Konev Victor

Pergamenchtchikov Serguei
email: serge.pergamenchtchikov@univ-rouen.fr.

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

Keywords: Semimartingale regression, Estimation from discrete data, Robust risk estimation, Model selection, Sharp oracle inequalities. AMS 2000 Subject Classifications: Primary: 62G08, Secondary: 62G05

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

Introduction

Consider a regression model in continuous time dy t = S(t)dt + dξ t , 0 ≤ t ≤ n , (1.1) where S is an unknown function which belongs to the linear space V 1 of 1periodic R → R cadlag functions; ξ = (ξ t ) t≥0 is an unobservable semimartingale noise with the values in the Skorokhod space D[0, n] such that for each function f from L 2 [0, n] the stochastic integral

I n (f ) = n 0 f (s)dξ s (1.2)
is well defined and has the properties:

E Q I n (f ) = 0 and E Q I 2 n (f ) ≤ κ Q n 0 f 2 (s)ds , (1.3)
where κ Q > 0 is some positive constant depending on the noise distribution Q on the space D[0, n]; E Q denotes the expectation under the distribution Q. The noise distribution Q is unknown and assumed to belong to some distribution family Q n specified below. All necessary tools concerning the stochastic calculus can be found, for example, in [START_REF] Jacod | Limit theorems for stochastic processes[END_REF]. The problem is to estimate the unknown function S in the model (1.1) on the basis of observations (y t j ) 0≤j≤np , t j = j∆ , ∆ = 1 p ,

where p ≥ 3 is an odd number depending on n. Such an assumption about discrete time observations arises if the continuous observation of the process (1.1) cannot be provided. There are many papers devoted to the similar nonparametric estimation problems for the regression model (1.1) and other continuous time processes on the basis of the discrete observations (1.4). Gobet, Hoffmann and Reiss [START_REF] Gobet | Nonparametric estimation of scalar diffusions based on low frequency data[END_REF] and Comte, Genon-Catalot and Rozenholc [START_REF] Comte | Nonparametric adaptive estimation for integrated diffusions[END_REF] studied the problem of estimating the coefficients of a diffusion process from discrete data. Hoffmann, Munk and Schmidt-Hieber [START_REF] Hoffmann | Adaptive wavelet estimation of the diffusion coefficient under additive error measurements[END_REF] investigate nonparametric estimation for diffusion coefficients from discrete data, when the observation are blurred by an additive noise. Comte and Genon-Catalot [START_REF] Comte | Estimation for Lévy processes from high frequency data within a long time interval[END_REF] studied nonparametric estimation problem for pure jumps Lévy process in the model (1.1) based on discrete time observations. In this paper we consider the estimation problem in an adaptive setting, i.e. when the regularity of S is unknown. The quality of an estimate S n (any one periodic R → R function constructed from discrete data (1.4)) will be measured with the quadratic risk

R Q ( S n , S) = E Q S n -S 2 , (1.5)
where f 2 = 1 0 f 2 (t)dt. 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) , (1.6)
which enables one 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 .

It will be noted that similar criteria in the problems of nonparametric estimation were used in a number of papers (see, for example, the papers by Galtchouk and Pergamenchtchikov in [START_REF] Galtchouk | Asymptotically efficient estimates for nonparametric regression models[END_REF], by the authors [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF] and the references therein). The goal of this paper is to develop a model selection method for estimating a continuous time semimartingale regression (1.1) from discrete data (1.4). The origin of the model selection method goes back to early seventies with the pioneering papers by Akaike [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] who suggested to use penalizing in a log-likelihood type criterion. Barron, Birgé, Massart [START_REF] Barron | Risk bounds for model selection via penalization[END_REF], Massart [START_REF] Massart | A non-asymptotic theory for model selection[END_REF] and Kneip [START_REF] Kneip | Ordered linear smoothers[END_REF] developed a non-asymptotic model selection method which enables one to derive non-asymptotic oracle inequalities for the nonparametric regression models with Gaussian disturbances. The model selection procedures for the regression schemes with dependent noises and unknown distributions were studied by Comte and Genon-Catalot [START_REF] Comte | Estimation for Lévy processes from high frequency data within a long time interval[END_REF], Fourdrinier and Pergamenchtchikov [START_REF] Fourdrinier | Improved model selection method for a regression function with dependent noise[END_REF], Galtchouk and Pergamenchtchikov [START_REF] Galtchouk | Nonparametric sequential estimation of the drift in diffusion processes via model selection[END_REF], and by the authors (see, [START_REF] Konev | General model selection estimation of a periodic regression with a Gaussian noise[END_REF], [START_REF] Konev | Nonparametric estimation in a semimartingale regression model. Part 1. Oracle inequalities[END_REF] for details and further references). The interest to the model selection procedures can be explained by the fact that they provide adaptive solutions to the non-asymptotic regression models by the technique of sharp non-asymptotic oracle inequalities. As is known (see, for example, [START_REF] Barron | Risk bounds for model selection via penalization[END_REF]), the oracle inequality gives an upper bound for the risk via the minimal risk corresponding to a chosen family of estimates. In constructing an adaptive model selection procedure, in this paper, we will use the approach close to that of the papers [START_REF] Galtchouk | Sharp non-asymptotic oracle inequalities for non-parametric heteroscedastic regression models[END_REF], [START_REF] Galtchouk | Adaptive asymptotically efficient estimation in heteroscedastic non-parametric regression[END_REF], [START_REF] Galtchouk | Adaptive asymptotically efficient estimation in heteroscedastic non-parametric regression via model selection[END_REF], developed for a heteroscedastic regression model in discrete time. The key idea of the method in [START_REF] Galtchouk | Sharp non-asymptotic oracle inequalities for non-parametric heteroscedastic regression models[END_REF] is to combine the Barron-Birgé-Massart non-asymptotic penalization method [START_REF] Barron | Risk bounds for model selection via penalization[END_REF] and the Pinsker weighted least squares method of minimizing the asymptotic risk [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 noise[END_REF]. The advantage of this method is two-fold: it deals with a robust risk and enables one to construct asymptotically efficient procedures.

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

Examples

2.1 Non-Gaussian Ornstein-Uhlenbeck process First we consider an example of the disturbance (ξ t ) t≥0 in (1.1) given by a non-Gaussian Ornstein-Uhlenbeck process with the Lévy subordinator. Such processes are used in the financial Black-Scholes market models with stepwise randomly evolving coefficients (see [START_REF] Barndorff-Nielsen | Non-Gaussian Ornstein-Uhlenbeck based models and some of their uses in financial economics[END_REF] for details and further references). Let the noise process in (1.1) obey the equation

dξ t = aξ t dt + du t , ξ 0 = 0 , (2.1)
where a ≤ 0, u t = 1 w t + 2 z t , 1 and 2 are unknown constants, (w t ) t≥0 is a standard Brownian motion, (z t ) t≥0 is a compound Poisson process of the form

z t = N t j=1 Y j (2.2)
where (N t ) t≥0 is a standard homogeneous Poisson process with unknown intensity λ > 0 and (Y j ) j≥1 is an i.i.d. sequence of random variables with

EY 1 = 0 , EY 2 1 = 1 and EY 4 1 < ∞ . (2.3)
Let (T k ) k≥1 denote the arrival times of the process (N t ) t≥0 , that is,

T k = inf{t ≥ 0 : N t = k} .
The parameters λ, a, 1 and 2 of the process (2.1) are assumed to satisfy the following conditions

   -a * ≤ a ≤ 0 , λ ≥ λ * , min ≤ 1 ≤ max , min ≤ * ≤ max , (2.4)
where * = 2 1 + λ 2 2 . For this noise model the family Q n consists of all distributions on the Skorokhod space D[0, n] of the process (2.1) with the parameters satisfying the inequalities (2.4) with unknown bounds λ * , a * , min and max . Remark 2.1. The estimation problem from the continuous data for the models of type (1.1), (2.1) has been studied in a parametric setting in [START_REF] Dehling | Drift estimation for a periodic mean reversion process[END_REF], [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] and [START_REF] Konev | Sequential estimation of the parameters in a trigonometric regression model with the Gaussian coloured noise[END_REF].

Martingale noise

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

dξ t = 1 (t)dw t + 2 (t)dz t , (2.5)
where 1 and 2 are positive two times continuously differentiable R + → R + nonrandom functions; the process (z t ) t≥0 is as defined in (2.2)-(2.3). We assume that λ ≥ λ * for some unknown λ * > 0. Moreover, denoting

(t) = 1 (t) + λ 2 (t) , (2.6
(t + 1) d dt (t) ≤ * , sup t≥0 d 2 dt 2 (t) ≤ * . (2.9)
In this case Q n is the family of all distributions of the process (2.5) on D[0, n] satisfying the conditions (2.7) and (2.9) for some fixed unknown parameters λ * > 0, min (•), max (•), * and * .

Oracle inequalities

In this section we construct a model selection procedure for estimating a function S in (1.1) by the discrete time observations (1.4) and establish the oracle inequalities for its risk. Let (φ j ) j≥1 be the standard trigonometric basis in L 2 [0, 1] defined as

φ 1 = 1 , φ j (x) =    √ 2 cos( j x) for even j ≥ 2; √ 2 sin( j x) for odd j ≥ 3 , (3.1)
where j = 2π[j/2], [x] denotes the integer part of x. The restrictions of the functions {φ 1 , . . . , φ p }, p ≥ 3, on the sampling lattice

T p = t 1 , . . . , t p , t j = j p ,
form an orthonormal basis in the Hilbert space R T p with the inner product (x, y) p = 1 p p j=1

x(t j )y(t j ) for x, y ∈ R T p (3.2)

and the norm x p = (x, x) p . It is clear that the space R T p is isometric to R p . One can check directly that for any odd p

(φ j , φ i ) p = χ {i=j} . (3.3)
This implies that the function S on the lattice T p coincides with its discrete Fourier transformation, i.e.

S(t) = p j=1 θ j,p φ j (t) , if t ∈ T p , (3.4)
where θ j,p = (S, φ j ) p . The first step in constructing the model selection procedure consists in estimating the coefficients θ j,p for S in (1.1) from the discrete data by the formulae θ j,p = 1 n n 0 ψ j,p (t) dy t , ψ j,p = D p (φ j ) .

(3.5)

Here D p stands for the linear mapping given by the equation

D p (f )(t) = np k=1 f (t k ) χ {t k-1 <t≤t k } . (3.6)
Furthermore, we note that the system of the functions (ψ j,p ) 1≤j≤p is orthonormal in L 2 [0, 1] because (ψ j,p , ψ i,p ) = 1 0 ψ j,p (t) ψ i,p (t)dt = (φ j , φ i ) p = χ {i=j} . In the sequel we need the Fourier coefficients for the function S with respect to these functions which can be written as

θ j,p = (S, ψ j,p ) = 1 0 S(t) ψ j,p (t) dt = θ j,p + h j,p , (3.7)
where h j,p = h j,p (S) = x 2 j .

(3.9)

We need the following conditions. C 1 ) There exists a variance proxy σ Q ≥ 0, which may be depend on n ≥ 1, such that the sequence ζ j,p = E Q ξ 2 j,p -σ Q satisfies the following inequality

L 1,Q (n) = sup p≥3 sup x∈H 1 p j=1 x j ζ j,p < ∞
where

H 1 = {x ∈ [-1, 1] p : #(x) ≤ n}. C 2 ) Assume that L 2,Q (n) = sup p≥3 sup x∈H 2 E Q   p j=1 x j ξ j,p   2 < ∞
where H 2 = {x ∈ R p : |x| ≤ 1 , #(x) ≤ n} and ξ j,p = ξ 2 j,p -E Q ξ 2 j,p . Remark 3.1. As is shown in the paper [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF], which considers the estimation problem for the model (1.1) under continuous observations, one needs some stability conditions of the noise variances in (3.8) and the boundedness of their deviations to derive the oracle inequalities for the robust risk. In the estimation problem from discrete data the noise properties depend highly on the frequency of observations as well. The additional supremum in the Conditions in C 1 ) and C 2 ) over p ≥ 3 assumes that the stability of the noise variances and their deviations hold uniformly in p. Further restrictions on the functionals L 1,Q (n) and L 2,Q (n) will be imposed in Section 3.2.

As will be shown in the Section 3.3, the conditions C 1 ) and C 2 ) are satisfied for the model (1.1) with the noises (2.1) and (2.5). Now we introduce a weight least squares estimate for S(t) as

S γ (t) = p j=1 γ(j) θ j,p ψ j,p (t) , (3.10)
where γ = (γ(1), . . . , γ(p)) ∈ [0, 1] p is the vector of weight coefficients. The model selection procedure will be chosen from a finite family of such estimates ( S γ ) γ∈Γ . The set of weight sequences Γ will be given below. We will need the following characteristic of this set and the corresponding system of weight sequences ν p = card Γ and

µ p = max γ∈Γ #(γ) . ( 3
Err(γ) = S γ -S 2 = p j=1 γ 2 (j) θ 2 j,p -2 p j=1 γ(j) θ j,p θ j,p + S 2 . (3.13)
Since the Fourier coefficients (θ j,p ) j≥1 are unknown, the weight coefficients (γ(j)) 1≤j≤p cannot be determined by minimizing this quantity. To circumvent this difficulty we replace the terms θ j,p θ j,p by

θ j,p = θ 2 j,p - σ n n , (3.14)
where σ n is an estimator for the variance proxy σ Q in the condition C 1 ). We will need the following characteristics of this estimate

r Q ( σ n ) = E Q | σ n -σ Q | and r * n ( σ n ) = sup Q∈Q n E Q | σ n -σ Q | . (3.15)
For replacing the terms θ j,p θ j,p by the estimates (3.14) on the right-hand side of the empirical squared error (3.13), one has to pay some penalty. Thus, one comes to the cost function of the form

J(γ) = p j=1 γ 2 (j) θ 2 j,p -2 p j=1 γ(j) θ j,p + ρ P (γ) (3.16)
where ρ is some positive constant and P (γ) is the penalty term defined as

P (γ) = σ n |γ| 2 n , (3.17)
where

|γ| 2 = p j=1 γ 2 (j). If σ Q is known, one can put σ n = σ Q and P (γ) = P Q (γ) = σ Q |γ| 2 n . (3.18)
Substituting in (3.10) the values of the weight coefficients γ(j), minimizing the cost function (3.16), that is

γ = argmin γ∈Γ J(γ) , (3.19)
yields the model selection procedure

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

Ψ Q (n, p) = 2 3L 1,Q (n) + ν p L 2,Q (n) σ Q -1 + ν p κ Q . ( 3
R Q ( S * , S) ≤ 1 + 3ρ 1 -3ρ min γ∈Γ R Q ( S γ , S) + 1 n Ψ Q (n, p) (1 -3ρ)ρ + 6µ p r Q ( σ n ) n(1 -3ρ) . (3.22)
The proof of Proposition 3.1 is given in Section 6.

