F Comte

V Genon-Catalot

Y Rozenholc

Penalized nonparametric mean square estimation of the coefficients of diffusion processes

Keywords: Adaptive estimation, Diffusion processes, Discrete time observations, Drift and diffusion coefficients, Mean square estimator, Model selection, Penalized contrast, Retrospective simulation

We consider a one-dimensional diffusion process (X t ) which is observed at n + 1 discrete times with regular sampling interval ∆. Assuming that (X t ) is strictly stationary, we propose nonparametric estimators of the drift and diffusion coefficients obtained by a penalized least square approach. Our estimators belong to a finite dimensional function space whose dimension is selected by a data-driven method. We provide non asymptotic risk bounds for the estimators. When the sampling interval tends to zero while the number of observations and the length of the observation time interval tend to infinity, we show that our estimators reach the minimax optimal rates of convergence. Numerical results based on exact simulations of diffusion processes are given for several examples of models and enlight the qualities of our estimation algorithms.

Introduction

In this paper, we consider the following problem. Let (X t ) t≥0 be a one-dimensional diffusion process with dynamics described by the following stochastic differential equation:

dX t = b(X t )dt + σ(X t )dW t , t ≥ 0, X 0 = η (1)
where (W t ) is a standard Brownian motion and η is a random variable independent of (W t ). Assuming that the process is strictly stationary (and ergodic), and that a discrete observation (X k∆ ) 1≤k≤n+1 of the sample path is available, we want to build nonparametric estimators of the drift function b and the (square of the) diffusion coefficient σ 2 .

Our aim is twofold: construct estimators that have optimal asymptotic properties and that can be implemented through feasible algorithms. Our asymptotic framework is such that the sampling interval ∆ = ∆ n tends to zero while n∆ n tends to infinity as n tends to infinity. Nevertheless, the risk bounds obtained below are non asymptotic in the sense that they are explicitly given as functions of ∆ or 1/(n∆) and fixed constants.

Nonparametric estimation of the coefficients of diffusion processes has been widely investigated in the last decades. The first estimators that have been proposed and studied are based on a continuous time observation of the sample path. Asymptotic results are given for ergodic models as the length of the observation time interval tends to infinity: see for instance the reference paper by [START_REF] Banon | Nonparametric identification for diffusion processes[END_REF], followed by more recent works of Prakasa [START_REF] Rao | Statistical inference for diffusion type processes[END_REF], [START_REF] Spokoiny | Adaptive drift estimation for nonparametric diffusion model[END_REF], [START_REF] Kutoyants | Statistical inference for ergodic diffusion processes[END_REF] or [START_REF] Dalalyan | Sharp adaptive estimation of the drift function for ergodic diffusions[END_REF].

Then discrete sampling of observations has been considered, with different asymptotic frameworks, implying different statistical strategies. It is now classical to distinguish between low-frequency and high-frequency data. In the former case, observations are taken at regularly spaced instants with fixed sampling interval ∆ and the asymptotic framework is that the number of observations tends to infinity. Then, only ergodic models are usually considered. Parametric estimation in this context has been studied by [START_REF] Bibby | Martingale estimation functions for discretely observed diffusion processes[END_REF], [START_REF] Kessler | Estimating equations based on eigenfunctions for a discretely observed diffusion process[END_REF], see also [START_REF] Bibby | Estimating functions for discretely sampled diffusion-type models[END_REF]. A nonparametric approach using spectral methods is investigated in [START_REF] Gobet | Nonparametric estimation of scalar diffusions based on low frequency data[END_REF], where non standard nonparametric rates are exhibited.

In high-frequency data, the sampling interval ∆ = ∆ n between two successive observations is assumed to tend to zero as the number of observations n tends to infinity. Taking 1 ∆ n = 1/n, so that the length of the observation time interval n∆ n = 1 is fixed, can only lead to estimating the diffusion coefficient consistently. This is done by [START_REF] Hoffmann | Adaptive estimation in diffusion processes[END_REF] who generalizes results by [START_REF] Jacod | Non-parametric Kernel Estimation of the Coefficient of a Diffusion[END_REF], Florens-Zmirou (1993) and Genon-Catalot et al. (1992). Now, estimating both drift and diffusion coefficients requires that the sampling interval ∆ n tends to zero while n∆ n tends to infinity. For ergodic diffusion models, Hoffmann (1999) proposes nonparametric estimators using projections on wavelet bases together with adaptive procedures. He exhibits minimax rates and shows that his estimators automatically reach these optimal rates up to logarithmic factors. Hoffmann's estimators are based on computations of some random times which make them difficult to implement.

In this paper, we propose simple nonparametric estimators based on a penalized mean square approach. The method is investigated in full details in [START_REF] Comte | Adaptive estimation of the spectrum of a stationary Gaussian sequence[END_REF]Rozenholc (2002, 2004) for regression models. We adapt it here to the case of discretized diffusion models.

The estimators are chosen to belong to finite dimensional spaces that include trigonometric, wavelet generated and piecewise polynomials spaces. The space dimension is chosen by a data driven method using a penalization device. Due to the construction of our estimators, we measure the risk of an estimator f of f (with f = b, σ 2 ) by E( ff 2 n ) where ff 2 n = n -1 n k=1 ( f (X k∆ )f (X k∆ )) 2 . We give bounds for this risk (see Theorem 3.1 and Theorem 4.1). Looking at these bounds as ∆ = ∆ n → 0 and n∆ n → +∞ shows that our estimators achieve the optimal nonparametric asymptotic rates obtained in [START_REF] Hoffmann | Adaptive estimation in diffusion processes[END_REF] without logarithmic loss (when the unknown functions belong to Besov balls). Then we proceed to numerical implementation on simulated data for several examples of models. We emphasize that our simulation method for diffusion processes is not based on approximations (like Euler schemes). Instead, we use the exact retrospective simulation method described in [START_REF] Beskos | Retrospective exact simulation of diffusion sample paths with applications[END_REF] and [START_REF] Beskos | Exact simulation of diffusions[END_REF]. Then we apply the algorithms developed in [START_REF] Comte | Adaptive estimation of the spectrum of a stationary Gaussian sequence[END_REF]Rozenholc (2002,2004) for nonparametric estimation using piecewise polynomials. The results are convincing even when some of the theoretical assumptions are not fulfilled.

The paper is organized as follows. In Section 2, we describe our framework (model, assumptions and spaces of approximation). Section 3 is devoted to drift estimation, Section 4 to diffusion coefficient estimation. In Section 5, we study examples and present numerical simulation results that illustrate the performances of estimators. Section 6 contains proofs.

In the Appendix (Section 7), a technical lemma is proved.

2 Framework and assumptions

Model assumptions

Let (X t ) t≥0 be a solution of (1) and assume that n + 1 observations X k∆ , k = 1, . . . , n + 1 with sampling interval ∆ are available. Throughout the paper, we assume that ∆ = ∆ n tends to 0 and n∆ n tends to infinity as n tends to infinity. To simplify notations, we only write ∆ without the subscript n. Nevertheless, when speaking of constants, we mean quantities that depend neither on n nor on ∆. We want to estimate the drift function b and the diffusion coefficient σ 2 when X is stationary and geometrically β-mixing. To this end, we consider the following assumptions:

[A1 ] (i) b ∈ C 1 (R) and there exists γ ≥ 0, such that, for all x ∈ R, |b ′ (x)| ≤ γ(1 + |x| γ ).
(ii) There exists b 0 , such that, for all x, |b(x)| ≤ b 0 (1 + |x|).

(iii) There exist d ≥ 0, r > 0, and R > 0, such that, for all |x| ≥ R, sgn(x)b(x) ≤ -r|x| d .

