E A Pchelintsev

Oracle inequalities for the stochastic differential equations *

Keywords: Non-parametric regression, Weighted least squares estimates, Improved non-asymptotic estimation, Robust quadratic risk, Lévy process, Ornstein -Uhlenbeck process, semi-Markov process, Model selection, Sharp oracle inequality, Adaptive estimation, Asymptotic efficiency AMS (2010) Subject Classification : primary 62G08; secondary 62G05

This paper is a survey of recent results on the adaptive robust non parametric methods for the continuous time regression model with the semi -martingale noises with jumps. The noises are modeled by the Lévy processes, the Ornstein -Uhlenbeck processes and semi-Markov processes. We represent the general model selection method and the sharp oracle inequalities methods which provide the robust efficient estimation in the adaptive setting. Moreover, we present the recent results on the improved model selection methods for the nonparametric estimation problems.

Introduction

This paper is a survey on the adaptive non parametric estimation methods for the general semi-martingale regression model in continuous time defined as d y t = S(t)d t + dξ t , 0 ≤ t ≤ n ,

where S(•) is an unknown 1 -periodic function, (ξ t ) 0≤t≤n is an unobservable noise defined by semimartingale with the values in the Skorokhod space D[0, n] such that, for any function f from L 2 [0, n], the stochastic integral

I n (f ) = n 0 f (s)dξ s
is well defined and has the following properties

E Q I n (f ) = 0 and E Q I 2 n (f ) ≤ κ Q n 0 f 2 (s)ds . (1.2)
We use E Q for the expectation with respect to the distribution Q in D[0, n] of the process (ξ t ) 0≤t≤n , which is assumed to belong to some probability family Q n and κ Q is some positive constant depending on the distribution Q. The problem consists to estimate the function S on the observations (y t ) 0≤t≤n . Note that if (ξ t ) 0≤t≤n is a brownian motion, then we obtain the well known "signal+white noise" model which is very popular in statistical radio-physics (see, for example, [START_REF] Ibragimov | Statistical Estimation: Asymptotic Theory[END_REF][START_REF] Yua | Estimation of the signal parameter in a Gaussian Noise[END_REF][START_REF] Yua | Parameter Estimation for Stochastic Processes[END_REF][START_REF] Pinsker | Optimal filtration of square integrable signals in Gaussian white noise[END_REF]). In this paper we assume that in addition to the intrinsic noise in the radio-electronic system, approximated usually by the Gaussian white or color noise, the useful signal S is distorted by the impulse flow described by the processes with jumps. The cause of the appearance of a pulse stream in the radio-electronic systems can be, for example, either external unintended (atmospheric) or intentional impulse noise and the errors in the demodulation and the channel decoding for the binary information symbols. Note that, for the first time the impulse noises for the detection signal problems have been introduced on the basis of the compound Poisson processes was introduced by Kassam in [START_REF] Kassam | Signal detection in non-Gaussian noise[END_REF]. Later, such processes was used in [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF][START_REF] Konev | Robust model selection for a semimartingale continuous time regression from discrete data[END_REF][START_REF] Konev | Estimation of a regression with the pulse type noise from discrete data[END_REF][START_REF] Pchelintsev | Improved estimation in a non-Gaussian parametric regression[END_REF] for the parametric and nonparametric signal estimation problems. However, the compound Poisson process can describe only the large impulses influence of fixed single frequency. Taking into account that in the telecommunication systems, the impulses are without limitations on frequencies one needs to extend the framework of the observation model by making use the Lévy processes (2.1) which is a particular case of the general semimartinagale regression model introduced in [START_REF] Konev | Nonparametric estimation in a semimartingale regression model. Part 1. Oracle Inequalities[END_REF]. Generally, we consider nonparametric estimation problems for the function S from L 2 under the condition that the distribution of the noise (ξ t ) 0≤t≤n is unknown. We know only that this distribution belongs to some distribution family Q n . In this case we use the robust estimation approach proposed in [START_REF] Galtchouk | Asymptotically efficient estimates for non parametric regression models[END_REF][START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF][START_REF] Konev | Robust model selection for a semimartingale continuous time regression from discrete data[END_REF] for the nonparametric estimation. According to this approach we have to construct an estimator S n (any function of (y t ) 0≤t≤n ) for S to minimize the robust risk defined as

R * n ( S n , S) = sup Q∈Q n R Q ( S n , S) , (1.3)
where

R Q (•, •) is the usual quadratic risk of the form R Q ( S n , S) := E Q S n -S 2 and S 2 = 1 0 S 2 (t)dt . (1.4)
It is clear that if we don't know the distribution of the observation one needs to find an estimator which will be optimal for all possible observation distributions. Moreover in this paper we consider the estimation problem in the adaptive setting, i.e. when the regularity of S is unknown. To this end we use the adaptive method based on the model selection approach. The interest to such statistical procedures is explained by the fact that they provide adaptive solutions for the nonparametric estimation through oracle inequalities which give the non-asymptotic upper bound for the quadratic risk including the minimal risk over chosen family of estimators. It should be noted that for the first time the model selection methods were proposed by Akaike [START_REF] Akaike | A new look at the statistical model identification[END_REF] and Mallows [START_REF] Mallows | Some comments on C p[END_REF] for parametric models. Then, these methods had been developed for the nonparametric estimation and the oracle inequalities for the quadratic risks was obtained by Barron, Birgé and Massart [START_REF] Barron | Risk bounds for model selection via penalization[END_REF], by Fourdrinier and Pergamenshchikov [START_REF] Fourdrinier | Improved selection model method for the regression with dependent noise[END_REF] for the regression models in discrete time and [START_REF] Konev | General model selection estimation of a periodic regression with a Gaussian noise[END_REF] in continuous time. Unfortunately, the oracle inequalities obtained in these papers can not provide the efficient estimation in the adaptive setting, since the upper bounds in these inequalities have some fixed coefficients in the main terms which are more than one. To obtain the efficiency property for estimation procedures one has to obtain the sharp oracle inequalities, i.e. in which the factor at the principal term on the right-hand side of the inequality is close to unity. The first result on sharp inequalities is most likely due to Kneip [START_REF] Kneip | Ordered linear smoothers[END_REF] who studied a Gaussian regression model in the discrete time. It will be observed that the derivation of oracle inequalities usually rests upon the fact that the initial model, by applying the Fourier transformation, can be reduced to the Gaussian independent observations. However, such transformation is possible only for Gaussian models with independent homogeneous observations or for inhomogeneous ones with known correlation characteristics. For the general non Gaussian observations one needs to use the approach proposed by Galtchouk and Pergamenshchikov [START_REF] Galtchouk | Sharp non-asymptotic oracle inequalities for nonparametric heteroscedastic regression models[END_REF][START_REF] Galtchouk | Adaptive asymptotically efficient estimation in heteroscedastic nonparametric regression[END_REF] for the heteroscedastic regression models in discrete time and developed then by Konev and Pergamenshchikov in [START_REF] Konev | Nonparametric estimation in a semimartingale regression model. Part 1. Oracle Inequalities[END_REF][START_REF] Konev | Nonparametric estimation in a semimartingale regression model. Part 2. Robust asymptotic efficiency[END_REF][START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF][START_REF] Konev | Robust model selection for a semimartingale continuous time regression from discrete data[END_REF] for semimartingale models in continuous time. In general the model selection is an adaptive rule α which choses an estimator S * = S α from an estimate family ( S α ) α∈A . The goal of this selection is to prove the following nonasymptotic oracle inequality: for any sufficient small δ > 0 and any observation duration n ≥ 1

R(S * , S) ≤ (1 + δ) min α∈A R( S α , S) + δ -1 B n , (1.5)
where the rest term B n is sufficiently small with respect to the minimax convergence rate. Such oracle inequalities are called sharp, since the coefficient in the main term 1 + δ is close to one for sufficiently small δ > 0. Moreover, in this paper we represent the new results on the improved estimation methods for the nonparametric models (1.1). Usually, the model selection procedures are based on the least squares estimators. But in [START_REF] Pchelintsev | Improved robust model selection methods for the Levy nonparametric regression in continuous time[END_REF] it is propose to use the improved least square estimators which enable to improve considerably the non asymptotic estimation accuracy. At the first time such idea was proposed in [START_REF] Fourdrinier | Improved selection model method for the regression with dependent noise[END_REF] for the regression in discrete time and in [START_REF] Konev | General model selection estimation of a periodic regression with a Gaussian noise[END_REF] for the Gaussian regression model in continuous time. In [START_REF] Pchelintsev | Improved robust model selection methods for the Levy nonparametric regression in continuous time[END_REF] these methods are developed for the non -Gaussion regression models in continuous time.

It should be noted that generally for the conditionally Gaussian regression models one can not use the well known improved estimators proposed in [START_REF] James | Estimation with quadratic loss[END_REF][START_REF] Fourdrinier | A paradox concerning shrinkage estimators: should a known scale parameter be replaced by an estimated value in the shrinkage factor?[END_REF] for Gaussian or spherically symmetric observations. To apply the improved estimation methods to the non Gaussian regression models in continuous time one needs to modify the well known James-Stein procedure in the way proposed in [START_REF] Pchelintsev | Improved estimation in a non-Gaussian parametric regression[END_REF][START_REF] Konev | Estimation of a regression with the pulse type noise from discrete data[END_REF]. For the improved model selection procedures the oracle inequality (1.5) is shown also. We note that this inequality allows us to provide the asymptotic efficiency without knowing the regularity of the function being estimated. The efficacy property for a nonparametric estimate

S * means lim n→∞ υ n sup S∈W k r R(S * , S) = lim n→∞ υ n inf S sup S∈W k r R( S, S) = l * (r) ,
where

l * (r) = ((1 + 2k)r) 1/(2k+1) k π(k + 1) 2k/(2k+1) , (1.6)
υ n is a normalizing coefficient (convergence rate), W k r is the Sobolev ball of a radius r > 0 and the regularity k ≥ 1. The limit (1.6) is called the Pinsker constant which is calculated by Pinsker in [START_REF] Pinsker | Optimal filtration of square integrable signals in Gaussian white noise[END_REF].

The rest of the paper is organized as follows. In the next section 2, we describe the Lévy, Ornstein-Uhlenbeck and semi-Markov processes as the examples of a semimartingale impusle noise in the model (1.1). In Section 3 we construct the model selection procedure based on the least square estimators and show the sharp oracle inequalities. In Section 4 we give the improved least squares estimators and we study the improvement effect for the semimartingale model (1.1). In Section 5 we construct the improved model selection procedure and show the sharp oracle inequalities. The asymptotic efficiency is studed in Section 6.

Examples

Lévy model

First we consider the model (1.1) with the Lévy noise process, i.e. we assume that the noise process (ξ t ) 0≤t≤n is defined as

ξ t = 1 w t + 2 z t and z t = x * (µ -µ) t , (2.1)
where 1 and 2 are some unknown constants, (w t ) t≥ 0 is a standard brownian motion, µ(ds dx) is a jump measure with deterministic compensator µ(ds dx) = dsΠ(dx), Π(•) is a Lévy measure, i.e. some positive measure on R * = R \ {0}, (see, for example, [START_REF] Jacod | Limit theorems for stochastic processes[END_REF][START_REF] Cont | Financial Modelling with Jump Processes[END_REF] for details) such that Π(x 2 ) = 1 and Π(x 6 ) < ∞ .

Here we use the notation Π(|x| m ) = R * |z| m Π(dz). Note that the Lévy measure Π(R * ) could be equal to +∞. One can check directly that for the process (2.1) the condition (1.2) holds with

κ Q = σ Q = 2 1 + 2 2 .
We assume that the nuisance parameters 1 and 2 of the process (ξ t ) 0≤t≤n satisfy the conditions 0

< ≤ 2 1 and σ Q ≤ ς * , (2.2)
where the bounds and ς * are functions of n, i.e. = The models (1.1) with the Lévy's type noise are used in different applied problems (see [START_REF] Bertoin | Lévy Processes[END_REF], for details). Such models naturally arise in the nonparametric functional statistics problems (see, for example, [START_REF] Ferraty | Nonparametric Functional Data Analysis : Theory and Practice[END_REF]).

Ornstein -Uhlenbeck model

Now we consider the noise process (ξ t ) t≥0 in (1.1) difened by a non-Gaussian Ornstein-Uhlenbeck process with the Lévy subordinator. Let the noise process in (1.1) obey the equation

dξ t = aξ t dt + du t , ξ 0 = 0 , (2.4)
where u t = 1 w t + 2 z t and the process z t is defied in (2.1). Here a ≤ 0, 1 and 2 are unknown parameters. We assume that the parameters 1 and 2 satisfy the conditions (2.2) and the parameter

-a max ≤ a ≤ 0 , (2.5)
where the bound a max > 0 is the function of n, i.e. a max = a max (n), such that for any δ > 0 lim

n→∞ a max n δ = 0 . (2.6)
In this case Q n is the family of all distributions of process (2.4) on the Skorokhod space D[0, n] satisfying the conditions (2.2), (2.5) and (2.6). Note also that the processes (2.1) and (2.4) are G -conditionally Gaussian square integrated semimartingales, where G = σ{z t , t ≥ 0}. Such processes are used in the financial Black-Scholes type markets with jumps (see, for example, [START_REF] Barndorff-Nielsen | Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial mathematics[END_REF][START_REF] Delong | Optimal investment and consumption in a Black-Scholes market with Lévy driven stochastic coefficients[END_REF] and the references therein). Note also that in the case when 2 = 0 for the parametric estimation problem such models are considered in [START_REF] Höpfner | On LAN for parametrized continuous periodic signals in a time inhomogeneous diffusion[END_REF][START_REF] Höpfner | Estimating discontinuous periodic signals in a time inhomogeneous diffusion[END_REF][START_REF] Konev | Sequential estimation of the parameters in a trigonometric regression model with the Gaussian coloured noise[END_REF].

