Adeline Samson
email: adeline.samson@parisdescartes.fr

Michèle Thieullen
email: michele.thieullen@upmc.fr

Contrast estimator for completely or partially observed hypoelliptic diffusion

Keywords: Hypoelliptic diffusion, Langevin system, Stochastic differential equations, Partial observations, Contrast estimator

Parametric estimation of two-dimensional hypoelliptic diffusions is considered when complete observations -both coordinates discretely observed-or partial observations -only one coordinate observed are available. Since the volatility matrix is degenerate, Euler contrast estimators cannot be used directly. For complete observations, we introduce an Euler contrast based on the second coordinate only. For partial observations, we define a contrast based on an integrated diffusion resulting from a transformation of the original one. A theoretical study proves that the estimators are consistent and asymptotically Gaussian. A numerical application to Langevin systems illustrates the nice properties of both complete and partial observations estimators.

Introduction

In this paper we consider parametric estimation for hypoelliptic diffusions. We focus on two dimensional diffusions, which are generalisations of systems called Langevin or hypoelliptic by different communities. They appear in many domains such as random mechanics, finance modeling, biology. Their common form is as follows:

dY t = g(Y t , Z t )dt dZ t = β(Y t , Z t )dt + α(Y t , Z t )dB t (1)
where g, β and α are real functions depending on unknown parameters θ. In these systems, noise acts directly on the "speed" Z t and on the "position" Y t only through Z t . We refer to [START_REF] Pokern | Parameter estimation for partially observed hypoelliptic diffusions[END_REF] for examples of such systems arising in applications.

In some applications, it is not possible to measure the two coordinates. Therefore, we consider two observations cases. The complete observations case assumes that both (Y t ) and (Z t ) are discretely observed. The partial observations case assumes that only the first coordinate (Y t ) is observed.

Statistical inference for discretely observed diffusion processes is complex and has been widely investigated [see e.g. [START_REF] Rao | Statistical inference from sampled data for stochastic processes[END_REF][START_REF] Yoshida | Estimation for diffusion processes from discrete observation[END_REF]. It is not possible in general to express the density of stochastic differential equation (SDE) explicitely. So different types of contrast estimators have been introduced for elliptic SDEs estimation, such as the multidimensional Euler contrast [START_REF] Genon-Catalot | On the estimation of the diffusion coefficient for multi-dimensional diffusion processes[END_REF][START_REF] Kessler | Estimation of an ergodic diffusion from discrete observations[END_REF]. However for the hypoelliptic system (1), Euler contrast methods are not directly applicable as the volatility matrix is non invertible. References on hypoelliptic estimations are few, even in the case of complete observations. The main paper is [START_REF] Pokern | Parameter estimation for partially observed hypoelliptic diffusions[END_REF]. They propose an empirical approximation of the likelihood based on Itô-Taylor expansion so that the variance matrix becomes invertible. They construct a Bayesian estimator of θ based on a Gibbs sampler. They consider both complete and partial observation cases. Their method is limited to g(Y t , Z t ) = Z t , a drift function β(Y t , Z t ) which is linear with respect to the parameter and a constant volatility function α(Y t , Z t ). In this paper, we consider more general models. We assume that g belongs to a family of functions such that it is possible to reduce to the case of integrated diffusions with a non-autonomous diffusion for (Z t ). Then, we propose to reduce to an Euler contrast based only on the second equation. This allows to consider general drift and volatility functions. We prove the consistency and the asymptotic normality of this contrast estimator when the number of observations n → ∞ and the time step between two observations ∆ n → 0.

The case of partial observations introduces more difficulties because (Y t ) is not Markovian while (Y t , Z t ) is Markovian. A maximum-likelihood estimation from discrete and partial observations of a two-dimensional linear system with a non-degenerate volatility function has been proposed [START_REF] Favetto | Parameter estimation for a bidimensional partially observed Ornstein-Uhlenbeck process with biological application[END_REF]. However, their approach can not be extended to a degenerate volatility function. Main references for partial observations of hypoelliptic diffusions are when the function g(Y t , Z t ) is equal to Z t . In this case, model [START_REF] Bally | The law of the Euler scheme for stochastic differential equations I. Convergence rate of the distribution function[END_REF] can be viewed as an integrated diffusion process. Parametric estimation methods have been proposed in this context under the additional condition that Z t satisfies an autonomous equation, meaning that the only coupling between Y t and Z t is through the identity Y t = t 0 Z s ds. Prediction-based estimating functions have been studied [START_REF] Ditlevsen | Inference for observations of integrated diffusion processes[END_REF]. [START_REF] Gloter | Parameter estimation for a discretely observed integrated diffusion process[END_REF] proposes an Euler contrast function and studies the properties of this estimator when the sampling interval ∆ n tends to zero [START_REF] Gloter | Parameter estimation for a discretely observed integrated diffusion process[END_REF]. However, their approaches are not adapted when Z t does not satisfy an autonomous equation and when g(Y t , Z t ) = Z t . In this paper, we extend the approach of [START_REF] Gloter | Parameter estimation for a discretely observed integrated diffusion process[END_REF] to this case. We prove the consistency and the asymptotic normality of this contrast estimator when ∆ n → 0 when n → ∞.

In order to establish asymptotic properties of our estimators we need existence and uniqueness of an invariant measure for system [START_REF] Bally | The law of the Euler scheme for stochastic differential equations I. Convergence rate of the distribution function[END_REF]. This is a major difference with respect to Gloter's work since in his framework the second component Z t satisfies an autonomous equation. Hence the invariant measure he introduces is that of a one dimensional diffusion. In our case, we need an invariant measure for the vector (Y t , Z t ). Ergodicity of Langevin systems has been widely studied, relying on the hypoellipticity of the system as well as a Lyapounov condition involving a Lyapounov function [START_REF] Lemaire | Estimation récursive de la mesure invariante d'un processus de diffusion[END_REF][START_REF] Mattingly | Ergodicity for sdes and approximations: locally lipschitz vector fields and degenerate noise[END_REF]. We detail these conditions and propose examples where our assumptions are verified. A numerical study is performed on these examples, to which we compare results obtained by [START_REF] Pokern | Parameter estimation for partially observed hypoelliptic diffusions[END_REF].

The paper is organized as follows. Section 2 presents the hypoelliptic system, general assumptions and more details for Langevin systems. Section 3 defines the two observations cases and the contrast estimators. The main results are presented, which consist in consistency and asymptotic normality of both estimators. Asymptotic properties of functionals of the processes are given in Section 4. Proofs of the estimator asymptotic properties are given in Section 5. Estimation methods are illustrated in Section 6 on simulated data. Section 7 presents some conclusions and discussions. Supplementary proofs are given in Appendix.

Hypoelliptic system and assumptions

2.1. The Model Let us consider system [START_REF] Bally | The law of the Euler scheme for stochastic differential equations I. Convergence rate of the distribution function[END_REF] and assume that the following condition holds

(C1) ∀(y, z) ∈ R × R, ∂ z g(y, z) = 0
Under assumption (C1), system (1) is hypoelliptic in the sense of stochastic calculus of variations [START_REF] Nualart | The Malliavin calculus and related topics, Probability and its Applications[END_REF]. Indeed, the Stratonovich form of (1) is

dY t = g(Y t , Z t )dt dZ t = β(Y t , Z t )dt + α(Y t , Z t ) • dB t (2)
with β(y, z) := β(y, z) -1 2 α(y, z)∂ z α(y, z). Writing the coefficients of (2) as vector fields

A 0 (y, z) = g(y, z) β(y, z) and A 1 (y, z) = 0 α(y, z)
and computing their Lie bracket leads to

[A 0 , A 1 ] = ∂ z g(y, z) γ(y, z)
The form of γ is explicit but not detailed here. Under condition (C1) the vectors A 1 and [A 0 , A 1 ] generate R 2 and system (1) is hypoelliptic. We will discuss the consequence of this property in Section 2.2.

Condition (C1) plays also a crucial role to reduce model (1) to an integrated diffusion. Indeed, by the change of variable X t := g(Y t , Z t ), the first equation of system (1) becomes dY t = X t dt which suggests that the process (Y t , X t ) should be an integrated diffusion. Condition (C1) enables us to apply the implicit function theorem which states that Z t can be uniquely defined as a function of (Y t , X t ). Consequently the vector (Y t , X t ) satisfies

dY t = X t dt dX t = b(Y t , X t )dt + a(Y t , X t )dB t (3)
where b and a result from the combination of the implicit function theorem and Itô formula. However the result of the implicit function theorem is only local and no explicit expression is available in general for Z t as a function of (Y t , X t ). Therefore in this paper, we assume that system (1) verifies the following condition

(C2)
The process (Y t , X t ) with X t := g(Y t , Z t ) satisfies a system of the form (3) with explicit functions b and a.

This includes in particular functions g for which an explicit function f is available such that Z t = f (Y t , X t ). Examples are g(y, z) = θ 1 y + θ 2 z or g(y, z) = φ θ (y) + θ 2 z for a function φ θ which depends on parameter θ. This condition is also satisfied for more general systems. The following system

dY t = -(θ 1 Y t -θ 2 Z 2 t )dt dZ t = -(θ 3 Z t + Z t F θ (Y t ))dt + α(Y t , Z t )dB t with F ∈ C ∞ (R, [0, +∞[)
a possibly non-linear function depending on parameter θ and α(Y t , Z t ) = σZ t is an example where the change of variables X t := g(Y t , Z t ) yields to explicit functions b and a, even if there exists no explicit function f such that

Z t = f (Y t , X t ). Variants with volatility func- tions α(Y t , Z t ) = σZ t /(1 + Z 2 t ) or α(Y t , Z t ) = σZ t F (Y t )
are other examples of systems that we consider in this paper.

In this paper, we focus on systems [START_REF] Bally | The law of the Euler scheme for stochastic differential equations I. Convergence rate of the distribution function[END_REF] for which conditions (C1) and (C2) hold. The first step of the estimation method consists in transforming system (1) into system (3). We denote µ and σ the parameters of functions b and a, respectively. These parameters include parameters of functions g, β and α of system (1). Our parameter is the vector (µ, σ 2 ) = θ. In the sequel, we denote b µ (Y t , X t ) and a σ (Y t , X t ) the drift and volatility functions.

Assumptions

We assume that the vector θ belongs to Θ = Θ 1 × Θ 2 for Θ 1 ⊂ R d 1 and Θ 2 ⊂ R d 2 two compact subsets. We now come to the assumptions regarding the drift and volatility functions. In this paper we work under conditions (C1)-(C2). In the present section we list our additional assumptions (A1) to (A4). Then we provide a set of sufficient conditions (S1) to (S3) ensuring that these assumptions are satisfied. We also examine the particular case of Langevin systems.

(A1) (a) there exists a constant c such that sup

σ∈Θ 2 |a -1 σ (y, x)| ≤ c(1+|y|+ |x|) (b) for all θ ∈ Θ, b µ and a σ belong to the class F of functions f ∈ C 2 (R 2 )
for which there exists a constant c such that the function, its first and second partial derivatives with respect to y and x are bounded by c(1 + |y| + |x|), for all x, y ∈ R, uniformly in θ.