[A2 ] (i) There exist σ 2 0 , σ 2 1 , such that, for all x, 0 < σ 2 0 ≤ σ 2 (x) ≤ σ 2 1 and there exists L such that for all (x, y Under the additional Assumption [A3], (X t ) is strictly stationary and ergodic. Moreover, it follows from Proposition 1 in Pardoux and Veretennikov (2001) that there exist constants K > 0, ν > 0 and θ > 0 such that:

) ∈ R 2 , |σ(x) -σ(y)| ≤ L|x -y| 1/2 . (ii) σ ∈ C 2 (R) and there exists γ ≥ 0, such that for all x ∈ R, |σ ′ (x)| + |σ"(x)| ≤ γ(1 + |x| γ ). Under [A1]-[A2],
E(exp(ν|X 0 |)) < +∞ and β X (t) ≤ Ke -θt , (2)
where β X (t) denotes the β-mixing coefficient of (X t ) and is given by

β X (t) = +∞ -∞ π(x)dx P t (x, dx ′ ) -π(x ′ )dx ′ T V .
The norm . T V is the total variation norm and P t denotes the transition probability. In particular, X 0 has moments of any (positive) order. Now, [A1] (i) ensures that for all t ≥ 0, h > 0, and k ≥ 1, there exists c = c(k, γ) such that

E( sup s∈[t,t+h] |b(X s ) -b(X t )| k |F t ) ≤ ch k/2 (1 + |X t | c ),
where F t = σ(X s , s ≤ t) (see e.g. [START_REF] Gloter | Discrete sampling of an integrated diffusion process and parameter estimation of the diffusion coefficient[END_REF], Proposition A). Thus, taking expectations, there exists c ′ such that

E( sup s∈[t,t+h] |b(X s ) -b(X t )| k ) ≤ c ′ h k/2 . ( 3
)
Functions b and σ 2 are estimated only on a compact set A. For simplicity and without loss of generality, we assume from now on that

A = [0, 1]. (4)
It follows from [A1], [A2](i) and [A3] that the stationary density π is bounded from below and above on any compact subset of R and we denote by π 0 , π 1 two positive real numbers such that

0 < π 0 ≤ π(x) ≤ π 1 , ∀x ∈ A = [0, 1]. ( 5
)

Spaces of approximation: piecewise polynomials

We aim at estimating functions b and σ 2 of Model (1) on [0, 1] using a data driven procedure. For that purpose, we consider families of finite dimensional linear subspaces of L 2 ([0, 1]) and compute for each space an associated least-squares estimator. Afterwards, an adaptive procedure chooses among the resulting collection of estimators the "best" one, in a sense that will be later specified, through a penalization device.

Several possible collections of spaces are available and discussed in Section 2.3. Now, to be consistent with the algorithm implemented in Section 5, we focus on a specific collection, namely the collection of dyadic regular piecewise polynomial spaces, denoted hereafter by [DP].

We fix an integer r ≥ 0. Let p ≥ 0 be an integer. On each subinterval I j = [(j -1)/2 p , j/2 p ], j = 1, . . . , 2 p , consider r + 1 polynomials of degree ℓ, ϕ j,ℓ (x), ℓ = 0, 1, . . . r and set ϕ j,ℓ (x) = 0 outside I j . The space S m , m = (p, r), is defined as generated by the D m = 2 p (r + 1) functions (ϕ j,ℓ ). A function t in S m may be written as

t(x) = 2 p j=1 r ℓ=0 t j,ℓ ϕ j,ℓ (x).
The collection of spaces (S m , m ∈ M n ) is such that

M n = {m = (p, r), p ∈ N, r ∈ {0, 1, . . . , r max }, 2 p (r max + 1) ≤ N n }. (6)
In other words, D m ≤ N n where N n ≤ n. The maximal dimension N n is subject to additional constraints given below. The role of N n is to bound all dimensions D m , even when m is random. In practice, it corresponds to the maximal number of coefficients to estimate. Thus it must not be too large.

More concretely, consider the orthogonal collection in L 2 ([-1, 1]) of Legendre polynomials (Q ℓ , ℓ ≥ 0), where the degree of Q ℓ is equal to ℓ, generating L 2 ([-1, 1]) (see [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF], p.774). They satisfy

|Q ℓ (x)| ≤ 1, ∀x ∈ [-1, 1], Q ℓ (1) = 1 and 1 -1 Q 2 ℓ (u)du = 2/(2ℓ + 1). Then we set P ℓ (x) = (2ℓ + 1) 1/2 Q ℓ (2x -1), to get an orthonormal basis of L 2 ([0, 1]). Finally, ϕ j,ℓ (x) = 2 p/2 P ℓ (2 p x -j + 1)1 I I j (x), j = 1, . . . , 2 p , ℓ = 0, 1, . . . , r.
The space S m has dimension D m = 2 p (r + 1) and its orthonormal basis described above satisfies

2 p j=1 r ℓ=0 ϕ 2 j,ℓ ∞ ≤ D m (r + 1) ≤ D m (r max + 1).
Hence, for all t ∈ S m , t ∞ ≤ (r max + 1) 1/2 D 1/2 m t , where t 2 = 1 0 t 2 (x)dx, for t in L 2 ([0, 1]), a property which is essential for the proofs.

Other spaces of approximation

From both theoretical and practical points of view, other spaces can be considered as, for example:

[T] Trigonometric spaces: S m is generated by { 1, 2 1/2 cos(2πjx), 2 1/2 sin(2πjx) for j = 1, . . . , m }, has dimension D m = 2m + 1 and m ∈ M n = {1, . . . , [n/2] -1}.

[W] Dyadic wavelet generated spaces with regularity r and compact support, as described e.g. in Daubechies (1992), [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF] or [START_REF] Hoffmann | Adaptive estimation in diffusion processes[END_REF].

The key properties that must be fulfilled to fit in our framework are the following:

(H 1 ) Norm connection: (S m ) m∈Mn is a collection of finite dimensional linear sub-spaces of L 2 ([0, 1]), with dimension dim(S m ) = D m such that D m ≤ N n ≤ n, ∀m ∈ M n
and satisfying:

There exists Φ 0 > 0, such that, for all m ∈ M n , for all t ∈ S m , t ∞ ≤ Φ 0 D 1/2 m t . (7)
An orthonormal basis of S m is denoted by (ϕ λ ) λ∈Λm where |Λ m | = D m . It follows from [START_REF] Birgé | From model selection to adaptive estimation[END_REF] that Property [START_REF] Barndorff-Nielsen | Hyperbolic distributions and distributions on hyperbolae[END_REF] in the context of (H 1 ) is equivalent to:

There exists Φ 0 > 0, λ∈Λm ϕ 2 λ ∞ ≤ Φ 2 0 D m . (8)
Thus, for collection [DP], (8) holds with Φ 2 0 = r max + 1. Moreover, for results concerning adaptive estimators, we need an additional assumption: (H 2 ) Nesting condition: (S m ) m∈Mn is a collection of models such that there exists a space denoted by S n , belonging to the collection, with ∀m ∈ M n , S m ⊂ S n . We denote by

N n the dimension of S n : dim(S n ) = N n (∀m ∈ M n , D m ≤ N n ).
As much as possible below, we keep general notations to allow extensions to other spaces of approximation than the collection [DP].

3 Drift estimation

Drift estimators: non adaptive case

Let

Y k∆ = X (k+1)∆ -X k∆ ∆ and Z k∆ = 1 ∆ (k+1)∆ k∆ σ(X s )dW s . (9)
The following standard regression-type decomposition holds:

Y k∆ = b(X k∆ ) + Z k∆ + 1 ∆ (k+1)∆ k∆ (b(X s ) -b(X k∆ ))ds
where b(X k∆ ) is the main term, Z k∆ the noise term and the last term is a negligible residual.

Now, for S m a space of the collection M n and for t ∈ S m , we consider the following regression contrast:

γ n (t) = 1 n n k=1 [Y k∆ -t(X k∆ )] 2 . ( 10
)
The estimator belonging to S m is defined as bm = arg min

t∈Sm γ n (t). (11)
A minimizer of γ n in S m , bm , always exists but may not be unique. Indeed in some common situations the minimization of γ n over S m leads to an affine space of solutions.

Consequently, it becomes impossible to consider a classical L 2 -risk for "the least-squares estimator" of b in S m . In contrast, the random R n -vector ( bm (X ∆ ), . . . , bm (X n∆ )) ′ is always uniquely defined. Indeed, let us denote by Π m the orthogonal projection (with respect to the inner product of R n ) onto the subspace

{(t(X ∆ ), . . . , t(X n∆ )) ′ , t ∈ S m } of R n . Then ( bm (X ∆ ), . . . , bm (X n∆ )) ′ = Π m Y where Y = (Y ∆ , . . . , Y n∆ ) ′
. This is the reason why we define the risk of bm by

E 1 n n k=1 { bm (X k∆ ) -b(X k∆ )} 2 = E( bm -b 2 n )
where

t 2 n = 1 n n k=1 t 2 (X k∆ ). ( 12
)
Thus, our risk is the expectation of an empirical norm. Note that, for a deterministic

function t, E( t 2 n ) = t 2 π = t 2 (x)dπ(x)
where π denotes the stationary law. In view of (5), the L 2 -norm, . , and the L 2 (π)-norm, . π , are equivalent for A-supported functions.

Risk of the non adaptive drift estimator

Using ( 9)-( 10)-( 12), we have:

γ n (t) -γ n (b) = t -b 2 n + 2 n n k=1 (b -t)(X k∆ )Z k∆ + 2 n∆ n k=1 (b -t)(X k∆ ) (k+1)∆ k∆ (b(X s ) -b(X k∆ ))ds
In view of this decomposition, we define the centered empirical process:

ν n (t) = 1 n n k=1 t(X k∆ )Z k∆ . (13)
Now denote by b m the orthogonal projection of b on S m . By definition of bm ,

γ n ( bm ) ≤ γ n (b m ). So, γ n ( bm ) -γ n (b) ≤ γ n (b m ) -γ n (b). This implies bm -b 2 n ≤ b m -b 2 n + 2ν n ( bm -b m ) + 2 n∆ n k=1 ( bm -b m )(X k∆ ) (k+1)∆ k∆ (b(X s ) -b(X k∆ ))ds
The functions bm and b m being A-supported, we can cancel the terms b1 I A c 2 n that appears in both sides of the inequality. This yields bm -b1

I A 2 n ≤ b m -b1 I A 2 n + 2ν n ( bm -b m ) + 2 n∆ n k=1 ( bm -b m )(X k∆ ) (k+1)∆ k∆ (b(X s ) -b(X k∆ ))ds (14)
On the basis of this inequality, we obtain the following result.