Semi -Markov model

In [START_REF] Barbu | Robust adaptive efficient estimation for semi -Markov nonparametric regression models[END_REF][START_REF] Barbu | Robust adaptive efficient estimation for a semi-Markov continuous time regression from discrete data[END_REF] it is introduced the regression model (1.1) in which the noise process describes by the equation

ξ t = 1 L t + 2 X t , (2.7)
and the Lévy process L t is defined as

L t = ˇ w t + 1 -ˇ 2 z t ,
where 0 ≤ ˇ ≤ 1 is an unknown constant. Moreover, we assume that the pure jump process (X t ) t≥ 0 in (2.7) is a semi-Markov process with the following form

X t = N t i=1 Y i , (2.8)
where (Y i ) i≥ 1 is an i.i.d. sequence of random variables with

E Y i = 0 , E Y 2 i = 1 and E Y 4 i < ∞ .
Here N t is a general counting process (see, for example, [START_REF] Mikosch | Non-Life Insurance Mathematics[END_REF]) defined as

N t = ∞ k=1 1 {T k ≤t} and T k = k l=1 τ l ,
where (τ l ) l≥ 1 is an i.i.d. sequence of positive integrated random variables with distribution η and mean τ = E τ 1 > 0. We assume that the processes (N t ) t≥0 and (Y i ) i≥ 1 are independent between them and are also independent of (L t ) t≥0 . Here, the family Q n is defined by of all distributions of process (2.7) on the Skorokhod space D[0, n] with the parameters 1 and 2 satisfying the conditions (2.2) and 0 ≤ ˇ ≤ 1. Note that the process (X t ) t≥ 0 is a special case of a semi-Markov process (see, e.g., [START_REF] Barbu | Semi-Markov Chains and Hidden Semi-Markov Models toward Applications -Their use in Reliability and DNA Analysis[END_REF] and [START_REF] Limnios | Semi-Markov Processes and Reliability[END_REF]). It should be noted that if τ j are exponential random variables, then (N t ) t≥0 is a Poisson process and, in this case, (ξ t ) t≥0 is a Lévy process. But, in the general case when the process (2.8) is not a Lévy process, this process has a memory and cannot be treated in the framework of semi-martingales with independent increments. In this case, we need to develop new tools based on renewal theory arguments from [START_REF] Goldie | Implicit renewal theory and tails of solutions of random equations[END_REF].

It should be noted that for ˇ > 0 the process (2.7) is G -conditionally Gaussian also. In this case G = σ{z t , X t , t ≥ 0}.

Model selection

Let (φ j ) j≥ 1 be an orthonormal uniformly bounded basis in L 2 [0, 1], i.e. for some constant φ * ≥ 1, which may be depend on n,

sup 0≤j≤n sup 0≤t≤1 |φ j (t)| ≤ φ * < ∞ .
For example, we can take the trigonometric basis defined as Tr 1 ≡ 1 and, for j ≥ 2,

Tr j (x) = √ 2    cos(2π[j/2]
x) for even j;

sin(2π[j/2]x) for odd j, (3.1)
where [x] denotes the integer part of x.

To estimate the function S we use here the model selection procedure for continuous time regression models from [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF] based on the Fourrier expansion. We recall that for any function S from L 2 [0, 1] we can write

S(t) = ∞ j=1 θ j φ j (t) and θ j = (S, φ j ) = 1 0 S(t)φ j (t)dt .
So, to estimate the function S it suffices to estimate the coefficients θ j and to replace them in this representation by their estimators. Using the fact that the function S and φ j are 1 -periodic we can write that

θ j = 1 n n 0 φ j (t) S(t)dt .
If we replace here the differential S(t)dt by the stochastic observed differential dy t then we obtain the natural estimate for θ j on the time interval [0, n]

θ j,n = 1 n n 0 φ j (t)d y t ,
which can be represented, in view of the model (1.1), as

θ j,n = θ j + 1 √ n ξ j,n and ξ j,n = 1 √ n I n (φ j ) .
We need to impose some stability conditions for the noise Fourier transform sequence (ξ j,n ) 1≤j≤n . To this end for some parameter σ Q > 0 we set the following function

L 1,n (Q) = n j=1 E Q ξ 2 j,n -σ Q . (3.2)
In [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF] the parameter σ Q is called proxy variance . C 1 ) There exists a variance proxy σ Q > 0 such that for any > 0

lim n→∞ L 1,n (Q) n = 0 .
Moreover, we define

L 2,n (Q) = sup |x|≤1 E Q   n j=1 x j ξ j,n   2 and ξ j,n = ξ 2 j,n -E Q ξ 2 j,n . C 2 ) Assume that for any > 0 lim n→∞ L 2,n (Q) n = 0 .
Remark 3.1. Note that the condition C 2 ) means the boundedness in L 2 of all partial sum of the sequence ( ξ j,n ) 1≤j≤n . In particular, if we set x j = 1/ √ n for 1 ≤ j ≤ n we obtain that the normalized sum

1 √ n n j=1 ξ j,n
has to be bounded in L 2 . Now (see, for example, [START_REF] Ibragimov | Statistical Estimation: Asymptotic Theory[END_REF]) we can estimate the function S by the projection estimators, i.e.

S m (t) = m j=1 θ j,n φ j (t) , 0 ≤ t ≤ 1 , (3.3)
for some number m → ∞ as n → ∞. It should be noted that Pinsker in [START_REF] Pinsker | Optimal filtration of square integrable signals in Gaussian white noise[END_REF] shows that the projection estimators of the form (3.3) are not efficient.

For obtaining efficient estimation one needs to use weighted least square estimators defined as

S λ (t) = n j=1 λ(j) θ j,n φ j (t) , (3.4)
where the coefficients λ = (λ(j)) 1≤j≤n belong to some finite set Λ from [0, 1] n . As it is shown in [START_REF] Pinsker | Optimal filtration of square integrable signals in Gaussian white noise[END_REF], in order to obtain efficient estimators, the coefficients λ(j) in (3.4) need to be chosen depending on the regularity of the unknown function S. Since we consider the adaptive case, i.e. we assume that the regularity of the function S is unknown, then we chose the weight coefficients on the basis of the model selection procedure proposed in [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF] for the general semi-martingale regression model in continuous time. To the end, first we set

ν = #(Λ) and |Λ| * = 1 + max λ∈Λ n j=1 λ(j) ,
where #(Λ) is the cardinal number of Λ. Now, to choose a weight sequence λ in the set Λ we use the empirical quadratic risk, defined as

Err n (λ) = S λ -S 2 ,
which in our case is equal to

Err n (λ) = n j=1 λ 2 (j) θ 2 j,n -2 n j=1 λ(j) θ j,n θ j + ∞ j=1 θ 2 j .
Since the Fourier coefficients (θ j ) j≥ 1 are unknown, we replace the terms θ j,n θ j,n by

θ j,n = θ 2 j,n - σ n n ,
where σ n is an estimate for the variance proxy σ Q defined in (3.2). If it is known, we take σ n = σ Q , otherwise, we can choose it, for example, as in [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF], i.e.

σ n = n j=[ √ n]+1 t 2 j,n , (3.5)
where t j,n are the estimators for the Fourier coefficients with respect to the trigonometric basis (3.1), i.e.

t j,n = 1 n n 0 T r j (t)dy t .
Finally, in order to choose the weights, we will minimize the following cost function

J n (λ) = n j=1 λ 2 (j) θ 2 j,n -2 n j=1 λ(j) θ j,n + δ P n (λ),
where δ > 0 is some threshold which will be specified later and the penalty term is

P n (λ) = σ n |λ| 2 n . (3.6)
We define the model selection procedure as

S * = S λ with λ = argmin λ∈Λ J n (λ) . (3.7)
We recall that the set Λ is finite so λ exists. In the case when λ is not unique, we take one of them. As is shown in [START_REF] Barbu | Robust adaptive efficient estimation for semi -Markov nonparametric regression models[END_REF][START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF][START_REF] Pchelintsev | Improved robust model selection methods for the Levy nonparametric regression in continuous time[END_REF] both Conditions C 1 ) and C 2 ) hold for the processes (2.1), (2.4) and (2.7). Proposition 3.1. If the conditions C 1 ) and C 2 ) hold for the distribution Q of the process ξ in (1.1), then, for any n ≥ 1 and 0 < δ < 1/3, the risk (1.4) of estimate (3.7) for S satisfies the oracle inequality

R Q ( S * , S) ≤ 1 + 3δ 1 -3δ min λ∈Λ R Q ( S λ , S) + B n (Q) δn , (3.8)
where

B n (Q) = U n (Q) 1 + |Λ| * E Q | σ n -σ Q | and the coefficient U n (Q) is such that for any > 0 lim n→∞ U n (Q) n = 0 . (3.9)
In the case, when the value of σ Q is known, one can take σ n = σ Q and

P n (λ) = σ Q |λ| 2 n n ,
then we can rewrite the oracle inequality (3.8) with B n (Q) = U n (Q). Also we study the accuracy properties for the estimator (3.5).

Proposition 3.2. Let in the model (1.1) the function S(•) is continuously differentiable. Then, for any n ≥ 2,

E Q | σ n -σ Q | ≤ κ n (Q)(1 + Ṡ 2 ) √ n ,
where the term κ n (Q) possesses the property (3.9) and Ṡ is the derivative of the function S.

To obtain the oracle inequality for the robust risk (1.3) we need some additional condition on the distribution family Q n . We set

ς * = ς * n = sup Q∈Q n σ Q and L * n = sup Q∈Q n (L 1,n (Q) + L 2,n (Q)) . ( 3
R * n ( S * , S) ≤ 1 + 3δ 1 -3δ min λ∈Λ R * n ( S λ , S) + 1 δn B * n (1 + Ṡ 2 ) ,
where the term B * n satisfies the property (3.9).

It should be noted that Propositions 3.1 -3.2 and Theorem 3.3 are shown in [START_REF] Barbu | Robust adaptive efficient estimation for semi -Markov nonparametric regression models[END_REF] and [START_REF] Konev | General model selection estimation of a periodic regression with a Gaussian noise[END_REF]. Now we specify the weight coefficients (λ(j)) j≥1 in the way proposed in [START_REF] Galtchouk | Sharp non-asymptotic oracle inequalities for nonparametric heteroscedastic regression models[END_REF] for a heteroscedastic regression model in discrete time. First we define the normalizing coefficient which defined the minimax convergence rate

v n = n ς * , (3.11)
where the upper proxy variance is ς * is defined in (3.10). Consider a numerical grid of the form

A n = {1, . . . , k * } × {r 1 , . . . , r m } ,
where

r i = iε and m = [1/ε 2 ]. Both parameters k * ≥ 1 and 0 < ε ≤ 1 are assumed to be functions of n, i.e. k * = k * (n) and ε = ε(n), such that for any δ > 0      lim n→∞ k * (n) = +∞ , lim n→∞ k * (n) ln n = 0 , lim n→∞ ε(n) = 0 and lim n→∞ n δ ε(n) = +∞ .
One can take, for example,

ε(n) = 1 ln(n + 1)
and k * (n) = ln(n + 1) .

For each α = (β, r) ∈ A n we introduce the weight sequence λ α = (λ α (j)) j≥1 as λ α (j) =

1 {1≤j≤d} + 1 -(j/ω α ) β 1 {d<j≤ωα} (3.12)
where 2β+1) and

d = d(α) = [ω α / ln(n + 1)], ω α = τ β r v n 1/(
τ β = (β + 1)(2β + 1) π 2β β . We set Λ = {λ α , α ∈ A n } . (3.13)
It will be noted that in this case the cardinal of the set Λ is ν = k * m. Moreover, taking into account that τ β < 1 for β ≥ 1 we obtain for the set (3.13)

|Λ| * ≤ 1 + sup α∈A ω α ≤ 1 + (υ n /ε) 1/3 .
Note that the form (3.12) for the weight coefficients was proposed by Pinsker in [START_REF] Pinsker | Optimal filtration of square integrable signals in Gaussian white noise[END_REF] for the efficient estimation in the nonadaptive case, i.e. when the regularity parameters of the function S are known. In the adaptive case these weight coefficients are used in [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF][START_REF] Konev | Robust model selection for a semimartingale continuous time regression from discrete data[END_REF] to show the asymptotic efficiency for model selection procedures.

Improved estimation

In this Section we consider the improved estimation method for the model (1.1). We impose the following additional condition on the noise distribution.

D 1 ) There exists n 0 ≥ 1 such that for any n ≥ n 0 there exists a σfield G n for which the random vector ξ d,n = (ξ j,n ) 1≤j≤d is the G n conditionally Gaussian in R d with the covariance matrix

G n = E ξ i,n ξ j,n |G n ) 1≤i,j≤d
and for some nonrandom constant l * n > 0 inf

Q∈Q n (tr G n -λ max (G n )) ≥ l * n ,
where λ max (A) is the maximal eigenvalue of the matrix A. Now, for the first d Fourier coefficients we use the improved estimation method proposed for parametric models in [START_REF] Pchelintsev | Improved estimation in a non-Gaussian parametric regression[END_REF]. To this end we set θ n = ( θ j,n ) 1≤j≤d . In the sequel we will use the norm |x| 2 d = d j=1 x 2 j for any vector x = (x j ) 1≤j≤d from R d . Now we define the shrinkage estimators as

θ * j,n = (1 -g(j)) θ j,n ,
where [START_REF] Pchelintsev | Improved estimation in a non-Gaussian parametric regression[END_REF] and [START_REF] Pchelintsev | Improved robust model selection methods for the Levy nonparametric regression in continuous time[END_REF].

g(j) = (c n /| θ n | d )1 {1≤j≤d} and c n = l * n r * n + d/v n n . ( 4
Remark 4.1. The inequality (4.4) means that non asymptotically, i.e. for any n ≥ 1, the estimate (4.3) outperforms in mean square accuracy the estimate (3.4). Moreover in the efficient weight coefficients d ≈ n δ as n → ∞ for some δ > 0. Therefore, in view of the definition (4.1) and the conditions (2.3) and (4.2) nc n → ∞ as n → ∞. This means that improvement is considerably may better than for the parametric regression when the parameter dimension d is fixed [START_REF] Pchelintsev | Improved estimation in a non-Gaussian parametric regression[END_REF].

Improved model selection

This Section gives the construction of a model selection procedure for estimating a function S in (1.1) on the basis of improved weighted least square estimates (S * λ ) λ∈Λ and states the sharp oracle inequality for the robust risk of proposed procedure.

