools concerning the stochastic calculus can be found, for example, in [18].

The problem is to estimate the unknown function S in the model (1.1) on the basis of observations

(y_t

j)_0≤j≤np, t_j =j∆, ∆ = 1

p, (1.4)

where p ≥ 3 is an odd number depending on n. Such an assumption about discrete time observations arises if the continuous observation of the process (1.1) cannot be provided. There are many papers devoted to the similar nonparametric estimation problems for the regression model (1.1) and other continuous time processes on the basis of the discrete observations (1.4).

Gobet, Hoffmann and Reiss [14] and Comte, Genon-Catalot and Rozenholc [5] studied the problem of estimating the coefficients of a diffusion process from discrete data. Hoffmann, Munk and Schmidt-Hieber [15] investigate nonparametric estimation for diffusion coefficients from discrete data, when the observation are blurred by an additive noise. Comte and Genon-Catalot

[4] studied nonparametric estimation problem for pure jumps L´evy process in the model (1.1) based on discrete time observations.

In this paper we consider the estimation problem in an adaptive setting, i.e. when the regularity ofS is unknown. The quality of an estimate Se_n (any one periodic R → R function constructed from discrete data (1.4)) will be measured with the quadratic risk

R_Q(Se_n, S) =E_QkSe_n−Sk², (1.5) wherekfk² =R1

0 f²(t)dt. Since the noise distributionQis unknown, it seems reasonable to introduce the robust risk of the form

R^∗_n(Se_n, S) = sup

Q∈Q_n

R_Q(Se_n, S), (1.6) which enables one to take into account the information that Q ∈ Q_n and ensures the quality of an estimate Se_n for all distributions in the family Q_n. It will be noted that similar criteria in the problems of nonparametric es- timation were used in a number of papers (see, for example, the papers by Galtchouk and Pergamenchtchikov in [10], by the authors [24] and the refer- ences therein).

The goal of this paper is to develop a model selection method for esti- mating a continuous time semimartingale regression (1.1) from discrete data (1.4). The origin of the model selection method goes back to early seventies with the pioneering papers by Akaike [1] and Mallows [25] who suggested to use penalizing in a log-likelihood type criterion. Barron, Birg´e, Massart [3], Massart [26] and Kneip [19] developed a non-asymptotic model selec- tion method which enables one to derive non-asymptotic oracle inequalities for the nonparametric regression models with Gaussian disturbances. The model selection procedures for the regression schemes with dependent noises and unknown distributions were studied by Comte and Genon-Catalot [4], Fourdrinier and Pergamenchtchikov [8], Galtchouk and Pergamenchtchikov [9], and by the authors (see, [23], [21] for details and further references). The interest to the model selection procedures can be explained by the fact that they provide adaptive solutions to the non-asymptotic regression models by the technique of sharp non-asymptotic oracle inequalities. As is known (see, for example, [3]), the oracle inequality gives an upper bound for the risk via the minimal risk corresponding to a chosen family of estimates. In con- structing an adaptive model selection procedure, in this paper, we will use

the approach close to that of the papers [12], [11], [13], developed for a het- eroscedastic regression model in discrete time. The key idea of the method in [12] is to combine the Barron-Birg´e-Massart non-asymptotic penalization method [3] and the Pinsker weighted least squares method of minimizing the asymptotic risk [27], [28]. The advantage of this method is two-fold: it deals with a robust risk and enables one to construct asymptotically efficient procedures.

The rest of the paper is organized as follows. Section 2 gives two specific examples of the noise in the model (1.1). First, a non - Gaussian Ornstein - Uhlenbeck process driven by a finite intensity L´evy process and second - a deterministic but time - varying linear combination of a Brownian motion and a compound Poisson process. In Section 3 we construct a model selec- tion procedure based on the weighted least squares estimators and, under general moment conditions on the distribution ξ in (1.1), obtain sharp non- asymptotic oracle inequalities for the risks (1.5) and (1.6). Moreover, we check these conditions for the examples given in Section 2. Section 4 illus- trates the performance of the proposed model selection procedure through numerical simulations. It turns out that the sample convergence rate is close to that obtained in [12] for heteroscedastic regression models in the discrete time. Section 5 deals with the asymptotic properties of the proposed proce- dure under the additional assumption that the unknown function S in (1.1) belongs to a Sobolev ball. In this section we obtain an asymptotic sharp lower bound for the robust risk (1.6) and we calculate the Pinsker constant for the model (1.1). Then we find the lower bound for the frequency sampling p which is needed for constructing efficient estimation procedures from dis- crete data. It is shown that the limit of the risk (1.6) for the proposed model selection procedure, normalized by the minimax rate, equals the Pinsker constant, i.e. the procedure is efficient. Section 6 gives the proofs of all the theorems. In the Appendix some technical results are established.