Proposition 3.1 Let ∆ = ∆ n be such that ∆ n → 0, n∆ n / ln 2 (n) → +∞ when n → +∞. Assume that [A1], [A2](i), [A3]
hold and consider a space S m in the collection [DP] with

N n = o(n∆/ ln 2 (n)) (N n is defined in (H 2 )). Then the estimator bm of b is such that E( bm -b A 2 n ) ≤ 7π 1 b m -b A 2 + K E(σ 2 (X 0 ))D m n∆ + K ′ ∆ + K" n∆ , (15)
where b A = b1 I A and K, K ′ and K" are some positive constants.

As a consequence, it is natural to select the dimension D m that leads to the best compromise between the squared bias term b mb A 2 and the variance term of order

D m /(n∆).
To compare the result of Proposition 3.1 with the optimal nonparametric rates exhibited by Hoffmann (1999), let us assume that b A belongs to a ball of some Besov

space, b A ∈ B α,2,∞ ([0, 1]), and that r + 1 ≥ α. Then, for b A α,2,∞ ≤ L, we have b A -b m 2 ≤ C(α, L)D -2α m . Thus, choosing D m = (n∆) 1/(2α+1) , we obtain E( bm -b A 2 n ) ≤ C(α, L)(n∆) -2α/(2α+1) + K ′ ∆ + K" n∆ . ( 16
)
The first term (n∆) -2α/(2α+1) is exactly the optimal nonparametric rate (see [START_REF] Hoffmann | Adaptive estimation in diffusion processes[END_REF]).

Moreover, under the standard condition ∆ = o(1/(n∆)), the last two terms in [START_REF] Comte | Adaptive estimation of the spectrum of a stationary Gaussian sequence[END_REF] are

O(1/(n∆))
which is negligible with respect to (n∆) -2α/(2α+1) . Proposition 3.1 holds for the wavelet basis [W] under the same assumptions. For the trigonometric basis [T], the additional constraint

N n ≤ O((n∆) 1/2 / ln(n)) is necessary.
Hence, when working with those bases, if b A ∈ B α,2,∞ ([0, 1]) as above, the optimal rate is reached for the same choice for D m , under the additional constraint that α > 1/2 for [T].

It is worth stressing that α > 1/2 automatically holds under [A1].

Adaptive drift estimator

As a second step, we must ensure an automatic selection of D m , which does not use any knowledge on b, and in particular which does not require to know α. This selection is standardly done by m = arg min

m∈Mn γ n ( bm ) + pen(m) , (17)
with pen(m) a penalty to be properly chosen. We denote by b m the resulting estimator and we need to determine pen(.) such that, ideally,

E( b m -b A 2 n ) ≤ C inf m∈Mn b A -b m 2 + E(σ 2 (X 0 ))D m n∆ + K ′ ∆ + K" n∆ ,
with C a constant which should not be too large. We almost reach this aim.

Theorem 3.1 Let ∆ = ∆ n be such that ∆ n → 0, n∆ n / ln 2 (n) → +∞ when n → +∞.
Assume that [A1]-[A2](i), [A3] hold and consider the nested collection of models [DP] with

maximal dimension N n = o(n∆/ ln 2 (n)). Let pen(m) ≥ κσ 2 1 D m n∆ , ( 18
)
where κ is a universal constant. Then the estimator b m of b with m defined in ( 17) is such that

E( b m -b A 2 n ) ≤ C inf m∈Mn b m -b A 2 + pen(m) + K ′ ∆ + K" n∆ . (19)
Some comments need to be made. It is possible to choose pen(m) = κσ 2 1 D m /(n∆), nevertheless this is not what is done in practice. It is better to calibrate additional terms. This is explained in Section 5.2. The constant κ in the penalty is numerical and must be calibrated for the problem. Its value is usually adapted by intensive simulation experiments. This point is also discussed in Section 5.2. From [START_REF] Comte | Adaptive estimation of the spectrum of a stationary Gaussian sequence[END_REF], one would expect to obtain E(σ 2 (X 0 )) instead of σ 2 1 in the penalty (18): we do not know if this is the consequence of technical problems or if this is a structural result. Another important point is that σ 2 1 is unknown. In practice, we just replace it by a rough estimator (see Section 5.2).

From [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF], we deduce that the adaptive estimator automatically realizes the biasvariance compromise: whenever b A belongs to some Besov ball (see [START_REF] Comte | Adaptive estimation of mean and volatility functions in (auto-)regressive models[END_REF]), if r + 1 ≥ α and n∆ 2 = o(1), b m achieves the optimal corresponding nonparametric rate, without logarithmic loss contrary to Hoffmann's adaptive estimator (see Hoffmann (1999, Theorem 5 p.159)). As mentioned above, Theorem 3.1 holds for the basis [W] and, if

N n = o((n∆) 1/2 / ln(n)), for [T] .

Adaptive estimation of the diffusion coefficient 4.1 Diffusion coefficient estimator: non adaptive case

To estimate σ 2 on A = [0, 1], we define

σ2 m = arg min t∈Sm γn (t), with γn (t) = 1 n n k=1 [U k∆ -t(X k∆ )] 2 , (20)
and

U k∆ = (X (k+1)∆ -X k∆ ) 2 ∆ . (21)
For diffusion coefficient estimation under our asymptotic framework, it is now well known that rates of convergence are faster than for drift estimation. This is the reason why the regression-type equation has to be more precise than for b. Let us set

ψ = 2σ ′ σb + [(σ ′ ) 2 + σσ"]σ 2 . ( 22
)
Some computations using Ito's formula and Fubini's theorem lead to

U k∆ = σ 2 (X k∆ ) + V k∆ + R k∆ where V k∆ = V (1) k∆ + V (2) k∆ + V (3)
k∆ with

V (1) k∆ = 1 ∆   (k+1)∆ k∆ σ(X s )dW s 2 - (k+1)∆ k∆ σ 2 (X s )ds   V (2) k∆ = 2 ∆ (k+1)∆ k∆ ((k + 1)∆ -s)σ ′ (X s )σ 2 (X s )dW s , V (3) k∆
= 2b(X k∆ ) (k+1)∆ k∆ σ(X s )dW s ,
and

R k∆ = 1 ∆ (k+1)∆ k∆ b(X s )ds 2 + 2 ∆ (k+1)∆ k∆ (b(X s ) -b(X k∆ ))ds (k+1)∆ k∆ σ(X s )dW s + 1 ∆ (k+1)∆ k∆ [(k + 1)∆ -s]ψ(X s )ds,
Obviously, the main noise term in the above decomposition must be V

k∆ as will be proved below.

Risk of the non adaptive estimator

As for the drift, we write:

γn (t) -γn (σ 2 ) = σ 2 -t 2 n + 2 n n k=1 (σ 2 -t)(X k∆ )V k∆ + 2 n n k=1 (σ 2 -t)(X k∆ )R k∆ .
We denote by σ 2 m the orthogonal projection of σ 2 on S m and define

νn (t) = 1 n n k=1 t(X k∆ )V k∆ . Again we use that γn (σ 2 m ) -γn (σ 2 ) ≤ γn (σ 2 m ) -γn (σ 2 ) to obtain σ2 m -σ 2 2 n ≤ σ 2 m -σ 2 2 n + 2ν n (σ 2 m -σ 2 m ) + 2 n n k=1 (σ 2 m -σ 2 m )(X k∆ )R k∆ .
Analogously to what was done for the drift, we can cancel on both sides the common term

σ 2 1 I A c 2 n . This yields σ2 m -σ 2 A 2 n ≤ σ 2 m -σ 2 A 2 n + 2ν n (σ 2 m -σ 2 m ) + 2 n n k=1 (σ 2 m -σ 2 m )(X k∆ )R k∆ . ( 23
)
And, we obtain the result

Proposition 4.1 Let ∆ = ∆ n be such that ∆ n → 0, n∆ n / ln 2 (n) → +∞ when n → +∞. Assume that [A1]-[A3] hold and consider a model S m in the collection [DP] with N n = o(n∆/ ln 2 (n))
where N n is defined in (H 2 ). Then the estimator σ2 m of σ 2 defined by ( 20) is such that

E( σ2 m -σ 2 A 2 n ) ≤ 7π 1 σ 2 m -σ 2 A 2 + K E(σ 4 (X 0 ))D m n + K ′ ∆ 2 + K" n , (24)
where σ 2 A = σ 2 1 I A , and K, K ′ , K" are some positive constants.