If the proxy variance σ Q in the Condition C 1 ) is known, then one comes to the following result.

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

R Q ( S * , S) ≤ 1 + 3ρ 1 -3ρ min γ∈Γ R Q ( S γ , S) + 1 n Ψ Q (n, p) (1 -3ρ)ρ . ( 3

Estimation of σ Q

In this section we will consider the case of an unknown proxy variance σ Q in the condition C 1 ) and derive the oracle inequalities for the continuous time estimate (3.10). We additionally assume that the unknown function S(t) in (1.1) has an absolutely integrable derivative. First we have to estimate σ Q and find an upper bound for r Q ( σ n ) in (3.22). One can use the following estimate for σ Q

σ n = n p p j=l 0 θ 2 j,p and p = min(p, n) , (3.24)
where l 0 ≥ 1 will be specified later. We set σ n = 0 for l 0 > p.

Lemma 3.3. Assume that the conditions (1.3), C 1 ) and C 2 ) hold and the unknown function S(t) is differentiable for 0 ≤ t ≤ 1 and such that for some r > 0

S 2 + Ṡ 2 = 1 0 |S(t)| 2 + | Ṡ(t)| 2 dt ≤ r . (3.25)
Then, for any n ≥ 1 and p ≥ 3, the quantity r Q ( σ n ), defined in (3.15), satisfies the inequality

r Q ( σ n ) ≤ K Q (n) n pl 0 + l 0 p + 1 √ p , (3.26)
where

K Q (n) = 16r + 8 √ rκ Q + L 1,Q (n) + σ Q + L 2,Q (n).
The proof of Lemma 3.3 is given in Section 6.

Minimizing the right hand side of the last inequality with respect to l 0 we find the appropriate value for l 0 in (3.24), namely

l 0 = √ n , (3.27)
which yields

r Q ( σ n ) ≤ 3 K Q (n) √ n p + 1 √ p := 3K Q (n) g * p,n . (3.28)
Combining Proposition 3.1 and Lemma 3.3 leads to the following result.

Proposition 3.4. Assume that the conditions (1.3), C 1 ) and C 2 ) hold and the function S satisfies the inequality (3.25) for some r > 0. Then, for any n ≥ 1, p ≥ 3, 0 < ρ < 1/3 and any set Γ with the property (3.12), the estimate (3.20) obeys the oracle inequality

R Q ( S * , S) ≤ 1 + 3ρ 1 -3ρ min γ∈Γ R Q ( S γ , S) + B Q (n, p) n(1 -3ρ)ρ , (3.29)
where

B Q (n, p) = Ψ Q (n, p) + 18K Q (n)µ p g * p,n . Remark 3.4.
Note that the oracle inequality (3.29) involves the term B Q (n, p) which depends on n and p. Our goal is to find conditions on the noise distribution Q and the frequency p providing the boundedness of this term by any power of n as n → ∞. As will be seen later, this property implies the efficiency of model selection procedure (3.20). To this end we need to study the robust risks for this estimate.

Robust estimation

In order to obtain the oracle inequality for the robust risk (1.6) of the estimate (3.10) we will impose additional conditions on the distribution family Q n of the noise ξ in the equation (1.1). Actually these conditions are stipulated also by the further studies of the asymptotic properties of the procedure (3.20) provided that both the number of observation periods n and the observation frequency p tend to infinity. Let P n denote the class of all probability measures on the space D[0, n] and P * n be its subclass defined as

P * n = Q ∈ P n : L 1,Q (n) ≤ L * 1,n , L 2,Q (n) ≤ L * 2,n , (3.30)
where L 1,Q (n) and L 2,Q (n) are functionals from the conditions C 1 ), C 2 ) and L * 1,n and L * 2,n are numerical sequences such that for any δ > 0

lim n→∞ L * 1,n + L * 2,n n δ = 0 . H 1 ) Assume that each distribution in the family Q n enters the class (3.30), i.e. Q n ⊆ P * n for each n ≥ 1. Besides, the constant κ Q in (1.
3) and the variance proxy σ Q from the condition C 1 ) are such that for n = 1, 2, ...

κ * (n) := sup Q∈Q n κ Q < ∞ , σ * (n) := sup Q∈Q n σ Q < ∞ , σ * (n) := inf Q∈Q n σ Q > 0 (3.31)
and for any δ > 0

lim n→∞ κ * (n) n δ = 0 , lim n→∞ σ * (n) n δ = 0 and lim n→∞ n δ σ * (n) = +∞ . (3.32)
Taking into account these notations, we get the following upper bounds for the functions Ψ Q (ρ) and K Q (n) in (3.21) and (3.26)

Ψ * n (p) = 2 2L * 1,n + ν p L * 2,n (σ * (n)) -1 + ν p κ * (n) and K * n = 16r + 8 rκ * (n) + L * 1,n + L * 2,n + σ * (n)
. Now proceeding from Theorem 3.4 we come to the following result. Theorem 3.5. Assume that the conditions (1.3) and H 1 ) hold and the function S satisfies the inequality (3.25) for some r > 0. Then, for any n ≥ 1, p ≥ 3, 0 < ρ < 1/3 and any set Γ with the property (3.12) the estimate (3.20) satisfies the oracle inequality for the robust risk (1.6), i.e.

R * n ( S * , S) ≤ 1 + 3ρ 1 -3ρ min γ∈Γ R * n ( S γ , S) + B * n (p) n(1 -3ρ)ρ , (3.33)
where

B * n (p) = Ψ * n (p) + 18K * n µ p g * p,n .

Specification of weights in the model selection

Now we will specify the weight coefficients (γ(j)) 1≤j≤p in the way proposed in [START_REF] Galtchouk | Sharp non-asymptotic oracle inequalities for non-parametric heteroscedastic regression models[END_REF]. Consider a numerical grid of the form

A = {1, . . . , k * } × {t 1 , . . . , t m } , (3.34)
where t i = iε and m = [1/ε 2 ]. We assume that both parameters k * ≥ 1 and

n -2 ≤ ε ≤ 1 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→∞ σ * (n)ε(n) = 0 and lim n→∞ n δ ε(n) = +∞ (3.35)
for any δ > 0. One can take, for example, for n ≥ 2

ε(n) = 1 (σ * (n) ln n) ∧ n 2 and k * (n) = k * 0 + √ ln n , (3.36)
where a ∧ b = min(a, b) and k * 0 ≥ 0 is some fixed constant. For each α = (β, t) ∈ A, we introduce the weight sequence

γ α = (γ α (j)) 1≤j≤p
with the elements

γ α (j) = 1 {1≤j<j * } + 1 -(j/ω α ) β 1 {j * ≤j≤ω α } (3.37)
where j * = [ω α / ln(n + 1)], ω α = (τ β t n) Taking into account that τ β < 1 for β ≥ 1 and that ε ≥ n -2 , we obtain ). The class of square integrable semimartingales, in our opinion, is the most appropriate one for the estimation problems with the quadratic risks (1.5) and (1.6). As an alternative, one could consider, for example, nonsemimartingale regression models based on the fractional Brownian motions with the Hurst parameter H = 1/2. However, in the latter case, the stochastic integral is defined not for functions from L 2 [0, n], but for some special spaces (see [START_REF] Duncan | Stochastic calculus for fractional brownian motion i. theory[END_REF] for details). This does not agree with the the definition

ω * = sup α∈A ω α ≤ (n/ε) 1/3 ≤ n . ( 3
B * n (p) = c 1 k * m + c 2 + c 3 µ p g * p,n (3.45)
for some positive fixed constants c 1 and c 2 .

When considering the estimation problem for the model (1.1) with the martingale noise (2.5), we set

L * 1,n = κ * (n) + * /2 + * (3.46)
and

L * 2,n = 8κ * (n) + 11(ln 2) 2 * 2 + 8 (κ * (n)) 2 E Y 4 1 λ * , (3.47)
where the parameters * , * are defined in (2.7) and κ

* (n) = sup 0≤t≤n | max (t)|. Moreover, we set σ * (n) = n -1 n 0 min (t)dt and σ * (n) = n -1 n 0 max (t)dt . ( 3
B * n (p) = c 1 1 + κ * (n) + 1 + (κ * (n)) 2 σ * (n) k * m + c 2 (1 + κ * (n)) µ p g * p,n
for some constants c 1 > 0 and c 2 > 0.

Remark 3.6. It will be noted that the equation for B * n (p) in the Theorem 3.8 reduces to the form (3.45) provided that the functions min (t) and max (t) are constant.

Now we write the oracle inequality for the "white noise" regression model, i.e. for the process (1.1), (2.5) with 1 ≡ 1 and 2 ≡ 0. In this case the distribution family Q n consists of a single distribution Q. Theorem 3.9. Let the model be given by (1.1)-(2.5) with 1 ≡ 1 and 2 ≡ 0. Then, for any n ≥ 3, p ≥ 3 and 0 < ρ < 1/3, the estimate (3.20) with the weight coefficients (3.38) satisfies the following oracle inequality

R Q ( S * , S) ≤ 1 + 3ρ 1 -3ρ min γ∈Γ R Q ( S γ , S) + 1 n 6k * m (1 -3ρ)ρ .
The proofs of Theorems 3.7-3.8 and Theorem 3.9 are given in Section 6.

Monte Carlo simulations

In this section we report the results of a Monte Carlo experiment to assess the performance of the proposed model selection procedure (3.20). We define S in (1.1) as 1 -periodic function with S(t) = t sin(2πt) + t 2 (1 -t) cos(4πt) on [0, 1] and then simulate the model as

dy t = S(t)dt + dξ t , (4.1)
where ξ t is a non-Gaussian Ornstein-Uhlenbeck process obeying the equation dξ t = -ξ t dt + 0.5 dw t + 0.5 dz t .

Here z t is a compound Poisson process (2.2) with the intensity λ = 0.1 and a Gaussian N (0, 1) sequence (Y j ) j≥1 .

The frequency of observations in (1.4) per period equals p = 100001. We use the weight sequence (3.37) in the procedure (3.20) as proposed in [START_REF] Galtchouk | Sharp non-asymptotic oracle inequalities for non-parametric heteroscedastic regression models[END_REF] for a discrete time model: k * = 100 + √ ln n, ε = 1/ ln n and m = [1/ε 2 ] and σ * (n) = 1. We used the cost function (3.16) with ρ = (3 + ln n) -2 , which satisfies the condition (5.4) below. We define the empirical risk as

R = 1 p p j=1 E S n (t j ) -S(t j ) 2 , (4.2)
and the relative quadratic risk

R * = R/ S 2 p . (4.3)
For the model (4.1) the empirical norm S 2 p = 0.18836. The expectations in (4.2) was taken as an average over N = 10000 replications, i.e.

E S n (•) -S(•) 2 = 1 N N l=1 S l n (•) -S(•) 2 .
The following

Table : Empirical risks

The Figures 123show the behavior of the regression function and its estimates by the model selection procedure (3.20) depending on the values of observation periods n.

Robust asymptotic efficiency

In this section we assume that the unknown function S in the model (1.1) belongs to the Sobolev ball W k r defined as

W k r = {f ∈ C k per [0, 1] : ∞ j=1 a j θ 2 j ≤ r} , θ j = 1 0 f (s) φ j (s)ds , (5.1)
where

C k per [0, 1] is the set of k times differentiable [0, 1] → R functions f such that the k -1 derivative f (k-1) is absolutely continuous and f (i) (0) = f (i) (1) for all 0 ≤ i ≤ k; a j = k i=0 j
2i , j is given in (3.1) and (θ j ) j≥1 are the Fourier coefficients of the function f . It is well known that the optimal (minimax) estimation convergence rate for the functions from the set (5.1) is n 2k/(2k+1) (see, for example, [START_REF] Pinsker | Optimal filtration of square-integrable signals in Gaussian noise[END_REF], [START_REF] Nussbaum | Spline smoothing in regression models and asymptotic efficiency in L 2[END_REF]). We will see that asymptotically the robust risk (1.6) normalized by this rate is bounded from below by the Pinsker constant R * k which in this case has the following form

R * k = ((2k + 1)r) 1/(2k+1) k (k + 1)π 2k/(2k+1)
.

(5.2)

To this end we denote by Π n the set of all estimators S n measurable with respect to the sigma-algebra σ{y t , 0 ≤ t ≤ n} generated by the process (1.1). Moreover, we denote by Q 0 the distribution of Wiener process with the scale parameter σ * (n) defined in (3.31).

H 2 ) Assume that the distribution Q 0 belongs to the family Q n .

Through this condition we obtain the following lower bound.

Theorem 5.1. Under the condition H 2 ) lim inf n→∞ (d n ) 2k/(2k+1) inf S n ∈Π n sup S∈W k r R * n ( S n , S) ≥ R * k , (5.3)
where

d n = n/σ * (n).
Proof of this theorem follows directly from Theorem 3.2 in [START_REF] Konev | Nonparametric estimation in a semimartingale regression model. Part 2. Robust asymptotic efficiency[END_REF]. Now we show that, under some conditions, the normalized robust risk for the model selection procedure is bounded from above by the same constant R * k . H 3 ) Assume that there exists δ > 0 such that

(d n ) 2k/(2k+1) inf S n ∈Π n sup S∈W k r R * n ( S n , S) = R * k .
(5.6)

Remark 5.1. It should be noted that the equality (5.6) means the robust efficiency holds with the convergence rate (d n ) 2k/(2k+1) . In view of the conditions (3.32) the sequence σ * n is slowly changing, i.e. asymptotically (as n → ∞) n -δ ≤ σ * n ≤ n δ for any δ > 0. So, if the σ * n = const, then one obtains the usual minimax rate for the Sobolev ball (see, for example, [START_REF] Pinsker | Optimal filtration of square-integrable signals in Gaussian noise[END_REF], [START_REF] Nussbaum | Spline smoothing in regression models and asymptotic efficiency in L 2[END_REF]). Furthermore, this rate becomes worse if the maximal value of the noise intensity σ * (n) → ∞ as n → ∞, and it improves if σ * (n) → 0 as n → ∞. For the regression model (1.1) with the martingale noise (2.5), the parameter σ * (n) is given in (3.48). If, for example, ρ max (t) = ln(t + 2), then σ * (n) → ∞ as n → ∞. In the case when ρ max (t) = 1/ ln(t + 2), one has σ * (n) → 0 as n → ∞.