(A2) (a) ∀k ∈]0, ∞[ sup t≥0 E(|X t | k + |Y t | k ) < ∞ (b) there exists a constant c such that ∀t ≥ 0, ∀δ ≥ 0, E( sup s∈[t,t+δ[ |X s | k |G t ) + E( sup s∈[t,t+δ[ |Y s | k |G t ) ≤ c(1 + |X t | k + |Y t | k )
where G t = σ(B s , s ≤ t).

(A3) (Y t , X t ) admits a unique invariant probability measure ν 0 with finite moments of any order i.e. ∀k > 0,

ν 0 (| • | k ) < ∞ (A4) (Y t , X t )
satisfies a weak version of the ergodic theorem namely

1 T T 0 f (Y s , X s )ds ---→ T →∞ ν 0 (f ) a.s.
for any continuous function f with polynomial growth at infinity.

Remark 1.

1. Actually we need assumption (A2) only for k ≤ 4 to prove the properties of our estimators.

2. We need (A4) for all f ∈ {a j , a j log a 2 , b k /a j , (∂b) k /a j , (∂ 2 b) k /a j , (∂a 2 ) k /a j , (∂ 2 a 2 ) k /a j , j ∈ {0, 1, 2, 4, 6}, k ∈ {0, 1, 2}}. These have indeed polynomial growth at infinity thanks to (A1).

We now provide a set of sufficient conditions, sometimes called stability conditions, for (A2) to (A4) to hold. They are based on the existence of a function V , called Lyapounov function. Lyapounov functions are efficient tools in the asymptotic study of systems; their use is classical for the Langevin systems that we consider in Section 6.

(S1) For all θ ∈ Θ,

(a) V (y, x) ≥ 1, lim ||(y,x)||→+∞ V (y, x) = +∞ (b) there exist c 1 > 0 and c 2 > 0 such that L θ V (y, x) ≤ -c 1 V (y, x) + c 2 , ∀(y, x) ∈ R 2
where L θ denotes the infinitesimal generator corresponding to (3).

(S2) (Y t , X t ) admits a unique invariant probability measure ν 0 .

(S3) For all θ ∈ Θ, ∃C > 0 such that (a σ (y,

x)∂ x V (y, x)) 2 ≤ CV (y, x) for all (y, x) ∈ R 2 .
Assumption (S1) implies existence and uniqueness of a solution to system (3) as well as existence of an invariant probability measure. Moreover, under (S1) the process S t := e c 1 t (V (Y t , X t ) -c 2 c 1 ) is a local submartingale, hence [cf. 17] for all k ≥ 1 and all t ≥ 0,

E sup s∈[t,t+δ[ |S s | k |G t ≤ k k -1 k E |S t+δ | k |G t ≤ k k -1 k |S t | k . ( 4
)
Uniqueness of the invariant probability measure is not guaranteed by (S1) and is the purpose of assumption (S2). We show now that, if there exists a polynomial Lyapounov function (or a Lyapounov function with polynomial growth at infinity), then (S1)-(S3) imply (A2)-(A4). The examples of Section 6 admit quadratic Lyapounov functions. So, let us assume here that a polynomial V in y and x satisfies (S1). An analogous argument can be used when V is dominated at infinity by a polynomial. From (4) and the polynomial character of V we deduce (A2) and (A3). From (S3), we know that for any λ < 2c 1 C , the function Ṽ (y, x) := exp(λV (y, x)) satisfies (S1) or in other words is a Lyapounov function, and also that ∀f ∈ C, 1

T T 0 f (Y s , X s )ds ---→ T →∞ ν 0 (f )
a.s. where C denotes the class of measurable functions f such that |f | is negligible w.r.t. exp( λ 2 V (y, x)). The class C contains all polynomials. As already mentioned, (S3) is satisfied when a σ is constant and V quadratic at infinity. The following assumption can also be used (S3') For all θ ∈ Θ, ∃C > 0 and ζ ∈ [0, 1[ such that (a σ (y, x)∂ x V (y, x)) 2 ≤ CV (y, x) 2-ζ for all (y, x) ∈ R 2 . In this case it is still possible to generate Lyapounov functions from V which are polynomials in V of bounded degree and a weak version of (A4) holds on a class C which contains all polynomials of degree smaller than some value. The reader can find more details about (S1)-(S3) and their consequences for the long time behaviour of (3) in [START_REF] Lemaire | Estimation récursive de la mesure invariante d'un processus de diffusion[END_REF].

We test our estimator numerically in Section 6 on particular Langevin systems. Such systems are defined by

dY t = X t dt dX t = [-γX t -F D (Y t )]dt + σdW t (5) with σ > 0, F ∈ C ∞ (R, [0, +∞[
) is a possibly non-linear function depending on parameter D and F denotes the derivative of F w.r.t. y.

For these systems the invariant probability ν 0 is unique and admits the density ρ(y, x) = C exp -γ σ 2 (x 2 -2F D (y)) where C is a multiplicative constant. Hence (A3) is fulfilled. Stability conditions for these systems are presented in [START_REF] Mattingly | Ergodicity for sdes and approximations: locally lipschitz vector fields and degenerate noise[END_REF]. If F D (y) ≥ 0, ∀y ∈ R and satisfies

βF D (y) - 1 2 F D (y)y + γ 2 8 β(2 -β) (1 -β) y 2 ≤ α (6)
for some β ∈]0, 1[ and α > 0, a Lyapounov function (which satisfies (S1)) is provided by

V (y, x) = 1 2 x 2 + F D (y) + γ 2 < y, x > + γ 2 4 y 2 + 1. ( 7
)
and condition (S3) is fulfilled. Note that the hypoelliptic property of these Langevin systems is exploited in [START_REF] Mattingly | Ergodicity for sdes and approximations: locally lipschitz vector fields and degenerate noise[END_REF] in order to establish their geometric ergodicity. In our numerical Section 6 we study respectively γ = 0, F D ≡ 0 which corresponds to our Model I , γ > 0, F D (y) ≡ D 2 y 2 in Model II and γ > 0, F D (y) ≡ -Σ n j=1 j -1 D j (cos y) j in Model III. In these three examples (A1) is satisfied as well as (A3) (cf. [START_REF] Mattingly | Ergodicity for sdes and approximations: locally lipschitz vector fields and degenerate noise[END_REF]). Moreover a σ is constant and V quadratic so (S3) holds which, as noticed previously, implies (A2) and (A4). Moreover, Models II and III satisfy (6).

Estimators and their properties

In this section, we first present the two observations frameworks. Then, for both frameworks, we introduce a discretized scheme of the system. The properties of these schemes are studied. They yield to the definition of the two contrast functions. Finally, we present the main results of the two contrast estimators, namely their consistency and the asymptotic normality.

Observations

We assume that (Y t , X t ) is the unique solution of the system

dY t = X t dt dX t = b µ 0 (Y t , X t )dt + a σ 0 (Y t , X t )dB t (8)
where θ 0 = (µ 0 , σ 0 ) is the true value of the parameter and functions b µ , a σ are such that assumptions (A1)-(A4) are fulfilled. We assume that θ 0 ∈ Θ.

From now on, we set b(y, x) = b µ 0 (y, x) and a(y, x) = a σ 0 (y, x). Now, we describe the two observations frameworks. The first case assumes that both components (Y t ) and (X t ) are observed at discrete times 0 = t 0 < t 1 < . . . < t n . The second case assumes that the process (X t ) t≥0 is hidden or not observed and that we only observe at discrete times t i the process (Y t ) t≥0 . In both cases, we assume that discrete times are equally spaced and denote ∆ n = t i -t i-1 the step size, so t i = i∆ n . We denote (Y i∆n , X i∆n ) the observation of the bidimensional process (Y t , X t ) t≥0 at time t i for the first case, and (Y i∆n ) the observation of the process (Y t ) t≥0 for the second case.

Our purpose is to estimate θ from the complete and partial observations. As for notation, in the sequel we use an upper index C (resp. P) for the case of complete (resp. partial) observations. The asymptotic behavior of the two estimators is studied for a step size ∆ n such that ∆ n → 0, as n → ∞, n∆ n → ∞ and n∆ 2 n → 0.

Contrast estimator for complete observations

When (Y t ) and (X t ) are both observed at discrete times (i∆ n ), we can consider the classical two-dimensional Euler-Maruyama approximation ( Y (i+1)∆n , X (i+1)∆n ) of (Y (i+1)∆n , X (i+1)∆n ) which is

Y (i+1)∆n X (i+1)∆n = Y i∆n X i∆n + ∆ n X i∆n b( Y i∆n , X i∆n ) + ∆ n Σ η 1 i η 2 i , (9)
Σ = 0 0 0 a( Y i∆n , X i∆n ) with (η 1 i , η 2
i ) independent identically distributed centered Gaussian vector. The two-dimensional Euler contrast can not be used directly to estimate parameters θ because Σ is not invertible. To circle this problem, [START_REF] Pokern | Parameter estimation for partially observed hypoelliptic diffusions[END_REF] considers an Itô-Taylor expansion of higher order, by adding the first non-zero noise term arising in the first coordinate. This yields to an invertible covariance matrix for some hypoelliptic models, which may be complex to calculate. On the contrary, our estimation approach remains based on the Euler scheme. As said previously, it can not be used directly. However, as we focus on parameter estimation of drift and volatility functions of the second coordinate which is observed in this subsection, we propose to consider a contrast based on the Euler-Maruyama approximation of this second equation. Dependence between successive terms (X i∆n ) are described in the following Proposition:

Proposition 1. Set G n i = G i∆n .
We have

X (i+1)∆n -X i∆n -∆ n b(Y i∆n , X i∆n ) = a(Y i∆n , X i∆n )η i,n + ε C i,n where η i,n is such that E(η 2k+1 i,n |G n i ) = 0 and E(η 2k i,n |G n i ) = (2k)!/(2 k k!)∆ k n for k ≥ 0; ε C i,n is such that E(|ε C i,n ||G n i ) ≤ c∆ 3/2 n (1 + |Y i∆n | + |X i∆n |) and E(|ε C i,n | k |G n i ) ≤ c∆ k/2+1 n (1 + |Y i∆n | k + |X i∆n | k ) for k ≥ 2.
This leads to the definition of the following estimation contrast

L C n (θ) = n-1 i=0 (X (i+1)∆n -X i∆n -∆ n b µ (Y i∆n , X i∆n )) 2 ∆ n a 2 σ (Y i∆n , X i∆n ) + log(a 2 σ (Y i∆n , X i∆n )) (10)
which is an extension of the classical Euler contrast for unidimensional SDE (see [START_REF] Kessler | Estimation of an ergodic diffusion from discrete observations[END_REF]) when drift b µ and volatility a σ depend on both Y and X. We define the minimum contrast estimator θC n for complete observations as θC n = arg min θ∈Θ L C n (θ)