Let us make some comments on the rates of convergence. If σ 2 A belongs to a ball of some Besov space, say σ 2 A ∈ B α,2,∞ ([0, 1]), and

σ 2 A α,2,∞ ≤ L, with r + 1 ≥ α, then σ 2 A -σ 2 m 2 ≤ C(α, L)D -2α
m . Therefore, if we choose D m = n 1/(2α+1) , we obtain

E( σ2 m -σ 2 A 2 n ) ≤ C(α, L)n -2α/(2α+1) + K ′ ∆ 2 + K" n . (25)
The first term n -2α/(2α+1) is the optimal nonparametric rate proved by Hoffmann (1999).

Moreover, under the standard condition ∆ 2 = o(1/n), the last two terms are O(1/n), i.e.

negligible with respect to n -2α/(2α+1) .

Adaptive diffusion coefficient estimator.

As previously, the second step is to ensure an automatic selection of D m , which does not use any knowledge on σ 2 . This selection is done by

m = arg min m∈Mn γn (σ 2 m ) + pen(m) . (26)
We denote by σ2 m the resulting estimator and we need to determine the penalty pen as for b. For simplicity, we use the same notation m in [START_REF] Kessler | Estimating equations based on eigenfunctions for a discretely observed diffusion process[END_REF] as in [START_REF] Comte | A new algorithm for fixed design regression and denoising[END_REF] although they are different. We can prove the following theorem. Theorem 4.1 Let ∆ = ∆ n be such that ∆ n → 0, n∆ n / ln 2 (n) → +∞ when n → +∞.

Assume that [A1]-[A3] hold. Consider the nested collection of models [DP] with maximal dimension

N n ≤ n∆/ ln 2 (n). Let pen(m) ≥ κσ 4 1 D m n , ( 27
)
where κ is a universal constant. Then, the estimator σ2 m of σ 2 with m defined by ( 26) is such that

E( σ2 m -σ 2 A 2 n ) ≤ C inf m∈Mn σ 2 m -σ 2 A 2 + pen(m) + K ′ ∆ 2 + K" n . ( 28
)
As for the drift, it is possible to choose pen(m) = κσ 4 1 D m /n, but this is not what is done in practice. Moreover, making such a choice, it follows from (28) that the adaptive estimator automatically realizes the bias-variance compromise. Whenever σ 2

A belongs to some Besov ball (see [START_REF] Jacod | Non-parametric Kernel Estimation of the Coefficient of a Diffusion[END_REF]), if n∆ 2 = o(1) and r + 1 ≥ α, σ2 m achieves the optimal corresponding nonparametric rate n -2α/(2α+1) , without logarithmic loss contrary to Hoffmann's adaptive estimator (see Hoffmann (1999, Theorem 6 p.160)). As mentioned for b, Proposition 4.1 and Theorem 4.1 hold for the basis [W] under the same assumptions on

N n . For [T], N n = o((n∆) 1/2 / ln(n)) is needed.

Examples and numerical simulation results

In this section, we consider examples of diffusions and implement the estimation algorithms on simulated data. To simulate sample paths of diffusion, we use the retrospective exact simulation algorithms proposed by [START_REF] Beskos | Retrospective exact simulation of diffusion sample paths with applications[END_REF] and [START_REF] Beskos | Exact simulation of diffusions[END_REF].

Contrary to the Euler scheme, these algorithms produce exact simulation of diffusions under some assumptions on the drift and diffusion coefficient. Therefore, we choose our examples in order to fit in these conditions in addition with our set of assumptions. For sake of simplicity, we focus on models that can be simulated by the simplest algorithm of [START_REF] Beskos | Retrospective exact simulation of diffusion sample paths with applications[END_REF], which is called EA1. More precisely, consider a diffusion model given by the stochastic differential equation

dX t = b(X t )dt + σ(X t )dW t . (29)
We assume that there is a C 2 one-to-one mapping F on R such that ξ t = F (X t ) satisfies

dξ t = α(ξ t )dt + dW t . (30)
To produce an exact realization of the random variable ξ ∆ , given that ξ 0 = x, the exact algorithm EA1 requires that α be C 1 , α 2 +α ′ be bounded from below and above. Moreover, setting A(ξ) = ξ α(u)du, the function

h(ξ) = exp (A(ξ) -(ξ -x) 2 /2∆) (31)
must be integrable on R and an exact realization of a random variable with density proportional to h must be possible. Provided that the process (ξ t ) admits a stationary distribution that is also possibly simulatable, using the Markov property, the algorithm can therefore produce an exact realization of a discrete sample (ξ k∆ , k = 0, 1, . . . , n + 1) in stationary regime. We deduce an exact realization of (X k∆ = F -1 (ξ k∆ ), k = 0, . . . , n + 1).

In all examples, we estimate the drift function α(ξ) and the constant 1 for models like [START_REF] Spokoiny | Adaptive drift estimation for nonparametric diffusion model[END_REF] or both the drift b(x) and the diffusion coefficient σ 2 (x) for models like [START_REF] Rao | Statistical inference for diffusion type processes[END_REF]. Let us note that Assumptions [A1]-[A2]-[A3] are fulfilled for all the models (ξ t ) below. For the models (X t ), the ergodicity and the exponential β-mixing property hold.

Examples of diffusions

Family 1

First, we consider (29) with

b(x) = -θx, σ(x) = c(1 + x 2 ) 1/2 . ( 32
)
Standard computations of the scale and speed densities show that the model is positive recurrent for θ + c 2 /2 > 0. In this case, its stationary distribution has density

π(x) ∝ 1 (1 + x 2 ) 1+θ/c 2 .
If X 0 = η has distribution π(x)dx, then, setting ν = 1 + 2θ/c 2 , ν 1/2 η has Student distribution t(ν) which can be easily simulated. Now, we consider F 1 (x) = x 0 1/(c(1 + x 2 ) 1/2 )dx = arg sinh(x)/c. By the Ito formula, ξ t = F 1 (X t ) satisfies [START_REF] Spokoiny | Adaptive drift estimation for nonparametric diffusion model[END_REF] with

α(ξ) = -(θ/c + c/2) tanh(cξ). ( 33
)
Assumptions [A1]-[A3] hold for (ξ t ) with ξ 0 = F 1 (X 0 ). Moreover,

α 2 (ξ) + α ′ (ξ) = {(θ/c + c/2) 2 + θ + c 2 /2} tanh 2 (cξ) -(θ + c 2 /2)
is bounded from below and above. And

A(ξ) = ξ 0 α(u)du = -(1/2 + θ/c 2 ) log(cosh(cξ)) ≤ 0,
so that exp(A(ξ)) ≤ 1. Therefore, function [START_REF] Viennet | Inequalities for absolutely regular sequences: application to density estimation[END_REF] is integrable for all x and by a simple rejection method, we can produce a realization of a random variable with density proportional to h(ξ) using a random variable with density N (x, ∆).

Note that model [START_REF] Rao | Statistical inference for diffusion type processes[END_REF] satisfies Assumptions [A1]-[A3] except that σ 2 (x) is not bounded from above. Nevertheless, since X t = F -1 1 (ξ t ) = sinh(c ξ t ), the process (X t ) is exponentially β-mixing. The upper bound σ 2 1 that appears explicitly in the penalty function must be replaced by an estimated upper bound.

-

Family 2

For the second family of models, we start with an equation of type [START_REF] Spokoiny | Adaptive drift estimation for nonparametric diffusion model[END_REF] where the drift is now (see [START_REF] Barndorff-Nielsen | Hyperbolic distributions and distributions on hyperbolae[END_REF]) The model for (ξ t ) is positive recurrent on R for θ > 0. Its stationary distribution is given by

α(ξ) = -θ ξ (1 + c 2 ξ 2 ) 1/2 . ( 34
) -3 -2 -1 0 1 2
π(ξ)dξ ∝ exp(-2 θ c 2 (1 + c 2 ξ 2 ) 1/2 ) = exp(-2θ|ξ|/c) × exp(ϕ(ξ)),
where exp ϕ(ξ) ≤ 1 so that a random variable with distribution π(ξ)dξ can be simulated by simple rejection method using a double exponential variable with distribution proportional to exp(-2θ|ξ|/c). The conditions required to perform an exact simulation of (ξ t ) hold. More precisely, α 2 + α ′ is bounded from below and above and 31) is integrable and we can produce a realization of a random variable with density proportional to [START_REF] Viennet | Inequalities for absolutely regular sequences: application to density estimation[END_REF]. Lastly, Assumptions [A1]-[A3] also hold for this model. Now, we consider X t = F 2 (ξ t ) = arg sinh(cξ t ) which satisfies a stochastic differential equation with coefficients:

A(ξ) = ξ 0 α(u)du = -(θ/c 2 )(1 + c 2 ξ 2 ) 1/2 . Hence exp(A(ξ)) ≤ 1, (
b(x) = -θ + c 2 2 cosh(x) sinh(x) cosh 2 (x) , σ(x) = c cosh(x) . ( 35
)
The process (X t ) is exponentially β-mixing as (ξ t ). The diffusion coefficient σ(x) is not bounded from below but has an upper bound.