To show Theprem 5.2 we need to estimate the function S from W k r in the nonadaptive setting, i.e. in the case when the parameters k ≥ 1, r > 0 and σ * (n) are known. To this end we set

γ 0 = γ α 0 and α 0 = (k, t 0 ) , (5.7)
where t 0 = [r/ε]ε, r = r/σ * (n) and ε satisfies the conditions in (3.35). Note that for sufficiently large n the pair α 0 belongs to the set (3.34).

Theorem 5.4. The estimator S γ 0 satisfies the following asymptotic upper bound lim sup

n→∞ d 2k/(2k+1) n sup S∈W k r R * n ( S γ 0 , S) ≤ R * k . ( 5
where θ j,p = θ j,p -θ j,p θ j,p . From (3.8) and (3.14), one finds

θ j,p = 1 √ n θ j,p ξ j,p + σ Q -σ n n + ζ j,p n + ξ j,p n ,
where the sequences ζ j,p and ξ j,p are defined in the conditions C 1 ) and C 2 ) respectively. Furthermore, for any x = (x 1 , . . . , x p ) ∈ R p we set

U(x) = p j=1 x j , V(x) = 1 √ n p j=1
x j θ j,p ξ j,p ,

B 1 (x) = p j=1
x j ζ j,p and B 2 (x) = p j=1

x j ξ j,p . (6.2)

Taking this into account, we rewrite (6.1) as

Err(γ) = J(γ) + 2V(γ) + 2(σ Q -σ n ) n U(γ) + 2B 1 (γ) n + 2 P Q (γ) B 2 (e(γ)) √ nσ Q + S 2 -ρ P (γ) (6.3)
where e(γ) = γ/|γ|. Now let γ 0 be a fixed weight vector in the set Γ and γ be as in (3.19). Substituting γ 0 and γ in (6.3), we arrive at the equation

Err( γ) -Err(γ 0 ) = J( γ) -J(γ 0 ) + 2(σ Q -σ n ) n U( x) + 2 n B 1 ( x) + 2 P Q ( γ) B 2 ( e) √ nσ Q -2 P Q (γ 0 ) B 2 (e 0 ) √ nσ Q + ρ P (γ 0 ) -ρ P ( γ) + 2V( x) (6.4)
where x = γ -γ 0 , e = e( γ) and e 0 = e(γ 0 ). From (3.11) it follows that for each γ ∈ Γ,

|U( x)| ≤ p j=1 (| γ(j)| + |γ 0 (j)|) ≤ 2µ p .
Similarly, in view of the condition C 1 ) we obtain

|B 1 ( x)| ≤ |B 1 ( γ)| + |B 1 (γ 0 )| ≤ 2L 1,Q (n) .
Therefore, by the definition (3.19) one gets

Err( γ) -Err(γ 0 ) ≤ 4| σ n -σ Q | n µ p + 4L 1,Q (n) n + 2 P Q ( γ) B 2 ( e) √ σ Q n -ρ P ( γ) + ρ P (γ 0 ) -2 P Q (γ 0 ) B 2 (e 0 ) √ nσ Q + 2V( x) . (6.5)
Further, by applying the elementary inequality

2|ab| ≤ εa 2 + ε -1 b 2 (6.6) with ε = ρ, one has 2 P Q (γ) |B 2 (e(γ))| √ nσ Q ≤ ρP Q (γ) + B 2 2 (e(γ)) nσ Q ρ .
Since 0 < ρ < 1, this inequality and (6.5) imply that

Err( γ) ≤ Err(γ 0 ) + 4L 1,Q (n) n + 2B * 2 nσ Q ρ + 1 n | σ n -σ Q | | γ| 2 + |γ 0 | 2 + 4µ p + 2ρP Q (γ 0 ) + 2V( x) ,
where B * 2 = sup γ∈Γ B 2 2 (e(γ)). Moreover, taking into account here that sup γ∈Γ |γ| 2 ≤ µ p , we get

Err( γ) ≤ Err(γ 0 ) + 4L 1,Q (n) n + 2B * 2 nσ Q ρ + 6µ p n | σ n -σ Q | + 2ρP Q (γ 0 ) + 2V( x) . (6.7)
By the condition C 2 ), one has

E Q B * 2 ≤ γ∈Γ E Q B 2 2 (e(γ)) ≤ ν p L 2,Q (n) , (6.8)
where ν p = card(Γ) is defined in (3.11). Now we examine the last term in the right side of (6.7). To this end for any x = (x 1 , . . . , x p ) ∈ R p we set

S x (t) = p j=1
x j θ j,p φ j (t) and S x (t) = p j=1

x j θ j,p φ j (t) . (6.9)

Using (1.3), one obtains that for any non-random vector

x from R p E Q V 2 (x) ≤ κ Q 1 n p j=1 x 2 j θ 2 j,p = κ Q n S x 2 . (6.10) Let denote Z * = max x∈Γ 1 nV 2 (x)/ S x 2 and Γ 1 = (Γ -γ 0 ) \ {0}.
The average of this quantity can be estimated as

E Q Z * ≤ x∈Γ 1 nE Q V 2 (x) S x 2 ≤ x∈Γ 1 κ Q ≤ κ Q ν p . (6.11)
By applying (6.6) one gets max

x∈Γ 1 2|V(x)| -ρ S x 2 ≤ Z * nρ . (6.12)
Now we need to find an upper bound for S x 2 . To this end note that for each x ∈ Γ 1 , one has the following estimate

S x 2 -S x 2 = p j=1 x 2 j (θ 2 j,p -θ 2 j,p ) ≤ -2V(x 2 ) ,
where

x 2 = (x 2 1 , . . . , x 2 p ). Since |x j | ≤ 1, one gets, for each x ∈ Γ 1 , E Q V 2 (x 2 ) ≤ κ Q p j=1 x 4 j θ 2 j,p n ≤ κ Q p j=1 x 2 j θ 2 j,p n = κ Q S x 2 n . Denoting Z * 1 = max x∈Γ 1 nV 2 (x 2 )/ S x 2 , one has E Q Z * 1 ≤ κ Q ν p . (6.13)
By the same argument as in (6.12), one shows that for any

x ∈ Γ 1 2|V(x 2 )| ≤ ρ S x 2 + Z * 1 nρ .
From here provided that 0 < ρ < 1, it follows that .14) Substituting this bound in (6.12) yields max

S x 2 ≤ S x 2 1 -ρ + Z * 1 nρ(1 -ρ) . ( 6
x∈Γ 1 2|V(x)| - ρ S x 2 1 -ρ ≤ Z * (1 -ρ) + ρZ * 1 nρ(1 -ρ) ≤ Z * + Z * 1 nρ(1 -ρ) .
Since the range of the random vector x is included in Γ 1 , the last inequality implies

2|V( x)| ≤ ρ S x 2 1 -ρ + Z * + Z * 1 nρ(1 -ρ) .
Note that the norm S x 2 can be estimated as

S x 2 = ( S γ -S) -(S -S γ 0 ) 2 ≤ 2(Err( γ) + Err(γ 0 )) , i.e. 2V( x) ≤ 2ρ(Err( γ) + Err(γ 0 )) 1 -ρ + Z * + Z * 1 nρ(1 -ρ) .
From here and (6.7), it follows that for 0 < ρ < 1/3

Err( γ) ≤ 1 + ρ 1 -3ρ Err(γ 0 ) + 4L 1,Q (n) (1 -3ρ)n + 2B * 2 ρσ Q (1 -3ρ)n + 6µ p | σ n -σ Q | (1 -3ρ)n + 2ρ 1 -3ρ P Q (γ 0 ) + Z * + Z * 1 ρ(1 -3ρ)n .
Taking the expectation and making use of (6.8), (6.11) and ( 6.13) we get

E Q S * -S 2 ≤ 1 + ρ 1 -3ρ E Q Err(γ 0 ) + 4L 1,Q (n) (1 -3ρ)n + 2ν p L 2,Q (n) ρσ Q (1 -3ρ)n + 6µ p r Q ( σ n ) (1 -3ρ)n + 2ρ 1 -3ρ P Q (γ 0 ) + 2ν p κ Q ρ(1 -3ρ)n ,
where r Q ( σ n ) is defined in (3.15). Lemma A.1 yields

E Q S * -S 2 ≤ 1 + 3ρ 1 -3ρ E Q Err(γ 0 ) + 2(2 + ρ)L 1,Q (n) (1 -3ρ)n + 2ν p L 2,Q (n) ρσ Q (1 -3ρ)n + 6µ p r Q ( σ n ) (1 -3ρ)n + 2ν p κ Q ρ(1 -3ρ)n .
From here, in view of the definition of the function Ψ Q (n, p) in (3.21), one comes to the inequality (3.22). Hence Proposition 3.1.

Proof of Lemma 3.3

It is clear that the inequality (3.26) holds for l 0 > p. Let now l 0 ≤ p. Setting x j = 1 {l 0 ≤j≤p} and substituting (3.8) in (3.24) yields

σ n = n p p j=l 0 (θ j,p ) 2 + 2n p V(x ) + 1 p p j=l 0 ξ 2 j,p . (6.15)
Furthermore, putting x j = p-1/2 1 {l 0 ≤j≤p} , one can write the last term on the right of (6.15) as

1 p p j=l 0 ξ 2 j,p = 1 √ p B 2 (x ) + 1 p B 1 (x ) + p -l 0 + 1 p σ Q ,
where the functions V(•), B 1 (•) and B 2 (•) are defined in (6.2). Note now that through the Cauchy inequality the condition (3.25) implies |S| 2 1 ≤ r and

| Ṡ| 2 1 ≤ r. Hence, Lemma A.3 yields E Q | σ n -σ Q | ≤ 8rn p ∞ j=l 0 j -2 + 2n p E Q V(x ) + E Q |B 2 (x )| 2 √ p + |B 1 (x )| p + l 0 -1 p σ Q .
Taking into account that j≥l 0 j -2 ≤ 2/l 0 and the conditions C 1 ) -C 2 ), one comes to the following upper bound

E Q | σ n -σ Q | ≤ 16 rn l 0 p + 2n p E Q V(x ) + L 2,Q (n) √ p + L 1,Q (n) + (l 0 -1)σ Q p . (6.16)
Moreover, in view of the inequality (6.10), one gets

E Q V(x ) ≤   κ Q n p j=l 0 θ 2 j,p   1/2 ≤ 4 rκ Q 1/2 n l 0 ≤ 4 rκ Q 1/2 l 0 .
This implies immediately Lemma 3.3 .

Proof of Theorem 3.7

One has to check the conditions (1.3), C 1 ) and C 2 ) for the model(1.1)-(2.1) and apply Theorem 3.6. The property (1.3) for this model is checked in [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF] with κ Q = 3 * and * = 2 1 + λ 2 2 . Therefore, from the definition of the distribution family in (2.4) we obtain κ Q = 3 max . A straightforward but tedious calculation shows that

E Q ξ 2 j,p = * (1 + τ j ) , (6.17)
where τ j = n -1 n 0 ψ j,p (t)ε j (t) dt and ε j (t) = a t 0 e a(t-s) ψ j,p (s) (1 + e 2as )ds. It is easy to check that |τ 1 | ≤ 2. For j ≥ 2 we represent τ j and ε j (t) as

τ j = τ 1,j + τ 2,j and ε j (t) = ε 1,j (t) + ε 2,j (t) ,
where ε 1,j (t) = a t 0 e a(t-s) ψ j,p (s) ds, ε 2,j (t) = a t 0 e a(t+s) ψ j,p (s) ds

τ 1,j = 1 n n 0 ψ j,p (t)ε 1,j (t) dt and τ 2,j = 1 n n 0 ψ j,p (t)ε 2,j (t) dt . (6.18)
In view of (3.6) τ 1,j can be represented as

τ 1,j = a n np k=1 k l=1 φ j (t k ) φ j (t l ) κ k,l ,
where

κ k,l = t l t l-1 t k t k-1
(e a(t-s) 1 {s≤t} )dt ds .

Integration yields κ k,k = κ * 0 (∆) = (e a∆ -1 -a∆)a -2 and for 1 ≤ l < k the coefficients κ k,l = κ * 1 (∆) e at k-l with κ * 1 (∆) = a -2 (e -a∆ -1) (1 -e a∆ ). Therefore, τ 1,j = aκ * 0 (∆)p + a κ * 1 (∆) p 2 A j,p , (6.19)
where

A j,p = 1 np 2 np-1 l=1 υ j,l , υ j,l = np k=l+1 φ j (t k ) φ j (t l ) e at k-l .
It should be noted that

0 ≤ κ * 0 (∆) ≤ ∆ 2 2 and 0 ≤ κ * 1 (∆) ≤ ∆ 2 e a * . (6.20)
From the definition of the functions (φ j ) 2≤j≤p it follows that

υ j,l = np-l k=1 cos( j t k ) e at k + (-1) j np-l k=1 cos( j t k+2l ) e at k := υ j,l + (-1) j υ j,l .
So, setting

A j,p = 1 np 2 np-1 l=1 υ j,l and A j,p = 1 np 2 np-1 l=1 υ j,l ,
we can represent A j,p in the form A j,p = A j,p + (-1) j A j,p , in which

A j,p = 1 p Re q j 1 -q j -B j,p and A j,p = 1 -e an np 2 B j . (6.21)
Here B j,p = 1 -e an np 2 Re q j (1 -q j ) 2 and B j = Re

q j q j (1 -q j )(1 -q j )
,

where q j = e a∆+i j ∆ and q j = e -a∆+i j ∆ . First, note that

Re 1 1 -q j = 1 -e a∆ cos( j ∆) 1 -2e a∆ cos( j ∆) + e 2a∆ = 1 -e a∆ + 2e a∆ sin 2 ( j ∆/2) (1 -e a∆ ) 2 + 4e a∆ sin 2 ( j ∆/2)
.