Contrast estimator for partial observations

Contrast [START_REF] Kessler | Estimation of an ergodic diffusion from discrete observations[END_REF] can not be used in the second case of observations, as (X i∆n ) is not observed. In the context of integrated diffusion, [START_REF] Gloter | Parameter estimation for a discretely observed integrated diffusion process[END_REF] proposes to approximate X i∆n by increments of (Y t ). We study the behavior of the process of increments in Proposition 2. The basic idea which consists in replacing directly X i∆n by Y i,n in contrast [START_REF] Kessler | Estimation of an ergodic diffusion from discrete observations[END_REF] leads to a biased estimator [see 8, for the case where (X t ) satisfies an autonomous diffusion]. This is due to the dependence between two successive terms of the rate process Y i,n (Proposition 3). The estimation contrast for partial observation must be corrected to take into account this correlation. Now, we present more precisely these ideas. First, we introduce the increment or rate process

Y i,n = Y (i+1)∆n -Y i∆n ∆ n (11)
Model (3) implies

Y i,n = 1 ∆ n (i+1)∆n i∆n X s ds
Thus, when ∆ n is small, Y i,n is close to X i∆n . More precisely, we have:

Proposition 2. Assume (A1)-(A2). We have Y i,n -X i∆n = ∆ 1/2 n a(Y i∆n , X i∆n )ξ i,n + e i,n
where there exists a constant c such that

|E(e i,n |G n i ) ≤ c∆ n (1+|X i∆n |+|Y i∆n |) and |E(e 2 i,n |G n i ) ≤ c∆ 2 n (1 + |X i∆n | 4 + |Y i∆n | 4 ). Furthermore, if k is a real number ≥ 1, then for all i, n, we have E Y i,n -X i∆n k |G n i ≤ c∆ k/2 n (1 + |X i∆n | k + |Y i∆n | k )
The link between two successive terms of the non-Markovian rate process Y i,n is studied in the following Proposition.

Proposition 3. Assume (A1)-(A2). Then Y i+1,n -Y i,n -∆ n b(Y i∆n , Y i,n ) = ∆ 1/2 n a(Y i∆n , X i∆n )U i,n + ε P i,n
where U i,n = ξ i,n + ξ i+1,n with

ξ i,n = 1 ∆ 3/2 n (i+1)∆n i∆n (s -i∆ n )dB s for i, n ≥ 0 ξ i+1,n = 1 ∆ 3/2 n (i+2)∆n (i+1)∆n ((i + 1)∆ n -s)dB s for i ≥ -1, n ≥ 0. If k is a real number ≥ 1, then for all i, n E Y i+1,n -Y i,n k |G n i ≤ c∆ k/2 n (1 + |X (i+1)∆n | k + |Y (i+1)∆n | k )
Moreover there exist constants c such that

E(ε P i,n |G n i ) ≤ c∆ 2 n (1 + |X (i+1)∆n | 3 + |Y (i+1)∆n | 3 ) E((ε P i,n ) 2 |G n i ) ≤ c∆ 2 n (1 + |X (i+1)∆n | 4 + |Y (i+1)∆n | 4 ) E((ε P i,n ) 4 |G n i ) ≤ c∆ 4 n (1 + |X (i+1)∆n | 8 + |Y (i+1)∆n | 8 ) E(ε P i,n U i,n |G n i ) ≤ c∆ 3/2 n (1 + |X (i+1)∆n | 2 + |Y (i+1)∆n | 2 ).
Remark that Proposition 3 implies that for any function f of the two variables

Y t and X t , f (Y i∆n , X i∆n ) and Y i+1,n -Y i,n -∆ n b(Y i∆n , Y i,n ) have a correlation of order ∆ 1/2
n . Moreover the variance of U i,n is 2/3∆ n , while the variance of η i,n in Proposition 1 is 1. [START_REF] Gloter | Parameter estimation for a discretely observed integrated diffusion process[END_REF] proposes a correction of the contrast by weighting the first sum in (10) by a factor 3/2. We extend this contrast to the case of drift and volatility functions depending on both processes (X t ) and (Y t ). Thus we consider the following contrast

L P n (θ) = n-2 i=1 3 2 Y i+1,n -Y i,n -∆ n b µ (Y (i-1)∆n , Y i-1,n ) 2 ∆ n a 2 σ (Y (i-1)∆n , Y i-1,n ) + log(a 2 σ (Y (i-1)∆n , Y i-1,n )) (12)
Remark that as (Y i,n ) is not markovian, we introduce a shift in the index of the drift and the diffusion functions to avoid a correlation term of order ∆

1/2 n between (Y i+1,n -Y i,n ) and functionals f (Y i∆n , Y i,n ).
We define the minimum contrast estimator for partial observations θP n as θP n = arg min θ∈Θ L P n (θ)

Main results

To simplify notations and proofs, we restrict to one-dimensional parameters µ and σ. This could easily be extended to multidimensional parameters [see remark 5 of 8]. Simulations (Section 6) illustrate this extension.

In this paper, we prove the consistency and asymptotic normality of both estimators under the following identifiability assumption

a σ (y, x) = a σ 0 (y, x) dν 0 (y, x) almost everywhere implies σ = σ 0 b µ (y, x) = b µ 0 (y, x) dν 0 (y, x) almost everywhere implies µ = µ 0
Classically, the consistency of the estimator θn requires ∆ n → 0.

Theorem 1. Under assumptions (A1)-(A4), the estimators θC n and θP n are consistent:

θC n P ---→ n→∞ θ 0 , and
θP n P ---→ n→∞ θ 0 .
The asymptotic distribution requires the additional condition n∆ 2 n → 0. The rate of convergence is different for μn and σ 2 n . The drift term is estimated with rate (n∆ n ) 1/2 and the diffusion term is estimated with rate n 1/2 .

Theorem 2. Set assumptions (A1)-(A4), n∆ 2 n ---→ n→∞ 0. In the complete observations case, √ n∆ n μC n -µ 0 , √ n σ 2 n C -σ 2 0 converges in distri- bution to N 0, ν 0 (∂ µ b) 2 (•, •) a 2 (•, •) -1 ⊗ N 0, 2 ν 0 (∂ σ 2 a 2 ) 2 (•, •) a 4 (•, •) -1
and in the partial observations case,

√ n∆ n μP n -µ 0 , √ n σ 2 n P -σ 2 0 converges in distribution to N 0, ν 0 (∂ µ b) 2 (•, •) a 2 (•, •) -1 ⊗ N 0, 9 4 ν 0 (∂ σ 2 a 2 ) 2 (•, •) a 4 (•, •) -1
Theorem 2 is an extension of several results. We first comment the complete observations case. When (X t ) is an autonomous diffusion, [START_REF] Kessler | Estimation of an ergodic diffusion from discrete observations[END_REF] proves that the asymptotic distribution is

N 0, ν X,0 (∂ µ b) 2 (•) a 2 (•) -1 ⊗ N 0, 2 ν X,0 (∂ σ 2 a 2 ) 2 (•) a 4 (•) -1
where the limit distribution ν X,0 is the stationary distribution of the diffusion (X t ) itself. In that case, observations of (Y t ) are not used. When (X t ) is not autonomous and the diffusion is bi-dimensional, this result can be generalized if the volatility matrix Σ is non degenerate. The asymptotic variance is then based on ν 0 , the stationary distribution of the vector (Y t , X t ). When the volatility matrix is degenerate as in model (3), the first assertion of Theorem 2 shows that reducing the contrast to the Euler approximation of the second coordinate yields to asymptotic normality of θn , the asymptotic variance involving the stationary distribution of the vector (Y t , X t ). This is a major difference with respect to the case of an autonomous diffusion for (X t ).

For the partial observations case, when (X t ) is autonomous, [START_REF] Gloter | Parameter estimation for a discretely observed integrated diffusion process[END_REF] proves that replacing X i∆n by Y i,n underestimates the asymptotic variance, as a consequence of Proposition 3. As in the complete observation case, when (X t ) is autonomous, the asymptotic variance is based on the stationary distribution ν X,0 . In model ( 3) where the diffusion is not autonomous, second assertion of Theorem 2 shows that the invariant measure of (Y t , X t ) is required in the asymptotic variance.

The estimation of µ is asymptotically efficient since ν 0

(∂µb) 2 (•,•) a 2 (•,•)
is the Fisher information of the continuous time model. This is not the case for σ 2 as its asymptotic variance is increased with a factor 9/16 instead of 1/2 for directly observed diffusion [START_REF] Kessler | Estimation of an ergodic diffusion from discrete observations[END_REF]. Proofs of Theorems 1 and 2 are given in Section 5. They are based on properties of functionals of (Y i∆n , X i∆n ) and (Y i∆n , Y i,n ), which are studied in Section 4.

4. Functionals of (Y i∆n , X i∆n ) and (Y i∆n , Y i,n )

Contrast properties rely on convergence results for functionals appearing in the contrast functions. These functionals are of different types: functional mean, variation and quadratic variation of X i∆n and Y i,n . We consider for the complete observations case, for a measurable function f , the three functionals:

ν C n (f ) = 1 n n-1 i=0 f (Y i∆n , X i∆n , θ), I C n (f ) = 1 n∆ n n-1 i=0 f (Y i∆n , X i∆n , θ)(X (i+1)∆n -X i∆n -∆ n b(Y i∆n , X i∆n )) Q C n (f ) = 1 n∆ n n-1 i=0 f (Y i∆n , X i∆n , θ) X (i+1)∆n -X i∆n 2
and for the partial observations case, the three functionals

ν P n (f ) = 1 n n-1 i=0 f (Y i∆n , Y i,n , θ) I P n (f ) = 1 n∆ n n-2 i=1 f (Y (i-1)∆n , Y i-1,n , θ)(Y i+1,n -Y i -∆ n b(Y (i-1)∆n , Y i-1,n )) Q P n (f ) = 1 n∆ n n-2 i=1 f (Y (i-1)∆n , Y i-1,n , θ) Y i+1,n -Y i 2
Note that in I P n (f ) and Q P n (f ), we introduce shifted processes Y (i-1)∆n and Y i-1,n in the function f as a consequence of the remark following Proposition 3. Consequently, the drift term b(Y (i-1)∆n , Y i-1,n ) in I P n (f ) and in the contrast L P n are also shifted so that, when the square quantity in L P n is developed, functionals to be studied have the proper index. Asymptotic study of these functionals is difficult because it involves (Y i∆n , Y i,n ) instead of the original Markovian process (Y i∆n , X i∆n ). We first study the uniform convergence of these functionals, then their convergence in distribution. In the following, we assume that f belongs to the class F introduced in Assumption (A1).

Uniform convergence

The first result concerns the empirical mean of the discretized process (X i∆n ) i≥0 and the rate process (Y i,n ) i≥1 . The limits involve the stationary distribution ν 0 of the vector (Y t , X t ). Proofs are given in the Appendix. They are essentially based on Propositions 2 and 3 and generalize the proofs of [START_REF] Gloter | Parameter estimation for a discretely observed integrated diffusion process[END_REF] to a non autonomous diffusion (X t ).