As in Section 3, the performance of any estimate S * λ will be measured by the empirical squared error

Err n (λ) = S * λ -S 2 .
In order to obtain a good estimate, we have to write a rule to choose a weight vector λ ∈ Λ in (4.3). It is obvious, that the best way is to minimise the empirical squared error with respect to λ. Making use the estimate definition (4.3) and the Fourier transformation of S implies

Err n (λ) = n j=1 λ 2 (j)(θ * j,n ) 2 -2 n j=1 λ(j)θ * j,n θ j + n j=1 θ 2 j .
Here one needs to replace the terms θ * j,n θ j by their estimators θj,n . We set θj,n = θ * j,n θ j,n -

σ n n ,
where σ n is defined in (3.5). For this change in the empirical squared error, one has to pay some penalty. Thus, one comes to the cost function of the form

J * n (λ) = n j=1 λ 2 (j)(θ * j,n ) 2 -2 n j=1 λ(j) θj,n + δ P n (λ) ,
where δ is some positive constant and the penalty term P n (λ) is defined in (3.6). Substituting the weight coefficients, minimizing the cost function

λ * = argmin λ∈Λ J * n (λ) , (5.1)
in (4.3) leads to the improved model selection procedure

S * = S * λ * . (5.2)
It will be noted that λ * exists because Λ is a finite set and also if the minimizing sequence in (5.1) λ * is not unique, one can take any minimizer. For this procedure in [START_REF] Pchelintsev | Improved robust model selection methods for the Levy nonparametric regression in continuous time[END_REF] it is obtained the following oracle inequality.

Theorem 5.1. If the conditions C 1 ) and C 2 ) hold for the distribution Q of the process ξ in (1.1), then, for any n ≥ 1 and 0 < δ < 1/3, the risk (1.4) of estimate (5.2) for S satisfies the oracle inequality

R Q (S * λ * , S) ≤ 1 + 3δ 1 -3δ min λ∈Λ R Q (S * λ , S) + Bn (Q) nδ ,
where

Bn (Q) = Ǔn (Q) 1 + |Λ| * E Q | σ n -σ Q | and the coefficient Ǔn (Q) sat-
isfies the property (3.9). Now Theorem 5.1 and Proposition 3.2 directly imply the following inequality for the robust risk (1.3) of the procedure (5.2).

Theorem 5.2. Assume that the conditions C * 1 ) and C * 2 ) hold and the function S is continuously differentiable. Then for any n ≥ 2 and 0 < δ < 1/3

R * n (S * λ * , S) ≤ 1 + 3δ 1 -3δ min λ∈Λ R * n (S * λ , S) + Ǔ * n (1 + Ṡ 2 ) nδ ,
where the coefficient Ǔ * n satisfies the property (3.9).

Asymptotic efficiency

In this section we present the results on the robust asymptotic efficiency obtained in [START_REF] Konev | Nonparametric estimation in a semimartingale regression model. Part 2. Robust asymptotic efficiency[END_REF] and [START_REF] Konev | General model selection estimation of a periodic regression with a Gaussian noise[END_REF] for the semimartinagale regression model (1.1). In order to study the asymptotic efficiency we define the following functional Sobolev ball

W k,r = {f ∈ C k p [0, 1] : k j=0 f (j) 2 ≤ r} ,
where r > 0 and k ≥ 1 are some unknown parameters, C k p [0, 1] is the space of k times differentiable 1 -periodic R → R functions such that f (i) (0) = f (i) (1) for any 0 ≤ i ≤ k -1. It is well known that for any S ∈ W k,r the optimal rate of convergence is n -2k/(2k+1) (see, for example, [START_REF] Pinsker | Optimal filtration of square integrable signals in Gaussian white noise[END_REF][START_REF] Nussbaum | Spline smoothing in regression models and asymptotic efficiency in L 2[END_REF]). On the basis of the model selection procedure we construct the adaptive procedure S * for which we obtain the following asymptotic upper bound for the quadratic risk, i.e. we show that the parameter (1.6) gives a lower bound for the asymptotic normalized risks. To this end we denote by Σ n of all estimators S n of S measurable with respect to the process (1.1), i.e. σ{y t , 0 ≤ t ≤ n}. We show that this lower bound is sharp in the following sense. Remark 6.1. Note that the equality (6.1) implies that the parameter (1.6) is the Pinsker constant in this case (cf. [START_REF] Pinsker | Optimal filtration of square integrable signals in Gaussian white noise[END_REF]). Moreover, it means that the robust efficiency holds with the convergence rate (v n ) 2k 2k+1 . It is well known that for the simple risks the optimal (minimax) estimation convergence rate for the functions from the set W k,r is n 2k/(2k+1) (see, for example, [START_REF] Ibragimov | Statistical Estimation: Asymptotic Theory[END_REF][START_REF] Nussbaum | Spline smoothing in regression models and asymptotic efficiency in L 2[END_REF][START_REF] Pinsker | Optimal filtration of square integrable signals in Gaussian white noise[END_REF]). So, if the distribution upper bound ς * → 0 as n → ∞ we obtain the more rapid rate, and if ς * → ∞ as n → ∞ we obtain the more slow rate. In the case when ς * is constant the robust rate is the same as the classical non robust convergence rate.

3 )

 3 For this example Q n is the family of all distributions of process (1.1) -(2.1) on the Skorokhod space D[0, n] satisfying the conditions (2.2) -(2.3).

.10) C * 1 ) 2 )Theorem 3 . 3 .

 1233 Assume that the conditions C 1 )-C 2 ) hold and for any > Now we impose the conditions on the set of the weight coefficients Λ. C * Assume that the set Λ is such that for any > Assume that the conditions C * 1 )-C * 2 ) hold. Then the robust risk (1.3) of the estimate (3.7) for continuously differentiable function S(t) satisfies for any n ≥ 2 and 0 < δ < 1/3 the oracle inequality

Proposition 4 . 1 . 1 . 4 . 2 .

 41142 Let in the model (1.1) the noise process describes by the Lévy process (2.1). Then the condition D 1 ) holds with l * n = (d -1) for any n ≥ Proposition Let in the model (1.1) the noise process describes by the Ornstein-Uhlenbeck process (2.4). Then the condition D 1 ) holds with l * n = (d -6) /2 for any n ≥ n 0 and d ≥ d 0 = inf{d ≥ 7 : 5 + ln d ≤ ǎd}, ǎ = (1 -e -a max )/(4a max ).

. 1 ) 3 )

 13 The positive parameter r * n is such that lim n→∞ r * n = ∞ and for any > v n defined in(3.11). Now we introduce a class of shrinkage weighted least squares estimates for S as We denote the difference of quadratic risks of the estimates (3.4) and(4.3) as ∆ Q (S) := R Q (S * λ , S) -R Q ( S λ , S) .Theorem 4.3. Let the observed process (y t ) 0≤t≤n describes by the equation (1.1) and the condition D 1 ) holds. Then for any n ≥ 1 sup Q∈Q n sup S ≤r * n ∆ Q (S) ≤ -c 2 n . (4.4) Propositions 4.1 -4.2 and Theorem 4.3 are shown in

Theorem 6 . 1 .

 61 The robust risk (1.3) admits the following asymptotic lower bound lim inf n→∞ inf S n ∈Σ n v 2k/(2k+1) n sup S∈W k,r R * n ( S n , S) ≥ l * (r) .

Theorem 6 . 2 .Corollary 6 . 3 .

 6263 The quadratic risk (1.3) for the estimating procedure S * has the following asymptotic upper boundlim sup n→∞ v 2k/(2k+1) n sup S∈W k,r R * n (S * , S) ≤ l * (r) .It is clear that Theorem 6.2 and Theorem 6.1 imply The model selection procedure S * is efficient, i.e. * , S) = l * (r) . (6.1)

Acknowledgements.

The first author is partially supported by the the RSF grant 17-11-01049. The last author is partially supported by the Russian Federal Professor program (project no. 1.472.2016/1.4, Ministry of Education and Science) and by the project XterM -Feder, University of Rouen, France. Moreover, the authors are grateful to the anonymous referee for very helpful comments and suggestions.

* This work was done under the Ministry of Education and Science of the Russian Federation in the framework of the research project no. 2.3208.2017/4.6, by RFBR Grant 16-01-00121 A and by the partial financial support of the RSF grant number 14-49-00079 (National Research University "MPEI" 14 Krasnokazarmennaya, 111250 Moscow, Russia).


Vlad Stefan Barbu
email: barbu@univ-rouen.fr

Slim Beltaief
email: slim.beltaief1@univ-rouen.fr

Serguei Pergamenshchikov
email: serge.pergamenshchikov@univ-rouen.fr

Robust adaptive efficient estimation for semi-Markov nonparametric regression models *

Keywords: primary 62G08, secondary 62G05 Non-asymptotic estimation, Robust risk, Model selection, Sharp oracle inequality, Asymptotic efficiency

HAL is

Introduction

Let us consider a regression model in continuous time

d y t = S(t)d t + d ξ t , 0 ≤ t ≤ n , (1.1)
where S(•) is an unknown 1-periodic function from L 2 [0, 1] defined on R with values in R, the noise process (ξ t ) t≥ 0 is defined as

ξ t = 1 L t + 2 z t , (1.2)
where 1 and 2 are unknown coefficients, (L t ) t≥ 0 is a Levy process and the pure jump process (z t ) t≥ 1 , defined in (2.3), is assumed to be a semi-Markov process (see, for example, [START_REF] Barbu | Semi-Markov Chains and Hidden Semi-Markov Models toward Applications -Their use in Reliability and DNA Analysis[END_REF]).

The problem is to estimate the unknown function S in the model (1.1) on the basis of observations (y t ) 0≤t≤n . Firstly, this problem was considered in the framework of the "signal+white noise" models (see, for example, [START_REF] Ibragimov | Statistical Estimation: Asymptotic Theory[END_REF] or [START_REF] Pinsker | Optimal filtration of square integrable signals in gaussian white noise[END_REF]). Later, in order to study dependent observations in continuous time, were introduced "sig-nal+color noise" regressions based on Ornstein-Uhlenbeck processes (cf. [START_REF] Höpfner | On LAN for parametrized continuous periodic signals in a time inhomogeneous diffusion[END_REF], [START_REF] Höpfner | Estimating discontinuous periodic signals in a time inhomogeneous diffusion[END_REF], [START_REF] Konev | Sequential estimation of the parameters in a trigonometric regression model with the gaussian coloured noise[END_REF], [START_REF] Konev | General model selection estimation of a periodic regression with a Gaussian noise[END_REF]).

Moreover, to include jumps in such models, the papers [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF] and [START_REF] Konev | Robust model selection for a semimartingale continuous time regression from discrete data[END_REF] used non Gaussian Ornstein-Uhlenbeck processes introduced in [START_REF] Barndorff-Nielsen | Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial mathematics[END_REF] for modeling of the risky assets in the stochastic volatility financial markets. Unfortunately, the dependence of the stable Ornstein-Uhlenbeck type decreases with a geometric rate. So, asymptotically when the duration of observations goes to infinity, we obtain very quickly the same "signal+white noise" model.

The main goal of this paper is to consider continuous time regression models with dependent observations for which the dependence does not disappear for a sufficient large duration of observations. To this end we define the noise in the model (1.1) through a semi-Markov process which keeps the dependence for any duration n. This type of models allows, for example, to estimate the signals observed under long impulse noise impact with a memory or "against signals".

In this paper we use the robust estimation approach introduced in [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF] for such problems. To this end, we denote by Q the distribution of (ξ t ) 0≤t≤n in the Skorokhod space D[0, n]. We assume that Q is unknown and belongs to some distribution family Q n specified in Section 4. In this paper we use the quadratic risk

R Q ( S n , S) = E Q,S S n -S 2 , (1.3)
where f 2 = 1 0 f 2 (s)ds and E Q,S is the expectation with respect to the distribution P Q,S of the process (1.1) corresponding to the noise distribution Q. Since the noise distribution Q is unknown, it seems reasonable to introduce the robust risk of the form R * n ( S n , S) = sup

Q∈Q n R Q ( S n , S) , (1.4)
which enables us to take into account the information that Q ∈ Q n and ensures the quality of an estimate S n for all distributions in the family Q n .

To summarize, the goal of this paper is to develop robust efficient model selection methods for the model (1.1) with the semi-Markov noise having unknown distribution, based on the approach proposed by Konev and Pergamenshchikov in [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF] and [START_REF] Konev | Robust model selection for a semimartingale continuous time regression from discrete data[END_REF] for continuous time regression models with semimartingale noises. Unfortunately, we cannot use directly this method for semi-Markov regression models, since their tool essentially uses the fact that the Ornstein-Uhlenbeck dependence decreases with geometrical rate and the "white noise" case is obtained sufficiently quickly.

Thus in the present paper we propose new analytical tools based on renewal methods to obtain the sharp non-asymptotic oracle inequalities. As a consequence, we obtain the robust efficiency for the proposed model selection procedures in the adaptive setting.

The rest of the paper is organized as follows. We start by introducing the main conditions in the next section. Then, in Section 3 we construct the model selection procedure on the basis of the weighted least squares estimates. The main results are stated in Section 4; here we also specify the set of admissible weight sequences in the model selection procedure. In Section 5 we derive some renewal results useful for obtaining other results of the paper. In Section 6 we develop stochastic calculus for semi Markov processes. In Section 7 we study some properties of the model (1.1). A numerical example is presented in Section 8. Most of the results of the paper are proved in Section 9. In Appendix some auxiliary propositions are given.