Taking into account that sin(x) ≥ 2x/π, 0 ≤ x ≤ π/2 , and j ≥ 2πj/3 , 2 ≤ j ≤ p , we obtain for 2 ≤ j ≤ p 2 sin( j ∆/2) ≥ j∆ . (

Therefore,

1 p Re 1 1 -q j ≤ |a|∆ 4pe a∆ sin 2 ( j ∆/2) + 1 2p ≤ a * e a * j 2 + 1 2p , (6.23)
where a * is given in (2.4). Now we estimate B j,p . We have B j,p = (1 -e an )e a∆ cos( j ∆) -2e a∆ + e 2a∆ cos( j ∆)

np 2 1 -2e a∆ cos( j ∆) + e 2a∆ 2 =
(1 -e an )e a∆ 1 -e a∆ 2 -2 1 + e 2a∆ sin 2 ( j ∆/2)

np 2 (1 -e a∆ ) 2 + 4e a∆ sin 2 ( j ∆/2) 2 .
Applying the inequality (6.22) yields

| B j,p | ≤ |a| 2 ∆ 2 16p 2 e a∆ sin 4 ( j ∆/2) + 1 4p 2 e a∆ sin 2 ( j ∆/2) ≤ e a * a 2 * j -4 + j -2 . (6.24)
By making use of the estimates (6.23) and (6.24) in (6.21) we obtain

sup n≥1 | A j,p | ≤ 1 2p + (a * + 1)e a * j 2 + a 2 * e a * j 4 . (6.25)
To estimate A j,p we represent B j as

B j = cos( j ∆) 2(cos( j ∆) -ch(a∆))
,

where ch(x) = (e x + e -x )/2. From here and (6.22), it follows that

| A j,p | ≤ 1 2np 2 1 ch(a∆) -cos( j ∆) ≤ 1 2np 2 1 1 -cos( j ∆) = 1 4np 2 sin 2 ( j ∆/2) ≤ 1 nj 2 . (6.26)
Combining (6.25) and (6.26) yields

p -1 sup n≥1 |A j,p | ≤ A * p -1 + j -2 , A * = 1 + (1 + a * + a 2 * )e a * .
This and (6.19), in view of (6.20), implies that

j -2 sup n≥1 |τ 1,j | ≤ τ * 1 p -1 + j -2 , τ * 1 = a * (1 + e a * A * ) . (6.27)
It remains to estimate τ 2,j in (6.18). First we note that

τ 2,j = a 2n ι 2 j,p
and ι j,p = n 0 ψ j,p (t) e at dt .

It is easy to check that for 2 ≤ j ≤ p

ι j,p = √ 2(e -a∆ -1) |a| (1 -e an ) Υ j 1 1 -q j -1 , (6.28)
where Υ j (z) = Re(z) for even j and Υ j (z) = Im(z) for odd j. For even j, in view of (6.23), one gets the inequality

|ι j,p | ≤ ι * p -1 + j -2 , ι * = √ 2 a * e 2a * + 3 e a * . (6.29)
For odd 3 ≤ j ≤ p one has the estimate

Im 1 1 -q j ≤ 1 j∆ ,
which implies that

|ι j,p | ≤ √ 2 e a * p -1 + j -1 ≤ ι * p -1 + j -1 . (6.30) Therefore, sup n≥1 |τ 2,j | ≤ a * ι 2 * p -2 + j -2 , 2 ≤ j ≤ p .
From here and the definition of τ j in (6.17), we obtain

sup n≥1 p j=1 |τ j | ≤ τ *   1 + j≥1 j -2   ≤ 3τ * ,
where τ * is defined in (3.44). This, in view of (6.17), shows that the condition C 1 ) holds with σ Q ≤ max . Since, for any Q from the noise distribution family Q n defined in (2.4), σ Q ≤ max , we obtain the upper bound for the functional

L 1,Q (n), i.e. L 1,Q (n) ≤ 3 * τ * ≤ 3 max τ * := L * 1,n
. Now we check the condition C 2 ) and find the upper bound L * 2,n for L 2,Q (n). We have

E Q   p j=1 x j ξ j,p   2 ≤ 2E Q ξ 2 1,p + 2E Q   p j=2 x j ξ j,p   2 and E Q   p j=2 x j ξ j,p   2 = 1 n 2 p j=2 p l=2 x j x l E Q I n (ψ j,p ) I n (ψ l,p ) , (6.31)
where

I n (f ) = I 2 n (f ) -E Q I 2 n (f ).
To study this term we will use the correlation measure introduced in [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF]. The correlation measure for two integrated [0, +∞) → R functions f and g is defined as

Υ f,g = max 0≤v≤n max 0≤t≤n-v | t 0 f (u + v)g(u)du|
and Υ * f,g = max Υ f,g , Υ g,f . Denoting υ j,l = Υ ψ j,p ,ψ l,p and υ * j,l = Υ * ψ j,p ,ψ l,p , and applying Theorem 4.4 from [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF], we obtain for any Q from the distribution family Q n the following inequality

|E Q I n (ψ j,p ) I n (ψ l,p )| ≤ 2nM Q 1 + υ * j,l + υ * j,1 + υ * l,1 ,
where

M Q = 116( * ) 2 + 33λ 4 2 EY 4 1 . It is clear, that υ * 1,1 = n, i.e. E Q ξ 2 1,p = 1 n 2 E Q I n (ψ 1,p ) 2 ≤ 8M Q .
Furthermore, note that for any t, v ≥ 0 and j ≥ 2 i.e. sup j≥2 υ * j,1 ≤ 1. Therefore, for j, l ≥ 2

|E Q I n (ψ j,p ) I n (ψ l,p )| ≤ 2nM Q 3 + υ * j,l . (6.32)
To estimate the term υ j,l note that for any 1 ≤ j, l ≤ p υ j,l = max

0≤v≤1 max 0≤t≤n-v [t]V j,l (v) + t [t] ψ j,p (u + v) ψ l,p (u) du , where V j,l (v) = 1 0 ψ j,p (u + v) ψ l,p (u) du. Therefore, υ j,l ≤ 2 + n max 0≤v≤1 V j,l (v) . (6.33)
Taking into account the definition (3.5), we find that for t s-1 ≤ v ≤ t s the function V j,l (v) can be represented as

V j,l (v) = (t s -v) p k=1 φ l (t k )φ j (t k + t s-1 ) + (v -t s-1 ) p k=1 φ l (t k )φ j (t k + t s ) . (6.34)
It is straightforward to verify that for any u, v ≥ 0 the trigonometric functions φ j satisfy the following equation

φ j (u + v) = k * j,1 (v)φ j-1 (u) + k * j,2 (v)φ j (u) + k * j,3 (v)φ j+1 (u) ,
with the bounded coefficients, because max j≥1 max 1≤i≤3 sup v≥0 |k * j,i (v)| ≤ 1. From here and (6.34), it follows that, V j,l (v) = 0 for |l -j| ≥ 2. So, taking into account in (6.33) that sup j,l≥1 |V j,l (v)| ≤ 2, we can represent the upper bound (6.33) as υ j,l ≤ 2 + 2n 1 {|l-j|≤1} and, therefore, υ * j,l ≤ 2 + 2n 1 {|l-j|≤1} . Substituting this estimate in (6.32) yields

|E Q I n (ψ j,p ) I n (ψ l,p )| ≤ 2nM Q 5 + 2n 1 {|l-j|≤1} .
Thus, taking into account that #(x) ≤ n and |x| ≤ 1, we estimate the right hand side of (6.31) as

E Q   p j=2 x j ξ j,p   2 ≤ 10M Q n p j=2 p l=2 x j x l 1 + n 1 {|l-j|≤1} ≤ 10M Q   1 n   p j=2 x j   2 + p j=2 x j x j-1 + p j=2 x 2 j + p-1 j=2 x j x j+1   ≤ 40M Q ,
which shows that L 2,Q (n) ≤ 96M Q and proves the validity of the condition C 2 ). For each noise distribution in the family Q n , defined in (2.4), we obtain that M Q ≤ M * . Therefore, the upper bound for L 2,Q (n) in the condition C 2 ) is equal to L * 2,n = M * . Hence Theorem 3.7.

Proof of Theorem 3.8

By making use of (2.5), (2.6) and applying Ito's formula, one obtains that for any square integrable R + → R functions f and g

E I t (f ) I t (g) = t 0 f (u)g(u) (u) du . (6.35)
Therefore, the condition (1.3) holds with

κ Q = κ Q (n) = * ,n = sup 0≤u≤n | (u)| ≤ max * ,n := κ * (n) .
By the condition (2.8) lim n→∞ 1 n δ κ * (n) = 0 for any δ > 0. Further we will show that the proxy σ Q in C 1 ) can be defined as

σ Q = 1 n n 0 (u)du . (6.36)
From (3.8) and (6.35), it follows that E ξ 2 1,p = σ Q . Moreover, taking into account that φ 2 j (t) = 1 + (-1) j cos(2 j t) for j ≥ 2, we obtain that E ξ 2 j,p = σ Q + (-1) j A j,p , where A j,p = n -1 pn k=1 cos(2 j t k )

t k t k-1 (u)du.
First we note that sup j≥1 |A j,p | ≤ κ * (n) . We use this estimate to check the condition C 1 ) for p ≤ 5, i.e.

p j=1 |E ξ 2 j,p -ς Q | ≤ 5 j=2 |A j,p | ≤ 4 κ * (n) . ( 6

.37)

Using the notations

A j,p = 1 2n sin( j ∆) np-1 k=1 b k sin j (2t k + ∆) and b k = t k+1 t k (u)du - t k t k-1 (u)du, one can represent A j,p as A j,p = 1 2n ∆ 0 ( (u + n -∆) -(u)) du -A j,p .
This implies that |A j,p | ≤ ∆ κ * (n)/n+|A j,p |. The last term can be rewritten as

A j,p = np k=2 (b k -b k-1 ) cos(2 j t k ) + b 1 cos(2 j ∆) -b np 4n sin 2 ( j ∆) .
In view of the conditions (2.9), we obtain sup

k≥1 |b k | ≤ * ∆ 2 and sup k≥2 |b k -b k-1 | ≤ * ∆ 3 . This yields |A j,p | ≤ ( * + 2 * )∆ 2 4 sin 2 ( j ∆) .
Note that 0 < j ∆ ≤ π/2 for 2 ≤ j ≤ ι * = 2[p/4] + 1 and π/2 < j ∆ ≤ π for ι * < j ≤ p. Therefore, similarly to (6.22) one gets sin( j ∆) ≥ j∆ for 2 ≤ j ≤ ι * . For ι * < j ≤ p -1, we get sin( j ∆) = sin(πj ∆) ≥ (p -j) ∆ . Thus, estimating |A p,n | from above κ * (n), we obtain that for p ≥ 9

p j=1 |E ξ 2 j,p -ς Q | ≤ 2κ * (n) + ι * j=2 |A j,p | + p-1 j=ι * +1 |A j,p | ≤ 2κ * (n) + ˇ *   ι * j=2 j -2 + p-1 j=ι * +1 (p -j) -2  
where ˇ * = ( * + 2 * )/4. From here taking into account that j≥1 j -2 ≤ 2, one gets

sup p≥1 p j=1 |E ξ 2 j,p -ς Q | ≤ L * 1,n .
Therefore, the condition C 1 ) holds. It remains to verify the conditions C 2 ).

We have

E   p j=1 x j ξ j,p   2 = 1 n 2 p i=1 p j=1 x i x j E I n (ψ i,p ) I n (ψ j,p ) , (6.38)
where

I t (f ) = I 2 t (f ) -E I 2 t (f )
. By applying Ito's formula one can calculate that for any bounded R + → R functions f and g

E I n (f ) I n (g) = 2 τ 2 f,g (n) + λE Y 4 1 n 0 f 2 (t)g 2 (t) 2 2 (t) dt , (6.39)
where τ f,g (t) = t 0 f (u) g(u) (u) du. First, we take f = g = ψ i,p and get

E I 2 n (ψ j,p ) ≤ 8 n 2 κ * (n) + 4 E Y 4 1 n (κ * (n)) 2 λ -1 * .
If f = ψ i,p and g = ψ j,p then

τ ψ i,p ,ψ j,p (n) = - np-1 k=1 t k t k-1 ( (u + ∆) -(u)) du k s=1 φ i (t s )φ j (t s ) .
Now, taking into account the orthogonality property (3.3) for i = j we get that for the k

= rp + ι with 1 ≤ ι < p k s=1 φ i (t s )φ j (t s ) = ι s=1 φ i (t s )φ j (t s ) ≤ p φ i p φ j p = p .
Thus, in view of the condition (2.9)

τ ψ i,p ,ψ j,p (n) ≤ np-1 k=1 t k t k-1 | (u + ∆) -(u)| ∆ du ≤ 2 n 0 d dt (t) dt ≤ 2 * ln(n+1) .
Therefore, for i = j and n ≥ 3

E I n (ψ i,p ) I n (ψ j,p ) ≤ 8 * 2 ln 2 (n + 1) + 4 E Y 4 1 n (κ * (n)) 2 λ -1 * ≤ 11(ln 2) 2 * 2 n + 4 E Y 4 1 n (κ * (n)) 2 λ -1 * .
This directly implies that the condition C 2 ) holds with the parameter L *

2,n

given in (3.47). Hence Theorem 3.8.

6.5 Proof of Theorem 3.9

One can check directly that in this case for 1 periodic functions f and g

E I n (f ) I n (g) = n (f, g) and E I n (f ) I n (g) = 2n 2 (f, g) 2 .
This implies the conditions C 1 ) and

C 2 ) with σ Q = 1, L 1,Q (n) = 0 and L 2,Q (n) = 2.
Therefore, through Corollary 3.2 we obtain Theorem 3.9.