Proposition 4. Under assumptions (A1)-(A4), we have uniformly in

θ ν C n (f ) P ---→ n→∞ ν 0 (f ), ν P n (f ) P ---→ n→∞ ν 0 (f ).
We see that replacing X i∆n by Y i,n in the partial observations case does not change the limit. The next result concerns the functionals I C n and I P n which involve the variations of the processes (X i∆n ) i≤0 and (Y i,n ) i≤0 , respectively. Theorem 3. Under assumptions (A1)-(A4), we have uniformly in θ

I C n (f ) P ---→ n→∞ 0, I P n (f ) P ---→ n→∞ 0. ( 13
)
The limit is the same for the complete and partial functionals. This is due to the introduction of the lag in the definition of

I P n (f ): f (Y (i-1)∆n , Y i-1,n , θ) and b(Y (i-1)∆n , Y i-1,n ) instead of f (Y i∆n , Y i,n , θ) and b(Y i∆n , Y i,n
). This enables us to avoid correlation terms of order ∆ 1/2 n . When no lag is introduced, the limit is not 0, see for instance [START_REF] Gloter | Parameter estimation for a discretely observed integrated diffusion process[END_REF]. The last result deals with the quadratic variations of (X i∆n ) i≥0 and (Y i,n ) i≥1 .

Theorem 4. Under assumptions (A1)-(A4), we have uniformly in θ

Q C n (f (•, •, θ)) P ---→ n→∞ ν 0 (f (•, •, θ)a 2 (•, •)) Q P n (f (•, •, θ)) P ---→ n→∞ 2 3 ν 0 (f (•, •, θ)a 2 (•, •)).
Theorem 4 is an extension of various results. It implies several comments which have already been partially addressed in Section 3.4. We first comment the complete observations case. When (X t ) is an autonomous diffusion, [START_REF] Kessler | Estimation of an ergodic diffusion from discrete observations[END_REF] proves that for a function f : R × Θ → R

1 n∆ n n-1 i=0 f (X i∆n , θ) X (i+1)∆n -X i∆n 2 P ---→ n→∞ ν X,0 (f (•, θ)a 2 (•))
where the limit distribution ν X,0 is the stationary distribution of the diffusion (X t ) itself. When the diffusion is two-dimensional and the volatility matrix Σ is non degenerate, the limit is then ν 0 (f (•, •, θ)ΣΣ (•, •)) where ν 0 is the stationary distribution of the vector (Y t , X t ). When the volatility matrix is degenerate as in model ( 3), the first assertion of Theorem 4 shows that the problem is reduced to the Euler approximation of the second equation of the system with the limit involving the stationary distribution of the vector (Y t , X t ).

For the partial observations case, when (X t ) is autonomous, [START_REF] Gloter | Parameter estimation for a discretely observed integrated diffusion process[END_REF] proves that replacing X i∆n by Y i,n modifies the result by underestimating ν X,0 (f (•, θ)a 2 (•)).

In the case of model (3) where the diffusion is not autonomous, the second assertion of Theorem 4 shows that the invariant measure of (Y t , X t ) is required.

Convergence in distribution of functionals of the process

In this section, we study some central limit theorems for the functionals I

n∆ n I C n (f ) D ---→ n→∞ N (0, ν 0 (f 2 a 2 )
)

n∆ n I P n (f ) D ---→ n→∞ N (0, ν 0 (f 2 a 2 ))
The condition

n∆ 2 n ---→ n→∞ 0 is classical [see 4]
. This condition imposes that the discretization step decreases to zero fast enough to ensure that the contribution of the error terms tends to 0 as n → ∞. The lag introduced in the definition of I P n makes the result very similar for both complete and partial observations case. We now present a central limit theorem for

Q P n and Q C n . Theorem 4 shows that Q P n underestimates ν 0 (f (•, •, θ)a 2 (•, •)).
The correction factor 2/3 is thus required in the associated central limit theorem: Theorem 6. Under assumptions (A1)-(A4) and n∆ 2 n ---→ n→∞ 0, we have

√ n Q C n (f ) -ν C n (f a 2 ) D ---→ n→∞ N (0, ν 0 (f 2 a 4 )) √ n Q P n (f ) - 2 3 ν P n (f a 2 ) D ---→ n→∞ N (0, ν 0 (f 2 a 4 ))
In the partial observations case, when we replace X i∆n by Y i,n , the asymptotic variance increases due to the factor 3/2. This can also be compared to the results of [START_REF] Gloter | Parameter estimation for a discretely observed integrated diffusion process[END_REF] when the diffusion (X t ) is autonomous.

Proofs of main results

In this section, asymptotic properties of estimators θC n and θP n are proved.

Proof of Theorem 1

We follow the proof of [START_REF] Kessler | Estimating equations based on eigenfunctions for a discretely observed diffusion process[END_REF]. We have to show that, uniformly in θ,

1 n L C n (θ) P ---→ n→∞ ν 0 a 2 σ 0 (y, x) a 2 σ (y, x) + log a 2 σ (y, x) (14)
1 n L P n (θ) P ---→ n→∞ ν 0 a 2 σ 0 (y, x) a 2 σ (y, x) + log a 2 σ (y, x) (15)
This ensures the convergence of σ2 n to σ 2 0 for both cases. Then, if we prove that

1 n∆ n (L C n (µ, σ) -L C n (µ 0 , σ)) P ---→ n→∞ ν 0 (b µ (y, x) -b µ 0 (y, x)) 2 a 2 σ (y, x) (16)
1 n∆ n (L P n (µ, σ) -L P n (µ 0 , σ)) P ---→ n→∞ 3 2 ν 0 (b µ (y, x) -b µ 0 (y, x)) 2 a 2 σ (y, x) (17)
this ensures the convergence of μn to µ 0 for both cases. We start by proving (14-15). In the complete observations case, we have

1 n L C n (θ) = Q C n (a -2 σ (•, •)) + ν C n (log a 2 σ (•, •)) -2∆ n I C n (a -2 σ (•, •)b µ (•, •)) +∆ n ν C n (a -2 σ (•, •) b 2 µ (•, •) -2b µ (•, •)b µ 0 (•, •) )
Using Proposition 4, Theorems 3 and 4, we easily prove [START_REF] Nualart | The Malliavin calculus and related topics, Probability and its Applications[END_REF]. In the partial observations case, we have

1 n L P n (θ) = 3 2 Q P n (a -2 σ (•, •)) + ν P n (log a 2 σ (•, •)) -3∆ n I P n (a -2 σ (•, •)b µ (•, •)) + 3 2 ∆ n ν P n (a -2 σ (•, •) b 2 µ (•, •) -2b µ (•, •)b µ 0 (•, •) )
Using Proposition 4, Theorems 3 and 4, we easily prove [START_REF] Pokern | Parameter estimation for partially observed hypoelliptic diffusions[END_REF]. For the proof of ( 16), we have

1 n∆ n L C n (µ, σ) -L C n (µ 0 , σ) = 2I C n b µ 0 a 2 σ (•, •) - b µ a 2 σ (•, •) +ν C n (b µ (•, •) -b µ 0 (•, •)) 2 a 2 σ (•, •)
We conclude with Proposition 4 and Theorem 3. For the proof of (17), we write

1 n∆ n L P n (µ, σ) -L P n (µ 0 , σ) = 3I P n b µ 0 a 2 σ (•, •) - b µ a 2 σ (•, •) + 3 2 ν P n (b µ (•, •) -b µ 0 (•, •)) 2 a 2 σ (•, •)
We conclude with Proposition 4 and Theorem 3.

Proof of Theorem 2

The scheme of the proof is the same for both complete and partial observations cases. Let θn and L n (θ) denote the estimator and contrast either for complete or partial observations. A Taylor's formula around θn yields:

D n = 1 0 C n (θ 0 + u( θn -θ 0 )du E n where D n = -( √ n∆ n ) -1 ∂ µ L n (θ 0 ) -( √ n) -1 ∂ σ L n (θ 0 ) , E n = √ n∆ n (μ n -µ 0 ) √ n σ 2 n -σ 2 0 , C n (θ) = 1 n∆n ∂ 2 µ 2 L n (θ) 1 n √ ∆n ∂ 2 σµ L n (θ) 1 n √ ∆n ∂ 2 µσ L n (θ) 1 n ∂ 2 σ 2 L n (θ)
.

Let now detail the two cases. In the complete observations case, we have

1 √ n∆ n ∂ µ L C n (θ 0 ) = -2 n∆ n I C n ∂ µ b µ 0 (•, •) a 2 σ 0 (•, •) 1 √ n ∂ σ 2 L C n (θ 0 ) = - √ n Q C n ∂ σ 2 (a 2 σ 0 (•, •)) a 4 σ 0 (•, •) -ν C n ∂ σ 2 (a 2 σ 0 (•, •)) a 2 σ 0 (•, •) + o P (1)
By Theorems 5 and 6, this yields

D C n D ---→ n→∞ N   0,   4ν 0 (∂µbµ 0 ) 2 (•,•) a 2 σ 0 (•,•) 0 0 2ν 0 (∂ σ 2 a 2 σ 0 ) 2 (•,•) a 4 σ 0 (•,•)    
The proof of the convergence in distribution of E n follows from the consistency of θC n and if we prove the uniform (with respect to θ) convergence in probability of C C n (θ). To prove the uniform convergence, we differentiate twice L C n . Proposition 4 and Theorem 3 show that C C n (θ) converges uniformly in θ in probability to C C (θ) where

C C (θ) = C C 11 (θ) 0 0 C C 22 (θ) C C 11 (θ) = 2ν 0 (∂ µ b µ ) 2 (•, •) a 2 σ (•, •) + ∂ 2 µ 2 b µ a 2 σ (•, •)(b µ (•, •) -b µ 0 (•, •)) C C 22 (θ) = ν 0 (∂ σ 2 a 2 σ ) 2 (•, •) 2a 2 σ 0 (•, •) a 6 σ (•, •) - 1 a 4 σ 0 (•, •) +ν 0 ∂ 2 σ 2 a 2 σ (•, •) 1 a 2 σ (•, •) - a 2 σ 0 (•, •) a 4 σ (•, •)
Hence the result for the complete observations case. In the partial observations case, we have

1 √ n∆ n ∂ µ L P n (θ 0 ) = -3 n∆ n I P n ∂ µ b µ 0 (•, •) a 2 σ 0 (•, •) 1 √ n ∂ σ 2 L P n (θ 0 ) = - 3 2 √ n Q P n ∂ σ 2 a 2 σ 0 (•, •) a 4 σ 0 (•, •) - 2 3 ν P n ∂ σ 2 a 2 σ 0 (•, •) a 2 σ 0 (•, •) + o P (1)
By Theorems 5 and 6, this yields

D P n D ---→ n→∞ N   0,   9ν 0 (∂µbµ 0 ) 2 (•,•) a 2 σ 0 (•,•) 0 0 9 4 ν 0 (∂ σ 2 a 2 σ 0 ) 2 (•,•) a 4 σ 0 (•,•)    
Proof of the convergence in distribution of E n follows from the consistency of θP n and the uniform (with respect to θ) convergence in probability of C P n (θ). To prove the uniform convergence, we differentiate twice L P n . Proposition 4 and Theorem 3 show that C P n (θ) converges uniformly in θ in probability to C P (θ) where

C P (θ) = C P 11 (θ) 0 0 C P 22 (θ) C P 11 (θ) = 3ν 0 (∂ µ b µ ) 2 (•, •) a 2 σ (•, •) + ∂ 2 µ 2 b µ a 2 σ (•, •)(b µ (•, •) -b µ 0 (•, •)) C P 22 (θ) = ν 0 (∂ σ 2 a 2 σ ) 2 (•, •) 2a 2 σ 0 (•, •) a 6 σ (•, •) - 1 a 4 σ 0 (•, •) +ν 0 ∂ 2 σ 2 a 2 σ (•, •) 1 a 2 σ (•, •) - a 2 σ 0 (•, •) a 4 σ (•, •)
Hence the result.