2 Examples

2.1 Non-Gaussian Ornstein-Uhlenbeck process

First we consider an example of the disturbance (ξ_t)_t≥0 in (1.1) given by a non-Gaussian Ornstein-Uhlenbeck process with the L´evy subordinator. Such processes are used in the financial Black-Scholes market models with stepwise

randomly evolving coefficients (see [2] for details and further references). Let the noise process in (1.1) obey the equation

dξ_t=aξ_tdt+ du_t, ξ₀ = 0, (2.1) where a ≤ 0, u_t = %₁w_t+%₂z_t, %₁ and %₂ are unknown constants, (w_t)_t≥0 is a standard Brownian motion, (z_t)_t≥0 is a compound Poisson process of the form

z_t=

N_t

X

j=1

Y_j (2.2)

where (N_t)_t≥0 is a standard homogeneous Poisson process with unknown intensity λ >0 and (Y_j)_j≥1 is an i.i.d. sequence of random variables with

EY₁ = 0, EY₁² = 1 and EY₁⁴ <∞. (2.3) Let (T_k)_k≥1 denote the arrival times of the process (N_t)_t≥0, that is,

T_k = inf{t≥0 : N_t=k}.

The parameters λ, a, %₁ and %₂ of the process (2.1) are assumed to satisfy the following conditions







−a_∗ ≤a≤0, λ ≥ λ_∗,

%_min ≤%₁ ≤%_max, %_min ≤%^∗ ≤%_max,

(2.4) where %^∗ = %²₁ +λ%²₂. For this noise model the family Q_n consists of all distributions on the Skorokhod space D[0, n] of the process (2.1) with the parameters satisfying the inequalities (2.4) with unknown bounds λ_∗, a_∗,

%_min and %_max.

Remark 2.1. The estimation problem from the continuous data for the mod- els of type (1.1), (2.1) has been studied in a parametric setting in [6], [16], [17] and [20].

2.2 Martingale noise

Next we consider a martingale noise in (1.1) obeying the equation dξ_t=p

%₁(t)dw_t+p

%₂(t)dz_t, (2.5)

where %₁ and %₂ are positive two times continuously differentiable R+ →R+

nonrandom functions; the process (z_t)_t≥0 is as defined in (2.2)–(2.3). We assume that λ≥λ_∗ for some unknown λ_∗ >0. Moreover, denoting

%(t) =e %₁(t) +λ%₂(t), (2.6) we assume that, there exist R+ → R+ continuous functions %_min(·) and

%_max(·) such that for t ≥0

%_min(t)≤%(t)e ≤%_max(t) (2.7) and, for any δ >0,

lim

t→∞ t^δ%_min(t) = +∞ and lim

t→∞

%_max(t)

t^δ = 0. (2.8) In addition, let the first two derivatives of the functions %(·) satisfy thee following conditions for some positive constants %⁰_∗ and %⁰⁰_∗

sup

t≥0

(t+ 1)

d dt%(t)e

≤%⁰_∗, sup

t≥0

d² dt² %(t)e

≤%⁰⁰_∗. (2.9) In this caseQ_nis the family of all distributions of the process (2.5) onD[0, n]

satisfying the conditions (2.7) and (2.9) for some fixed unknown parameters λ_∗ >0, %_min(·),%_max(·),%⁰_∗ and %⁰⁰_∗.

3 Oracle inequalities

In this section we construct a model selection procedure for estimating a function S in (1.1) by the discrete time observations (1.4) and establish the oracle inequalities for its risk. Let (φ_j)_j≥1 be the standard trigonometric basis in L₂[0,1] defined as

φ₁ = 1, φ_j(x) =







√2 cos($_jx) for even j ≥2;

√2 sin($_jx) for odd j ≥3,

(3.1) where$_j = 2π[j/2], [x] denotes the integer part ofx. The restrictions of the functions {φ₁, . . . , φ_p}, p≥3, on the sampling lattice

T_p =

t₁, . . . , t_p , t_j = j p,

form an orthonormal basis in the Hilbert space R^Tp with the inner product (x, y)_p = 1

p

p

X

j=1

x(t_j)y(t_j) for x, y ∈R^Tp (3.2)

and the norm kxk_p =q

(x, x)_p. It is clear that the space R^Tp is isometric to R^p. One can check directly that for any odd p

(φ_j, φ_i)_p =χ_{i=j}. (3.3) This implies that the function S on the lattice T_p coincides with its discrete Fourier transformation, i.e.