6.6 Proof of Theorem 5.4

First, we note that in view of (3.8) one can represent the quadratic risk with the norm (3.2) for the estimate (3.10) as

E Q S γ 0 -S 2 p = 1 n p j=1 γ 2 0 (j) E Q ξ 2 j,p + Θ p ,
where Θ p = p j=1 θ j,p -γ 0 (j) θ j,p 2 . The first term can be estimated by the condition H 1 ) as sup

Q∈Q n E Q p j=1 γ 2 0 (j) ξ 2 j,p ≤ σ * (n) p j=1 γ 2 0 (j) + L * 1,n .
Therefore, taking into account that

d n = n/σ * (n), we get sup Q∈Q n E Q S γ 0 -S 2 p ≤ 1 d n p j=1 γ 2 0 (j) + L * 1,n n + Θ p .
To examine the first term on the right hand of this inequality, we note that where Θ p = p j=1 (1 -γ 0 (j)) 2 θ 2 j,p . In view of the definition (3.37), we can represent this term as Θ p = [ω 0 ] j=ι 0 (1 -γ 0 (j)) 2 θ 2 j,p + p j=[ω 0 ]+1 θ 2 j,p := Θ 1,p + Θ 2,p , where ι 0 = j * (α 0 ), ω 0 = ω α 0 = (τ k t 0 n) 1/(2k+1) and t 0 = [r/σ * (n)ε] ε. By applying Lemma A.5 and the inequality (6.6) yield Θ 1,p ≤ (1 + ε)

lim n→∞ 1 d 1/(2k+1) n p j=1 γ 2 0 (j) = 2(τ k r)
[ω 0 ] j=l (1 -γ 0 (j)) 2 θ 2 j + 4π 2 r(1 + ε -1 ) ω 3 0 p -2 .
Similarly, through Lemma A.4 Θ 2,p ≤ (1 + ε)

j≥[ω 0 ]+1 θ 2 j + (1 + ε -1 ) r p -2 .
Hence,

Θ p ≤ (1 + ε) Θ * ι 0 + (1 + ε -1 ) 4π 2 rω 3 0 + r p -2 ,
where Θ * l = j≥l (1 -γ 0 (j)) 2 θ 2 j . To estimate the last term on the right side of (6.41) we note that sup (1 -γ 0 (j)) 2 /a j , where the sequence (a j ) j≥1 is defined in (5.1). This leads to the inequality sup

Conclusion

In this paper, we have developed a model selection method for estimating a continuous time regression with semimartingale noises from discrete data. The discreteness of observations is a key factor which seriously complicates the estimation problem because the efficient estimation procedures proposed in papers [START_REF] Konev | Nonparametric estimation in a semimartingale regression model. Part 1. Oracle inequalities[END_REF] - [START_REF] Konev | Efficient robust nonparametric estimation in a semimartingale regression model[END_REF] for the continuously observed regression models do not work in this case. Moreover we are faced with the new problem of having to specify the sampling frequency which allows one to construct a robust adaptive efficient procedure. To this end we have derived non-asymptotic oracle inequalities for the robust quadratic risks under some general conditions C 1 ) and C 2 ).

S(t)dt

where Υ j = Re(z) for even j and Υ j = Im(z) for odd j, q j = e i j ∆ , ∆ = 1/p. Taking into account that the function S is 1 -periodic we transform the sum as

p k=1 q k j t k t k-1
S(t)dt = q j (1 -q j ) -1 p k=1 q k j a k , where a k = a k (S) = | Ṡ(u)|du , we obtain

|θ j,p | ≤ √ 2 |1 -q j | -1 p k=1 |a k | ≤ 2 √ 2∆| Ṡ| 1 |1 -q j | -1 .
Applying the inequalities (6.22) yields |1 -q j | = 2 sin( j ∆/2) ≥ j∆ and we obtain the second upper bound in (A.1). Hence Lemma A.3.

. 21 ) 3 . 1 .

 2131 Proposition Let the conditions (1.3), C 1 ) and C 2 ) imposed on the noise distribution (ξ t ) t≥0 in (1.1) hold. Then, for any n ≥ 2, p ≥ 3, 0 < ρ < 1/3 and any set Γ with the property (3.12), the estimator (3.20) satisfies the following oracle inequality

τ

  (1.5) and, moreover, the first limiting relation in(3.32) fails to hold. Now we consider the model (1.1)-(2.1) with the parameters satisfying the conditions (2.4) and derive the oracle inequality for the model selection procedure (3.20). To this end we set * = a * 1 + e a * + 2 ( √ 2a * e a * + 3) 2 e 2a * . (3.44) Theorem 3.7. Let the model be given by (1.1)-(2.1) with the parameters satisfying the conditions (2.4). Then, for any n ≥ 3, p ≥ 3 and 0 < ρ < 1/3, the estimate (3.20) with the weight coefficients (3.38) satisfies the oracle inequality (3.33) in which κ * (n) = 3 max , σ * (n) = max , σ * (n) = min , L * 1,n = 3 max τ * , L * 2,n = 96M * and the term B * n (p) has the property (3.43) and can be represented as

.48) Theorem 3 . 8 .

 38 Let the model be given by (1.1)-(2.5) with the parameters satisfying the conditions (2.7), (2.8) and (2.9). Then, for any n ≥ 3, p ≥ 3 and 0 < ρ < 1/3, the estimate (3.20) with the weight coefficients (3.38) satisfies the oracle inequality (3.33) with the parameters (3.46) -(3.48), in which the term B * n (p) possesses the property (3.43) and has the form

Figure 1

 1 Figure 1: n=20

Figure 2 1 .

 21 Figure 2: n=200

Theorem 5 . 2 .Corollary 5 . 3 .

 5253 assume that in the procedure(3.20) the parameter ρ = ρ(n) is a function of n such that lim n→∞ Assume that the conditions H 1 ) and H 3 ) hold. Then the robust risk (1.6) for the model selection procedure S * , defined in (3.20) with the weight coefficients (3.38), admits the following asymptotic upper boundlim sup n→∞ (d n ) 2k/(2k+1) sup S∈W k r R * n ( S * , S) ≤ R * k .Under the conditions H 1 ), H 2 ) and H 3 ) lim n→∞

. 8 )For

 8 Now taking into account the upper bound (5.8) in the oracle inequality (3.33) and using the property (3.43) we come to Theorem 5.2. each estimate (3.10) with γ ∈ Γ, combining (3.13) and (3.16) yields Err(γ) = J(γ) + 2 p j=1 γ(j)θ j,p + S 2 -ρ P (γ) ,

t 0 ψ 1 0

 01 j,p (u + v) du = v+t v+[t] ψ j,p (u) du ≤ |ψ j,p (u)| du ≤ ψ j,p = 1 ,

.

  ≤ Ṡ 2 p -2 ≤ r p -2 .Moreover, W k r ⊆ W 2 r for any k ≥ 2. From here and Lemma A.6 one has for any k ≥ 1 the inequalitysup ≤ r p -1 1 {k=1} + 3p -2 1 {k≥2} .Moreover, in view of the conditions (3.32) and H 3 )lim n→∞ d 2k/(2k+1) n p -1 1 {k=1} + ω 3 0 p -2 = 0 . So, lim sup n→∞ d 2k/(2k+1) n sup S∈W k r Θ p ≤ lim sup n→∞ d 2k/(2k+1)To estimate the term Θ * ι 0 we set υ n = d 2k/(2k+1) n sup j≥ι 0

a j θ 2 jr 1 /E

 21 ≤ υ n r .The conditions(3.35) imply lim n→∞ σ * (n)t 0 = r, and therefore lim supn→∞ υ n ≤ π -2k (τ k r) -2k/(2k+1) ,where the coefficient τ k is given in(3.37). This implies immediately (2k+1)π 2k (τ k ) 2k/(2k+1) . k r) 1/(2k+1) k 2 (k + 1)(2k + 1) + r 1/(2k+1) π 2k (τ k ) 2k/(2k+1)and applying (6.40) and (6Q S γ 0 -S 2 p ≤ R * k . (6.43)Moreover, Lemma A.2 yields that for any ε > 0 supS∈W k r R * n ( S γ 0 , S) ≤ (1 + ε) sup S∈W k r sup Q∈Q n E Q S γ 0 -S 2 p + (1 + ε -1 )r p -2 .So, in view of the condition H 3 ), we derive the desired inequality lim n→∞

  )we assume that, there exist R + → R + continuous functions min (•) and

	max (•) such that for t ≥ 0
			min (t) ≤ (t) ≤ max (t)			(2.7)
	and, for any δ > 0,
	lim t→∞	t δ	min (t) = +∞ and lim t→∞	max (t) t δ	= 0 .	(2.8)

In addition, let the first two derivatives of the functions (•) satisfy the following conditions for some positive constants * and * sup t≥0

  .23) Remark 3.3. It should be noted that Comte and Genon-Catalot in[START_REF] Comte | Estimation for Lévy processes from high frequency data within a long time interval[END_REF] use the oracle inequality of type(3.22) to find an adaptive convergence rate for the density estimation problem for the Lévy processes from discrete data. In this paper we will use this inequality to study efficiency properties for the procedure(3.19).

  These conditions have been verified in two important examples.Noting that, for t l-1 < t ≤ t l , one has the estimate|f (t l ) -f (t)| 2 ≤ | ḟ (u)| 2 du ,one comes to the first inequality. Similarly, one can verify the second inequality. Hence Lemma A.2.A.3 Properties of the Fourier coefficients.Lemma A.3. Let the function S(t) in (1.1) be absolutely continuous and have an absolutely integrable derivative. Then the coefficients (θ j,p ) 1≤j≤p defined in (3.8) satisfy the inequalities |θ 1,p | ≤ |S| 1 and max If j ≥ 2, then the coefficients θ j,p can be represented as θ j,p =

	t l	| ḟ (u)|du	2	≤ p -1	t l
	t l-1					t l-1
				√	2Υ j	p k=1 q k j	t k t k-1

2≤j≤p j|θ j,p | ≤ 2 √ 2 | Ṡ| 1 , (A.1)

where

|S| 1 = 1 0 |S(t)|dt.

Proof. We note, that if j = 1, then

|θ 1,p | = | 1 0 S(t)dt| ≤ |S| 1 .

Acknowledgment. The authors are grateful to the anonymous referees and to the AE for careful reading and for helpful comments.

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

Appendix

A.1 Upper bound for the penalty term (3.18) Lemma A.1. Assume that the condition C 1 ) holds. Then for any n ≥ 1 and γ ∈ Γ,

Proof. Taking into account (3.8) and the condition C 1 ) in (3.13) we obtain

Hence Lemma A.1.

A.2 Relations between the norms • and • p .

Lemma A.2. Let f be an absolutely continuous [0, 1] → R function with ḟ < ∞ and g be a simple [0, 1] → R function of the form g(t) = p j=1 c j χ (t j-1 ,t j ] (t), where c j are some constants. Then ∀ ε > 0 the function ∆ = f -g satisfies the following inequalities

Proof. Applying the inequality (6.6) we obtain

Lemma A.4. For any p ≥ 2, 1 ≤ N ≤ p and r > 0, the coefficients (θ j,p ) 1≤j≤p of functions S from the class W 1 r satisfy, for any ε > 0, the following inequality

j≥N θ j φ j (t). By applying Lemma A.2 and taking into account that f N 2 = j≥N θ 2 j and ḟN 2 ≤ r, we obtain the bound (A.2). Hence Lemma A.4

Lemma A.5. For any p ≥ 2 and r > 0, the coefficients (θ j,p ) 1≤j≤p of functions S from the class W 1 r satisfy the following inequality

Proof. First of all we note that θ 1,p = θ 1 . Therefore, it suffices to check that the upper bound (A.3) holds for 2 ≤ j ≤ p. Moreover, note that if S ∈ W 1 r then the function S is absolutely continuous. Therefore, the coefficients (3.4) and (5.1) satisfy the inequalities

Using here that

Lemma A.6. For any p ≥ 2 and r > 0 the correction coefficients h j,p 1≤j≤p

for the functions S from the class W


Yuri Kabanov
email: youri.kabanov@univ-fcomte.fr

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

Keywords: Life annuity insurance, Financial markets, Smoothness of the exit probability, Exponential functional of a Brownian motion, Autoregression with random coefficients Mathematics Subject Classification (2000) 60G44 JEL Classification G22, G23

We investigate models of the life annuity insurance when the company invests its reserve into a risky asset with price following a geometric Brownian motion. Our main result is an exact asymptotic of the ruin probabilities for the case of exponentially distributed benefits. As in the case of non-life insurance with exponential claims, the ruin probabilities are either decreasing with a rate given by a power function (the case of small volatility) or equal to unit identically (the case of large volatility). The result allows us to quantify the share of reserve to invest into such a risky asset to avoid a catastrophic outcome: the ruin with probability one. We address also the question of smoothness of the ruin probabilities as a function of the initial reserve for generally distributed jumps.

Introduction

In the modern world insurance companies operate in a financial environment. In particular, they invest their reserves into various assets and this may add more risk to the business. To our knowledge, the first model of an insurance company investing its capital into a risky asset has appeared in the short note [START_REF] Frolova | Some mathematical models of risk theory[END_REF] where the author provided arguments showing that the asymptotic behavior of ruin probability is radically different from that in the classical Lundberg-Cramér model. A rigorous analysis in [START_REF] Frolova | In the insurance business risky investments are dangerous[END_REF] confirmed the conjecture. Using Kalashnikov's estimates for linear finite difference equations with random coefficients, [START_REF] Kalashnikov | Ruin probability under random interest rate[END_REF], it was shown that, independently of the safety loading, the ruin is imminent with unit probability when the volatility σ of the stock price is large with respect to the instantaneous rate of return a (namely, when 2a/σ 2 < 1), and the ruin probability is decreasing as a power function, when the volatility is small (namely, when 2a/σ 2 > 1). For the model with exponentially distributed claims the exact asymptotic was found. Threshold case 2a/σ 2 = 1 was studied in the papers [START_REF] Pergamenshchikov | Ruin probability in the presence of risky investments[END_REF], [START_REF] Pergamenshchikov | Ruin probability in the presence of risky investments[END_REF] where it was shown, using techniques based on a renewal theorem, that the ruin is imminent with unit probability. The setting of [START_REF] Frolova | In the insurance business risky investments are dangerous[END_REF] and [START_REF] Pergamenshchikov | Ruin probability in the presence of risky investments[END_REF] is of the so-called non-life insurance: the company receives a flow of contributions and pay primes. The ruin occurs when a new claim arrives and its value is too large to be covered by the reserve: the risk process exits from the positive half-axis by a jump. The model can be studied in the discrete-time framework but the method of differential equations happens to be more efficient in the case of exponential claims where it allows to get the exact asymptotic.