Simulation study

We consider three models of simulation, which are those proposed by [START_REF] Pokern | Parameter estimation for partially observed hypoelliptic diffusions[END_REF]. Their general form is given as the Langevin system ( 5) where F D is some (possibly non-linear) force function parameterized by D. Model I corresponds to a simple linear stochastic growth with γ = 0, F D ≡ 0. Model II coresponds to a linear oscillator subject to noise and damping with γ > 0, F D (y) ≡ D 2 y 2 . Model III is a non-linear oscillator subject to noise and damping with γ > 0, F D (y) ≡ -Σ n j=1 j -1 D j (cos y) j . Stability conditions for these models have been detailed in Section 2.

Model I: stochastic growth

We consider the following simple model

dY t = X t dt dX t = σ 0 dB t (18)
The process has one unknown parameter, σ 0 , that describes the size of the fluctuations. Model (18) has a matricial form dU t = AU t dt + ΓdB t where U t = (Y t , X t ) t and

A = 0 1 0 0 , Γ = 0 0 0 σ 0
This model has an explicit solution U t = e A(t-t 0 ) U 0 + t t 0 e A(t-s) ΓdB s . Given the fact that e At = I + At for this simple model, the process (U t ) is Gaussian with expectation vector and covariance matrix

E(U t |U 0 ) = 1 t 0 1 U 0 , V ar(U t |U 0 ) = σ 2 0 Σ t = σ 2 0 t 3 /3 t 2 /2 t 2 /2 t
The covariance matrix Σ t is invertible. Note that the process (U t ) has no stationary probability distribution. It is usual to consider the Lebesgue measure, which is not a probability measure, as its invariant measure. Although the theory developed in this paper has to be extended to the existence of an invariant measure which is not a probability measure, this is beyond the scope of this paper. Nevertheless, this example illustrates that estimators have good properties in that case. An exact discrete sampling scheme can be deduced from the exact distribution of (U t )

Y (i+1)∆n X (i+1)∆n = Y i∆n + ∆ n X i∆n X i∆n + σ 0 Σ 1/2 ∆n ε (1) i∆n ε (2) i∆n , (19)
As the exact distribution is available and easily computable, the estimation of σ can be obtained from the exact likelihood when complete observations are available. The exact maximum likelihood estimator (MLE) is thus

σ M LE = 1 2n n-1 i=0 (U (i+1)∆n -e A∆n U i∆n ) (Σ ∆n ) -1 (U (i+1)∆n -e A∆n U i∆n )
The fact that the MLE is explicit is very specific to this simple model. It is also interesting to study the two contrast estimators, which are defined for more general models. The estimator for the complete observations case is equal to

σ C = 1 ∆ n n n-1 i=0 (X (i+1)∆n -X i∆n ) 2 .
When partial observations are available, the estimator is

σ P = 3 2 1 ∆ n (n -2) n-2 i=1 (Y i+1,n -Y i,n ) 2 .
The behavior of these three estimators are compared on simulated data. Three designs (∆ n , n) of simulations are considered: ∆ n = 0.1, n = 100; ∆ n = 0.1, n = 1000 and ∆ n = 0.01, n = 1000. A thousand of datasets are simulated for each design with the exact discrete scheme (19), the true parameter value σ 0 = 1 and U 0 = (1, 1) . The three estimators are computed on each dataset. Kernel estimations of the density of these estimators are represented in Figure 1. The three estimators are unbiased for the three designs. Their variances are small and decrease when n increases, whatever the value of ∆ n . The maximum likelihood estimator σ M LE has a smaller variance than the two contrast estimators σ C and σ P , whatever the values of n and ∆ n . This is expected as the MLE is based on the exact distribution of the diffusion, while σ C and σ P are based on Euler approximation. The two contrast estimators behave very similarly. Empirical means and standard deviations of the three estimators for the three designs are presented in Table 1. Means and standard deviations obtained by [START_REF] Pokern | Parameter estimation for partially observed hypoelliptic diffusions[END_REF] on the same example are also reported. With complete observations, the MLE and the contrast estimator have similar estimate means and are unbiased. The standard deviations of σ C are three times larger than for σ M LE . With partial observations, the contrast estimator σ P has similar mean than the one of [START_REF] Pokern | Parameter estimation for partially observed hypoelliptic diffusions[END_REF], but twice greater standard deviations. [START_REF] Pokern | Parameter estimation for partially observed hypoelliptic diffusions[END_REF] with partial observations.

Design ∆ n = 0.1 ∆ n = 0.1 ∆ n = 0.

Model II: harmonic oscillator

We consider an harmonic oscillator that is driven by a white noise forcing:

dY t = X t dt dX t = (-D 0 Y t -γ 0 X t )dt + σ 0 dB t (20)
with γ 0 > 0 and D 0 > 0. The process has three unknown parameters (D 0 , γ 0 , σ 0 ). Model (20) has a matricial form dU t = AU t dt + ΓdB t where

U t = (Y t , X t ) t , A = 0 1 -D 0 -γ 0 , Γ = 0 0 0 σ 0
The stationary distribution of (U t ) is Gaussian with zero mean and an explicit variance matrix [START_REF] Gardiner | Handbook of Stochastic Methods[END_REF] V ar

(U t ) = Σ U = 1 -2tr(A)det(A) (det(A)ΓΓ +(A-tr(A)I 2 )ΓΓ (A-tr(A)I 2 )
where tr(A) and det(A) are the trace and the determinant of A.

The estimator for the complete observations case is defined as

θ C = arg min θ n-1 i=0 (X (i+1)∆n -X i∆n + ∆ n (DY i∆n + γX i∆n )) 2 ∆ n σ 2 + n log σ 2 .
When only partial observations (Y i∆n ) are available, the constrast is

θ P = arg min θ 3 2 n-2 i=1 (Y i+1,n -Y i,n + ∆ n (DY (i-1)∆n + γY i-1,n )) 2 ∆ n σ 2 + (n -2) log σ 2
The behavior of these two estimators is compared on simulated data. Three designs (∆ n , n) of simulations are considered: ∆ n = 0.1, n = 1000; ∆ n = 0.1, n = 100 and ∆ n = 0.01, n = 1000. A thousand of datasets are simulated for each design with the exact stationary distribution, the true parameter values D 0 = 4, γ 0 = 0.5 and σ 0 = 1 and U 0 = (1, 0) . The two estimators θ C and θ P are computed on each dataset. Empirical mean and standard deviations of the estimators are reported on Table 2 (simultaneous estimation of the three parameters). Results obtained by [START_REF] Pokern | Parameter estimation for partially observed hypoelliptic diffusions[END_REF] when estimating only σ P are reported in Table 2. In their paper, the authors do not detail their results for drift parameters when using the Gibbs loop, which corresponds to a simultaneous estimation. So we only report results for σ. Parameter σ is estimated with very small bias whatever the design and the kind of observations. The bias of σ is slightly less than the one of [START_REF] Pokern | Parameter estimation for partially observed hypoelliptic diffusions[END_REF]. Its standard deviation decreases with n. The drift parameters D and γ are estimated with bias for the two first designs. The bias decrease when n = 1000, ∆ n = 0.01. This is corroborated by the theoretical results, as the asymptotic conditions are not the same for drift and volatility parameter estimation. The bias is very small for D but still remains for γ when n = 1000, ∆ n = 0.01. The bias for drift parameters obtained with partial observations are larger than with complete observations. For example, with n = 100, ∆ n = 0.1, the mean estimated value for parameter D is 3.588 with partial observations and 3.567 for complete observations, to be compared to the true value 4. When n increases and ∆ n decreases, this difference decreases and the bias is small.

Model III: trigonometric oscillator

We consider the dynamics of a particle moving in a trigonometric potential [see 15]. The model is

dY t = X t dt dX t = -γ 0 X t -c j=1 D 0,j sin(Y t ) cos j-1 (Y t ) dt + σ 0 dB t ( 21
)
with parameters θ = (γ 0 , D 0j , j = 1, . . . , c, σ 0 ). This system is non-linear.

No explicit closed form expression for the solution is known. The estimator for the complete observations case is defined as

θ C = arg min θ [n log σ 2 + n-1 i=0 (X(i+1)∆ n -X i∆n +∆n(γX i∆n + P c j=1 D j sin(Y i∆n ) cos j-1 (Y i∆n ))) 2 ∆nσ 2
When only partial observations (Y i∆n ) are available, the estimator is

θ P = arg min θ [(n -2) log σ 2 + 3 2 n-2 i=1 (Y i+1,n -Y i,n +∆n(γY i-1,n + P c j=1 D j sin(Y (i-1)∆n ) cos j-1 (Y (i-1)∆n ))) 2 ∆nσ 2
The behavior of these estimators is compared on simulated data. estimators are reported on Table 3. [START_REF] Pokern | Parameter estimation for partially observed hypoelliptic diffusions[END_REF]'s results are presented as figures and are not reported here. Bias and standard deviations of drift and volatility parameters decrease when n increases and ∆ n decreases. For example, for σ with complete observations, the mean estimated value is 0.886 when n = 100, ∆ n = 0.1 and 0.714 when n = 1000, ∆ n = 0.01, to be compared to the true value 0.7. For γ with complete observations, the mean estimated values is 0.638 when n = 100, ∆ n = 0.1 and 0.522 when n = 1000, ∆ n = 0.01, to be compared to the true value 0.5. Estimators obtained from partial observations have greater bias than those obtained from complete observations. For example, when n = 100, ∆ n = 0.1, for σ with complete observations, the mean estimated value for σ is 0.886 with complete observations and 1.012 with partial observations, to be compared to the true value 0.7.