Main conditions

In the model (1.2) we assume that the Levy process L t is defined as

L t = ˇ w t + 1 -ˇ 2 Ľt , Ľt = x * (µ -µ) t , (2.1)
where, 0 ≤ ˇ ≤ 1 is an unknown constant, (w t ) t≥ 0 is a standard Brownian motion, µ(ds, dx) is the jump measure with the deterministic compensator µ(ds dx) = dsΠ(dx), where Π(•) is some positive measure on R (see, for example [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF][START_REF] Cont | Financial Modelling with Jump Processes[END_REF] for details) for which we assume that

Π(x 2 ) = 1 and Π(x 8 ) < ∞ , (2.2)
where we use the usual notation Π(|x| m ) = R |z| m Π(dz) for any m > 0. Note that Π(R) may be equal to +∞. Moreover, we assume that the pure jump process (z t ) t≥ 0 in (1.2) is a semi-Markov process with the following form

z t = N t i=1 Y i , (2.3)
where (Y i ) i≥ 1 is an i.i.d. sequence of random variables with

E Y i = 0 , E Y 2 i = 1 and E Y 4 i < ∞ .
Here N t is a general counting process (see, for example, [START_REF] Mikosch | Non-Life Insurance Mathematics. An Introduction with Stochastic Processes[END_REF]) defined as

N t = ∞ k=1 1 {T k ≤t} and T k = k l=1 τ l , (2.4)
where (τ l ) l≥ 1 is an i.i.d. sequence of positive integrated random variables with distribution η and mean τ = E τ 1 > 0. We assume that the processes (N t ) t≥0 and (Y i ) i≥ 1 are independent between them and are also independent of (L t ) t≥0 . Note that the process (z t ) t≥ 0 is a special case of a semi-Markov process (see, e.g., [START_REF] Barbu | Semi-Markov Chains and Hidden Semi-Markov Models toward Applications -Their use in Reliability and DNA Analysis[END_REF] and [START_REF] Limnios | Semi-Markov Processes and Reliability[END_REF]).

Remark 2.1. It should be noted that if τ j are exponential random variables, then (N t ) t≥0 is a Poisson process and, in this case, (ξ t ) t≥0 is a Levy process for which this model has been studied in [START_REF] Konev | Nonparametric estimation in a semimartingale regression model. Part 1. Oracle Inequalities[END_REF], [START_REF] Konev | Nonparametric estimation in a semimartingale regression model. Part 2. Robust asymptotic efficiency[END_REF] and [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF]. But, in the general case when the process (2.3) is not a Levy process, this process has a memory and cannot be treated in the framework of semi-martingales with independent increments. In this case, we need to develop new tools based on renewal theory arguments, what we do in Section 5. This tools will be intensively used in the proofs of the main results of this paper.

Note that for any function

f from L 2 [0, n], f : [0, n] → R, for the noise process (ξ t ) t≥ 0 defined in (1.2), with (z t ) t≥ 0 given in (2.3), the integral I n (f ) = n 0 f (s)dξ s (2.5)
is well defined with E Q I n (f ) = 0. Moreover, as it is shown in Lemma 6.2,

E Q I 2 n (f ) ≤ κ Q n 0 f 2 (s)d s , (2.6)
where

κ Q = 2 1 + 2 2 |ρ| * and |ρ| * = sup t≥0 |ρ(t)| < ∞.
Here ρ is the density of the renewal measure η defined as

η = ∞ l=1 η (l) , (2.7)
where η (l) is the lth convolution power for η. To study the series (2.7) we assume that the measure η has a density g which satisfies the following conditions.

H 1 ) Assume that, for any x ∈ R, there exist the finite limits

g(x-) = lim z→x- g(z) and g(x+) = lim z→x+ g(z)
and, for any K > 0, there exists δ = δ(K) > 0 for which

sup |x|≤K δ 0 |g(x + t) + g(x -t) -g(x+) -g(x-)| t dt < ∞. H 2 ) For any γ > 0, sup z≥0 z γ |2g(z) -g(z-) -g(z+)| < ∞.
H 3 ) There exists β > 0 such that R e βx g(x) dx < ∞.

Remark 2.2. It should be noted that the condition H 3 ) means that there exists an exponential moment for the random variable (τ j ) j≥1 , i.e. these random variables are not too large. This is a natural constraint since these random variables define the intervals between jumps, i.e., the frequency of the jumps. So, to study the influence of the jumps in the model (1.1) one needs to consider the noise process (1.2) with "small" interval between jumps or large jump frequency.

For the next condition we need to introduce the Fourier transform of any function f from L 1 (R), f : R → R, defined as

f (θ) = 1 2π R e iθx f (x) dx. (2.

8)

H 4 ) There exists t * > 0 such that the function g(θ -it) belongs to L 1 (R) for any 0 ≤ t ≤ t * .

It is clear that Conditions H 1 )-H 4 ) hold true for any continuously differentiable function g, for example for the exponential density. Now we define the family of the noise distributions for the model (1.1) which is used in the robust risk (1.4). Note that any distribution Q from Q n is defined by the unknown parameters in (1.2) and (2.1). We assume that

ς * ≤ 2 1 ≤ ς * , 0 ≤ ˇ ≤ 1 and ς * ≤ σ Q ≤ ς * , (2.9)
where

σ Q = 2 1 + 2 2 /τ , the unknown bounds 0 < ς * ≤ ς * are functions of n, i.e. ς * = ς * (n) and ς * = ς * (n), such that for any ˇ > 0, lim n→∞ n ˇ ς * (n) = +∞ and lim n→∞ ς * (n) n ˇ = 0 . (2.10)
Remark 2.3. As we will see later, the parameter σ Q is the limit for the Fourier transform of the noise process (1.2). Such limit is called variance proxy (see [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF]).

Remark 2.4. Note that, generally (but it is not necessary) the parameters 1 and 2 can be dependent on n. The conditions (2.10) means that we consider all possible cases, i.e. these parameters may go to the infinity or be constant or to the zero as well. See, for example, the conditions (3.32) in [START_REF] Konev | Robust model selection for a semimartingale continuous time regression from discrete data[END_REF].

Model selection

Let (φ j ) j≥ 1 be an orthonormal uniformly bounded basis in L 2 [0, 1], i.e., for some constant φ * ≥ 1, which may be depend on n,

sup 0≤j≤n sup 0≤t≤1 |φ j (t)| ≤ φ * < ∞ . (3.1)
We extend the functions φ j (t) by periodicity, i.e., we set φ j (t) := φ j ({t}), where {t} is the fractional part of t ≥ 0. For example, we can take the trigonometric basis defined as Tr 1 ≡ 1 and, for j ≥ 2,

Tr j (x) = √ 2    cos(2π[j/2]x) for even j; sin(2π[j/2]x) for odd j, (3.2)
where [x] denotes the integer part of x.

To estimate the function S we use here the model selection procedure for continuous time regression models from [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF] based on the Fourrier expansion. We recall that for any function S from L 2 [0, 1] we can write

S(t) = ∞ j=1 θ j φ j (t) and θ j = (S, φ j ) = 1 0 S(t)φ j (t)dt . (3.3)
So, to estimate the function S it suffices to estimate the coefficients θ j and to replace them in this representation by their estimators. Using the fact that the function S and φ j are 1 -periodic we can write that

θ j = 1 n n 0 φ j (t) S(t)dt .
If we replace here the differential S(t)dt by the stochastic observed differential dy t we obtain the natural estimate for θ j on the time interval [0, n]

θ j,n = 1 n n 0 φ j (t)d y t , (3.4)
which can be represented, in view of the model (1.1), as

θ j,n = θ j + 1 √ n ξ j,n , ξ j,n = 1 √ n I n (φ j ) . (3.5)
Now (see, for example, [START_REF] Ibragimov | Statistical Estimation: Asymptotic Theory[END_REF]) we can estimate the function S by the projection estimators, i.e.

S m (t) = m j=1 θ j,n φ j (t) , 0 ≤ t ≤ 1 , (3.6)
for some number m → ∞ as n → ∞. It should be noted that Pinsker in [START_REF] Pinsker | Optimal filtration of square integrable signals in gaussian white noise[END_REF] shows that the projection estimators of the form (3.6) are not efficient. For obtaining efficient estimation one needs to use weighted least square estimators defined as

S λ (t) = n j=1 λ(j) θ j,n φ j (t) , (3.7)
where the coefficients λ = (λ(j)) 1≤j≤n belong to some finite set Λ from [0, 1] n . As it is shown in [START_REF] Pinsker | Optimal filtration of square integrable signals in gaussian white noise[END_REF], in order to obtain efficient estimators, the coefficients λ(j) in (3.7) need to be chosen depending on the regularity of the unknown function S.

In this paper we consider the adaptive case, i.e. we assume that the regularity of the function S is unknown. In this case we chose the weight coefficients on the basis of the model selection procedure proposed in [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF] for the general semi-martingale regression model in continuous time. These coefficients will be obtained later in (3.20). To the end, first we set

ι = #(Λ) and |Λ| * = 1 + max λ∈Λ Ľ(λ) , (3.8)
where #(Λ) is the cardinal number of Λ and Ľ(λ) = n j=1 λ(j). Now, to choose a weight sequence λ in the set Λ we use the empirical quadratic risk, defined as Err n (λ) = S λ -S 2 , which in our case is equal to

Err n (λ) = n j=1 λ 2 (j) θ 2 j,n -2 n j=1 λ(j) θ j,n θ j + ∞ j=1 θ 2 j . (3.9)
Since the Fourier coefficients (θ j ) j≥ 1 are unknown, we replace the terms θ j,n θ j,n by

θ j,n = θ 2 j,n - σ n n , (3.10)
where σ n is an estimate for the variance proxy σ Q defined in (2.9). If it is known, we take σ n = σ Q ; otherwise, we can choose it, for example, as in [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF], i.e.

σ n = n j=[ √ n]+1 t 2 j,n , (3.11)
where t j,n are the estimators for the Fourier coefficients with respect to the trigonometric basis (3.2), i.e.

t j,n = 1 n n 0 T r j (t)dy t . (3.12)
Finally, in order to choose the weights, we will minimize the following cost function

J n (λ) = n j=1 λ 2 (j) θ 2 j,n -2 n j=1 λ(j) θ j,n + δ P n (λ), (3.13)
where δ > 0 is some threshold which will be specified later and the penalty term is

P n (λ) = σ n |λ| 2 n . (3.14)
We define the model selection procedure as

S * = S λ , (3.15)
where λ = argmin λ∈Λ J n (λ).

(3.16)

We recall that the set Λ is finite so λ exists. In the case when λ is not unique, we take one of them.

Let us now specify the weight coefficients (λ(j)) 1≤j≤n . Consider, for some fixed 0 < ε < 1, a numerical grid of the form

A = {1, . . . , k * } × {ε, . . . , mε} , (3.17)
where m = [1/ε 2 ]. We assume that both parameters k * ≥ 1 and ε are functions of n, i.e.

k * = k * (n) and ε = ε(n), such that        lim n→∞ k * (n) = +∞ , lim n→∞ k * (n) ln n = 0 , lim n→∞ ε(n) = 0 and lim n→∞ n δ ε(n) = +∞ (3.18)
for any δ > 0. One can take, for example, for n ≥ 2

ε(n) = 1 ln n and k * (n) = k * 0 + √ ln n , (3.19)
where k * 0 ≥ 0 is some fixed constant and the threshold ς * (n) is introduced in (2.9). For each α = (β, l) ∈ A, we introduce the weight sequence

λ α = (λ α (j)) 1≤j≤n
with the elements

λ α (j) = 1 {1≤j<j * } + 1 -(j/ω α ) β 1 {j * ≤j≤ω α } , (3.20)
where

j * = 1 + [ln υ n ], ω α = (d β lυ n ) 1/(2β+1) , d β = (β + 1)(2β + 1) π 2β β and υ n = n/ς * .
Now we define the set Λ as

Λ = {λ α , α ∈ A} . (3.21)
It will be noted that in this case the cardinal of the set Λ is

ι = k * m . (3.22)
Moreover, taking into account that d β < 1 for β ≥ 1 we obtain for the set (3.21)

|Λ| * ≤ 1 + sup α∈A ω α ≤ 1 + (υ n /ε) 1/3 . (3.23)
Remark 3.1. Note that the form (3.20) for the weight coefficients in (3.7) was proposed by Pinsker in [START_REF] Pinsker | Optimal filtration of square integrable signals in gaussian white noise[END_REF] for the efficient estimation in the nonadaptive case, i.e. when the regularity parameters of the function S are known. In the adaptive case these weight coefficients are used in [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF][START_REF] Konev | Robust model selection for a semimartingale continuous time regression from discrete data[END_REF] to show the asymptotic efficiency for model selection procedures.

Main results

In this section we obtain in Theorem 4.3 the non-asymptotic oracle inequality for the quadratic risk (1.

3) for the model selection procedure (3.15) and in Theorem 4.4 the non-asymptotic oracle inequality for the robust risk (1.4) for the same model selection procedure (3.15), considered with the coefficients (3.20). We give the lower and upper bound for the robust risk in Theorems 4.5 and 4.7, and also the optimal convergence rate in Corollary 4.8.

Before stating the non-asymptotic oracle inequality, let us first introduce the following parameters which will be used for describing the rest term in the oracle inequalities. For the renewal density ρ defined in (2.7) we set

Υ(x) = ρ(x) - 1 τ and Υ 1 = +∞ 0 |Υ(x)| dx , (4.1)
where τ = E τ 1 . In Proposition 5.2 we show that |ρ| * = sup t≥0 |ρ(t)| < ∞ and Υ 1 < ∞. So, using this, we can introduce the following parameters

Ψ Q = 4κ Q ι + 5 + 4ι σ Q σ Q τ φ 2 max Υ 1 + φ 4 max (1 + σ 2 Q ) 3 ľ (4.2) and c * Q = σ Q + 2κ Q + σ Q τ φ 2 max Υ 1 + φ 4 max (1 + σ 2 Q ) 2 ľ , (4.3)
where ľ = (4τ

2 + 8) Υ 1 + 5 + 13(1 + τ ) 2 (1 + |ρ| 2 * )(EY 4 1 ) + 4Π(x 4
). First, let us state the non-asymptotic oracle inequality for the quadratic risk (1.3) for the model selection procedure (3.15). Theorem 4.1. Assume that Conditions H 1 )-H 4 ) hold. Then, for any n ≥ 1 and 0 < δ < 1/6, the estimator of S given in (3.15) satisfies the following oracle inequality

R Q ( S * , S) ≤ 1 + 3δ 1 -3δ min λ∈Λ R Q ( S λ , S) + Ψ Q + 10|Λ| * E S | σ n -σ Q | nδ . (4.4)
Now we study the estimate (3.11).