In the present note we investigate the setting of life annuity, or pension, insurance, when the company pays to the policyholder a rent and earns a premium when the insured person dies, see, e.g., [START_REF] Grandell | Aspects of Risk Theory[END_REF], p. 8 or [START_REF] Saxén | On the probability of ruin in the collective risk theory for insurance enterprises with only negative risk sums[END_REF]; in the classical literature such models are called models with negative risk sums. The highly stylized model of the collective risk theory may describe the situation where annuitant, entering the contract, pays a lump sum, e.g., the savings during his working life, getting in exchange regular payments until the death. One can interpret the premium as the reserve release. If the company has a large "stationary" portfolio of such contracts, with new arriving customers, then it is reasonable to think that the dates of reserve releases form a Poisson process and the sizes of premiums are independent random variables of the same distribution. A special type of life annuity, called le viager exists in France. This practice is, basically, an exchange of property, a house or an apartment, for lifelong regularly payments. The annuitant enjoys a regularly income without having to give a house. For the company having a portfolio of viagers the amount of premiums depends on the real estate market.

For this classical model, where the reserve process has jumps upwards and may leave the positive half-axis only in a continuous way, the ruin problem can be easily reduced to the ruin problem for the non-life insurance using the so-called duality method, [START_REF] Asmussen | Ruin Probabilities[END_REF]. Its idea is to define the "dual" process by replacing the line segments of the original process between consecutive jumps by downward jumps of the same depth and the upward jumps by line segments of the same height with positive fixed slope. Note that in the literature the models with upward jumps are often referred to as dual models, [START_REF] Avanzi | Optimal dividends in the dual model[END_REF], [START_REF] Albrecher | On the dual risk model with taxation[END_REF].

The duality method does not work in our setting where the capital of the company or its fraction is invested into a risky asset. The change of two signs to the opposite ones in the equation defining the dynamics of the reserve leads to technical complications. In particular, now the ruin may happen before the instant of the first jump and the latter is no more the instant of the regeneration, after which the process starts afresh provided being strictly positive.

Nevertheless, a suitable modification of arguments of [START_REF] Frolova | In the insurance business risky investments are dangerous[END_REF] and [START_REF] Pergamenshchikov | Ruin probability in the presence of risky investments[END_REF] combined with some ideas allows us to obtain the asymptotic of the ruin probability as the initial reserve tends to infinity. Expectedly, it is the same as in the non-life insurance case. In many countries there are rules allowing insurance companies to invest only a small share of their reserves into risky assets. Our simple model confirms that it is reasonable and even provides a quantitative answer. To avoid the situation when the ruin happens with probability one, the proportion of investment into the risky asset should be strictly less than 2a/σ 2 .

It should be emphasized that the life insurance case is rather different and its study is not a straightforward exercise. The main difficulty in deriving the integro-differential equation is to prove the smoothness of the ruin probability and the integrability of derivatives. This issue is already delicate in the non-life insurance case. Unfortunately, it was not discussed it in [START_REF] Frolova | In the insurance business risky investments are dangerous[END_REF] where the reader was directed towards the literature. Now we are not sure that we disposed at that time a clean reference. The smoothness of the exit probability is discussed in many papers, see, e.g., an interesting article [START_REF] Wang | Distributions for the risk process with a stochastic return on investments[END_REF] where it the explicit formula for an exponential functional of Brownian motion due to Marc Yor is used. Unfortunately, the needed smoothness property was established only under constraints on coefficients. One of the authors of the present note had a fruitful discussion with Marc on a possibility to deduce smoothness of the ruin probability function and the integrability of its derivatives without using complicated explicit formulae. His suggestions are realized here for the life insurance case. Note that in the literature on non-life insurance one can find other methods to establish the smoothness, for example an approach similar to verification theorems in the stochastic control theory, see [START_REF] Belkina | Risky investments for insurers and sufficiency theorems for the survival probability[END_REF].

The structure of the paper is as follows. Section 2 contains the formulation of the main results. In Section 3 we establish an upper asymptotic bound for the exit probability (from ]0, ∞[) for a solution of a non-homogeneous linear stochastic equation and a lower asymptotic bound for a small volatility case. As a tool we use the Dufresne theorem rather than Goldie's renewal theorem from [START_REF] Goldie | Implicit renewal theory and tails of solutions of random equation[END_REF], which was the kew ingredient of arguments in [START_REF] Pergamenshchikov | Ruin probability in the presence of risky investments[END_REF]. The proof of Theorem 2 asserting that in the case of large volatility the ruin is imminent is given in Section 4. The regularity of the non-ruin probability Φ is studied in Section 5 using a method based on integral representations. At the end of this section we derive the integrodifferential equation for Φ. Section 6 contains the proof of the main theorem. Finally, in the appendix we provide a formulation of an ergodic theorem for an autoregression with random coefficients proven in [START_REF] Pergamenshchikov | Ruin probability in the presence of risky investments[END_REF].

Kalashnikov's approach (developed further in his joint work with Ragnar Norberg [START_REF] Kalashnikov | Power tailed ruin probabilities in the presence of risky investments[END_REF]) plays an important role in our study. Our techniques is elementary. More profound and general results can be found in [START_REF] Paulsen | Sharp conditions for certain ruin in a risk process with stochastic return on investments[END_REF], [START_REF] Paulsen | Stochastic Calculus with Applications to Risk Theory[END_REF], [START_REF] Paulsen | Ruin theory with stochastic return on investments[END_REF], [START_REF] Nyrhinen | On the ruin probabilities in a general economic environment[END_REF], [START_REF] Nyrhinen | Finite and infinite time ruin probabilities in a stochastic economic environment[END_REF] et al.

The model

We are given a stochastic basis (Ω, F, F = (F t ) t≥0 , P) with a Wiener process W independent of the integer-valued random measure p(dt, dx) with the compensator p(dt, dx).

Let us consider a process X = X u of the form

X t = u + a t 0 Xsds + σ t 0 XsdWs -ct + t 0 xp(ds, dx), (1)
where a and σ are arbitrary constants and c ≥ 0. We shall assume that p(dt, dx) = αdtF (dx) where F (dx) is a probability distribution on ]0, ∞[. In this case the integral with respect to the jump measure is simply a compound Poisson process. It can be written as Nt i=1 ξ i where N is a Poisson process with intensity α and ξ i are random variables with common distribution F ; the random variables W , N , ξ i , i ∈ N, are independent. We denote by Tn the successive jumps of N ; the inter-arrival time intervals T i -T i-1 are independent and exponentially distributed with the parameter α.

In our main result (Theorem 1) we assume that F is the exponential distribution with parameter µ.

Let τ u := inf{t : X u t ≤ 0} (the instant of ruin), Ψ (u) := P (τ u < ∞) (the ruin probability), and Φ(u) := 1 -Ψ (u) (the survival probability).

The parameter values a = 0, σ = 0, correspond to the life insurance ( i.e. the dual) version of the Lundberg-Cramér model for which the risk process is usually written as

r t := u -ct + Nt i=1 ξ i .
(2)

In the considered case the capital evolves due to continuously outgoing cash flow with rate c and incoming random payoffs ξ i at times forming an independent Poisson process N with intensity α. For the classical model with positive safety loading and F having a "non-heavy" tail, the Lundberg inequality provides an encouraging information: the ruin probability decreases exponentially as the initial capital u tends to infinity. Moreover, for the exponentially distributed claims the ruin probability admits an explicit expression, see [START_REF] Asmussen | Ruin Probabilities[END_REF] or [START_REF] Grandell | Aspects of Risk Theory[END_REF].

The more realistic case a > 0, σ = 0, corresponding to non-risky investments, does not pose any problem.

We study here the case σ > 0. Now the equation ( 1) describes the evolution of the reserve of an insurance company which pays a rent and continuously reinvests its capital into an asset with the price following a geometric Brownian motion. The same model can be used for the description of the capital of a venture company funding R&D and selling innovations, [START_REF] Bayraktar | Optimizing venture capital investments in a jump diffusion model[END_REF]. Notations. Throughout the paper we shall use the following abbreviations:

κ := a - 1 2 σ 2 , β := 2κ σ 2 = 2a σ 2 -1, η t := κt + σW t .
The solution of the linear stochastic equation ( 1) can be written using the Cauchy formula:

X t = e ηt u + [0,t]
e -ηs drs .

(3)

Theorem 1 Let F (x) = 1 -e -x/µ , x > 0. Assume that σ > 0. (i) If β > 0, then for some K > 0 Ψ (u) = Ku -β (1 + o(1)), u → ∞. ( 4
) (ii) If β ≤ 0, then Ψ (u) = 1 for all u > 0.
The formulation of this theorem is exactly the same as in [START_REF] Frolova | In the insurance business risky investments are dangerous[END_REF] for the non-life insurance model (the case β = 0 was analyzed in [START_REF] Pergamenshchikov | Ruin probability in the presence of risky investments[END_REF], [START_REF] Pergamenshchikov | Ruin probability in the presence of risky investments[END_REF]).

One must admit that in the life insurance models, especially, in the case of a company keeping a portfolio of viager contracts, the hypothesis that the benefits (i.e. prices of houses) follow the exponential distribution, are highly unrealistic. Nevertheless, we can claim, without any assumption on the distribution, that the ruin probabilities lay between two power functions, see Propositions 1 and 2.

The next result (implying the statement (ii) above) says that for δ > 0 the ruin is imminent. It requires only that the distribution F has moments of positive order.

Theorem 2 Assume that there is δ > 0 such that Eξ δ 1 < ∞. If β ≤ 0, then Ψ (u) = 1 for any u > 0.
The same model serves well in the situation where only a fixed part γ ∈]0, 1] of the capital is invested in the risky asset: one should only replace the parameters a and σ in (1) by aγ and σγ. Theorem 1 implies that the ruin with probability one will be avoided only if 2aγ/(σγ) 2 > 1, i.e. when the share of investment into the risky asset is strictly less than 2a/σ 2 .

It is worth mentioning that our conclusion are robust and holds for more general models. The reader may contest that intensity of outgoing payments c is constant. Indeed, after the death of the annuitant the payments stop and the intensity must decrease while with a new customer it increases. Easy comparison arguments show that the above statements hold if

c = (c t ) is a random process, such that 0 < C 1 ≤ c ≤ C 2 where C 1 and C 2 are constants.
The crucial part of the asymptotic analysis in Theorem 1 is based on the fact that for the Markov process given by (1) the non-exit probability Φ(u) is smooth and satisfies the following equation:

1 2 σ 2 u 2 Φ (u) + (au -c)Φ (u) -αΦ(u) + α ∞ 0 Φ(u + y)dF (y) = 0. (5)
With σ > 0 this equation is of the second order and, hence, requires two boundary conditions -in contrast to the classical case (a = 0, σ = 0) where it degenerates to an equation of the first order requiring a single boundary condition, see [START_REF] Grandell | Aspects of Risk Theory[END_REF]. The estimate given in Proposition 1 shows that Φ(∞) = 1.

There is an extensive literature concerning of the regularity of the survival probability for process with jumps in the context of non-life insurance models and models based on the Lévy processes, see, e.g., [START_REF] Wang | Distributions for the risk process with a stochastic return on investments[END_REF], [START_REF] Gaier | Ruin probabilities and investment under interest force in the presence of regularly varying tails[END_REF], [START_REF] Granditz | A Karamata-type theorem and ruin probabilities for an insurer investing proportionally in the stock market[END_REF], [START_REF] Belkina | Singular boundary value problem for the integrodifferential equation in an insurance model with stochastic premiums: Analysis and numerical solution[END_REF], [START_REF] Belkina | Singular problems for integro-differential equations in dynamic insurance models[END_REF]. Our Theorem 3 requiring only smoothness of F and the integrability of F seems to be the first result on the regularity of the survival probability in the considered setting.

3 Asymptotic bounds for the small volatility case

Upper asymptotic bound

Let us consider the exit probability problem for a more general process X = X u of the form

X t = u + a t 0 Xsds + σ t 0 Xs dWs -ct + Z t , (6)
where a, σ, c > 0 are arbitrary constants and Z = (Z t ) t≥0 is an increasing adapted càdlàg process starting from zero.

Proposition 1 Assume that in the general model (1) the parameter β > 0. Then

lim sup u→∞ u β Ψ (u) ≤ 2 β c β σ 2β βΓ (β) .
Proof. Let Y be a solution of the linear stochastic equation

Y t = u + a t 0 Ys ds + σ t 0 Ys dWs -ct.
Introducing the notation

R t := c t 0 e -ηv dv (7)
we express the solution as

Y t := e ηt u -R t ).
The difference X -Y satisfies the linear equation with zero initial condition. Since Z is increasing, we have the inequality X ≥ Y showing that the exit of X from ]0, ∞[ implies the exit of Y . Thus,

Ψ (u) ≤ P (R∞ > u). (8)
and the asymptotic behavior of the ruin probability for this general model can be estimated by the tail behavior of the distribution function of R∞. Using the change of variable v = (4/σ 2 )t and observing that B t := -(1/2)σW (4/σ 2 )t is a Wiener process we obtain the representation

R∞ = c ∞ 0 e -(a-σ 2 /2)v-σWv dv = 4c σ 2 ∞ 0 e -2βt+2Bt dt =: 4c σ 2 A (-β) ∞ .
The Dufresne theorem (see, [START_REF] Dufresne | The distribution of a perpetuity, with application to risk theory and pension funding[END_REF] or [START_REF] Matsumoto | Exponential functionals of Brownian motion, I: Probability laws at fixed time[END_REF], Theorem 6.2) claims that

A (-β) ∞
is distributed as the random variable 1/(2γ) where γ has the gamma distribution with parameter β. Thus,

P(R∞ > u) = P(2c/(σ 2 γ) > u)) = P(γ < 2c/(σ 2 u)) = 1 Γ (β) 2c/(σ 2 u) 0 x β-1 e -x dx ∼ 2 β c β σ 2β βΓ (β) u -β
and the result follows from (8). 2

Lower asymptotic bound

The next result shows that the ruin probability decreases as the initial capital tends to infinity not faster than a certain power function.