S(t) =

p

X

j=1

θ_j,pφ_j(t), if t∈ T_p, (3.4) where θ_j,p = (S, φ_j)_p. The first step in constructing the model selection procedure consists in estimating the coefficients θ_j,p for S in (1.1) from the discrete data by the formulae

θb_j,p = 1 n

Z n 0

ψ_j,p(t) dy_t, ψ_j,p =D_p(φ_j). (3.5) Here D_p stands for the linear mapping given by the equation

D_p(f)(t) =

np

X

k=1

f(t_k)χ_{t

k−1<t≤t_k}. (3.6)

Furthermore, we note that the system of the functions (ψ_j,p)_1≤j≤p is orthonor- mal inL2[0,1] because (ψ_j,p, ψ_i,p) = R1

0 ψ_j,p(t)ψ_i,p(t)dt= (φ_j, φ_i)_p =χ_{i=j}. In the sequel we need the Fourier coefficients for the function S with respect to these functions which can be written as

θ_j,p = (S, ψ_j,p) = Z 1

0

S(t)ψ_j,p(t) dt=θ_j,p+h_j,p, (3.7) where h_j,p = h_j,p(S) = Pp

l=1

Rt_l

t_l−1 φ_j(t_l)(S(t)−S(t_l))dt. Substituting (1.1) in (3.5) yields

θb_j,p =θ_j,p+ 1

√nξ_j,p and ξ_j,p = I_n(ψ_j,p)

√n . (3.8)

Besides we will impose the additional conditions on the distribution of the noise ξ_t in (1.1). For any x= (x₁, . . . , x_p) from R^p, we set

#(x) =

p

X

j=1

1_{|x

j|>0} and |x|² =

p

X

j=1

x²_j. (3.9) We need the following conditions.

C₁) There exists a variance proxy σ_Q ≥ 0, which may be depend on n ≥ 1, such that the sequence ζ_j,p =E_Qξ_j,p² −σ_Q satisfies the following inequality

L_1,Q(n) = sup

p≥3

sup

x∈H₁

p

X

j=1

x_jζ_j,p

<∞ where H₁ ={x∈[−1,1]^p : #(x)≤n}.

C₂) Assume that

L_2,Q(n) = sup

p≥3

sup

x∈H₂

E_Q





p

X

j=1

x_jξe_j,p





2

<∞

where H₂ ={x∈R^p : |x| ≤1, #(x)≤n} and ξe_j,p=ξ_j,p² −E_Qξ_j,p² .

Remark 3.1. As is shown in the paper [24], which considers the estimation problem for the model (1.1) under continuous observations, one needs some stability conditions of the noise variances in (3.8) and the boundedness of their deviations to derive the oracle inequalities for the robust risk. In the estimation problem from discrete data the noise properties depend highly on the frequency of observations as well. The additional supremum in the Con- ditions in C₁) and C₂) over p ≥ 3 assumes that the stability of the noise variances and their deviations hold uniformly in p. Further restrictions on the functionals L_1,Q(n) and L_2,Q(n) will be imposed in Section 3.2.

As will be shown in the Section 3.3, the conditionsC₁) and C₂) are satisfied for the model (1.1) with the noises (2.1) and (2.5). Now we introduce a weight least squares estimate for S(t) as

Sb_γ(t) =

p

X

j=1

γ(j)bθ_j,pψ_j,p(t), (3.10)

where γ = (γ(1), . . . , γ(p))⁰ ∈ [0,1]^p is the vector of weight coefficients. The model selection procedure will be chosen from a finite family of such estimates (Sb_γ)_γ∈Γ. The set of weight sequences Γ will be given below. We will need the following characteristic of this set and the corresponding system of weight sequences

ν_p =cardΓ and µ_p = max

γ∈Γ #(γ). (3.11)

We assume that

µ_p ≤n . (3.12)

Remark 3.2. The last inequality means that we cannot use more than n nonzero terms in the estimate (3.10). We need this condition because the noise variance in (3.8) tends to zero as n⁻¹.