Proposition 4.2. Assume that Conditions H 1 )-H 4 ) hold and that the function S(•) is continuously differentiable. Then, for any n ≥ 2,

E Q,S | σ n -σ Q | ≤ 6 Ṡ 2 + c * Q √ n . ( 4
R Q ( S * , S) ≤ 1 + 3δ 1 -3δ min λ∈Λ R Q ( S λ , S) + 60 Λ n Ṡ 2 + Ψ Q,n nδ , (4.6)
where

Ψ Q,n = 10 Λ n c * Q + Ψ Q and Λ n = |Λ| * / √ n. Remark 4.1. Note that the coefficient κ Q can be estimated as κ Q ≤ (1+τ |ρ| * )σ Q .
Therefore,taking into account that φ 4 max ≥ 1, the remainder term in (4.6) can be estimated as

Ψ Q,n ≤ C * 1 + σ 6 Q + 1 σ Q (1 + Λ n )ιφ 4 max , (4.7)
where C * > 0 is some constant which is independent of the distribution Q.

Furthermore, let us study the robust risk (1.4) for the procedure (3.15). In this case, the distribution family Q n consists in all distributions on the Skorokhod space D[0, n] of the process (1.2) with the parameters satisfying the conditions (2.9) and (2.10).

Moreover, we assume also that the number of the weight vectors and the upper bound for the basis functions in (3.1) may depend on n ≥ 1, i.e. ι = ι(n) and

φ * = φ * (n), such that for any ˇ > 0 lim n→∞ ι(n) n ˇ = 0 and lim n→∞ φ * (n) n ˇ = 0 . (4.8)
The next result presents the non-asymptotic oracle inequality for the robust risk (1.4) for the model selection procedure (3.15), considered with the coefficients (3.20).

Theorem 4.4. Assume that Conditions H 1 ) -H 4 ) hold and that the unknown function S is continuously differentiable. Then, for the robust risk defined in (1.4) through the distribution family (2.9) -(2.10), the procedure (3.15) with the coefficients (3.20) for any n ≥ 1 and 0 < δ < 1/6, satisfies the following oracle inequality

R * ( S * , S) ≤ 1 + 3δ 1 -3δ min λ∈Λ R * ( S λ , S) + U * n (S) nδ , (4.9)
where the sequence U * n (S) > 0 is such that, under the conditions (2.10), (3.18) and (4.8), for any r > 0 and δ > 0,

lim n→∞ sup Ṡ ≤r U * n (S) n δ = 0. (4.10)
Now we study the asymptotic efficiency for the procedure (3.15) with the coefficients (3.20), with respect to the robust risk (1.4) defined by the distribution family (2.9)-(2.10). To this end, we assume that the unknown function S in the model (1.1) belongs to the Sobolev ball

W k r = {f ∈ C k per [0, 1] : k j=0 f (j) 2 ≤ r} , (4.11)
where r > 0 and k ≥ 1 are some unknown parameters,

C k per [0, 1] is the set of k times continuously differentiable functions f : [0, 1] → R such that f (i) (0) = f (i) (1) for all 0 ≤ i ≤ k. The function class W k
r can be written as an ellipsoid in L 2 [0, 1], i.e.,

W k r = {f ∈ C k per [0, 1] : ∞ j=1 a j θ 2 j ≤ r}, (4.12)
where

a j = k i=0 (2π[j/2]) 2i and θ j = 1 0 f (v)Tr j (v)dv.
We recall that the trigonometric basis (Tr j ) j≥1 is defined in (3.2).

Similarly to [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF][START_REF] Konev | Robust model selection for a semimartingale continuous time regression from discrete data[END_REF] we will show here that the asymptotic sharp lower bound for the robust risk (1.4) is given by

r * k = ((2k + 1)r) 1/(2k+1) k (k + 1)π 2k/(2k+1) . (4.13)
Note that this is the well-known Pinsker constant obtained for the nonadaptive filtration problem in "signal + small white noise" model (see, for example, [START_REF] Pinsker | Optimal filtration of square integrable signals in gaussian white noise[END_REF]). Let Π n be the set of all estimators S n measurable with respect to the σ -field σ{y t , 0 ≤ t ≤ n} generated by the process (1.1).

The following two results give the lower and upper bound for the robust risk in our case. Theorem 4.5. Under Conditions (2.9) and (2.10),

lim inf n→∞ υ 2k/(2k+1) n inf S n ∈Π n sup S∈W k r R * n ( S n , S) ≥ r * k , (4.14)
where υ n = n/ς * .

Note that if the parameters r and k are known, i.e. for the non adaptive estimation case, then to obtain the efficient estimation for the "signal+white noise" model Pinsker in [START_REF] Pinsker | Optimal filtration of square integrable signals in gaussian white noise[END_REF] proposed to use the estimate S λ 0 defined in (3.7) with the weights (3.20) in which

λ 0 = λ α 0 and α 0 = (k, l 0 ) , (4.15)
where

l 0 = [r/ε]ε. For the model (1.1) -(1.
2) we show the same result.

Proposition 4.6. The estimator S λ 0 satisfies the following asymptotic upper bound

lim n→∞ υ 2k/(2k+1) n sup S∈W k r R * n ( S λ 0 , S) ≤ r * k .
For the adaptive estimation we user the model selection procedure

lim n→∞ υ 2k/(2k+1) n inf S n ∈Π n sup S∈W k r R * n ( S n , S) = r * k . (4.18) Remark 4.2.
It is well known that the optimal (minimax) risk convergence rate for the Sobolev ball W k r is n 2k/(2k+1) (see, for example, [START_REF] Pinsker | Optimal filtration of square integrable signals in gaussian white noise[END_REF], [START_REF] Nussbaum | Spline smoothing in regression models and asymptotic efficiency in L 2[END_REF]). We see here that the efficient robust rate is υ 2k/(2k+1) n , i.e., if the distribution upper bound ς * → 0 as n → ∞, we obtain a faster rate with respect to n 2k/(2k+1) , and, if ς * → ∞ as n → ∞, we obtain a slower rate. In the case when ς * is constant, than the robust rate is the same as the classical non robust convergence rate.

Renewal density

This section is concerned with results related to the renewal measure (2.7). We start with the following lemma.

Lemma 5.1. Let τ be a positive random variable with a density g, such that Ee βτ < ∞ for some β > 0. Then there exists a constant

β 1 , 0 < β 1 ≤ β for which, Ee (β 1 +iω)τ = 1 ∀ω ∈ R .
Proof. We will show this lemma by the contradiction, i.e. assume there exists some sequence of positive numbers going to zero (γ k ) k≥1 and a sequence

(w k ) k≥1 such that Ee (γ k +iω k )τ = 1 (5.1)
for any k ≥ 1. Firstly assume that lim sup k→∞ w k = +∞. Note that in this case, for any Since for some β > 0 the integral +∞ 0 e βt g(t)dt < ∞, we get

N ≥ 1, N 0 e γ k t cos(w k t) g(t)dt ≤ N 0 cos(w k t) g(t)dt + N 0 (e γ k t -1) cos(w k t) g(t)dt ,
lim k→∞ +∞ 0 e γ k t cos(w k t) g(t)dt = 0 . Let now lim sup k→∞ w k = ω ∞ = 0 and 0 < |ω ∞ | < ∞.
In this case there exists a sequence (l k ) k≥1 such that lim k→∞ w l k = ω ∞ , i.e.

1 = lim sup k→∞ Ee γ l k τ cos(τ w l k ) = E cos(τ w ∞ ) .
It is clear that, for random variables having density, the last equality is possible if and only if w ∞ = 0. In this case, i.e. when lim sup k→∞ w l k = 0, the equation (5.1) implies

lim sup k→∞ E e γ l k τ sin(τ w l k ) w l k = E τ = 0 .
Proposition 5.2. Let τ be a positive random variable with the distribution η having a density g which satisfies Conditions H 1 )-H 4 ). Then the renewal measure (2.7) is absolutely continuous with density ρ, for which

ρ(x) = 1 τ + Υ(x) , (5.2)
where τ = Eτ 1 and Υ(•) is some function defined on R + with values in R such that sup

x≥0 x γ |Υ(x)| < ∞ for all γ > 0 .
Proof. First note, that we can represent the renewal measure η as η = η * η 0 and η 0 = ∞ j=0 η (j) . It is clear that in this case the density ρ of η can be written as

ρ(x) = x 0 g(x -y) n≥0 g (n) (y)dy . (5.3)
Now we use the arguments proposed in the proof of Lemma 9.5 from [START_REF] Goldie | Implicit renewal theory and tails of solutions of random equations[END_REF]. For any 0 < < 1 we set

ρ (x) = x 0 g(x -y)   n≥0 (1 -) n g (n) (y) - ( 1
-) τ g 0 (y)   dy -g(x) , (5.4)
where g 0 (y) = e -y/τ 1 {y>0} . It is easy to deduce that for any

x ∈ R lim →0 ρ (x) = ρ(x) - 1 τ x 0 g(z) dz -g(x) . (5.5)
Moreover, in view of the condition H 1 ) we obtain that the function ρ (x) satisfies the condition D) from Section A.2. So, through Proposition A.5 we get

ρ (x+) + ρ (x-) = 1 π R e -ixθ ρ (θ) dθ , where ρ (θ) = R e iθx ρ (x)dx. Note that | g(θ)| = R e iθx g(x)dx ≤ R g(x)dx = 1 , i.e. for any 0 < < 1 we have |1 -(1 -) g(θ)| < 1 and therefore ∞ n=0 (1 -) n ( g(θ)) n = 1 1 -(1 -) g(θ)
.

From this and, taking into account that

g 0 (θ) = R e iθx g 0 (x)dx = τ -iτ θ ,
we obtain

ρ (θ) = g(θ) ∞ n=0 ( 1
-) n ( g(θ)) n - 1 - τ g(θ) g 0 (θ) -g(θ) = g(θ)G (θ) and G (θ) = 1 1 -(1 -) g(θ) - 1 -iτ θ -iτ θ , i.e. ρ (x-) + ρ (x+) = 1 π R e -ixθ g(θ)G (θ) dθ . (5.6)
One can check directly that

sup 0< <1,θ∈R |G (θ)| < ∞ .
Therefore, using the condition H 3 ) and the Lebesgue's dominated convergence theorem, we can pass to limit as → 0 in (5.6), i.e., we obtain that

ρ(x+) + ρ(x-) - 2 τ x 0 g(z) dz -g(x+) -g(x-) = 1 π R e -ixθ g(θ)G 0 (θ) dθ , where G 0 (θ) = 1 1 -g(θ) + 1 -iτ θ iτ θ .
Using here again Proposition A.5 we deduce that

ρ(x+) + ρ(x-) = 2 τ x 0 g(z) dz + 1 π R e -ixθ g(θ) Ǧ(θ) dθ (5.7)
and

Ǧ(θ) = 1 1 -g(θ) + 1 iτ θ
.

Note now that we can represent the density (5.3) as

ρ(x) = g * n≥0 g (n) = n≥1 g (n) (x) = g(x) + n≥2 g (n) (x) =: g(x) + ρ c (x)
and the function ρ c (x) is continuous for all x ∈ R. This means that

ρ(x) = ρ(x+) + ρ(x-) 2 -ρ(x) = g(x+) + g(x-) 2 -g(x)
and, therefore, the condition H 2 ) implies that, for any γ > 0,

sup x≥0 x γ | ρ(x)| < ∞.
Now we can rewrite (5.7) as

ρ(x) = 1 τ x 0 g(z) dz + 1 2π R e -ixθ g(θ) Ǧ(θ) dθ -ρ(x). (5.8)
Taking into account that E e βτ < ∞ for some β > 0 we can obtain that

sup x≥0 x γ +∞ x g(z) dz < ∞ .
To study the second term in (5.8) we will use Proposition A.3. Indeed, the condition H 3 ) implies the first limit equality in (A.1). The second one follows directly from Lemma A.4. Therefore, in view of Proposition A.3, there exists some β * > 0 such that, for any 0

≤ β 0 ≤ β * , R e -ixθ g(θ) Ǧ(θ) dθ = e -β 0 x R e -ixθ g(θ -iβ 0 ) Ǧ(θ -iβ 0 ) dθ .
Note that, due to Lemma 5.1, the function 1 -g(θ) has no zeros on the line {z ∈ C : Im(z) = -β 1 }. Moreover, one can check directly that θ = 0 is an isolated zero. So, this means that for any N > 1 there can be only finitely many zeros in {z ∈ C :

-β 1 < Im(z) < 0 , |Re(z)| < N } of the function 1-g(θ)
. Moreover, note that in view of lemma A.4 for any r > 0 lim Re(θ)→∞,|Im(θ)|≤r g(θ) = 0 .

This means that there exists N > 0 such that the function

1 -g(θ) = 0 for θ ∈ {z ∈ C : -β 1 < Im(z) < 0 , |Re(z)| ≥ N }.
So, there can be only finitely many zeros of the function 1 -g(θ) in {z ∈ C : -β 1 < Im(z) < 0} for some fixed 0 < β 1 < β. Therefore, there exists some β 0 > 0 for which the function 1 -g(θ) has no zeros in {z ∈ C : -β 0 < Im(z) < 0}, i.e. the function Ǧ(θ) will be bounded in this set and we obtain that

sup x≥0 e β 0 x R e -ixθ g(θ) Ǧ(θ) dθ < ∞ .
This the conclusion follows.

2 Using this proposition we can study the renewal process (N t ) t≥0 introduced in (2.4).

Corollary 5.3. Assume that Conditions H 1 )-H 4 ) hold true. Then, for any t > 0,

E N t ≤ |ρ| * t and E N 2 t ≤ |ρ| * t + |ρ| 2 * t 2 .
(5.9)

Proof. First, by means of Proposition 5.2, note that we get

E N t = E k≥1 1 {T k ≤t} = t 0 ρ(v) dv ≤ |ρ| * t .
Regarding the last bound in (5.9), we use the same reasoning as in the previous inequality, i.e., we obtain

E N 2 t = E k≥1 1 {T k ≤t} + 2E k≥1 1 {T k ≤t} j=k+1 1 {T j ≤t} = E N t + 2E k≥1 1 {T k ≤t} Θ(T k ) = E N t + t 0 Θ(v) ρ(v) dv ,
where, for 0 ≤ v ≤ t, we defined the function

Θ(v) = E N t-v ≤ |ρ| * (t -v). 2
6 Stochastic calculus for semi-Markov processes

In this section we give some results of stochastic calculus for the process (ξ t ) t≥ 0 given in (1.2), needed all along this paper. As the process ξ t is the combination of a Levy process and a semi-Markov process, these results are not standard and need to be provided. Lemma 6.1. Let f and g be any non-random functions from L 2 [0, n] and (I t (f )) t≥ 0 be the process defined in (2.5). Then, for any 0 ≤ t ≤ n,

E I t (f )I t (g) = 2 1 (f, g) t + 2 2 (f, gρ) t , (6.1)
where (f, g) t = t 0 f (s) g(s)ds and ρ is the density defined in (2.7).