Discussion

We consider two cases of observations (partial and complete) of a hypoelliptic two-dimensional diffusion, with non-autonomous equations. The contrast estimators are based on Euler approximations of the second coordinate. We prove their consistency and give their asymptotic distribution. The case of complete observations leads to efficient estimator. On the contrary, in the case of partial observations, our estimator is not efficient. This extends the results of [START_REF] Gloter | Parameter estimation for a discretely observed integrated diffusion process[END_REF] to non-autonomous diffusion. We compare our estimators to [START_REF] Pokern | Parameter estimation for partially observed hypoelliptic diffusions[END_REF]'s estimator. [START_REF] Pokern | Parameter estimation for partially observed hypoelliptic diffusions[END_REF] limit their study to linear drift and constant diffusion coefficient. Their estimator is based on a hybrid Gibbs sampler in a Bayesian framework. Their algorithm may be time consuming. Our estimator has the advantage to be simple to compute. For example, on the three examples considered in the simulation study, which are the same than those handled by [START_REF] Pokern | Parameter estimation for partially observed hypoelliptic diffusions[END_REF], our estimators are explicit and thus computed in less than one second.

Only second-order hypoelliptic systems have been considered in this paper. The estimation method proposed by [START_REF] Pokern | Parameter estimation for partially observed hypoelliptic diffusions[END_REF] works for larger order. The extension of our approach to these higher order hypoelliptic systems would require higher order approximation schemes, as Runge Kutta schemes. Although Model (1) involves a function g in the first coordinate, we reduce to the case dY t = X t for the definition of the contrast functions as explained in Section 2. Our estimation procedure could be used to estimate parameters of function g. Numerical study of such models would be explored in future works. This could have great usefulness to consider more complex models and real data.

Appendix A. Proofs

Proposition A of [START_REF] Gloter | Discrete sampling of an integrated diffusion process and parameter estimation of the diffusion coefficient[END_REF] can be extended to drift and volatility depending both on y and x:

Proposition 5. Let f ∈ C 1 . If ∃c, ∀y, x such that |f y (y, x)| + |f x (y, x)| ≤ c(1 + |y| + |x|) then, for all integer k ≥ 1, we have E sup t∈[i∆n,(i+1)∆n[ |f (Y t , X t ) -f (Y i∆n , X i∆n )| k |G n i ≤ c∆ k/2 n (1 + |Y i∆n | k + |X i∆n | k ) Proof. With start with f (y, x) = x. Let δ i,n = sup t∈[i∆n,(i+1)∆n[ |X t -X i∆n |.
Using the Burkholder inequality, we get

E(δ k i,n |G n i ) ≤ c E   (i+1)∆n i∆n |b(Y t , X t )|dt k |G n i   + c E   (i+1)∆n i∆n |a 2 (Y t , X t )|dt k/2 |G n i  
Using Assumption (A2), we get

E(δ k i,n |G n i ) ≤ c ∆ k n E sup t∈[i∆n,(i+1)∆n[ |b k (Y t , X t )||G n i + c ∆ k/2 n E sup t∈[i∆n,(i+1)∆n[ |a k (Y t , X t )||G n i + c ∆ k/2 n (1 + |Y i∆n | k + |X i∆n | k ) Now for a general f , we study f (Y t , X t ) -f (Y i∆n , X i∆n ) = f (Y t , X t ) - f (Y i∆n , X t ) + f (Y i∆n , X t ) -f (Y i∆n , X i∆n ).
We have

E sup t∈[i∆n,(i+1)∆n[ |f (Y t , X t ) -f (Y i∆n , X i∆n )| k |G n i ≤ 2 k-1 E sup t∈[i∆n,(i+1)∆n[ |f (Y t , X t ) -f (Y i∆n , X t )| k |G n i +2 k-1 E sup t∈[i∆n,(i+1)∆n[ |f (Y i∆n , X t ) -f (Y i∆n , X i∆n )| k |G n i We first study f (Y t , X t ) -f (Y i∆n , X t ). Burkholder inequality yields E( sup t∈[i∆n,(i+1)∆n[ |f (Y t , X t ) -f (Y i∆n , X t )| k |G n i ) ≤ c ∆ k/2 n (1 + |Y i∆n | k + |X i∆n | k )
We then study f (Y i∆n , X t ) -f (Y i∆n , X i∆n ). Burkholder inequality yields the result.

Proof of Proposition 1. We have

X (i+1)∆n -X i∆n -∆ n b(Y i∆n , X i∆n ) = a(Y i∆n , X i∆n )η i,n + α in + β in where η i,n = (i+1)∆n i∆n dB s , α i,n = (i+1)∆n i∆n
(a(Y s , X s ) -a(Y i∆n , X i∆n ))dB s and

β i,n = (i+1)∆n i∆n (b(Y s , X s ) -b(Y i∆n , X i∆n ))ds.
Properties of η i,n are directly deduced from properties of the Brownian motion. Let

ε C i,n = α i,n + β i,n . Assumptions (A1)-(A2) lead to |E(β i,n |G n i )| ≤ c∆ 3/2 n (1 + |Y i∆n | + |X i∆n |). Proposition 5 provides E(|β i,n | k |G n i ) ≤ c∆ k/2 n (1 + |Y i∆n | k + |X i∆n | k ) for k ≥ 2. Burkholder inequality gives E(|α i,n | k |G n i ) ≤ c∆ k n (1 + |Y i∆n | k + |X i∆n | k ) for k ≥ 2. Proof of Proposition 2. We have Y i,n -X i∆n = 1 ∆n (i+1)∆n i∆n (X v -X i∆n )dv and X v -X i∆n = v i∆n b(Y s , X s )ds + v i∆n a(Y s , X s )dB s . By the Fubini theorem, we get Y i,n -X i∆n = ∆ 1/2
n a(Y i∆n , X i∆n )ξ i,n + e i,n where e i,n = α i,n + β i,n and

α i,n = 1 ∆ n (i+1)∆n i∆n (a(Y v , X v ) -a(Y i∆n , X i∆n )) ((i + 1)∆ n -v)dB v β i,n = 1 ∆ n (
i,n | k |G n i ≤ c∆ k n (1 + |Y i∆n | k + |X i∆n | k ).
For k ≥ 2, applying the Burkholder-Davis-Gundy and the Jensen inequalities yields:

E α k i,n |G n i ≤ c (i+1)∆n i∆n E |a(Y s , X s ) -a(Y i∆n , X i∆n )| k |G n i ds
By proposition 5 and assumption (A1), we get

E α k i,n | ≤ c∆ k/2+1 n (1 + |Y i∆n | k + |X i∆n | k ). Finally, we get |E(e 2 i,n |G n i ) ≤ c∆ 2 n (1 + |X i∆n | 2 + |Y i∆n | 2 ). Using Proposition 5, we have E sup s∈[i∆n,(i+1)∆n[ |X s -X i∆n | k |G n i ≤ ∆ k/2 n (1 + |Y i∆n | k + |X i∆n | k )
thus we directly deduce

E Y i,n -X i∆n k |G n i = E   1 ∆ n (i+1)∆n i∆n (X s -X i∆n )ds k |G n i   ≤ ∆ k/2 n (1 + |Y i∆n | k + |X i∆n | k )
Proof of Proposition 3. We have

Y i+1,n -Y i,n = 1 ∆ n (i+1)∆n i∆n s+∆n s a(Y v , X v )dB v ds A i + 1 ∆ n (i+1)∆n i∆n s+∆n s b(Y v , X v )dvds B i
By Fubini theorem, we have

A i = (i+1)∆n i∆n a(Y v , X v )(v -i∆ n )dB v + (i+2)∆n (i+1)∆n a(Y v , X v )((i + 2)∆ n -v)dB v B i = (i+1)∆n i∆n b(Y v , X v )(v -i∆ n )dv + (i+2)∆n (i+1)∆n b(Y v , X v )((i + 2)∆ n -v)dv
We can rewrite A i as

A i = a(Y i∆n , X i∆n )∆ 3/2 n (ξ i,n + ξ i+1,n ) + a i,n + a i+1,n where a i,n = (i+1)∆n i∆n (a(Y v , X v ) -a(Y i∆n , X i∆n )) (v -i∆ n )dB v and a i+1,n = (i+2)∆n (i+1)∆n (a(Y v , X v ) -a(Y i∆n , X i∆n )) ((i + 2)∆ n -v)dB v . Similarly, B i = b(Y i∆n , X i∆n )∆ 2 n + b i,n + b i+1,n where b i,n = (i+1)∆n i∆n b(Y v , X v ) -b(Y i∆n , Y i,n ) (v -i∆ n )dB v and b i+1,n = (i+2)∆n (i+1)∆n b(Y v , X v ) -b(Y i∆n , Y i,n ) ((i + 2)∆ n -v)dB v . Therefore, this yields Y i+1,n -Y i,n -∆ n b(Y i∆n , Y i,n ) = a(Y i∆n , X i∆n )∆ 1/2 n (ξ i,n + ξ i+1,n ) + ε P i,n with ε P i,n = a i,n ∆n + a i+1,n ∆n + b i,n ∆n + b i+1,n ∆n . • Let us prove |E(ε P i,n |G n i )| ≤ c∆ 2 n (1 + |X (i+1)∆n | 3 + |Y (i+1)∆n | 3 ). We have E(a i,n |G n i ) = E(a i+1,n |G n i ) = 0 and E b i,n ∆ n |G n i = 1 ∆ n (i+1)∆n i∆n (v -i∆ n )E(b(Y v , X v ) -b(Y i∆n , X i∆n )|G n i )dv + 1 ∆ n (i+1)∆n i∆n (v -i∆ n )E(b(Y i∆n , X i∆n ) -b(Y i∆n , Y i,n )|G n i )dv
By Itô's formula, assumptions (A1)-(A2) and Proposition 5, we get

sup v∈[i∆n,(i+1)∆n[ |E (b(Y v , X v ) -b(Y i∆n , X i∆n )|G n i )| ≤ ∆ n c(1 + |Y i∆n | 2 + |X i∆n | 2 )
By Taylor's formula of order two, there exists

Z ∈ (Y i,n , X i∆n ) such that b(Y i∆n , Y i,n )-b(Y i∆n , X i∆n ) = b x (Y i∆n , X i∆n )(Y i,n -X i∆n )+ 1 2 b x 2 (Y i∆n , Z)(Y i,n -X i∆n ) 2
Using the Cauchy Schwartz inequality, we get

E b(Y i∆n , Y i,n ) -b(Y i∆n , X i∆n )|G n i ≤ c∆ n (1 + |Y i∆n | 2 + |X i∆n | 2 )| Hence sup v∈[i∆n,(i+1)∆n[ E b(Y v , X v ) -b(Y i∆n , Y i,n )|G n i ≤ ∆ n c(1 + |Y i∆n | 2 + |X i∆n | 2 )
and