Now note that in view of the definition (3.7), the empirical squared error of the estimator (3.10) can be represented as

Err(γ) =kSb_γ−Sk² =

p

X

j=1

γ²(j)bθ²_j,p−2

p

X

j=1

γ(j)bθ_j,pθ_j,p + kSk². (3.13)

Since the Fourier coefficients (θ_j,p)_j≥1 are unknown, the weight coefficients (γ(j))_1≤j≤pcannot be determined by minimizing this quantity. To circumvent this difficulty we replace the terms θb_j,pθ_j,p by

θe_j,p =bθ²_j,p−σb_n

n , (3.14)

whereσb_n is an estimator for the variance proxy σ_Q in the conditionC₁). We will need the following characteristics of this estimate

r_Q(bσ_n) = E_Q|σb_n−σ_Q| and r^∗_n(σb_n) = sup

Q∈Q_n

E_Q|bσ_n−σ_Q|. (3.15) For replacing the termsbθ_j,pθ_j,p by the estimates (3.14) on the right-hand side of the empirical squared error (3.13), one has to pay some penalty. Thus, one comes to the cost function of the form

J(γ) =

p

X

j=1

γ²(j)bθ_j,p² −2

p

X

j=1

γ(j)θe_j,p +ρPb(γ) (3.16)

where ρ is some positive constant and Pb(γ) is the penalty term defined as Pb(γ) = bσ_n|γ|²

n , (3.17)

where |γ|² =Pp

j=1γ²(j). If σ_Q is known, one can put bσ_n =σ_Q and Pb(γ) =P_Q(γ) = σ_Q|γ|²

n . (3.18)

Substituting in (3.10) the values of the weight coefficients γ(j), minimizing the cost function (3.16), that is

bγ = argmin_γ∈ΓJ(γ), (3.19) yields the model selection procedure

Sb_∗ =Sb

bγ. (3.20)

Our first goal is to obtain the oracle inequalities for the quadratic risk of the estimate (3.20) defined in (1.5). To state the result we introduce the sequence

Ψ_Q(n, p) = 2

3L_1,Q(n) +ν_pL_2,Q(n) σ_Q−1

+ν_pκQ

. (3.21)

Proposition 3.1. Let the conditions (1.3), C₁) and C₂) imposed on the noise distribution (ξ_t)_t≥0 in (1.1) hold. Then, for anyn ≥2, p≥3, 0< ρ <

1/3and any set Γ with the property (3.12), the estimator (3.20) satisfies the following oracle inequality

R_Q(Sb_∗, S)≤ 1 + 3ρ 1−3ρmin

γ∈Γ

R_Q(Sb_γ, S) + 1 n

Ψ_Q(n, p) (1−3ρ)ρ +6µ_pr_Q(bσ_n)

n(1−3ρ) . (3.22)

The proof of Proposition 3.1 is given in Section 6.

If the proxy variance σ_Q in the Condition C₁) is known, then one comes to the following result.

Corollary 3.2. Suppose that the conditions of Proposition 3.1 hold with known σ_Q >0. Then, for any n≥2, p≥3, 0< ρ <1/3 and any set Γ with the property (3.12), the estimator (3.20) satisfies the oracle inequality

R_Q(Sb_∗, S)≤ 1 + 3ρ 1−3ρmin

γ∈Γ R_Q(Sb_γ, S) + 1 n

Ψ_Q(n, p)

(1−3ρ)ρ. (3.23) Remark 3.3. It should be noted that Comte and Genon-Catalot in [4] use the oracle inequality of type (3.22) to find an adaptive convergence rate for the density estimation problem for the L´evy processes from discrete data. In this paper we will use this inequality to study efficiency properties for the procedure (3.19).

3.1 Estimation of σ_Q

In this section we will consider the case of an unknown proxy varianceσ_Q in the condition C₁) and derive the oracle inequalities for the continuous time estimate (3.10). We additionally assume that the unknown function S(t) in (1.1) has an absolutely integrable derivative. First we have to estimate σ_Q and find an upper bound for r_Q(bσ_n) in (3.22). One can use the following estimate for σ_Q

σb_n = n ˇ p

ˇ p

X

j=l₀

θb_j,p² and pˇ= min(p, n), (3.24) where l₀ ≥1 will be specified later. We set σb_n = 0 for l₀ >p.ˇ

Lemma 3.3. Assume that the conditions (1.3), C₁) and C₂) hold and the unknown function S(t)is differentiable for 0≤t ≤1and such that for some r >0

kSk² +kSk˙ ² = Z 1

0

|S(t)|²+|S(t)|˙ ²

dt ≤ r . (3.25) Then, for any n ≥ 1 and p ≥ 3, the quantity r_Q(bσ_n), defined in (3.15), satisfies the inequality

r_Q(bσ_n)≤ K_Q(n) n

ˇ pl₀ +l₀

ˇ p + 1

√pˇ

, (3.26)

where K_Q(n) = 16r+ 8√

rκQ+L_1,Q(n) +σ_Q+q

L_2,Q(n).