Proof. First, note that we can represent the stochastic integral I t (f ) as

I t (f ) = 1 I L t (f ) + 2 I z t (f ) , (6.2)
where

I L t (f ) = t 0 f (s)dL s and I z t (f ) = t 0 f (s)dz s .
Note that the mutual covariation for the martingales I L t (f ) and I L t (g) (see, for example, [START_REF] Sh | Theory of Martingales[END_REF]) may be calculated as

[I L (f ), I L (g)] t = ˇ 2 t 0 f (s)g(s)ds + (1 -ˇ 2 ) 0≤s≤t f (s)g(s) ∆ Ľs 2 , (6.3)
where ∆ Ľs = Ľs -Ľs-. Taking into account that E I L t (f ) I L t (g) = E [I L (f ), I L (g)] t and that in view of the first condition in (2.2) Π(x 2 ) = 1, we obtain that

E I L t (f ) I L t (g) = ˇ 2 t 0 f (s)g(s)ds + (1 -ˇ 2 ) Π(x 2 ) t 0 f (s) g(s)ds = t 0 f (s) g(s)ds . (6.4)
Moreover, note that

EI z t (f )I z t (g) = E ∞ l=1 f (T l )g(T l )Y 2 l 1 {T l ≤t} = E ∞ l=1 f (T l )g(T l )1 {T l ≤t} = t 0 f (s)g(s)ρ(s)ds .
Hence the conclusion follows. 2 Lemma 6.2. Assume that Conditions H 1 )-H 4 ) hold true. Then, for any n ≥ 1 and for any non random function f from L 2 [0, n], the stochastic integral (2.5) exists and satisfies the properties (2.6) with the coefficient κ Q given in (2.6).

Proof. This lemma follows directly from Lemma 6.1 with f = g and Proposition 5.2. 2 Lemma 6.3. Let f and g be bounded functions defined on [0, ∞) × R. Then, for any k ≥ 1,

E I T k-(f ) I T k-(g) | G = 2 1 (f , g) T k + 2 2 k-1 l=1 f (T l ) g(T l ),
where G is the σ-field generated by the sequence (T l ) l≥1 , i.e., G = σ{T l , l ≥ 1}.

Proof. Using (6.2), (6.4) and, taking into account that the process (L t ) t≥0 is independent of the G, we obtain

E I T k-(f ) I T k-(g) | G = 2 1 (f , g) T k + E I z T k- (f ) I z T k- (g) | G .
Moreover,

E I z T k- (f ) I z T k- (g) | G = E k-1 l=1 f (T l )Y l k-1 l=1 g(T l )Y l | G = k-1 l=1 f (T l ) g(T l ) .
This we obtain the desired result. 2 Lemma 6.4. Assume that Conditions H 1 )-H 4 ) hold true. Then, for any measurable bounded non-random functions f and g, we have

E n 0 I 2 t-(f ) g(t) dm t ≤ 2 2 2 |g| * |f | 2 * Υ 1 n.
Proof. Using the definition of the process (m t ) t≥0 we can represent this integral as

n 0 I 2 t-(f ) g(t) dm t = k≥1 I 2 T k -(f ) g(T k ) Y 2 k 1 {T k ≤n} - n 0 I 2 t (f ) g(t) ρ(t) dt =: V n -U n . (6.5)
Note now that

E V n = E k≥1 g(T k ) E I 2 T k -(f ) | G 1 {T k ≤n} .
Now, using Lemma 6.3 we can represent the last expectation as

E V n = 2 1 E V n + 2 2 E V n , (6.6)
where

V n = k≥1 g(T k ) f 2 T k 1 {T k ≤n} and V n = k≥2 g(T k ) 1 {T k ≤n} k-1 l=1 f 2 (T l ) .
The first term in (6.6) can be represented as

E V n = n 0 g(t) f 2 t ρ(t)dt .
To estimate the last expectation in (6.6), note that

E V n = E l≥1 f 2 (T l ) ḡ(T l )1 {T l ≤n} = n 0 f 2 (v) ḡ(v) ρ(v)dv ,
where

ḡ(v) = E k≥1 g(v + T k ) 1 {T k ≤n-v} = n v g(t) ρ(t -v)dt .
Moreover, using now the representation (6.1), we calculate the expectation of the last term in (6.5)

E U n = 2 1 n 0 f 2 t g(t) ρ(t) dt + 2 2 n 0 f (t) g(t) ρ(t) dt ,
where f (t) = t 0 f 2 (s) ρ(s) ds. This implies that

E n 0 I 2 t-(f ) g(t) dm t = 2 2 n 0 g(t) δ(t)dt , where δ(t) = t 0 f 2 (v) (ρ(t -v) -ρ(t)) ρ(v) dv.
Note that, in view of Proposition 5.2, the function δ can be estimated as

|δ(t)| ≤ |f | 2 * |ρ| * t 0 |Υ(t -v) -Υ(t)| dv ≤ |f | 2 * |ρ| * ( Υ 1 + t|Υ(t)|) .
Therefore,

E n 0 I 2 t-(f ) g(t) dm t ≤ 2 2 2 |g| * |f | 2 * Υ 1 n
and this finishes the proof. 2 Lemma 6.5. Assume that Conditions H 1 )-H 4 ) hold true. Then, for any measurable bounded non-random functions f and g, one has

E n 0 I 2 t-(f )I t-(g)g(t)dξ t = 0.
Proof. First, note that n 0

I 2 t-(f )I t-(g)g(t)dξ t = 1 n 0 I 2 t (f )I t (g)g(t)dL t + 2 n 0 I 2 t-(f )I t-(g)g(t)dz t .
Second, we will show that

E n 0 I 2 t-(f )I t-(g)g(t)dL t = 0 . (6.7)
Using the notations (6.2), we set

J 1 = n 0 I 2 t (f )I L t (g)g(t)dL t and J 2 = n 0 I 2 t (f )I z t (g)g(t)dL t ,
we obtain that

n 0 I 2 t (f )I t (g)g(t)dL t = 1 J 1 + 2 J 2 . (6.8)
Now let us recall the Novikov inequalities, [START_REF] Novikov | On discontinuous martingales[END_REF], also referred to as the Bichteler-Jacod inequalities (see [START_REF] Bichteler | Calcul de Malliavin pour les diffusions avec sauts: existence d'une densité dans le cas unidimensionnel[END_REF][START_REF] Marinelli | On maximal inequalities for purely discontinuous martingales in infinite dimensions[END_REF]) providing bound moments of supremum of purely discontinuous local martingales for any predictable function h and any p ≥ 2

E sup 0≤t≤n [0,t]×R h d(µ -ν) p ≤ C * p E Jp,n (h) , (6.9)
where C * p is some positive constant and

Jp,n (h) = [0,n]×R h 2 dν p/2 + [0,n]×R h p dν .
By applying this inequality for the non-random function h = (s, x) = g(s)x, and, recalling that Π(x 8 ) < ∞, we obtain,

sup 0≤t≤n E I Ľ t (g) 8 < ∞ .
Taking into account that, for any non random square integrated function f, the integral t 0 f (s)dw s is Gaussian with the parameters 0, t 0 f 2 (s)ds , we obtain

sup 0≤t≤n E I L t (g) 8 < ∞.
Finally, by using the Cauchy inequality, we can estimate for any 0 < t ≤ the following expectation as

E (I L t (f )) 4 (I L t (g)) 2 < E (I L t (f )) 8 E (I L t (f )) 4 i.e., sup 0≤t≤n E (I L t (f )) 4 (I L t (g)) 2 < ∞ .
Moreover, taking into account that the processes (L t ) t≥0 and (z t ) t≥0 are independent, we obtain that

E (I z t (f )) 4 (I L t (g)) 2 = E (I z t (f )) 4 E (I L t (g)) 2 = Π(x 2 ) t 0 g 2 (s)ds E (I z t (f )) 4 .
One can check directly here that, for t > 0,

E |I z t (f )| 4 ≤ |f | 4 * E Y 4 1 EN 2 t .
Note that the last bound in Corollary 5.3 yields sup 0≤t≤n E (I z t (f )) 4 < ∞ and, therefore,

sup 0≤t≤n E (I t (f )) 4 (I L t (g)) 2 < ∞ .
It follows directly that EJ 1 = 0. Now we study the last term in (6.8). To this end, first note that similarly to the previous reasoning we obtain that

E n 0 (I L t (f )) 2 I z t (g)g(t)dL t = 0 and E n 0 I L t (f )I z t (f )I z t (g)g(t)dL t = 0 .
Therefore, to show (6.7) one needs to show that

E n 0 (I z t (f )) 2 I z t (g)g(t) dL t = 0 . (6.10)
To check this note that, for any 0 < t ≤ n and for any bounded function f,

I z t (f ) = ∞ k=1 f (T k ) Y k 1 {T k ≤t} = N n k=1 f (T k ) Y k 1 {T k ≤t} , i.e., n 0 (I z t (f )) 2 I z t (g)g(t) dL t = N n k=1 N n l=1 N n j=1 f (T k ) f (T l ) g(T j ) Y j Y l Y k I klj ,
where

I klj = n 0 1 {T k ≤t} 1 {T l ≤t} 1 {T j ≤t} dL t .
Taking into account that the (L t ) t≥0 is independent of the field G z = σ{z t , t ≥ 0}, we obtain that E I klj |G z = 0. Therefore,

E n 0 (I z t (f )) 2 I z t (g)g(t) dL t = E N n k=1 N n l=1 N n j=1 f (T k ) f (T l ) g(T j ) Y j Y l Y k E I klj |G z = 0.
So, we obtain (6.10) and hence the proof is achieved. 2

7 Properties of the regression model (1.1)

In order to prove the oracle inequalities we need to study the conditions introduced in [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF] for the general semi-martingale model (1.1). To this end we set for any x ∈ R n the functions

B 1,Q,n (x) = n j=1 x j E Q ξ 2 j,n -σ Q and B 2,Q,n (x) = n j=1
x j ξ j,n , (7.1)

where σ Q is defined in (2.9) and ξ j,n = ξ 2 j,n -E Q ξ 2 j,n .

Proposition 7.1. Assume that Conditions H 1 )-H 4 ) hold. Then

sup x∈[-1,1] n |B 1,Q,n (x)| ≤ C 1,Q,n , (7.2)
where

C 1,Q,n = σ Q τ φ 2 max Υ 1 .
Proof. First, note that from (6.2) we have

ξ j,n = 1 √ n I L n (φ j ) + 2 √ n I z n (φ j ) .
So, using (6.4) we can write that

Eξ 2 j,n = 2 1 n n 0 φ 2 j (t)d t + 2 2 n E ∞ l=1 φ 2 j (T l )1 {T l ≤n} . (7.3) Proposition 5.2 implies E ∞ l=1 φ 2 j (T l )1 {T l ≤n} = n 0 φ 2 j (x) ρ(x)dx = 1 τ n 0 φ 2 j (x)d x + n 0 φ 2 j (x)Υ(x)d x .
Note that n 0 φ 2 j (t)d t = n. So, in view of the condition (3.1), we obtain

Eξ 2 j,n -σ Q = 2 2 n n 0 φ 2 j (x)Υ(x)d x ≤ 2 2 n φ 2 max Υ 1 . (7.4)
Estimating here 2 2 by σ Q τ we obtain the inequality (7.2) and hence the conclusion follows.