To obtain a different shape for the diffusion coefficient, showing two bumps, we consider X t = G(ξ t ) = arg sinh(ξ t -5) + arg sinh(ξ t + 5) where (ξ t ) is as in ( 30)-(34). The function

dX t = -θ + c 2 /(2 cosh(X t )) (sinh(X t )/ cosh 2 (X t ))dt + (c/ cosh(X t ))dW t , n = 5000, ∆ = 1/20, θ = 3, c = 2,
dotted line: true, full line: estimate. The algorithm selects (p, r) equal to (0, 2) for the drift, (0, 3) for σ 2 . G(.) is invertible and its inverse has the following explicit expression,

G -1 (x) = 1 2 1/2 sinh(x) 49 sinh 2 (x) + 100 + cosh(x)(sinh 2 (x) -100) 1/2 .
The diffusion coefficient of (X t ) is given by

σ(x) = 1 (1 + (G -1 (x) -5) 2 ) 1/2 + 1 (1 + (G -1 (x) + 5) 2 ) 1/2 . ( 36
)
The drift is given by b

(x) = G ′ (G -1 (x))α(G -1 (x)) + 1 2 G"(G -1 (x)).

Estimation algorithms and numerical results

We do not give here a complete Monte-Carlo study but we illustrate how the algorithm works and what kind of estimate it delivers visually.

We consider the regular collection [DP] (see Section 2.2) . The algorithm minimizes the mean-square contrast and selects the space of approximation in the sense that it selects p and r for integers p and r such that 2 p (r + 1) ≤ N n ≤ n∆/ ln 2 (n) and r ∈ {0, 1, . . . , r max }.

Note that the degree is global in the sense that it is the same on all the intervals of the subdivision. We take r max = 9 in practice. Moreover, additive (but negligible) correcting terms are classically involved in the penalty (see [START_REF] Comte | A new algorithm for fixed design regression and denoising[END_REF]). Such terms avoid under-penalization and are in accordance with the fact that the theorems provide lower bounds for the penalty. The correcting terms are asymptotically negligible so they do not affect the rate of convergence. Thus, both penalties contain additional logarithmic terms which have been calibrated in other contexts by intensive simulation experiments (see [START_REF] Comte | Adaptive estimation of the spectrum of a stationary Gaussian sequence[END_REF]Rozenholc (2002, 2004)).

The constant κ in both penalties pen(m) and pen(m) has been set equal to 4.

We kept the idea that the adequate term in the penalty was E(σ 2 (X 0 ))/∆ for b and E(σ 4 (X 0 )) for σ 2 , instead of those obtained (σ 2 1 /∆ and σ 4 1 respectively). Indeed, in classical regression models, the corresponding coefficient is the variance of the noise. This variance is usually unknown and replaced by a rough estimate. Therefore, in penalties, σ 2 1 /∆ and σ 4 1 are replaced by empirical variances computed using initial estimators b, σ2 chosen in the collection and corresponding to a space with medium dimension: σ 2 1 /∆ for pen(.) is replaced ŝ2 1 = γ n ( b) (see [START_REF] Beskos | Exact simulation of diffusions[END_REF]); and σ 4 1 for the other penalty is replaced by ŝ2 2 = γn (σ 2 ) (see [START_REF] Florens-Zmirou | On estimating the diffusion coefficient from discrete observations[END_REF]).

Finally, for m = (p, r), the penalties pen(m) for i = 1 and pen(m) for i = 2 are given by 4 ŝ2

i n 2 p r + 1 + ln 2.5 (r + 1) . The simulation of sample paths does not rely on Euler schemes as in the estimation method. Therefore, the data simulation method is disconnected with the estimation procedures and cannot be suspected of being favorable to our estimation algorithm.

6 Proofs

Proof of Proposition 3.1

We recall that for A-supported functions, t 2 π = A t 2 (x)π(x)dx. Starting from ( 13)-( 14), we obtain bm

-b A 2 n ≤ b m -b A 2 n + 2 bm -b m π sup t∈Sm, t π =1 |ν n (t)| +2 bm -b m n   1 n∆ 2 n k=1 (k+1)∆ k∆ (b(X s ) -b(X k∆ ))ds 2   1/2 ≤ b m -b A 2 n + 1 8 bm -b m 2 π + 8 sup t∈Sm, t π =1 [ν n (t)] 2 + 1 8 bm -b m 2 n + 8 n∆ 2 n k=1 (k+1)∆ k∆ (b(X s ) -b(X k∆ ))ds 2
Because the L 2 -norm, . π , and the empirical norm [START_REF] Bibby | Martingale estimation functions for discretely observed diffusion processes[END_REF] are not equivalent, we must introduce a set on which they are and afterwards, prove that this set has small probability.

Let us define (see [START_REF] Barlow | Semi-martingale inequalities via the Garsia-Rodemich-Rumsey Lemma and applications to local times[END_REF])

Ω n = ω/ t 2 n t 2 π -1 ≤ 1 2 , ∀t ∈ ∪ m,m ′ ∈Mn (S m + S m ′ )/{0} . ( 37
) On Ω n , bm -b m 2 π ≤ 2 bm -b m 2 n , and bm -b m 2 n ≤ 2( bm -b A 2 n + b m -b A 2 n ).
Hence, some elementary computations yield:

1 4 bm -b A 2 n 1 I Ωn ≤ 7 4 b m -b A 2 n +8 sup t∈Sm, t π =1 [ν n (t)] 2 + 8 n∆ 2 n k=1 (k+1)∆ k∆ (b(X s ) -b(X k∆ ))ds 2
Now, using (3), we get

E (k+1)∆ k∆ (b(X s ) -b(X k∆ ))ds 2 ≤ ∆ (k+1)∆ k∆ E[(b(X s ) -b(X k∆ )) 2 ]ds ≤ c ′ ∆ 3 .
Consequently,

E( bm -b A 2 n 1 I Ωn ) ≤ 7 b m -b A 2 π + 32 E sup t∈Sm, t π =1 [ν n (t)] 2 + 32c ′ ∆. (38)
Next, using ( 5)-( 7)-( 8)-( 9)-( 13), it is easy to see that, since

t π = 1 ⇒ t 2 ≤ 1/π 0 , E sup t∈Sm, t π =1 [ν n (t)] 2 ≤ 1 π 0 E sup t∈Sm, t ≤1 [ν n (t)] 2 ≤ 1 π 0 λ∈Λm E[ν 2 n (ϕ λ )] = 1 π 0 n 2 ∆ 2 n k=1 E    λ∈Λm ϕ 2 λ (X k∆ ) (k+1)∆ k∆ σ 2 (X s )ds    ≤ Φ 2 0 D m π 0 n 2 ∆ 2 n k=1 E (k+1)∆ k∆ σ 2 (X s )ds = Φ 2 0 D m π 0 n∆ 2 E ∆ 0 σ 2 (X s )ds = Φ 2 0 E(σ 2 (X 0 ))D m π 0 n∆ .
Gathering bounds, and using the upper bound π 1 defined in (5), we get

E( bm -b A 2 n 1 I Ωn ) ≤ 7π 1 b m -b A 2 + 32 Φ 2 0 E(σ 2 (X 0 ))D m π 0 n∆ + 32c ′ ∆. Now, it remains to deal with Ω c n . Since bm -b A 2 n ≤ bm -b 2 n , it is enough to check that E( bm -b 2 n 1 I Ω c n ) ≤ c/n. Write the regression model as Y k∆ = b(X k∆ ) + ε k∆ with ε k∆ = 1 ∆ (k+1)∆ k∆ [b(X s ) -b(X k∆ )]ds + 1 ∆ (k+1)∆ k∆ σ(X s )dW s .
Recall that Π m denotes the orthogonal projection (with respect to the inner product of R n ) onto the subspace

{(t(X ∆ ), . . . , t(X n∆ )) ′ , t ∈ S m } of R n . We have ( bm (X ∆ ), . . . , bm (X n∆ )) ′ = Π m Y where Y = (Y ∆ , . . . , Y n∆ ) ′ .
Using the same notation for the function t and the vector (t(X ∆ ), . . . , t(X n∆ )) ′ , we see that b -bm