The proof of Lemma 3.3 is given in Section 6.

Minimizing the right hand side of the last inequality with respect to l₀ we find the appropriate value for l₀ in (3.24), namely

l₀ =√ n

, (3.27)

which yields

r_Q(bσ_n)≤ 3K_Q(n) √

n ˇ p + 1

√pˇ

:= 3K_Q(n)g^∗_p,n. (3.28) Combining Proposition 3.1 and Lemma 3.3 leads to the following result.

Proposition 3.4. Assume that the conditions (1.3), C₁) and C₂) hold and the function S satisfies the inequality (3.25) for some r > 0. Then, for any n ≥ 1, p ≥ 3, 0 < ρ < 1/3 and any set Γ with the property (3.12), the estimate (3.20) obeys the oracle inequality

R_Q(Sb_∗, S) ≤ 1 + 3ρ 1−3ρmin

γ∈ΓR_Q(Sb_γ, S) + B_Q(n, p)

n(1−3ρ)ρ, (3.29) where B_Q(n, p) = Ψ_Q(n, p) + 18K_Q(n)µ_pg^∗_p,n.

Remark 3.4. Note that the oracle inequality (3.29)involves the termB_Q(n, p) which depends on n and p. Our goal is to find conditions on the noise dis- tribution Q and the frequency p providing the boundedness of this term by any power of n as n → ∞. As will be seen later, this property implies the efficiency of model selection procedure (3.20). To this end we need to study the robust risks for this estimate.

3.2 Robust estimation

In order to obtain the oracle inequality for the robust risk (1.6) of the esti- mate (3.10) we will impose additional conditions on the distribution family Q_n of the noise ξ in the equation (1.1). Actually these conditions are stipu- lated also by the further studies of the asymptotic properties of the procedure (3.20) provided that both the number of observation periodsnand the obser- vation frequency ptend to infinity. LetP_ndenote the class of all probability measures on the space D[0, n] and P_n^∗ be its subclass defined as

P_n^∗ =n

Q∈ P_n : L_1,Q(n)≤ L^∗_1,n, L_2,Q(n)≤ L^∗_2,no

, (3.30)

where L_1,Q(n) andL_2,Q(n) are functionals from the conditions C₁),C₂) and L^∗_1,n and L^∗_2,n are numerical sequences such that for any δ >0

lim

n→∞

L^∗_1,n+L^∗_2,n n^δ = 0.

H₁) Assume that each distribution in the family Q_n enters the class (3.30), i.e. Q_n ⊆ P_n^∗ for each n ≥ 1. Besides, the constant κQ in (1.3) and the variance proxy σ_Q from the condition C₁) are such that for n= 1,2, ...

( κ^∗(n) := sup_Q∈Q

nκQ <∞, σ^∗(n) := sup_Q∈Q

nσ_Q <∞, σ_∗(n) := inf_Q∈Q

nσ_Q >0 (3.31)

and for any δ >0 lim

n→∞

κ^∗(n)

n^δ = 0, lim

n→∞

σ^∗(n)

n^δ = 0 and lim

n→∞ n^δσ_∗(n) = +∞. (3.32) Taking into account these notations, we get the following upper bounds for the functions Ψ_Q(ρ) andK_Q(n) in (3.21) and (3.26)

Ψ^∗_n(p) = 2

2L^∗_1,n+ν_pL^∗_2,n(σ_∗(n))⁻¹+ν_pκ^∗(n) and

K^∗_n= 16r+ 8p

rκ^∗(n) +L^∗_1,n+q

L^∗_2,n+σ^∗(n).

Now proceeding from Theorem 3.4 we come to the following result.

Theorem 3.5. Assume that the conditions (1.3)andH₁) hold and the func- tion S satisfies the inequality (3.25) for some r > 0. Then, for any n ≥ 1, p ≥ 3, 0 < ρ < 1/3 and any set Γ with the property (3.12) the estimate (3.20) satisfies the oracle inequality for the robust risk (1.6), i.e.