2

Proposition 7.2. Assume that Conditions H 1 )-H 4 ) hold. Then

sup |x|≤1 E Q B 2 2,Q,n (x) ≤ C 2,Q,n , (7.5)
where

C 2,Q,n = φ 4 max (1 + σ 2 Q ) 3 ľ and ľ is given in (4.
3). Proof. By Ito's formula one gets

dI 2 t (f ) = 2I t-(f )dI t (f ) + 2 1 ˇ 2 f 2 (t)d t + 0≤s≤t f 2 (s)(∆ξ d s ) 2 , (7.6)
where ξ d t = 3 Ľt + 2 z t and 3 = 1 1 -ˇ 2 . Taking into account that the processes ( Ľt ) t≥0 and (z t ) t≥0 are independent and the time of jumps T k defined in (2.4) has a density, we have ∆z s ∆ Ľs = 0 a.s. for any s ≥ 0. Therefore, we can rewrite the differential (7.6) as

dI 2 t (f ) =2I t-(f )dI t (f ) + 2 1 ˇ 2 f 2 (t)d t + 2 3 d 0≤s≤t f 2 (s)(∆ Ľs ) 2 + 2 2 d 0≤s≤t f 2 (s)(∆z s ) 2 . (7.7)
From Lemma 6.2 it follows that

EI 2 t (f ) = 2 1 t 0 f 2 (s)ds + 2 2 t 0 f 2 (s)ρ(s)ds .
Therefore, putting

I t (f ) = I 2 t (f ) -EI 2 t (f ) , (7.8)
we obtain

d I t (f ) = 2I t-(f )f (t)dξ t + f 2 (t)d m t , m t = 2 3 mt + 2 2 m t ,
where mt = 0≤s≤t (∆ Ľs ) 2 -t and m t = 0≤s≤t (∆z s ) 2 -t 0 ρ(s)ds. For any non-random vector x = (x j ) 1≤j≤n with n j=1 x 2 j ≤ 1, we set

Īt (x) = n j=1
x j I t (φ j ). (7.9)

Denoting A t (x) = n j=1
x j I t (φ j )φ j (t) and B t (x) = n j=1

x j φ 2 j (t) , (7.10)

we get the following stochastic differential equation for (7.9)

d Īt (x) = 2A t-(x)dξ t + B t (x)d m t , Ī0 (x) = 0 .
Applying the Ito's formula one obtains

E Ī2 n (x) =2E n 0 Īt-(x)d Īt (x) + 4 2 1 ˇ 2 E n 0 A 2 t (x)d t + 2 3 E Ďn (x) + 2 2 E D n (x) , (7.11)
where

Ďn (x) = 0≤t≤n 2A t-(x)∆ Ľt + 2 3 B t (x)(∆ Ľt ) 2 2 and D n (x) = +∞ k=1 2A T k -(x)Y k + 2 B T k -(x)Y 2 k 2 1 {T k ≤n} . Let us now show that E n 0 Īt-(x)d Īt (x) ≤ 2 4 2 φ 3 max Υ 1 n 2 . (7.12)
To this end, note that

n 0 Īt-(x)d Īt (x) =2 1≤j,l≤ n x j x l n 0 I t-(φ j ) I t-(φ l )φ l (t)dξ t + n j=1 x j n 0 I t-(φ j )B t (x)d m t .
Using here Lemma 6.5, we get

E n 0 I t-(φ j ) I t-(φ i )φ i (t)dξ t = 0. Moreover, the process ( mt ) t≥0 is a martingale, i.e. E n 0 I t-(φ j )B t (x)d m t = 0. Therefore, E n 0 Īt-(x)d Īt (x) = 2 2 n j=1 x j E n 0 I t-(φ j )B t (x)dm t .
Taking into account here that for any non-random bounded function f

E n 0 f (t)dm t = 0, we obtain E n 0 I t-(φ j ) B t (x) dm t = E n 0 I 2 t-(φ j ) B t (x) dm t . So, Lemma 6.4 yields E n 0 I t-(φ j ) B t (x)dm t = n l=1 x l E n 0 I 2 t-(φ j )φ 2 l (t)dm t ≤ 2 2 2 φ 3 max Υ 1 n l=1 |x l |n .
Therefore,

E n 0 Īt-(x)d Īt (x) ≤ 2 4 2 φ 3 max Υ 1 n 1≤l,j≤ n |x l | |x j | = 2 4 2 φ 3 max Υ 1 n n l=1 |x l | 2 .
Taking into account here that

n l=1 |x l | 2 ≤ n l≥ 1 x 2 l
≤ n, we obtain (7.12). Reminding that Π(x 2 ) = 1 we can calculate directly that

E Ďn (x) = 4 E n 0 A 2 t (x)dt + 4 3 Π(x 4 ) n 0 B 2 t (x)dt . (7.13)
Note that, thanks to Lemma 6.1, we obtain that

E n 0 A 2 t (x)d t = i,j x i x j n 0 φ i (t)φ j (t)EI t φ i (t)I t φ j (t)d t = i,j x i x j n 0 φ i (t)φ j (t) t 0 φ i (v)φ j (v)( 2 1 + 2 2 ρ(v))dv = 1 2 2 1 i,j x i x j n 0 φ i (t)φ j (t)dt 2 + 2 2 A 1,n (x) = n 2 2 2 1 + 2 2 A 1,n (x) ,
where

A 1,n (x) = i,j x i x j n 0 φ i (t)φ j (t) t 0 φ i (v)φ j (v) ρ(v)dv dt.
This term can be estimated through Proposition 5.2 as

A 1,n (x) = n 2 2τ + i,j x i x j n 0 φ i (t)φ j (t) t 0 φ i (v)φ j (v) Υ(v)dv dt ≤ n 2 2τ + n φ 4 max Υ 1 i,j |x i ||x j | ≤ 1 2τ + φ 4 max Υ 1 n 2 .
So, reminding that σ Q = 2 1 + 2 2 /τ and that φ max ≥ 1, we obtain that

E n 0 A 2 t (x)d t ≤ σ Q 2 + φ 4 max Υ 1 n 2 ≤ 1 4 + Υ 1 φ 4 max (1 + σ 2 Q ) n 2 . (7.14)
Taking into account that

sup t≥0 B 2 t (x) ≤ φ 4 max   n j=1 |x j |   2 ≤ φ 4 max n , (7.15)
that φ max ≥ 1, and that 4 1 ≤ σ 2 Q we estimate the expectation in (7.13) as

E Ďn ≤ 4φ 4 max (1 + σ 2 Q ) 1 + Υ 1 + Π(x 4 ) n 2 . ( 7.16)
Moreover, taking into account that the random variable Y k is independent of A T k -(x) and of the field G = σ{T j , j ≥ 1} and that E A T k -(x) |G = 0, we get

E +∞ k=1 B T k -(x) A T k -(x)Y 3 k 1 {T k ≤n} = +∞ k=1 EE B T k -(x) A T k -(x)Y 3 k 1 {T k ≤n} |G = E Y 3 1 E +∞ k=1 B T k -(x)1 {T k ≤n} E(A T k -(x)|G) = 0 .
Therefore,

E D n (x) = 2 2 EY 4 1 D 1,n (x) + 4D 2,n (x), (7.17)
where

D 1,n (x) = +∞ k=1 E B 2 T k -(x)1 {T k ≤n} and D 2,n (x) = +∞ k=1 E A 2 T k -(x)1 {T k ≤n} .
Using the bound (7.15) we can estimate the term D 1,n as D 1,n (x) ≤ φ 4 max nE N n . Using here Corollary 5.3, we obtain

D 1,n (x) ≤ |ρ| * φ 4 max n 2 . (7.18)
Now, to estimate the last term in (7.17), note that the process A t (x) can be rewritten as

A t (x) = t 0 Q x (t, s)dξ s , with Q x (t, s) = n j=1
x j φ j (s)φ j (t).

Applying Lemma 6.3 again, we obtain for any k ≥ 1

E A 2 T k -(x)|G = 2 1 T k 0 Q 2 x (T k , s)ds + 2 2 k-1 j=1 Q 2 x (T k , T j ) .
So, we can represent the last term in (7.17) as

D 2,n = 2 1 D (1) 2,n + 2 2 D (2) 2,n , (7.20)
where

D (1) 2,n = +∞ k=1 E 1 {T k ≤n} T k 0 Q 2 x (T k , s)ds and D (2) 2,n = +∞ k=1 E 1 {T k ≤n} k-1 j=1 Q 2 x (T k , T j ) .
Thanks to Proposition 5.2 we obtain

D (1) 2,n = n 0 t 0 Q 2 x (t, s)ds ρ(t) dt ≤ |ρ| * n 0 n 0 Q 2 x (t, s)ds dt .
In view of the definition of Q x in (7. [START_REF] Mikosch | Non-Life Insurance Mathematics. An Introduction with Stochastic Processes[END_REF], we can rewrite the last integral as

n 0 Q 2 x (t, s)ds = 1≤i,j≤n x i x j φ i (t) φ j (t) n 0 φ i (s) φ j (s) ds = n i=1 x 2 i φ 2 i (t) n 0 φ 2 i (s) ds = n n i=1 x 2 i φ 2 i (t) .
Since n j=1 x 2 j ≤ 1, we obtain that,

n 0 Q 2 x (t, s)ds ≤ φ 2 max n and D (1) 2,n ≤ φ 2 max |ρ| * n 2 . (7.21)
Let us estimate now the last term in (7.20). First, note that we can represent this term as

D (2) 2,n = +∞ k=1 E 1 {T k ≤n} k-1 j=1 Q 2 x (T k , T j ) = ∞ j=1 1 {T j ≤n} G(T j ) = n 0 G(t) ρ(t)dt ,
where

G(t) = +∞ k=1 E 1 {T k ≤n} Q 2 x ((t + T k ), t) = n 0 Q 2 x (t + v, t) ρ(v)dv = n+t t Q 2 x (u, t) ρ(u -t)du .
It is clear that, for any

0 ≤ t ≤ n, n+t t Q 2 x (u, u -t) ρ(u) du ≤ |ρ| * 2n 0 Q 2 x (v, t) dv .
In view of the inequality (7.21) we obtain

2n 0 Q 2 x (u, t) du = 2n 0 Q 2 x (t, u) du ≤ 2φ 2 max n .
Therefore,

max 0≤t≤n G(t) ≤ 2|ρ| * φ 2 max n and D (2) 2,n ≤ 2|ρ| 2 * φ 2 max n 2 .
So, estimating 2 2 by τ σ Q and taking into account that EY 4 1 ≥ 1, we obtain that we obtain that

E D n (x) ≤ 13 (1 + τ )φ 4 max EY 4 1 (1 + |ρ| 2 * ) n 2 σ Q .
Using all these bound in (7.11) we obtain (7.5) and thus the conclusion follows. 2 Remark 7.1. The properties (7.2) and (7.5) are used to obtain the oracle inequalities given in Section 4 (see, for example, [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF]).

Simulation

In this section we report the results of a Monte Carlo experiment in order to assess the performance of the proposed model selection procedure (3.15). In (1.1) we chose a 1-periodic function which is defined, for 0 ≤ t ≤ 1, as S(t) = t sin(2πt) + t 2 (1 -t) cos(4πt) .

(8.1)

We simulate the model dy t = S(t)dt + dξ t , where ξ t = 0.5dwt + 0.5dz t . Here z t is the semi-Markov process defined in (2.3) with a Gaussian N (0, 1) sequence (Y j ) j≥1 and (τ k ) k≥1 used in (2.4) taken as τ k ∼ χ 2 3 . We use the model selection procedure (3.15) with the weights (3.20) in which k * = 100 + ln(n), t i = i/ ln(n), m = [ln 2 (n)] and δ = (3 + ln(n)) -2 . We define the empirical risk as

R = 1 p p j=1 Ê S n (t j ) -S(t j ) 2 , (8.2)
where the observation frequency p = 100001 and the expectation was taken as an average over N = 10000 replications, i.e., Ê S n (.) -S(.)

2 = 1 N N l=1 S l n (.) -S(.) 2 . n R R * 20
R * = R/||S|| 2 p , with ||S|| 2 p = 1 p p j=0 S 2 (t j ) . (8.3)
In our case ||S|| 2 p = 0.1883601. Table 1 gives the values for the sample risks (8.2) and (8.3) for different numbers of observations n.

Figures 1234show the behaviour of the regression function and its estimates by the model selection procedure (3.15) depending on the values of observation periods n. The black full line is the regression function (8.1) and the red dotted line is the associated estimator. From numerical simulations of the procedure (3.15) with various observation numbers n we may conclude that the quality of the proposed procedure: (i) is good for practical needs, i.e. for reasonable (non large) number of observations; (ii) is improving as the number of observations increases.

Proofs

We will prove here most of the results of this paper.

9.1 Proof of Theorem 4.1

First, note that we can rewrite the empirical squared error in (3.9) as follows

Err n (λ) = J n (λ) + 2 ∞ j=1 λ(j) θj,n + ||S|| 2 -δP n (λ), (9.1)
where θj,n = θ j,n -θ j θ j,n . Using the definition of θ j,n in (3.10) we obtain that

θj,n = 1 √ n θ j ξ j,n + 1 n ξ j,n + 1 n ς j,n + σ Q -σ n n ,
where ς j,n = E Q ξ 2 j,n -σ Q and ξ j,n = ξ 2 j,n -E Q ξ 2 j,n . Putting

M (λ) = 1 √ n n j=1
λ(j)θ j ξ j,n and

P 0 n = σ Q |λ| 2 n , (9.2)
we can rewrite (9.1) as Let λ 0 = (λ 0 (j)) 1≤j≤ n be a fixed sequence in Λ and λ be as in (3.16). Substituting λ 0 and λ in Equation ( 9.3), we obtain

Err n (λ) =J n (λ) + 2 σ Q -σ n n Ľ(λ) + 2M (λ) + 2 n B 1,Q,n (λ) + 2 P 0 n (λ) B 2,Q,n (e(λ) √ σ Q n + S 2 -ρP n (λ), (9.3
Err n ( λ) -Err n (λ 0 ) =J( λ) -J(λ 0 ) + 2 σ Q -σ Q n Ľ( ) + 2 n B 1,Q,n ( ) + 2M ( ) + 2 P 0 n ( λ) B 2,Q,n ( e) √ σ Q n -2 P 0 n (λ 0 ) B 2,Q,n (e 0 ) √ σ Q n -δP n ( λ) + δP n (λ 0 ), (9.4)
where = λ -λ 0 , e = e( λ) and e 0 = e(λ 0 ). Note that, (3.8),

| Ľ(x)| ≤ Ľ( λ) + Ľ(λ) ≤ 2|Λ| * .
Applying the inequality 2|ab| ≤ δa 2 + δ -1 b 2 (9.5) implies that, for any λ ∈ Λ,

2 P 0 n (λ) |B 2,Q,n (e(λ))| √ σ Q n ≤ δP 0 n (λ) + B 2 2,Q,n (e(λ)) δσ Q n .
Taking into account the bound (7.2), we get

Err n ( λ) ≤ Err n (λ 0 ) + 2M ( ) + 2C 1,Q,n n + 2B * 2,Q,n δσ Q n + 1 n | σ -σ Q |(| λ| 2 + |λ 0 | 2 ) + 2δP n (λ 0 ) ,
where B * 2,Q,n = sup λ∈Λ B 2 2,Q,n ((e(λ)). Moreover, noting that in view of (3.8) sup λ∈Λ |λ| 2 ≤ |Λ| * , we can rewrite the previous bound as

Err n ( λ) ≤Err n (λ 0 ) + 2M ( ) + 2C 1,Q,n n + 2B * 2,Q,n δσ Q n + 4|Λ| * n | σ -σ Q | + 2δP n (λ 0 ). (9.6)
To estimate the second term in the right side of this inequality we set

S x = n j=1
x(j)θ j φ j , x = (x(j)) 1≤j≤n ∈ R n .