2 n = b -Π m b 2 n + Π m ε 2 n ≤ b 2 n + n -1 n i=1 ε 2 i∆ .
Therefore,

E b -bm 2 n 1 I Ω c n ≤ E b 2 n 1 I Ω c n + 1 n n k=1 E ε 2 k∆ 1 I Ω c n ≤ E 1/2 (b 4 (X 0 )) + E 1/2 (ε 4 ∆ ) P 1/2 (Ω c n ). By [A1](ii), we have E(b 4 (X 0 )) ≤ c(1 + E(X 4 0 )) = K.
With the Burholder-Davis-Gundy inequality, we find

E(ε 4 ∆ ) ≤ 2 3 1 ∆ ∆ 0 E[(b(X s ) -b(X ∆ )) 4 ]ds + 36 ∆ 3 E ∆ 0 σ 4 (X s )ds . Under [A1]-[A2](i)-[A3] and (3), we obtain E(ε 4 ∆ ) ≤ C(1 + σ 4 1 /∆ 2 ) := C ′ /∆ 2 .
The next lemma enables us to complete the proof. Lemma 6.1 Let Ω n be defined by (37) and assume that

n∆ n / ln 2 (n) → +∞ when n → +∞. Then, if N n ≤ O(n∆ n / ln 2 (n)) for collections [DP] and [W], and if N n ≤ O((n∆ n ) 1/2 / ln(n))
for collection [T],

P(Ω c n ) ≤ c n 4 . (39)
The proof of Lemma 6.1 is given in the appendix. Now, we gather all terms and use (39) to get (15).

Proof of Theorem 3.1

The proof relies on the following Bernstein-type inequality: Lemma 6.2 Under the assumptions of Theorem 3.1, for any positive numbers ǫ and v, we have

P n k=1 t(X k∆ )Z k∆ ≥ nǫ, t 2 n ≤ v 2 ≤ exp - n∆ǫ 2 2σ 2 1 v 2 .
Proof of Lemma 6.2. We use that n k=1 t(X k∆ )Z k∆ can be written as a stochastic integral. Consider the process:

H n u = H u = n k=1 1 I [k∆,(k+1)∆[ (u)t(X k∆ )σ(X u ) which satisfies H 2 u ≤ σ 2 1 t 2
∞ for all u ≥ 0. Then, denoting by M s = s 0 H u dW u , we get that

M (n+1)∆ = n k=1 t(X k∆ ) (k+1)∆ k∆ σ(X s )dW s , M (n+1)∆ = n k=1 t 2 (X k∆ ) (k+1)∆ k∆ σ 2 (X s )ds. Moreover, M s = s 0 H 2 u du ≤ nσ 2 1 ∆ t 2
n , ∀s ≥ 0, so that (M s ) and exp(λM sλ 2 M s /2) are martingales with respect to the filtration F s = σ(X u , u ≤ s). Therefore, for all s ≥ 0, c > 0, d > 0, λ > 0,

P(M s ≥ c, M s ≤ d) ≤ P e λMs-λ 2 2 M s ≥ e λc-λ 2 2 d ≤ e -(λc-λ 2 2 d) .
Therefore,

P(M s ≥ c, M s ≤ d) ≤ inf λ>0 e -(λc-λ 2 2 d) = e -c 2 2d .
Finally,

P n k=1 t(X k∆ )Z k∆ ≥ nǫ, t 2 n ≤ v 2 = P(M (n+1)∆ ≥ n∆ǫ, M (n+1)∆ ≤ nv 2 σ 2 1 ∆) ≤ exp - (n∆ǫ) 2 2nv 2 σ 2 1 ∆ = exp - nǫ 2 ∆ 2v 2 σ 2
E( b m -b A 2 n 1 I Ωn ) ≤ 7π 1 b m -b A 2 + 4pen(m) + 32E sup t∈Sm+S m, t π =1 [ν n (t)] 2 1 I Ωn -4E(pen( m)) + 32c ′ ∆.
The main problem here is to control the supremum of ν n (t) on a random ball (which depends on the random m). This is done by using the martingale property of ν n (t).

Let us introduce the notation

G m (m ′ ) = sup t∈Sm+S m ′ , t π =1
|ν n (t)|. Now, we plug in a function p(m, m ′ ), which will in turn fix the penalty:

G 2 m ( m)1 I Ωn ≤ [(G 2 m ( m) -p(m, m))1 I Ωn ] + + p(m, m) ≤ m ′ ∈Mn [(G 2 m (m ′ ) -p(m, m ′ ))1 I Ωn ] + + p(m, m).
And pen is chosen such that 8p(m, m ′ ) ≤ pen(m) + pen(m ′ ). More precisely, the next proposition determines the choice of p(m, m ′ ).

Proposition 6.1 Under the assumptions of Theorem 3.1, there exists a numerical con-

stant κ 1 such that, for p(m, m ′ ) = κ 1 σ 2 1 (D m + D m ′ )/(n∆), we have E[(G 2 m (m ′ ) -p(m, m ′ ))1 I Ωn ] + ≤ cσ 2 1 e -D m ′ n∆ .
Proof of Proposition 6.1. The result of Proposition 6.1 follows from the inequality of It is easy to see that the result of Theorem 3.1 follows from Proposition 6.1 with pen(m) ≥ κσ 2 1 D m /(n∆) and κ = 8κ 1 .

Proof of Proposition 4.1

First, we prove that

E( 1 n n k=1 R 2 k∆ ) ≤ K∆ 2 . ( 40
)
Proof of (40). With obvious convention, let R k∆ = R

(1)

k∆ + R (2) k∆ + R (3)
k∆ so that (40

) holds if E[(R (i) k∆ ) 2 ] ≤ K i ∆ 2 for i = 1, 2, 3. Using [A1], E[(R (1) k∆ ) 2 ] ≤ E (k+1)∆ k∆ b 2 (X s )ds 2 ≤ ∆E (k+1)∆ k∆ b 4 (X s )ds ≤ ∆ 2 E(b 4 (X 0 )) ≤ c∆ 2 . E[(R (2) k∆ ) 2 ] ≤ 1 ∆ 2   E (k+1)∆ k∆ (b(X s ) -b(X k∆ ))ds 4 E (k+1)∆ k∆ σ(X s )dW s 4   1/2 Using (3), we get E[(R (2) k∆ ) 2 ] ≤ c ′ ∆ 2 .
Lastly, using [A1]-[A2] and [START_REF] Gloter | Discrete sampling of an integrated diffusion process and parameter estimation of the diffusion coefficient[END_REF],

E[(R (3) k∆ ) 2 ] ≤ 1 ∆ E (k+1)∆ k∆ ((k + 1)∆ -s) 2 ψ 2 (X s )ds ≤ E(ψ 2 (X 0 )) ∆ 2 3 ≤ c"∆ 2 .
Therefore (40) is proved. Now, we turn back to [START_REF] Gobet | Nonparametric estimation of scalar diffusions based on low frequency data[END_REF] and recall that Ω n is defined by (37). The study is close to the one done for the drift estimator.

On Ω n , σ2 m -σ 2 m 2 π ≤ 2 σ2 m -σ 2 m 2 n , σ2 m -σ 2 A 2 n ≤ σ 2 m -σ 2 A 2 n + 1 8 σ2 m -σ 2 m 2 π + 8 sup t∈Sm, t π =1 ν2 n (t) + 1 8 σ2 m -σ2 m 2 n + 8 n n k=1 R 2 k∆ ≤ σ 2 m -σ 2 A 2 n + 3 8 σ2 m -σ 2 m 2 n + 8 sup t∈Sm, t π =1 ν2 n (t) + 8 n n k=1 R 2 k∆ .
Setting B m (0, 1) = {t ∈ S m , t ≤ 1} and B π m (0, 1) = {t ∈ S m , t π ≤ 1}, the following holds on Ω n :

1 4 σ2 m -σ 2 A 2 n ≤ 7 4 σ 2 m -σ 2 A 2 n + 8 sup t∈B π m (0,1) ν2 n (t) + 8 n n k=1 R 2 k∆ . Moreover E( sup t∈B π m (0,1) ν2 n (t)) ≤ 1 π 0 E( sup t∈Bm(0,1) ν2 n (t)) ≤ 1 π 0 λ∈Λm E(ν 2 n (ϕ λ )) ≤ 1 π 0 n 2 λ∈Λm E n k=1 ϕ 2 λ (X k∆ )V 2 k∆ ≤ Φ 2 0 D m π 0 n {12E(σ 4 (X 0 )) + 4∆C b,σ } where C b,σ = E((σ ′ σ 2 ) 2 (X 0 )) + σ 2 1 E(b 2 (X 0 ))
. Now using the condition on N n , we have ∆D m /n ≤ ∆N n /n ≤ ∆ 2 / ln 2 (n). This yields the first three terms of the right-hand-side of [START_REF] Hoffmann | Adaptive estimation in diffusion processes[END_REF].