E b i,n ∆n |G n i ≤ ∆ 2 n c(1 + |Y i∆n | 2 + |X i∆n | 2 ). Similarly, E b i+1,n ∆n |G n i ≤ ∆ 2 n c(1 + |Y (i+1)∆n | 2 + |X (i+1)∆n | 2 )
|G n i ≤ c ∆ 4 n E (i+1)∆n i∆n (a(Y v , X v ) -a(Y i∆n , X i∆n )) 4 dv (i+1)∆n i∆n (v -i∆ n ) 4 dv |G n i ≤ c∆ 4 n (1 + |Y i∆n | 4 + |X i∆n | 4 )
Similarly, we obtain E a i+1,n ∆n 4

|G n i ≤ c∆ 4 n (1 + |Y 4 (i+1)∆n + |X (i+1)∆n | 4 ). This completes the proof for the bound of |E(ε 4 i,n |G n i )|. • We now proof |E(ε i,n U i,n |G n i ) ≤ c∆ 3/2 n (1 + |X i∆n | 2 + |Y i∆n | 2 ).
From the definitions of (a i,n , a i+1,n , b i,n , b i+1,n ), we can prove the following inequalities

|E(a i,n ξ i,n |G n i )| ≤ ∆ 5/2 n c(1 + |Y i∆n | + |X i∆n |), |E(a i+1,n ξ i,n |G n )| = 0 |E(b i,n ξ i,n |G n i )| ≤ ∆ 5/2 n c(1 + |Y i∆n | + |X i∆n |), |E(b i+1,n ξ i,n |G n i )| ≤ ∆ 5/2 n c(1 + |Y (i+1)∆n | + |X (i+1)∆n |) Hence the results for |E(ε i,n U i,n |G n i ). • Proposition 5 yields E sup s∈[i∆n,(i+2)∆n[ |X s -X i∆n | k ≤ c∆ k/2 n (1 + |Y (i+1)∆n | k + |X (i+1)∆n | k ) which provides E Y i+1,n -Y i,n k |G n i ≤ c∆ k/2 n (1 + |Y (i+1)∆n | k + |X (i+1)∆n | k )
Proof of Proposition 4. The first assertion in the complete observations case is based on the convergence of the Euler scheme [START_REF] Bally | The law of the Euler scheme for stochastic differential equations I. Convergence rate of the distribution function[END_REF]. For partial observations case, Taylor's expansion ensures that there exists s ∈ (Y i,n , X i∆n ) such that

f (Y i∆n , Y i,n , θ) = f (Y i∆n , X i∆n , θ) + f x (Y i∆n , X s , θ)(Y i,n -X i∆n ). Thus we deduce that E(sup |f (Y i∆n , Y i,n , θ)-f (Y i∆n , X i∆n , θ)||G n i ) ≤ c∆ 1/2 n (1+ |X i∆n | + |Y i∆n |). Hence, the L 1 convergence of sup 1 n n i=0 |f (Y i∆n , Y i,n , θ) - f (Y i∆n , X i∆n , θ)| is proved.
The results yields by applying Proposition 2.

Proof of Theorem 3. The scheme of the proof is the same for both complete and partial observations cases, but the arguments are simpler for the complete observations case. We only detail the second case. Set ĨP n (f ) =

1 n∆n n-2 i=0 f (Y (i-1)∆n , Y i-1,n , θ) Y i+1,n -Y i -∆ n b(Y i∆n , Y i,n ) . We can write I P n (f ) = ĨP n (f )+ 1 n∆ n n-1 i=1 ∆ n f (Y (i-1)∆n , Y i-1,n , θ)(b(Y i∆n , Y i )-b(Y (i-1)∆n , Y i-1,n ))
we first study the convergence of ĨP n (f ) and then we deduce the result for

I P n (f ). We have ĨP n (f ) = 1 n∆n n-1 i=0 Z i,n (θ) with Z i,n (θ) = f (Y (i-1)∆n , Y i-1,n , θ) Y i+1,n -Y i -∆ n b(Y i∆n , Y i,n ) . The random variable Y i,n is G n i+1 -measurable and Z i,n (θ) is G n i+2 -measurable. We split ĨP n (f ) into the sum of three terms ĨP n (f ) = 1 n∆ n n-1 i=0 Z 3i,n (θ) + n-1 i=0 Z 3i+1,n (θ) + n-1 i=0 Z 3i+2,n (θ)
To prove [START_REF] Mattingly | Ergodicity for sdes and approximations: locally lipschitz vector fields and degenerate noise[END_REF], it is enough to show that 1

Z i,n (θ) = z (2) i,n (θ) + z (2) i,n (θ) with z (1) i,n (θ) = f (Y (i-1)∆n , Y i-1,n , θ)∆ 1/2 n a(Y i∆n , X i∆n )U i,n z (2) i,n (θ) = f (Y (i-1)∆n , Y i-1,n , θ)ε P i,n To prove 1 n∆n n-1 i=0 z (j) 3i,n (θ) P ---→ n→∞ 0 for j = 1, 2, we use lemma A2 of [8]. It is thus enough to prove 1 n∆ n n-1 i=0 E(z (j) 3i,n (θ)|G n 3i ) P ---→ n→∞ 0 1 n 2 ∆ 2 n n-1 i=0 E((z (j) 3i,n (θ)) 2 |G n 3i ) P ---→ n→∞ 0 As Y 3i-1,n is G n 3i measurable and E(U 3i,n |G n 3i ) = 0, we have E(z (1) i,n (θ)|G n 3i ) = 0. Using E(U 2 3i,n |G n 3i ) = 2/6, we get E((z (1) 3i,n (θ)) 2 |G n 3i ) = 2 6 ∆ n f 2 (Y (3i-1)∆n , Y 3i-1,n , θ)a 2 (Y 3i∆n , X 3i∆n ) Assumptions (A1)-(A2) yields 1 n 2 ∆ 2 n n-1 i=0 E((z (1) 3i,n (θ)) 2 |G n 3i ) P ---→ n→∞ 0. For z (2)
3i,n (θ), using proposition 3, we get E(z

(1) 3i,n (θ)|G n 3i ) ≤ cf (Y (3i-1)∆n , Y 3i-1,n , θ)∆ 2 n (1+ |Y (3i-1)∆n | 3 + |X (3i-1)∆n | 3 ) and thus 1 n∆n n-1 i=0 E(z (2) 3i,n (θ)|G n 3i ) P ---→ n→∞ 0.
Similarly, we have E((z

(2) 3i,n 2 |G n 3i ) ≤ cf 2 (Y 3i∆n , Y 3i,n , θ)∆ 2 n (1+|Y 3i∆n | 4 + |X 3i∆n | 4 ) and thus 1 n 2 ∆ 2 n n-1 i=0 E((z (2) 3i,n (θ)) 2 |G n 3i ) P ---→ n→∞ 0.
This gives the convergence in probability of ĨP n (f ) for all θ. To obtain uniformity with respect to θ, we use the proposition 51 of [START_REF] Gloter | Parameter estimation for a discretely observed integrated diffusion process[END_REF]

. It is enough to show sup n∈N E sup θ |∂ θ ĨP n (f )| < ∞. We have ∂ θ ĨP n (f ) = 1 n∆ n n-1 i=1 ∂ θ f (Y (i-1)∆n , Y i-1,n , θ) ∆ 1/2 n a(Y i∆n , X i∆n )U i,n + ε i,n . As E(U i , n|G n i ) = 0 and E(ε i,n |G n i ) ≤ c∆ 2 n (1 + |X i∆n | 3 + |Y i∆n | 3 ), we have E ∂ θ f θ (Y (i-1)∆n , Y i-1,n ) ∆ 1/2 n a(Y i∆n , X i∆n )U i,n + ε i,n |G n i ≤ c∆ n (1+|X i∆n | 3 + |Y i∆n | 3 ). With assumption (A2), it implies E(∂ θ f (Y (i-1)∆n , Y i-1,n , θ) ∆ 1/2 n a(Y i∆n , X i∆n )U i,n + ε i,n |G n i ) ≤ c∆ n .
Hence, sup n∈N E sup θ |∂ ĨP n (f )| < ∞ and uniformity in θ follows. We can now deduce the result for I P n (f ). Taylor's formula gives the existence of s 1 and s

2 such that b(Y i∆n , Y i ) -b(Y (i-1)∆n , Y i-1,n ) = b y (Y s 1 , Y i )(Y i∆n -Y (i-1)∆n ) +b x (Y (i-1)∆n , X s 2 )(Y i -Y i-1,n ) fact that Y 3i-1,n is G n
3i -measurable, we obtain:

E(w (1) 3i (θ)|G n 3i ) = 2∆ n 3 a 2 (Y 3i∆n , X 3i∆n )f (Y (3i-1)∆n , Y 3i-1,n , θ) E( w (1) 3i (θ) 2 |G n 3i ) = 4∆ 2 n 3 a 4 (Y 3i∆n , X 3i∆n )f ( 2 Y (3i-1)∆n , Y 3i-1,n , θ)
Thus, applying lemma A1 of [START_REF] Gloter | Parameter estimation for a discretely observed integrated diffusion process[END_REF], we get

(n∆ n ) -1 n-1 i=0 E(w (1) 3i (θ)|G n 3i ) P ---→ n→∞ 2 3 ν 0 (f (•, •, θ)a 2 (•, •))
and by assumption (A2), we get E E( w

3i (θ) due to proposition 51 of [START_REF] Gloter | Parameter estimation for a discretely observed integrated diffusion process[END_REF]. This is easily obtained using proposition 2 and assumption (A2).

2 |G n 3i ) ≤ c∆ 2 n and therefore, (n∆ n ) -1 n-1 i=0 E( w (1) 3i (θ)
Proof of Theorem 5. We only detail the partial observations case. We have

√ n∆ n I P n (f ) = √ n∆ n ĨP n (f ) + 1 √ n∆n n-1 i=2 f (Y (i-1)∆n , Y i-1,n , θ)∆ n (b(Y i∆n , Y i ) -b(Y (i-1)∆n , Y i-1,n )).
We first study the distribution convergence of = f (Y (i-1)∆n , Y (i-1)∆n , θ)a(Y i∆n , X i∆n )ξ i,n +f (Y (i-2)∆n , Y (i-2)∆n , θ)a(Y (i-1)∆n , X (i-1)∆n )ξ i,n ).

√ n∆ n ĨP n (
We have E(s

in |G n i ) = 0 and we compute the conditional variance E[(s

(1) in ) 2 |G n i ]: E[(s (1)
in ) 2 |G n i ] = 1 3 {f 2 (Y (i-1)∆n , Y (i-1)∆n , θ)a 2 (Y i∆n , X i∆n ) +f (Y (i-1)∆n , Y (i-1)∆n , θ)f (Y (i-2)∆n , Y (i-2)∆n , θ)a(Y i∆n , X i∆n )a(Y (i-1)∆n , X (i-1)∆n ) +f 2 (Y (i-2)∆n , Y (i-2)∆n , θ)a 2 (Y (i-1)∆n , X (i-1)∆n )}.