Thanks to (2.6) we estimate the term M (x) for any x ∈ R n as

E Q M 2 (x) ≤ κ Q 1 n n j=1 x 2 (j)θ 2 j = κ Q 1 n ||S x || 2 . ( 9.7)
To estimate this function for a random vector x ∈ R n we set

Z * = sup xεΛ 1 nM 2 (x) ||S x || 2 , Λ 1 = Λ -λ 0 .
So, through Inequality (9.5), we get

2|M (x)| ≤ δ||S x || 2 + Z * nδ . (9.8)
It is clear that the last term here can be estimated as

E Q Z * ≤ x∈Λ 1 nE Q M 2 (x) ||S x || 2 ≤ x∈Λ 1 κ Q = κ Q ι , (9.9)
where ι = card(Λ). Moreover, note that, for any x ∈ Λ 1 ,

||S x || 2 -|| S x || 2 = n j=1 x 2 (j)(θ 2 j -θ 2 j ) ≤ -2M 1 (x), (9.10)
where M 1 (x) = n -1/2 n j=1 x 2 (j)θ j ξ j,n . Taking into account that, for any x ∈ Λ 1 the components |x(j)| ≤ 1 , we can estimate this term as in (9.7), i.e.,

E Q M 2 1 (x) ≤ κ Q ||S x || 2 n .
Similarly to the previous reasoning we set

Z * 1 = sup xεΛ 1 nM 2 1 (x) ||S x || 2 and we get E Q Z * 1 ≤ κ Q ι . (9.11)
Using the same type of arguments as in (9.8), we can derive

2|M 1 (x)| ≤ δ||S x || 2 + Z * 1 nδ . (9.12)
From here and (9.10), we get

||S x || 2 ≤ || S x || 2 1 -δ + Z * 1 nδ(1 -δ) (9.13)
for any 0 < δ < 1. Using this bound in (9.8) yields

2M (x) ≤ δ|| S x || 2 1 -δ + Z * + Z * 1 nδ(1 -δ) .
Taking into account that S 2 ≤ 2 (Err n ( λ) + Err n (λ 0 )), we obtain

2M ( ) ≤ 2δ(Err n ( λ) + Err n (λ 0 )) 1 -δ + Z * + Z * 1 nδ(1 -δ) .
Using this bound in (9.6) we obtain

Err n ( λ) ≤ 1 + δ 1 -3δ Err n (λ 0 ) + Z * + Z * 1 nδ(1 -3δ) + 2C 1,Q,n n(1 -3δ) + 2B * 2,Q,n δ(1 -3δ)σ Q n + (4|Λ| * + 2) n(1 -3δ) | σ -σ Q | + 2δ (1 -3δ) P 0 n (λ 0 ).
Moreover, for 0 < δ < 1/6, we can rewrite this inequality as

Err n ( λ) ≤ 1 + δ 1 -3δ Err n (λ 0 ) + 2(Z * + Z * 1 ) nδ + 4C 1,Q,n n + 4B * 2,Q,n δσ Q n + (8|Λ| * + 2) n | σ n -σ Q | + 2δ (1 -3δ) P 0 n (λ 0 ).
In view of Proposition 7.2 we estimate the expectation of the term B * 2,Q,n in (9.6) as

E Q B * 2,Q,n ≤ λ∈Λ E Q B 2 2,Q,n (e(λ)) ≤ ιC 2,Q,n .
Taking into account that |Λ| * ≥ 1, we get

R( S * , S) ≤ 1 + δ 1 -3δ R( S λ 0 , S) + 4κ Q ι nδ + 4C 1,Q,n n + 4ιC 2,Q,n δσ Q n + 10|Λ| * n E Q | σ -σ Q | + 2δ (1 -3δ) P 0 n (λ 0 ).
Using the upper bound for P n (λ 0 ) in Lemma A.1, one obtains (4.1), that finishes the proof. We use here the same method as in [START_REF] Konev | Nonparametric estimation in a semimartingale regression model. Part 1. Oracle Inequalities[END_REF]. First of all note that the definition (3.12) implies that

t j,n = t j + 1 √ n η j,n , (9.14)
where

t j = 1 0 S(t) T r j (t)dt and η j,n = 1 √ n n 0 Tr j (t) dξ t .
So, we have

σ n = n j=[ √ n]+1 t 2 j + 2M n + 1 n n j=[ √ n]+1 η 2 j,n , (9.15)
where

M n = 1 √ n n j=[ √ n]+1 t j η j,n .
Note that, for continuously differentiable functions (see, for example, Lemma A.6 in [START_REF] Konev | Nonparametric estimation in a semimartingale regression model. Part 1. Oracle Inequalities[END_REF]), the Fourier coefficients (t j ) satisfy the following inequality, for any n ≥ 1,

∞ j=[ √ n]+1 t 2 j ≤ 4 1 0 | Ṡ(t)|dt 2 √ n ≤ 4 Ṡ 2 √ n . (9.16)
In the same way as in (9.7) we estimate the term M n , i.e.,

E Q M 2 n ≤ κ Q n n j=[ √ n]+1 t 2 j ≤ 4κ Q Ṡ 2 n √ n ,
while the absolute value of this term for n ≥ 1 can be estimated as

|E Q M n | ≤ κ Q + Ṡ 2 √ n .
Moreover, using Propositions 7.1 and 7.2 we can represent the last term in (9.15) as

1 n n j=[ √ n]+1 η 2 j,n = σ Q (n - √ n) n + B 1,Q,n (x ) n + B 2,Q,n (x ) √ n , with x j = 1 { √ n<j≤n} and x j = 1 { √ n<j≤n} / √ n. Therefore, E Q 1 n n j=[ √ n]+1 η 2 j,n -σ Q ≤ σ Q √ n + C 1,Q,n n + C 2,Q,n √ n .
Taking into account that C Putting λ 0 (j) = 0 for j ≥ n we can represent the quadratic risk for the estimator (3.7) as

S λ 0 -S 2 = ∞ j=1 (1 -λ 0 (j)) 2 θ 2 j -2H n + 1 n n j=1 λ 2 0 (j)ξ 2 j,n ,
where

H n = n -1/2 n j=1 (1 -λ 0 (j))λ 0 (j)θ j ξ j,n . Note that E Q H n = 0 for any Q ∈ Q n , therefore, E Q S λ 0 -S 2 = ∞ j=1 (1 -λ 0 (j)) 2 θ 2 j + 1 n E Q n j=1
λ 2 0 (j)ξ 2 j,n .

Proposition 7.1 and the last inequality in (2.9) imply that for any (1 -λ 0 (j)) 2 /a j , with a j defined in (4.12), we estimate the first term in (9.17 Taking into account that a j /(π 2k j 2k ) → 1 as j → ∞ and l 0 → r as ε → 0 and using the definition of ω α 0 in (3.20), we obtain that .

Q ∈ Q n E Q n j=1 λ 2
So, taking into account that ω α 0 /υ 1/(2k+1) n → (d k r) 1/(2k+1) as n → ∞, the limit of Υ 2,n can be calculated as Therefore, letting N → ∞ in (A.3) we obtain (A.2). Hence we get the desired result. 2

The following technical lemma is also needed in the present paper.

  (3.15) with the parameter δ defined as a function of n satisfying lim n δ n = 0 and lim n n δ δ n = 0 (4.16) for any δ > 0. For example, we can take δ n = (6 + ln n) -1 . Theorem 4.7. Assume that Conditions H 1 )-H 4 ) hold true. Then the robust risk defined in (1.4) through the distribution family (2.9)-(2.10) for the procedure (3.15) based on the trigonometric basis (3.2) with the coefficients (3.20) and the parameter δ = δ n satisfying (4.16) has the following asymptotic upper bound lim sup n→∞ υ 2k/(2k+1) n sup S∈W k r R * n ( S * , S) ≤ r * k . (4.17) Theorem 4.5 and Theorem 4.7 allow us to compute the optimal convergence rate. Corollary 4.8. Under the assumptions of Theorem 4.7, we have

  i.e., in view of Lemma A.4, for any fixed N ≥ 1 lim sup k→N N 0 e γ k t cos(w k t) g(t)dt = 0 .

Figure 1 :

 1 Figure 1: Estimator of S for n = 20

Figure 2 :Figure 3 :

 23 Figure 2: Estimator of S for n = 100

Figure 4 :

 4 Figure 4: Estimator of S for n = 1000

  ) where e(λ) = λ/|λ|, the function Ľ(•) is defined in (3.8) and the functions B 1,Q,n (•) and B 2,Q,n (•) are given in (7.1).

2

 2

9. 2

 2 Proof of Proposition 4.2

  where j * and υ n are defined in(3.20). SettingΥ 1,n (S) = υ 2k/(2k+1) λ 0 , S) ≤ Υ 1,n (S) + Υ 2,n + Čn ,(9.17) where Čn = υ 2k/(2k+1) n C * 1,n /n. Note, that the conditions (2.10) and (4.8) imply that C * 1,n = o(n δ ) as n → ∞ for any δ > 0; therefore, Čn → 0 as n → ∞. Putting u n = υ 2k/(2k+1) n sup j≥j *

a

  j θ j ≤ u n r .

( 1 -

 1 n (S) ≤ r 1/(2k+1) π 2k (d k ) 2k/(2k+1) =: Υ * 1 . t k ) 2 dt = 2k 2 (k + 1)(2k + 1)

2 9. 6 7 Combining 1 γe

 2671 lim n→∞ Υ 2,n = 2(d k r) 1/(2k+1) k 2 (k + 1)(2k + 1) =: Υ * 2 .Moreover, sinceΥ * 1 + Υ * 2 =: r * k , we obtain lim n→∞ υ 2k/(2k+1) n sup S∈W k r R * n ( S λ 0 , S) ≤ r *k and get the desired result. Proof of Theorem 4.Proposition 4.6 and Theorem 4.4 yields Theorem 4.6. 2Now, in view of the Cauchy theorem, we obtain that for any N ≥ izx g(z)dz= N +it)x g(N + it) dt + -N N e i(-iβ+θ)x g(-iβ + θ)dθ + 0 -β e i(-N +it)x g(-N + it)dt = 0 . N +it)x g(N + it) dt = lim N →∞ 0 -βe i(-N +it)x g(-N + it) dt = 0 .

Lemma A. 4 .(A. 4 ) 2

 442 Let g : [a, b] → R be a function from L 1 [a, b]. Then, for any fixed -∞ ≤ a < b ≤ +∞, cos(N x)dx = 0 . Proof. Let first -∞ < a < b < +∞. Assume that g is continuously differentiable, i.e. g ∈ C 1 [a, b]. Then integrating by parts gives us b a g(x) sin(N x) dx = 1 N g(b) sin(N b) -g(a) sin(N a) -b a g (x) cos(N x) dx . So, from this we obtain that b a g(x) sin(N x) dx ≤ |g(a)| + |g(a)| + (b -a) max a≤x≤b |g (x)| N . This implies the first limit in (A.4) for this case. The second one is obtained similarly. Let now g be any absolutely integrated function on [a, b], i.e. g ∈ L 1 [a, b]. In this case there exists a sequence g n ∈ C 1 [a, b] such that lim n→∞ b a |g(x) -g n (x)|dx = 0 . Therefore, taking into account that for any n ≥ 1 lim N →∞ b a g n (x) sin(N x)dx = 0 , sin(N x)dx| ≤ b a |g(x) -g n (x)|dx .So, letting in this inequality n → ∞ we obtain the first limit in (A.4) and, similarly, we obtain the second one. Let now b = +∞ and a = -∞. In this case we obtain that for any-∞ < a < b < +∞ +∞ -∞ g(x) sin(N x)dx ≤ +∞ -∞ g(x) sin(N x)dx +Using here the previous results we obtain that for any -∞ < a < b < +∞ lim supPassing here to limit as b → +∞ and a → -∞ we obtain the first limit in (A.4). Similarly, we can obtain the second one.Let us now study the inverse Fourier transform. To this end, we need the following local Dini condition. D) Assume that, for some fixed x ∈ R, there exist the finite limitsg(x-) = lim z→x-g(z) and g(x+) = lim z→x+ g(z)and there exists δ = δ(x) > 0 for whichδ 0 |g(x + t) + g(x -t) -g(x+) -g(x-)| t dt < ∞.Proposition A.5. Let g : R → R be a function from L 1 (R). If, for some x ∈ R, this function satisfies the condition D, then g(x+) + g(x-) = 1 π R e -iθx g(θ) dθ , (A.5)

  .5) Theorem 4.1 and Proposition 4.2 implies the following result.

Theorem 4.3. Assume that Conditions H 1 )-H 4 ) hold and that the function S is continuously differentiable. Then, for any n ≥ 1 and 0 < δ ≤ 1/6, the procedure (3.15),

(3.11)

satisfies the following oracle inequality

Table 1 :

 1 Empirical risksWe set the relative quadratic risk as

		0.04430	0.235
	100	0.01290	0.068
	200	0.00812	0.043
	1000	0.00196	0.010

Acknowledgments. This research was partially supported by the Ministry of Education and Science of the Russian Federation, project (No 2.3208.2017/PCH), by the Russian Federal Professor program of the Ministry of Education and Science of the Russian Federation, (project No 1.472.2016/FPM) and by the Academic D.I. Mendeleev Fund Program of the Tomsk State University (research project NU 8.1.55.2015 L).

* This work was done under partial financial support of the grant of RSF number 14-49-00079 (National Research University "MPEI" 14 Krasnokazarmennaya, 111250 Moscow, Russia) and of the RFBR Grant 16-01-00121.

Appendix

A.1 Property of the penalty term Lemma A.1. For any n ≥ 1 and λ ∈ Λ,

where the coefficient P 0 n (λ) was defined in (9.2).

Proof. By the definition of Err n (λ) one has

.

In view of Proposition 7.1, this leads to the desired result

2

A.2 Properties of the Fourier transform Theorem A.2. Cauchy (1825) Let U be a simply connected open subset of C, let g : U → C be a holomorphic function, and let γ be a rectifiable path in U whose start point is equal to its end point. Then

Then, for any x ∈ R and for any 0

Proof. First note that the conditions of this theorem imply that R e iθx g(θ) dθ = lim

e iθx g(θ) dθ .

We fix now 0 < β < β 1 and we set for any N ≥ 1

where

Proof. First, for any fixed N > 0 we set

Taking into account that for any N > 0 the integral

and denoting B(x) = (g(x+) + g(x-))/2, we obtain that Hence we have the desired result. 2

Now we represent the last integral as