The study on Ω c n is the same as for b with the regression model

U k∆ = σ 2 (X k∆ ) + η k∆ , where η k∆ = V k∆ +R k∆ . By standard inequalities, E(η 4 ∆ ) ≤ K{∆ 4 E(b 8 (X 0 ))+E(σ 8 (X 0 ))}. Hence, E(η 4
∆ ) is bounded. Moreover, using Lemma 6.1, P(Ω c n ) ≤ c/n 2 .

Proof of Theorem 4.1

This proof follows the same lines as the proof of Theorem 3.1. We start with a Bernsteintype inequality.

Lemma 6.3 Under the assumptions of Theorem 4.1,

P n k=1 t(X k∆ )V (1) k∆ ≥ nǫ, t 2 n ≤ v 2 ≤ exp -Cn ǫ 2 /2 2σ 4 1 v 2 + ǫ t ∞ σ 2 1 v and P 1 n n k=1 t(X k∆ )V (1) k∆ ≥ vσ 2 1 (2x) 1/2 + σ 2 1 t ∞ x, t 2 n ≤ v 2 ≤ exp(-Cnx). (41)
The non trivial link between the above two inequalities is enhanced by Birgé and Massart (1998) so that we just prove the first one.

Proof of Lemma 6.3. First we note that:

E e ut(X n∆ )V (1) n∆ |F n∆ = 1 + +∞ p=2 u p p! E (t(X n ∆)V (1) n∆ ) p |F n∆ ≤ 1 + +∞ p=2 u p p! |t(X n∆ )| p E |V (1) n∆ | p |F n∆ .
Next we apply successively the Hölder inequality and the Burkholder-Davis-Gundy inequality with best constant (Proposition 4.2 of [START_REF] Barlow | Semi-martingale inequalities via the Garsia-Rodemich-Rumsey Lemma and applications to local times[END_REF]). For a continuous martingale (M t ), with

M 0 = 0, for k ≥ 2, M * t = sup s≤t |M s | satisfies M * k ≤ ck 1/2 M 1/2 k
, with c a universal constant. And we obtain:

E(|V (1) n∆ | p |F n∆ ) ≤ 2 p-1 ∆ p    E   (n+1)∆ n∆ σ(X s )dW s 2p |F n∆   + E (n+1)∆ n∆ σ 2 (X s )ds p |F n∆    ≤ 2 p-1 ∆ p (c 2p (2p) p ∆ p σ 2p 1 + ∆ p σ 2p 1 ) ≤ (2σ 1 c) 2p p p .
Therefore,

E e ut(X n∆ )V (1) n∆ |F n∆ ≤ 1 + ∞ k=2 p p p! (4uσ 2 1 c 2 ) p |t(X n∆ )| p .
Using p p /p! ≤ e p-1 , we find

E e ut(X n∆ )V (1) n∆ |F n∆ ≤ 1 + e -1 ∞ k=2 (4uσ 2 1 c 2 e) p |t(X n∆ )| p ≤ 1 + e -1 (4uσ 2 1 c 2 e) 2 t 2 (X n∆ ) 1 -(4uσ 2 1 c 2 e t ∞ ) . Now, let us set a = e(4σ 2 1 c 2 ) 2 and b = 4σ 2 1 c 2 e t ∞ .
Since for x ≥ 0, 1 + x ≤ e x , we get, for all u such that bu < 1,

E e ut(X n∆ )V (1) n∆ |F n∆ ≤ 1 + au 2 t 2 (X n∆ ) 1 -bu ≤ exp au 2 t 2 (X n∆ ) 1 -bu .
This can also be written:

E exp ut(X n∆ )V (1) n∆ - au 2 t 2 (X n∆ ) 1 -bu |F n∆ ≤ 1.
Therefore, iterating conditional expectations yields

E exp n k=1 ut(X k∆ )V (1)
k∆ -

au 2 t 2 (X k∆ ) 1 -bu ≤ 1.
Then, we deduce that

P n k=1 t(X k∆ )V (1) k∆ ≥ nǫ, t 2 n ≤ v 2 ≤ e -nuǫ E 1 I t 2 n ≤v 2 exp u n k=1 t(X k∆ )V (1) k∆ ≤ e -nuǫ E 1 I t 2 n ≤v 2 exp n k=1 (ut(X k∆ )V (1) k∆ - au 2 t 2 (X k∆ ) 1 -bu ) e (au 2 )/(1-bu) P n k=1 t 2 (X k∆ ) ≤ e -nuǫ e (nau 2 v 2 )/(1-bu) E exp n k=1 (ut(X k∆ )V (1) k∆ - au 2 t 2 (X k∆ ) 1 -bu ) ≤ e -nuǫ e (nau 2 v 2 )/(1-bu) .
The inequality holds for any u such that bu < 1. In particular, u = ǫ/(2av 2 + ǫb) gives

-uǫ + av 2 u 2 /(1 -bu) = -(1/2)(ǫ 2 /(2av 2 + ǫb) and therefore P n k=1 t(X k∆ )V (1) k∆ ≥ nǫ, t 2 n ≤ v 2 ≤ exp -n ǫ 2 /2 2av 2 + ǫb .
As for b m, we introduce the additional penalty terms and obtain that the risk satisfies

E( σ2 m -σ 2 A 2 n 1 I Ωn ) ≤ 7π 1 σ 2 m -σ 2 A 2 + 4 pen(m) + 32E sup t∈B π m, m(0,1) (ν n (t)) 2 1 I Ωn -4E( pen( m)) + K ′ ∆ 2 (42)
where

B π m,m ′ (0, 1) = {t ∈ S m + S m ′ , t π = 1}. Let us denote by Gm (m ′ ) = sup t∈B π m,m ′ (0,1)
|ν (1) n (t)| the main quantity to be studied, where ν( 1)

n (t) = 1 n n k=1 t(X k∆ )V (1)
k∆ , ν( 2)

n (t) = 1 n n k=1 t(X k∆ )(V (2)
k∆ + V

k∆ ).

As for the drift, we write

E( G2 m ( m)) ≤ E[( G2 m ( m) -p(m, m))1 I Ωn ] + + E(p(m, m)) ≤ m ′ ∈Mn E[( G2 m (m ′ ) -p(m, m ′ ))1 I Ωn ] + + E(p(m, m)).
Now we have the following statement.

D m + D m ′ n + Φ 2 0 π 0 D m + D m ′ n 2 ,
where κ * is a numerical constant, we have

E[( G2 m (m ′ ) -p(m, m ′ ))1 I Ωn ] + ≤ cσ 4 1 e -D m ′ n .
The result of Proposition 6.2 is obtained from inequality (41) of Lemma 6. Note that there is a difference between Propositions 6.1 and 6.2 which comes from the additional term t ∞ appearing in Lemma 6.

(ν (2) n (t)) 2 ≤ 1 π 0 E sup t∈Sn, t =1 (ν (2) n (t)) 2 ≤ KC b,σ Φ 2 0 ∆N n π 0 n ≤ K ′ ∆ 2 .
The result of Theorem 4.1 follows.

7 Appendix. Proof of Lemma 6.1

Using Baraud et al. (2001a), we prove that, for all n and ∆ > 0,

P(Ω c n ) ≤ 2nβ X (q n ∆) + 2n 2 exp(-C 0 n q n L n (φ) ) ( 43
)
where C 0 is a constant depending on π 0 , π 1 , q n is an integer such that q n < n and L n (φ) is a quantity depending on the basis of the largest nesting space S n of the collection and is defined below. We recall that N n = dim(S n ).

Proof of (43). We use Berbee's coupling method as in Viennet (1997), Proposition 5.1 and its proof p.484. We assume that n = 2p n q n . Then there exist random variables X * i∆ , i = 1, ..., n satisfying the following properties:

• For ℓ = 1, ..., p n , the random vectors U ℓ,1 = (X [2(ℓ-1)qn+1]∆ , ..., X (2ℓ-1)qn∆ ) ′ and U * ℓ,1 = (X * [2(ℓ-1)qn+1]∆ , ..., X * (2ℓ-1)qn∆ ) ′ have the same distribution, and so have the vectors U ℓ,2 = (X [(2ℓ-1)qn+1]∆ , ..., X 2ℓqn∆ ) ′ and U * ℓ,2 = (X * [(2ℓ-1)qn+1]∆ , ..., X * 2ℓqn∆ ) ′ .

• For ℓ = 1, ..., p n , P( U ℓ,1 = U * ℓ,1 ) ≤ β X (q n ∆) and P( U ℓ,2 = U * ℓ,2 ) ≤ β X (q n ∆).

• For each δ ∈ {1, 2}, the random vectors U * 1,δ , ..., U * pn,δ are independent.

Let us define Ω * = {X i∆ = X * i∆ , i = 1, . . . , n}. We have P(Ω c n ) ≤ P(Ω c n ∩ Ω * ) + P(Ω * c ) and clearly P(Ω * c ) ≤ 2p n β X (q n ∆) ≤ nβ X (q n ∆).