Proposition 2 Assume that β > 0. Then there exists β * > 0 such that

lim inf u→∞ u β * Ψ (u) > 0 .
Proof. Let Y = (Y k ) k≥1 be the imbedded discrete-time process, that is the sequence of random variables defined recursively as follows:

Y k = M k Y k-1 + Q k , Y 0 = u, (9)
where

M k = e η T k -η T k-1 and Q k = ξ k -c T k T k-1 e η T k -ηv dv . Let θ u := inf{k : Y k ≤ 0}. It is clear that X T k = Y k for any k ≥ 1.
So, for any u > 0

Ψ (u) = P(τ u < ∞) ≥ P(θ u < ∞). (10)
Take ∈]0, 1[ and chose B sufficiently large to ensure that

B 1 = B - 1 2 (1 -) > 0.
Define the sets

Γ k = {M k ≤ } ∩ {Q k ≤ -1 } , D k = {M k ≤ -1 } ∩ {Q k ≤ -B}. (11)
On the set ∩ n k=1 Γ k we have

Yn = u n j=1 M j + n k=1 Q k n j=k+1 M j ≤ u n + 1 (1 -) .
Therefore, on the set

∩ n k=1 Γ k ∩ D n+1 Y n+1 = M n+1 Yn + Q n+1 ≤ u n-1 + 1 2 (1 -) -B = u n-1 -B 1 .
It is easy to check that u n-1 ≤ B 1 , when u > B 1 and

n = 3 + ln(u/B 1 ) | ln | ,
where [...] means the integer part. Therefore,

P(θ u < ∞) ≥ P ∩ n k=1 Γ k ∩ D n+1 = (P(Γ 1 )) n P(D 1 ).
Taking into account that P(Γ 1 ) > 0 and P(D 1 ) > 0, we obtain that

lim u→∞ u β * P(θ u < ∞) = ∞
for any

β * > ln P(Γ 1 ) ln .
This implies the claim. 2

4 Large volatility: proof of Theorem 2

We consider separately two cases: β < 0 and β = 0 and show that in both cases the ruin probability is equal to one.

Proposition 3 Assume that β < 0 and E ξ δ 1 < ∞ for some δ > 0. Then Ψ (u) = 1 for all u > 0.

Proof. As in the proof of Proposition 2 we consider the imbedded discrete-time process Y = (Y k ) k≥1 defined by [START_REF] Dufresne | The distribution of a perpetuity, with application to risk theory and pension funding[END_REF]. By virtue of [START_REF] Fedoryuk | Asymptotic Analysis: Linear Ordinary Differential Equations[END_REF] it is sufficient to show that P(θ u < ∞) = 1.

Note that for δ ∈]0, -β[

Ee δηt = Ee δ(κt+σWt) = e δt(β+δ)σ 2 /2 < 1,
and, therefore,

EM δ 1 = ∞ 0 Ee δηt αe -αt dt < 1. (12)
According to ( [START_REF] Borodin | Handbook of Brownian Motion -Facts and Formulae[END_REF], p. 250) if a random variable ν is independent of W and has the exponential distribution with parameter α, then

E e q max v≤ν (µv+Wv) = 2α + µ 2 -µ 2α + µ 2 -µ -q . ( 13
)
provided that

2α + µ 2 -µ -q > 0.
Changing the variable and estimating the integrand by its maximal value we get that

E T1 0 e η T 1 -ηv dv δ = E T1 0 e κv+σWv dv δ ≤ E T δ 1 e δ max v≤T 1 (κv+σWv) = ∞ 0 t δ E e δ max v≤t (κv+σWv) αe -αt d t ≤ 2 sup t≥0 {t δ e -αt/2 }E e δ max v≤ν (κv+σWv) = 2e δ(ln(2δ/α)-1) E e δ max v≤ν (κv+σWv) ,
where ν is an exponential random variable with parameter α/2. In view of the equality ( 13)

E e δ max v≤ν (κv+σWv) = √ σ 2 α + κ 2 -κ √ σ 2 α + κ 2 -κ -δσ 2 , provided that δ < √ σ 2 α + κ 2 -κ σ 2 ,
i.e. for such δ > 0 we have the bound

E T1 0 e η T 1 -ηv dv δ ≤ 2 e δ(ln(2δ/α)-1) √ σ 2 α + κ 2 -κ √ σ 2 α + κ 2 -κ -δσ 2 . ( 14
)
Using these estimates and the assumption of the proposition we conclude that E |Q 1 | δ < ∞ for sufficiently small δ > 0. Thus, the hypothesis of the ergodic theorem for autoregression with random coefficients is fulfilled (see Proposition 8). The latter claims that for any bounded uniformly continuous function f

P-lim N 1 N N k=1 f (Y k ) = E f (ζ), (15)
where

ζ = Q 1 + ∞ k=2 Q k k-1 j=1 M j .
Let us represent ζ in the form

ζ = ξ 1 - T1 0 e η T 1 -ηv dv + e η T 1 ζ 1 , ζ 1 := ∞ k=2 Q k k-1 j=2 M j .
Clearly, the random variables ξ, ζ 1 and (η T1 , T1 0 e η T 1 -ηv dv) are independent. Moreover, Lemma 1 given after the proof implies that the support of conditional distribution of the integral t 0 e ηt-ηv dv given η t = x is unbounded from above. From this we easily infer that the support of distribution of ζ is unbounded from below. Thus, for

f (x) = 1 {x≤-1} + |x| 1 {-1<x<0}
the right-hand side of ( 15) is strictly positive and, therefore,

P inf k≥1 Y k < 0 = 1. 2 Lemma 1 Let σ > 0.
Then the support of conditional distribution of the random variable

I = 1 0 e σWv d v
given W 1 = y is unbounded from above.

Proof. It is well-known (see, e.g., [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF]) that the conditional distribution of the Wiener process (Wv) v≤1 given W 1 = y coincides with the distribution of the Brownian bridge B y with B y t = W t +t(y-W 1 ). Thus, the conditional distribution of I is the same as the unconditional distribution of

Ĩ := 1 0 e σ(Wv+v(y-W1)) d v.
Since the Wiener measure has the full support in the space C 0 [0, 1] of continuous functions on [0, 1] with zero initial value, the support of distribution of Ĩ is unbounded from above. 2 Proposition 4 Assume that β = 0 and E ξ δ 1 < ∞ for some δ > 0. Then Ψ (u) = 1 for all u > 0.

Proof. In the considered case the imbedded discrete-time process is defined by ( 9) with

M k = e σ∆V k and Q k = ξ k -c T k T k-1 e σ(W T k -Wv) dv,
where

V k = W T k and ∆V k := V k -V k-1 .
To study the asymptotic properties of the equation ( 9) we use the approach proposed in [START_REF] Pergamenshchikov | Ruin probability in the presence of risky investments[END_REF] for the non-life insurance models. To this end, define recursively the sequence of random variables putting θ 0 := 0 and

θn := inf{k > θ n-1 : V k -V θn-1 < 0}, n ≥ 1. (16)
Note that θn = n j=1 ∆θ j where (∆θ j ) j≥1 is a sequence of i.i.d. random variables distributed as θ 1 . It is known (see, e.g., XII.7, Theorem 1a in [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF]) that

C := sup n≥1 n 1/2 P(θ 1 > n) < ∞ . ( 17
)
Putting y k = Y θ k , we obtain from (9) that

y k = a k y k-1 + b k , y 0 = u, (18)
where

a k = ∆θ k j=1 M θ k-1 +j = e σ(V θ k -V θ k-1 ) and b k = ∆θ k l=1   ∆θ k j=l+1 M θ k-1 +j   Q θ k-1 +l .
It is clear that a k < 1 a.s. Moreover, the first condition in Theorem 2 and the inequality ( 14) with κ = 0 implies that E|Q 1 | δ < ∞ for any sufficiently small δ. Now, taking into account that

|b 1 | ≤ ∆θ1 l=1   ∆θ1 j=l+1 M j   |Q l | = ∆θ1 l=1 a 1 l j=1 M j |Q l | ≤ ∆θ1 l=1 |Q l |,
we can get, for r ∈]0, 1[ and an increasing sequence of integers ln, that

E |b 1 | r ≤ 1 + r n≥1 1 n 1-r P(|b 1 | > n) ≤ 1 + r n≥1 1 n 1-r P   ln j=1 |Q j | > n   + r n≥1 1 n 1-r P(|t 1 | > ln) ≤ 1 + r E |Q 1 | δ n≥1 ln n 1-r+δ + r C n≥1 1 n 1-r l 1/2 n .
Putting here ln = [n 4r ] we obtain that E |b 1 | r < ∞ for any r ∈]0, δ/5[. Therefore, due to Proposition 8, we obtain that for any bounded uniformly continuous function f

P-lim N 1 N N k=1 f (y k ) = Ef (ζ),
where

ζ = b 1 + ∞ k=2 b k k-1 j=1 a j . ( 19
)
Now we show that P(ζ < -x) > 0 for any

x > 0.
Indeed, the random variable ( 19) can be represented as

ζ = b 1 + a 1 ζ 1 , ζ 1 = ∞ k=2 b k k-1 j=2 a j .
It is clear that the ζ 1 is independent on b 1 and a 1 . Note that on the set {∆θ 1 = 1} we have a 1 = M 1 and b 1 = Q 1 . Therefore, for any x > 0

P(ζ < -x) ≥ P(b 1 + a 1 ζ 1 < -x, ∆θ 1 = 1) = P(Q 1 + M 1 ζ < -x , w T1 < 0)
and we conclude as in the previous proposition. 2 [START_REF] Belkina | Risky investments for insurers and sufficiency theorems for the survival probability[END_REF] Regularity of the ruin probability

Integral representations

The proof of smoothness of a function H admitting an integral representation is based on a simple idea which merits to be explained.

First, we recall the classical result on differentiability of the integral

H(u) = f (u, z)dz where f (u, .) ∈ L 1 for each u from open subset U ∈ R. If f (., z) is differentiable on an open interval ]u 0 -ε, u 0 + ε[ for
almost all z and, on this interval, |∂f (., z)/∂u| ≤ g(z) (a.e.) where g ∈ L 1 . Then H is differentiable at u 0 and H (u 0 ) = ∂f (u 0 , z)/∂u dz.

Suppose that we are given a bounded measurable function h(z) and a Gaussian random

variable ζ ∼ N (0, 1). Let H(u) = Eh(u + ζ) = h(u + x)ϕ 0,1 (x)dx.
Then H is differentiable and even of the class C ∞ . Of course, the above result cannot be applied directly. But using the change of variable we get the representation

H(u) = h(u + x)ϕ 0,1 (x)dx = h(z)ϕ 0,1 (z -u)dz.
Now the parameter u appears only in the function ϕ 0,1 , the integrand is differentiable in u and we can apply the classical sufficient condition.

The issues here are: an integral representation, the smoothness of the density, and the integrability of its derivatives. In the case of survival probability Φ the integral representation is obtained from the strong Markov property. Unfortunately, the structure of the representation is rather complicated, the random variable standing for ζ is not a Gaussian one, and its density is not given by a simple formula. Nevertheless, the idea of using a change of variable to move the parameter from the unknown function on which we have only a limited information (essentially, the boundedness and measurability) to the density still does work. The main difficulty is to check the smoothness of the density and find appropriate bounds for its derivatives.

Theorem 3 Assume that the distribution function of ξ 1 has a density f differentiable on R + and such that f ∈ L 1 (R + ). Then Φ(u) is two times continuously differentiable on ]0, ∞[.

Proof. We again consider the process

Y u t = e ηt (u -R t ) , (20)
where R t is defined in [START_REF] Belkina | Singular problems for integro-differential equations in dynamic insurance models[END_REF]. Put

θ u := inf{t ≥ 0 : Y u t ≤ 0}. (21)
By virtue of the strong Markov property of

X u Φ(u) = EΦ(X u θ u ∧T1 ). (22)
Note that the process Y u is strictly positive before the time θ u , zero at θ u , and strictly negative afterwards. Due to the independence of the Wiener process and the instants of jumps, θ u = T 1 a.s. Thus, {Y u T1 > 0} = {θ u > T 1 } a.s. Taking into account that Φ(0) = 0, we get that

Φ(u) = EI {Y u T 1 >0} Φ(X u T1 ) = EI {Y u T 1 >0} Φ(Y u T1 + ξ 1 ) = Φ 1 (u) + Φ 2 (u), Lemma 2 The random variable ζ x has a density ρ(x, .) ∈ C ∞ such that for any n ≥ 1 sup y≥0 ∂ n ∂y n ρ(x, y) ≤ Cne Cn|x| (26)
with some constant Cn and (∂ n /∂y n )ρ(x, 0) = 0.

Proof. We obtain the result using again the integral representation. Let us introduce the random process

Ds := (Ws -2sW 1/2 ) + s(W 1/2 -W 1 ) I {s≤1/2} + (1 -s)(Ws -W 1/2 )) -s(W 1 -Ws) I {s>1/2} ,
and the piecewise linear function

γs := sI {s≤1/2} + (1 -s)I {s>1/2} , s ∈ [0, 1].
The following identity is obvious:

Ws -sW 1 = Ds + γsW 1/2 .
Since Ds and W 1/2 are independent random variables, for any bounded Borel function g we have:

E g(ζ x ) = E g c 1 0 e -(κs+sx+σDs+σγsv) ds ϕ 0,1/2 (v)dv
Let v(x, .) be the inverse of the continuous strictly decreasing function

y → c 1 0
e -(κs+sx+σDs+σγsy) ds depending on the parameter x (and also on ω omitted as usual). Note that v(x, 0+) = ∞ and v(x, ∞) = 0. After the change of variable, we obtain, using the notation

K(x, z) := cσ 1 0 γse -(κs+sx+σDs+σγsz) ds, that Eg(ζ x ) = ∞ 0 g(y)ρ(x, y)dy, where ρ(x, .) := E ϕ 0,1/2 (v(x, .)) K(x, v(x, .)) . (27)
Thus, ρ(x, .) is the density of distribution of the random variable ζ x . It remains to check that it is infinitely differentiable and find appropriate bounds for its derivatives. Put

Q (0) (x, z) := ϕ 0,1/2 (z) K(x, z) , Q (n) (x, z) := - Q (n-1) z (x, z) K(x, z) , n ≥ 1.
Then

∂ ∂y Q (0) (x, v(x, y)) = Q (0) z (x, v(x, y))vy(x, y) = - Q (0) z (x, v(x, y)) K(x, v(x, y)) = Q (1) (x, v(x, y))
and, similarly,

∂ n ∂y n Q (0) (x, v(x, y)) = - Q (n-1) z (x, v(x, y)) K(x, v(x, y)) = Q (n) (x, v(x, y)).
It is easily seen that