We want to prove that 1 A Taylor development and the Cauchy-Schwarz inequality provide the convergence in L 1 towards 0 of sup θ 1 n n-1 i=2 |f 2 (Y (i-1)∆n , Y (i-1)∆n , θ)a 2 (Y i∆n , X i∆n )f 2 (Y (i-1)∆n , X (i-1)∆n , θ)a 2 (Y i∆n , X i∆n )| using assumptions (A1)-(A2). The terms f (Y (i-1)∆n , Y (i-1)∆n , θ)f (Y (i-2)∆n , Y (i-2)∆n , θ)a(Y i∆n , X i∆n ) a(Y (i-1)∆n , X (i-1)∆n ) and f 2 (Y (i-2)∆n , Y (i-2)∆n , θ)a 2 (Y (i-1)∆n , X (i-1)∆n ) are similar. We easily bound E[(s Proof of Theorem 6. We only detail the partial observations case. We use the same notations as in Theorem 4. Set M n (f ) = √ n Q P n (f ) -2 3 ν n (f a 2 ) and β(y, x) = a 2 (y, x)f (y, x, θ). We have

M n (f ) = √ n 1 n∆n n-1 i=2 ∆ n f (Y i∆n , Y i , θ)a 2 (Y i∆n , X i∆n )U 2 i +f (Y i∆n , Y i , θ)(ε P i,n + ∆ n b(Y i∆n , Y i )) 2 + 2∆ 1/2 n f (Y i∆n , Y i , θ)a(Y i∆n , X i∆n )U i (ε P i,n +∆ n b(Y i∆n , Y i )) -2 3n n-1 i=2 f (Y i∆n , Y i , θ)a 2 (Y i∆n , Y i )
To prove the convergence of M and so does M

n . By lemma A2 of [START_REF] Gloter | Parameter estimation for a discretely observed integrated diffusion process[END_REF], we have

n .

Theorem 5 .

 5 probability to 0 (Theorem 3), they also satisfy a central limit theorems as follows: Under assumptions (A1)-(A4) and n∆ 2 n ---→ n→∞ 0, we have

Figure 1 :

 1 Figure 1: Model I: Stochastic Growth. Estimator densities of parameter σ computed on 1000 simulated datasets for three designs ∆ n = 0.1, n = 100 (a), ∆ n = 0.1, n = 1000 (b) and ∆ n = 0.01, n = 1000 (c). True value of σ is 1 (vertical line). Three estimators are compared: MLE (dotted line), complete observations contrast estimator σ C (plain line), partial observations contrast estimator σ P (bold line).

  : Harmonic Growth, estimation of the three parameters D, γ, σ. Mean and standard error of parameter estimators D, γ and σ computed on 1000 simulated datasets for three designs ∆ n = 0.1, n = 100 (a), ∆ n = 0.1, n = 1000 (b) and ∆ n = 0.01, n = 1000 (c). Three estimators are compared: complete observations contrast estimator θ C , partial observations contrast estimator θ P and Gibbs estimator obtained by[START_REF] Pokern | Parameter estimation for partially observed hypoelliptic diffusions[END_REF] with partial observations.

  Four designs (∆ n , n) of simulations are considered: ∆ n = 0.1, n = 100; ∆ n = 0.1, n = 1000; ∆ n = 0.01, n = 1000 and ∆ n = 0.01, n = 10000. A thousand of datasets are simulated for each design with the exact stationary distribution and the true parameter values proposed by [15] D 01 = 1, D 02 = -8, D 03 = 8, γ 0 = 0.5 and σ 0 = 0.7 and U 0 = (1, 1

D 1 ,

 1 D 2 , D 3 , γ, σ. Mean and standard error of parameter estimators D, γ and σ computed on 1000 simulated datasets for four designs ∆ n = 0.1, n = 100, ∆ n = 0.1, n = 1000, ∆ n = 0.01, n = 1000 and ∆ n = 0.01, n = 10000. Two estimators are compared: complete observations contrast estimator θ C and partial observations contrast estimator θ P .

  3i+2,n (θ) is analogous). Using Proposition 3, we set

2 3E

 2 ν 0 (f (•, •, θ)a 2 (•, •)) in probability. Using proposition 3 and lemma A2 of [8], we easily prove that Q (Y i+1,n -Y i,n ) 2 sup θ ∂ θ f (Y i∆n , Y i,n , θ) < ∞

  f ) then we deduce the result for √ n∆ n I P n (f ). Using the same notations as in Theorem 4Y(i-1)∆n , Y (i-1)∆n , θ) a(Y i∆n , X i∆n )(ξ i,n + ξ i+1,n ) and N Y (i-1)∆n , Y (i-1)∆n , θ)ε in . First, we study N (1)n . In order to use a martingale central limit theorem, 0 , Y 0 , θ)a(Y ∆n , X ∆n )ξ 0,n (n-2)∆n , Y (n-2)∆n , θ)a(Y (n-1)∆n , X (n-1)∆n )ξ n,n

i=2 f 2 f 2

 22 (Y (i-1)∆n , Y (i-1)∆n , θ)a 2 (Y i∆n , X i∆n ). By the ergodic theorem, (Y (i-1)∆n , X (i-1)∆n , θ)a 2 (Y i∆n , X i∆n ) P ---→ n→∞ ν 0 (f 2 a 2 ).

∆ 1 √

 1 ν 0 (f 2 a 2 )). By (A.1), we deduce N ν 0 (f 2 a 2 )).We now have to prove the convergence to 0 of N f (Y (i-1)∆n , Y (i-1)∆n , θ)ε in |G n i ] in probability. Similarly, we obtain that1 n∆n n-1 i=1 E[f 2 (Y (i-1)∆n , Y (i-1)∆n , θ)ε 2in |G n i ] converges to 0 in probability. Thus, using proposition 5, we getN n f (Y (i-1)∆n , Y i-1,n , θ)(b(Y i∆n , Y i ) -b(Y (i-1)∆n , Y i-1,n )) gives the convergence in distribution of √ n∆ n ĨP n (f ). To deduce the results for √ n∆ n I P n (f ), we remark that b(Y i∆n , Y i ) -b(Y (i-1)∆n , Y i-1,n ) = b(Y i∆n , Y i ) -b(Y (i-1)∆n , Y i ) + b(Y (i-1)∆n , Y i ) -b(Y (i-1)∆n , Y i-1,n ).Taylor's development givesE[|b(Y i∆n , Y i,n ) -b(Y i∆n , Y i-1,n )|G n i ] ≤ c ∆ n (1 + |X i∆n | + |Y i∆n |). Using b(Y i∆n , Y i )-b(Y (i-1)∆n , Y i ) = i∆n (i-1)∆n b y (Y s , Y i-1,n )(X i + s (i-1)∆n b(V u , X u )du+ s (i-1)∆n a(V u , X u )dB u )ds and the Burkholder inequality, this yieldsE[|b(Y i∆n , Y i ) -b(Y (i-1)∆n , Y i )| |G n i ] ≤ c ∆ n (1 + |X i∆n | + |Y i∆n |).Using assumptions (A1)-(A4), we deduce the convergence to 0 in probability ofn∆n n-1 i=2 f (Y (i-1)∆n , Y i-1,n , θ)∆ n (b(Y i∆n , Y i ) -b(Y (i-1)∆n , Y i-1,n )). We deduce that √ n∆ n I P n (f ) -√ n∆ n ĨP n (f ) = o P (1).

  ν 0 (β 2 ))

  l = 2, . . . , 5. This holds true using that n∆ 2 n → 0 and the hypothesis (A1) for M

Table 1 :

 1 Model I: Stochastic Growth. True value is σ = 1. Mean and standard error of estimators of parameter σ computed on 1000 simulated datasets for three designs ∆ n = 0.1, n = 100, ∆ n = 0.1, n = 1000 and ∆ n = 0.01, n = 1000. Four estimators are compared: MLE with complete observations, complete observations contrast estimator σ C , partial observations contrast estimator σ P and Gibbs estimates obtained by

	01

Table 2 :

 2 Model II

  ) . The two estimators θ C and θ P are computed on each dataset. Simultaneous estimation of the five parameters is performed. Empirical mean and standard deviations of the

						Design
	Estimator True	∆ n = 0.1	∆ n = 0.1	∆ n = 0.01	∆ n = 0.01
			value	n = 100	n = 1000	n = 1000	n = 10000
	σ C		0.7	0.886 (0.110) 0.861 (0.032) 0.713 (0.019) 0.714 (0.006)
	σ P			1.012 (0.118) 1.021 (0.034) 0.873 (0.024) 0.784 (0.008)
	D 1	C	1	0.987 (0.414) 1.003 (0.125) 1.043 (0.381) 1.010 (0.111)
	D 1	P		1.002 (0.378) 1.002 (0.116) 1.036 (0.378) 1.005 (0.110)
	D 2	C	-8	-8.221 (1.451) -8.020 (0.367) -8.082 (1.878) -8.042 (0.498)
	D 2	P		-7.340 (1.382) -7.251 (0.339) -8.019 (1.859) -7.998 (0.495)
	D 3	C	8	8.271 (2.424) 8.001 (0.597) 7.722 (3.589) 8.010 (0.764)
	D 3	P		7.068 (2.235) 7.007 (0.565) 7.641 (3.559) 7.964 (0.758)
	γ C		0.5	0.638 (0.290) 0.524 (0.074) 0.671 (0.384) 0.522 (0.099)
	γ P			0.889 (0.304) 0.763 (0.074) 0.701 (0.384) 0.548 (0.100)

Table 3 :

 3 Model III: Trigonometric Growth, estimation of the five parameters

  A1), we get|β i,n | ≤ c∆ n (1 + sup s∈[i∆n,(i+1)∆n[ (|Y s | + |X s |)). As E(α i,n |G n i ) = 0, we get |E(e i,n |G n i ) ≤ c∆ n (1+ |X i∆n | + |Y i∆n |). By assumption (A2), for all k ≥ 0, we get E |β

	i+1)∆n	v
		b(Y s , X s )dsdv
	i∆n	i∆n
	By Assumption (

Preprint submitted to Stochastic Process and their ApplicationsDecember 16, 2011

Acknowledgements

This work was supported by Agence Nationale de la Recherche through the project MANDy ANR-09-BLAN-0008-01

Assumptions (A1)-(A2), Cauchy-Schwarz inequality and Y

This implies

Hence the result.

Proof of Theorem 4. We only detail the partial observations case. We set

We split the sum into the sum of three terms W 3i,n , W 3i+1,n and W 3i+2,n . Given this partition, it is enough to show that

As the expression of Q P n is slightly different from [START_REF] Gloter | Parameter estimation for a discretely observed integrated diffusion process[END_REF], we are able to write W 3i,n (θ) as the sum of only three terms (instead four). Using Taylor's formula, there exists

We set Q

3i,n (θ), for j = 1, 2, 3. We start by studying Q

and the

By Taylor expansion, there exists

)

n . Reordering terms to obtain a triangular array of martingale increments, we get

)β(Y (n-1)∆n , X (n-1)∆n )

where

i+1 measurable and centered conditionally to G n i . Furthermore, using the properties of (ξ i,n , ξ i,n ), we deduce E(s 2 in |G n i ) = 2 9 β 2 (Y i∆n , X i∆n ) + 2 9 β 2 (Y (i-1)∆n , X (i-1)∆n ) + 4 3 ξ 2 i-1,n β 2 (Y (i-1)∆n , X (i-1)∆n ) + 1 9 β(Y (i-1)∆n , X (i-1)∆n )β(Y i∆n , X i∆n )