R^∗_n(Sb_∗, S) ≤ 1 + 3ρ 1−3ρmin

γ∈ΓR^∗_n(Sb_γ, S) + B_n^∗(p)

n(1−3ρ)ρ, (3.33) where B^∗_n(p) = Ψ^∗_n(p) + 18K^∗_nµ_pg^∗_p,n.

3.3 Specification of weights in the model selection

Now we will specify the weight coefficients (γ(j))_1≤j≤p in the way proposed in [12]. Consider a numerical grid of the form

A={1, . . . , k^∗} × {t₁, . . . , t_m}, (3.34) where t_i =iε and m = [1/ε²]. We assume that both parameters k^∗ ≥1 and n⁻² ≤ε≤1 are functions ofn, i.e. k^∗ =k^∗(n) and ε=ε(n), such that







lim_n→∞ k^∗(n) = +∞, lim_n→∞ k^∗(n) lnn = 0,

lim_n→∞σ^∗(n)ε(n) = 0 and lim_n→∞ n^δε(n) = +∞

(3.35)

for any δ >0. One can take, for example, for n ≥2

ε(n) = 1

(σ^∗(n) lnn)∧n² and k^∗(n) = k₀^∗+

√

lnn , (3.36) where a∧b = min(a, b) and k₀^∗ ≥ 0 is some fixed constant. For each α = (β, t)∈ A, we introduce the weight sequence

γ_α = (γ_α(j))_1≤j≤p with the elements

γ_α(j) = 1_{1≤j<j

∗}+ 1−(j/ωα)^β 1_{j

∗≤j≤ω_α} (3.37) where j_∗ = [ω_α/ln(n+ 1)], ω_α = (τ_βt n)^1/(2β+1) and

τ_β = (β+ 1)(2β+ 1)/π^2ββ. Now we specify the set Γ in (3.19) as

Γ = {γ_α, α∈ A}. (3.38)

Note that for any p ≥ 1 the number of elements in Γ equals ν_p = k^∗m.

Therefore, in view of the assumptions (3.35), for any δ >0, one has lim

n→∞

k^∗m

n^δ = 0. (3.39)

Taking into account that τ_β <1 for β≥1 and that ε≥n⁻², we obtain ω^∗ = sup

α∈A

ω_α ≤(n/ε)^1/3 ≤n . (3.40)

Therefore, for any p≥1 and n ≥1 µ_p = sup

α∈A p

X

j=1

1_{γ

α(j)>0} ≤ω^∗ ≤ n ,

i.e. the condition (3.12) holds. Moreover, the last condition in (3.35) yields lim

n→∞

sup_p≥1 µ_p

n^1/3+δ = 0 for any δ >0.

Our main goal is to bound asymptotically (as n → ∞) the term B_n^∗(p) in (3.33) by any power of n. To this end we note that if

p ≥ p_∗ = 2[n^5/6/2] + 3, (3.41) then the upper bound (3.28) can be estimated as g_p,n^∗ ≤2n^−1/3, i.e.

lim

n→∞

1 n^δ sup

p≥p_∗

µ_pg^∗_p,n= 0, ∀ δ >0. (3.42) Therefore, using these properties in (3.33) one comes to the following result.

Theorem 3.6. Assume that the conditions (1.3) andH₁) hold, the function S satisfies the inequality (3.25) for some r >0. Then, for any n ≥3, p≥3 and 0 < ρ < 1/3, the estimate (3.20) with the weight coefficients (3.38) satisfies the oracle inequality (3.33), in which, for any δ >0,

lim

n→∞

1 n^δ sup

p≥p_∗

B_n^∗(p) = 0. (3.43)

Remark 3.5. It will be noted that the assertion of Theorem 3.6 holds true, generally speaking, for any random process (ξ_t)_t≥0 in (1.1) for which the stochastic integral (1.2) is well defined and satisfies the conditions (1.3) and H₁). The class of square integrable semimartingales, in our opinion, is the most appropriate one for the estimation problems with the quadratic risks (1.5) and (1.6). As an alternative, one could consider, for example, non- semimartingale regression models based on the fractional Brownian motions with the Hurst parameterH 6= 1/2. However, in the latter case, the stochastic integral is defined not for functions from L₂[0, n], but for some special spaces (see [7] for details). This does not agree with the the definition (1.5) and, moreover, the first limiting relation in (3.32) fails to hold.