Q (n) (x, z) = ϕ 0,1/2 (z) K n+1 (x, z) n k=0 P k (z)R n-k (x, z), (28)
where P k (z) is a polynomial of order k and R n-k (x, z) is a linear combination of products of derivatives of K(x, z) in variable z. Note that for any x, y ∈ [0, 1] we have the bounds

-|x| -σ|z| -3σW * 1 ≤ κs + sx + σDs + σγsz ≤ κ + |x| + σ|z| + 3σW * 1
where W * 1 = sup s≤1 |Ws|. It follows that there exists a constant Cn > 0 such that

|Q (n) (x, z)| ≤ Cne Cn|x| (1 + z n )e -z 2 e Cn W * 1 . (29)
Since E e Cn W * 1 < ∞, for each x, y and n the derivative (∂ n /∂y n ) Q (0) (x, v(x, y)) admits a P-integrable bound. Thus, we can differentiate under the sign of expectation and obtain that

∂ n ∂y n ρ(x, y) = E ∂ n ∂y n ϕ 0,1/2 (v(x, y)) K(x, v(x, y)) = E ∂ n ∂y n Q (0) (x, v(x, y)) = EQ (n) (x, v(x, y)).
Moreover, the bound [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF] ensures that, for some constant Cn, Proof. Putting H(u, z) := H(e κ+σx z)ρ(x, u -z)ϕ 0,1 (x) dx, we rewrite the formula (25) as

sup y≥0 E ∂ n ∂y n Q (0) (x, v(x, y)) ≤ E sup z∈R |Q (n) (x, z)| ≤ Cne
Φ 2 (u) = u 0 H(u, z) dz. (30)
Clearly, the function H is continuous on ]0, ∞[×]0, ∞[. Using Lemma 2 we obtain that

∂ ∂u H(u, z) = H(e κ+σx z) ∂ ∂u ρ(x, u -z)ϕ 0,1 (x) dx and sup u,z ∂ ∂u H(u, z) < ∞.
By induction, for every n ≥ 1,

∂ n ∂u n H(u, z) = H(e κ+σx z) ∂ n ∂u n ρ(x, u -z)ϕ 0,1 (x) dx and sup u,z ∂ n ∂u n H(u, z) < ∞.
By virtue of Lemma 2 ρ(x, 0) = 0, i.e. H(u, u) = 0. So,

d du Φ 2 (u) = H(u, u) + u 0 ∂ ∂u H(u, z) dz = u 0 ∂ ∂u H(u, z) dz.
In the same way we check that

d n du n Φ 2 (u) = u 0 ∂ n ∂u n H(u, z) dz.
for any n ≥ 1. 2

Smoothness of Φ 1

Arguing in the same spirit as in the previous subsection but taking this time the conditional expectation with respect to W t we obtain that

E1 {Rt<u} h(e κt+σWt (u -R t )) = 1 √ t E 1 {ζ t,x <u} h(u, t, x)ϕ 0,1 x √ t dx
where we use the abbreviations It is easily seen that the random variable ζ t,x has infinitely differentiable density (the same as of ζ x defined in [START_REF] Paulsen | Stochastic Calculus with Applications to Risk Theory[END_REF] but with the parameters ct, κt, and σt 1/2 ). Unfortunately, derivatives of this density have non-integrable singularities as t tends to zero (see Remark at the end of previous subsection). By this reason we cannot use the strategy of proof used for Φ 2 . Nevertheless, the hypothesis on the distribution of ξ 1 allows us to establish the claimed result.

Note that the function x → ζ t,x is strictly decreasing and maps R onto R + . Let denote z(t, .) its inverse. The derivative of the latter is given by the formula

zx(t, x) = - t L(t, z(t, x)) ,
Inspecting the formula (31) defining L(t, z) we conclude that there exist positive constants C 0 ("small") and C 1 ("large") such that

max t∈]0,2] (L(t, z) + t|Lz(t, z)|) ≤ C 1 e C1(|z|+W * t )
and for any t ∈]0, 2]

|L(t, z)| ≥ C 0 t 2 e -C1(|z|+W * t ) ,
where W * t := max v≤t |Wv|. Taking this into account we obtain that for some C > 0

|Θ(u, t, z)| ≤ C t -6 e C(|z|+W * t ) . (36)
Using a generic notation C for a constant (which may vary even within a single formula) we obtain, for any t ∈]0, 2]

|Hu(t, u)| ≤ Ct -7 e CW * t ∞ z(t,u) e C|z|-z 2 2t L(t, z) dz ≤ Ct -7 e C W * 2 e -z 2 (t,u) 4t ∞ 0 e C|z|-z 2 4 dz.
So, we have the bound

|Hu(t, u)| ≤ Ct -7 e CW * 2 e -z 2 (t,u) 4t .
For any u ≥ u 0 > 0 and t ∈]0, 2]

u 0 ≤ u = ζ t,z(t,u) ≤ ct e 2|κ|+σ|z(t,u)|+σW * t , i.e. |z(t, u)| ≥ 1 σ ln e -2|κ| u 0 tc -W * t .
Put

t 0 := min e -2|κ|-3σ u 0 c , 2 , Γ := {W * t ≤ t 1/4 }.
Thus, for t ∈]0, t 0 ] on the set Γ we have the inequality

|z(t, u)| ≥ 1.
Taking into account that E e aW * t < ∞ for any a and t > 0, we obtain that

E max t∈[t0,2] sup u≥0 |Hu(t, u)| ≤ Ct -7 0 E e CW * 2 < ∞. For t ∈]0, t 0 ] we have E max u≥u0 |Hu(t, u)| ≤ Ct -7 e -1 4t + Ee CW * t 1 Γ c ≤ Ct -7 e -1 4t + E e CW * 2 P W * t ≥ t 1/4 .
By the Chebyshev inequality we have:

P W * t ≥ t 1/4 ≤ e -t -1/4 E e W * t √ t = e -t -1/4 E e W * 1 , that is sup t≥0 e t -1/4 P W * t ≥ t 1/4 < ∞.
This implies that

max t∈]0,t0] E max u≥u0 |Hu(t, u)| < ∞.
Therefore,

max t∈]0,2] E max u≥u0 |Hu(t, u)| < ∞.
Differentiating (35) we find that

Huu(t, u) = h(0) Θ(u, t, z(t, u)) ϕ z(t, u) √ t + u 0 Υ (u, t, z(t, x)) ϕ z(t, x) √ t dx,
where Υ (u, t, z) = hu(u, t, z)Θ(u, t, z) + h(u, t, z)Θu(u, t, z).

By assumption, the distribution function F has the density f whose derivative f is a continuous function on R + integrable with respect to the Lebesgue measure. Changing variable we get that

h (y) = d dy ∞ 0 Φ(y + x)f (x)dx = d dy ∞ y Φ(x)f (x -y)dx = -Φ(y) f (0) - ∞ y Φ(z)f (z -y) dz, i.e. sup x≥0 |h (x)| < ∞ .
Similarly to (36) we obtain that

|Υ t (u, z)| ≤ C t -8 e C(|z|+W * t ) .
Therefore,

|Huu(t, u)| ≤ Ct -9 e C W * t e -z 2 (t,u) 4t , implying that max t∈]0,2] E max u≥u0 |Huu(t, u)| < ∞.
Proposition 6 is proven. 2

Integro-differential equation for the survival probability

Proposition 7 Suppose that Φ ∈ C 2 . Then Φ satisfies the equation [START_REF] Belkina | Risky investments for insurers and sufficiency theorems for the survival probability[END_REF].

Proof. For h > 0 and > 0 assumed to be small enough to ensure that u ∈] , -1 [ we put

τ h := inf t ≥ 0 : Y u t / ∈ [ , -1 ] ∧ h.
Using the Ito formula and taking into account that on the interval on [0, T 1 [ the process X u coincides with Y u we obtain the representation

Φ(X u τ h ∧T1 ) = Φ(u) + σ τ h ∧T1 0 Φ (Y u s ) dWs + τ h ∧T1 0 1 2 σ 2 (Y u s ) 2 Φ (Y u s ) + (aY u s -c)Φ (Y u s ) ds +(Φ(Y u T1 + ξ 1 ) -Φ(Y u T1 ))I {T1≤τ h } .
Due to the strong Markov property Φ(u) = E Φ(X u τ h ∧T1 ). For every > 0 the integrands above are bounded by constants and, hence, the expectation of the stochastic integral is zero. Moreover, τ h ∧ T 1 = h when h is sufficiently small (the threshold below which we have this equality, of course, depends on ω).

It follows that, independently of ,

1 h E τ h ∧T1 0 1 2 σ 2 (Y u s ) 2 Φ (Y u s ) + aY u s Φ (Y u s ) -c ds → 1 2 σ 2 u 2 Φ (u) + (au -c)Φ (u).
Finally,

1 h E(Φ(Y u T1 + ξ 1 ) -Φ(Y u T1 ))I {T1≤τ h } = αE 1 h τ h 0 (Φ(Y u t + ξ 1 ) -Φ(Y u t
))e -αt dt.

The right-hand converges to α(EΦ(u + ξ 1 ) -Φ(u)) as h → 0 in virtue of the Lebesgue theorem on dominated convergence. It follows that Φ satisfies the equation (5). 2

6 Proof of Theorem 1

Assume that the claims are exponentially distributed, i.e. F (x) = 1 -e -x/µ . Similarly to the classical case, this assumption allows us to obtain for the ruin probability an ordinary differential equation (but of a higher order). Indeed, now the integro-differential equation (5) has the form Differentiating (37) and adding to it the obtained identity multiplied by µ we exclude the integral term and arrive at a third order ordinary differential equation. It does not contain Since Φ + (u) → ∞ as u → ∞, we obtain immediately that C 1 = 0. For the case β > 0 we know from Proposition 1 that Ψ (∞) = 0. Thus, C 0 = 0 and

Ψ (u) = C 2 ∞ u x -2a/σ 2 (1 + δ(x)) dx,
where δ(x) → 0 as x → ∞ and C 2 > 0. The integral decreases at infinity as the power function u -β /β and we obtain Theorem 1. Proposition 8 Assume that there exists δ ∈]0, 1] such that

E |a 1 | δ < 1, E |b 1 | δ < ∞.
Then for any bounded uniformly continuous function f The proof is given in [START_REF] Pergamenshchikov | Ruin probability in the presence of risky investments[END_REF].

P-lim N 1 N N n=1 f (xn) = E f (ζ),

Cn|x| 2 Remark 1 Proposition 5

 215 and the bound[START_REF] Paulsen | Ruin theory with stochastic return on investments[END_REF] holds.Since v(x, 0+) = ∞, the bound[START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF] implies that (∂ n /∂y n )ρ(x, 0) = 0. It is worth to trace in these arguments the dependence of the constant Cn on the parameters c and σ when they are approaching zero. From the formula[START_REF] Pergamenshchikov | Ruin probability in the presence of risky investments[END_REF] it is clear that Cn should be proportional to (cσ) -n . The function Φ 2 (u) belongs to C ∞ (]0, ∞[).

h

  (u, t, x) := h(e κt+σx (u -ζ t,x )), h(y) = E Φ (y + ξ 1 ) and ζ t,x := c t 0 e -(sx/t+κs+σ(Ws-(s/t)Wt) ds.

1 2 σ 2 u 2 Φµ ∞ 0 Φ

 20 (u) + (au -c)Φ (u) -αΦ(u) + α µ ∞ 0Φ(u + y)e -y/µ dy = 0. + y)e -y/µ dy = -Φ(u) + 1 (u + y)e -y/µ dy.

2 7

 2 Appendix: ergodic theorem for an autoregression with random coefficients Let (an , bn) n≥1 be an i.i.d. sequence of random variable in R 2 and let x 0 be an arbitrary constant. Define the sequence (xn) recursively by the formula xn = an x n-1 + bn, n ≥ 1.

Acknowledgements The research is funded by the grant of the Government of Russian Federation n • 14.A.12.31.0007. The second author is partially supported by the department of Mathematics and Mechanics of National Research Tomsk State University .

We analyze separately the smoothness of Φ 1 and Φ 2 using for this function appropriate integral representations.

Smoothness of Φ 2

We start with a simpler case of Φ 2 and show that this function is infinitely differentiable without any assumptions on the distributions of ξ 1 .

From the representation

we obtain, using the independence of Y u 1 and the process (ηs -η 1 ) s≥1 , that

where

e ηt-ηs ds .

Substituting the expression for Y u 1 given by [START_REF] Nyrhinen | On the ruin probabilities in a general economic environment[END_REF] we have:

where H is a function taking values in [0, 1] and given by the formula

Taking into account that the conditional distribution of the process (Ws) s≤1 given W 1 = x is the same as of the Brownian bridge

where

Lemma 2 below asserts that for every x the random variable ζ x has a density ρ(x, .) on ]0, ∞[ and we easily obtain from ( 23) by changing variable that

where

Changing the variable we obtain that

where

.

Summarizing, we get that

where

Proposition 6 Under the conditions of Theorem 3 the function H(t, u) defined in [START_REF] Yor | On some exponential functionals of Brownian motion[END_REF] for any fixed u 0 > 0 satisfies the following inequality

Proof. In virtue of the hypothesis the function h(y) = EΦ(y + ξ 1 ) is differentiable. Differentiating [START_REF] Yor | On some exponential functionals of Brownian motion[END_REF] we get that

where h(0) = EΦ(ξ 1 ) and hu(u, t, z(t, x)) = h e κt+σz(t,x) (u -x) e κt+σz(t,x) .

Note that

∂ ∂x h(u, t, z(t, x)) = hu(u, t, z(t, x)) σzx(t, x)(u -x) -1 .

Therefore,

Integrating by parts and taking into account that z(t, 0+) = ∞ we get that

where

the function itself and, therefore, is reduced to a second order differential equation for the function G = Φ which can be easily transformed to the form

where

The substitution

eliminates the first derivative and leads to the equation

where

with certain constants A i which are of no importance in our asymptotic analysis. It is easy to check that

where x 0 = sup{x ≥ 1 : q(x) ≤ 0} + 1.

According to [START_REF] Fedoryuk | Asymptotic Analysis: Linear Ordinary Differential Equations[END_REF], pp. 54-55, the equation (39) has two fundamental solutions Z + and

we obtain that (38) admits, as solutions, functions with the following asymptotics:

The differential equation of the third order for Φ has three solutions: Φ 0 (u) = 1,

The ruin probability Ψ := 1 -Φ is the linear combination of these functions, i.e.