(44)

Thus, (43) holds if we prove

P(Ω c n ∩ Ω * ) ≤ 2N 2 n exp -A 0 π 2 0 π 1 n q n L n (φ) ( 45
)
where L n (φ) is defined as follows. Let (ϕ λ ) λ∈Λn be a L 2 (A, dx)-orthonormal basis of S n and as in [START_REF] Baraud | Adaptive estimation in an autoregression and a geometrical β-mixing regression framework[END_REF], define the matrices: By choosing x = (ρ 0 π 0 ) 2 /(16π 1 L n (φ)) and ρ 0 = 1/2, and reminding that π 0 ≤ π 1 , we obtain that sup

V = A ϕ 2 λ (x)ϕ 2 λ ′ (x)dx
P λ∈Λn a 2 λ ≤1 λλ ′ |a λ a λ ′ ||v n (ϕ λ ϕ λ ′ )| ≤ ρ 0 π 0 = π 0 2 .

  Equation (1) has a unique strong solution. Note that [A2](ii) is only used for the estimation of σ 2 and not for b. Elementary computations show that the scale density s(x) = exp -2 x 0 b(u) σ 2 (u) du satisfies -∞ s(x)dx = +∞ = +∞ s(x)dx and the speed density m(x) = 1/(σ 2 (x)s(x)) satisfies +∞ -∞ m(x)dx = M < +∞. Hence, Model (1) admits a unique invariant probability π(x)dx with π(x) = M -1 m(x). Now we assume that [A3 ] X 0 = η has distribution π.

Figure 1 :

 1 Figure 1: dX t = -(θ/c + c/2) tanh(cX t ) + dW t , n = 5000, ∆ = 1/20, θ = 6, c = 2, dotted line: true, full line: estimate. The algorithm selects (p, r) equal to (0, 1) for the drift, (0, 2) for σ 2 .

Figure 2 :

 2 Figure 2:dX t = -θX t dt + c(1 + X 2 t ) 1/2 dW t , n = 5000, ∆ = 1/20, θ = 6, c = 2, dotted line: true, full line: estimate. The algorithm selects (p, r) equal to (0, 1) for the drift, (0, 2) for σ 2 .

Figure 3 :

 3 Figure 3:dX t =θ + c 2 /(2 cosh(X t )) (sinh(X t )/ cosh 2 (X t ))dt + (c/ cosh(X t ))dW t , n =

Figure 4 :

 4 Figure 4: Two paths for the two-bumps diffusion coefficient model X t = G(ξ t ), dξ t = -θξ t /(1 + c 2 ξ 2 t ) 1/2 dt + dW t , G(x) = arg sinh(x -5) + arg sinh(x + 5), n = 5000, ∆ = 1/20, θ = 1, c = 10, dotted line: true, full line: estimate. The algorithm selects (p, r) equal to (0, 3)(above) and (2, 0) (below) for the drift, (0, 6) (above) and (1, 3) (below) for σ 2 .

Figures 1 -

 1 Figures 1-4 illustrate our simulation results. We have plotted the data points (X k∆ , Y k∆ ) (see[START_REF] Beskos | Retrospective exact simulation of diffusion sample paths with applications[END_REF]) and (X k∆ , U k∆ ) (see[START_REF] Genon-Catalot | Nonparametric estimation of the diffusion coefficient by wavelet methods[END_REF]), the true functions b and σ 2 and the estimated functions based on 95% of data points. Parameters have been chosen in the admissible range of ergodicity. The sample size n = 5000 and the step ∆ = 1/20 are in accordance with the asymptotic context (great n's and small ∆'s) and may be relevant for applications in finance. It clearly appears that the estimated functions stick very well to the true ones.

1 . 2 n = b m -b A 2 n 1 I 2 n 1 I

 122121 Now we turn to the proof of Theorem 3.1. As in the proof of Proposition 3.1, we have to split b mb A Ωn + b mb A Ω c n . For the study on Ω c n , the end of the proof of Proposition 3.1 can be used. Now, we focus on what happens on Ω n . From the definition of b m, we have, ∀m ∈ M n , γ n ( b m) + pen( m) ≤ γ n (b m ) + pen(m). We proceed as in the proof of Proposition 3.1 with the additional penalty terms (see (38)) and obtain

Lemma 6 .

 6 2 by the L 2 -chaining technique used in Baraud et al. (2001b) (see Section 7 p.44-47, Lemma 7.1, with s 2 = σ 2 1 /∆).

Proposition 6 . 2

 62 Under the assumptions of Theorem 4.1, for p(m, m ′ ) = κ * σ 4 1

  3 by a L 2 (π)-L ∞ chaining technique. For a description of this method, in a more general setting, we refer to Proposition 2-4 pp.282-287 in Comte (2001), to Theorem 5 in Birgé and Massart (1998) and to Proposition 7, Theorem 8 and Theorem 9 in Barron et al. (1999).

3 .-σ 2 A 2 n ) ≤ 7π 1 σ 2 m -σ 2 A 2 +-σ 2 A 2 n 1 I

 322222221 For this reason, we need to use thatλ∈Λm β λ ψ λ ∞ / sup λ∈Λm |β λ | ≤ |ψ λ | ∞ ≤ (r max + 1)Dfor (ψ λ ) λ∈Λm an L 2 (π)-orthonormal basis built by orthonormalisation of the (ϕ λ )'s. This explains the additional term appearing in p(m, m ′ ).Choosing pen(m) ≥ κσ 4 1 D m /n with κ = 16κ * , we deduce from (42), Proposition 6.2 andD m ≤ N n ≤ n∆/ ln 2 (n) that, E( σ2 m n (t)) 2 + K ′ ∆ 2 + E( σ2 m Ω c n ). The bound for E( σ2 mσ 2 2 n 1 I Ω c n ) isthe same as the one given in the end of the proof of Proposition 4.1. It is less than c/n provided that N n ≤ n∆/ ln 2 (n) for [DP] and [W] and N 2 n ≤ n∆/ ln 2 (n) for [T]. And since the spaces all contained in a space denoted by S n with dimension N n bounded as right above, we have E sup t∈B π m, m(0,1)

1 / 2 λv n (t 2 )λ∈Λn a 2 λP λ∈Λn a 2 λ

 12222 ,λ ′ ∈Λn×Λn , B = ( ϕ λ ϕ λ ′ ∞ ) λ,λ ′ ∈Λn×Λn . Then we set L n (φ) = max{ρ 2 (V ), ρ(B)}, where, for any symmetricmatrix M = (M λ,λ ′ ), ρ(M ) = sup {a λ }, P λ a 2 λ ≤1 λ,λ ′ |a λ ||a λ ′ ||M λ,λ ′ |. Now we prove (45). Let P * (•) := P(•∩Ω * ). We use Baraud (2002, Claim 2 in Proposition 4.2). Consider v n (t) = (1/n) n i=1 [t(X i∆ ) -E(t(X i∆ ))], B π (0, 1) = {t ∈ S n , t π ≤ 1} and B(0, 1) = {t ∈ S n , t ≤ 1}. As, on A, π 0 ≤ π(x) ≤ π 1 ≥ ρ 0 ≤ P * sup t∈B(0,1) v n (t 2 ) ≥ π 0 ρ 0 ≤1 λ,λ ′ ∈Λn |a λ a λ ′ ||v n (ϕ λ ϕ λ ′ )| ≥ π 0 ρ 0  On the set ∀(λ, λ ′ ) ∈ Λ 2 n , |v n (ϕ λ ϕ λ ′ )| ≤ 2V λλ ′ (2π 1 x) 1/2 + 3B λλ ′ x , we have sup ≤1 λ,λ ′ ∈Λn |a λ a λ ′ ||v n (ϕ λ ϕ λ ′ )| ≤ 2ρ(V )(2π 1 x) 1/2 + 3ρ(B)x.

Acknowledgements. The authors wish to thank the Associate Editor and anonymous referees for comments that helped to significantly improve the paper.

This leads to

v n (t 2 ) ≥ 1 2

The proof of Inequality (45) is then achieved by using the following claim, which is exactly Claim 6 p.872 in the proof of Proposition 7 of Baraud et al. (2001a).

Claim 7.1 Let (ϕ λ ) λ∈Λn be an L 2 (A, dx) orthonormal basis of S n . Then, for all x ≥ 0 and all integer q, 1 ≤ q ≤ n,

Claim 7.1 implies that

, 43) to complete the proof of Lemma 6.1. By assumption, the diffusion process X is geometrically β-mixing. So, for some constant θ, β X (q n ∆) ≤ e -θqn∆ . Provided that ∆ = ∆ n satisfies ln(n)/(n∆) → 0, it is possible to take q n = [5 ln(n)/(θ∆)] + 1. This yields

The above constraint on ∆ must be strengthened. Indeed, to ensure (39), we need that