Fabienne Comte

Valentine Genon-Catalot

Yves Rozenholc

Non-parametric estimation of the coefficients of ergodic diffusion processes based on high-frequency data

Introduction

The content of this chapter is directly inspired by [START_REF] Comte | Nonparametric estimation of a discretely observed integrated diffusion model[END_REF][START_REF] Comte | Penalized nonparametric mean square estimation of the coefficients of diffusion processes[END_REF]. We consider non-parametric estimation of the drift and diffusion coefficients of a one-dimensional diffusion process. The main assumption on the diffusion model is that it is ergodic and geometrically βmixing. The sample path is assumed to be discretely observed with a small regular sampling interval ∆. The estimation method that we develop is based on a penalized mean square approach. This point of view is fully investigated for regression models in [START_REF] Comte | Adaptive estimation of the spectrum of a stationary Gaussian sequence[END_REF]Rozenholc (2002, 2004). We adapt it to discretized diffusion models.

Model and assumptions

Let (X t ) t≥0 be a one-dimensional diffusion process with dynamics described by the stochastic differential equation:

dX t = b(X t )dt + σ(X t )dW t ,t ≥ 0,X 0 = ⌘, (5.1)
where (W t ) is a standard Brownian motion and ⌘ is a random variable independent of (W t ). Consider the following assumptions:

[A1] -1  l<r +1, b and σ belong to C 1 ((l, r)), σ(x) > 0 for all x 2 (l, r).

[A2] For x 0 ,x 2 (l, r), let s(x)=exp(-2 R x x 0 b(u)/σ 2 (u)du) denote the scale density and m(x)=1/[σ 2 (x)s(x)] the speed density. We assume

Z l s(x)dx =+1 = Z r s(x)dx, Z r l m(x)dx = M<1.
[A3] The initial random variable ⌘ has distribution ⇡(x)dx = m(x)/M 1 (l,r) (x)dx.

Under [A1] -[A2], equation (5.1) admits a unique strong solution with state space the open interval (l, r) of the real line. Moreover, it is positive recurrent on this interval and admits as unique stationary distribution the normalized speed density ⇡. With the additional assumption [A3], the process (X t ) is strictly stationary, with marginal distribution ⇡(x)dx, ergodic and β-mixing, i.e. lim t!+1 β X (t)=0where β X (t) denotes the β-mixing coefficient of (X t ). For stationary Markov processes such as (X t ), the β-mixing coefficient has the following explicit expression

β X (t)= Z r l ⇡(x)dxkP t (x, dx 0 ) -⇡(x 0 )dx 0 k TV .
(5.2)

The norm k.k TV is the total variation norm and P t denotes the transition probability (see e.g. [START_REF] Genon-Catalot | Stochastic volatility models as hidden markov models and statistical applications[END_REF] for a review).

The statistical study relies on a stronger mixing condition which is satisfied in most standard models.

[A4] There exist constants K>0 and ✓>0 such that:

β X (t)  Ke -✓t .

(5.3)

In some cases, assumption [A4] can be checked directly using formula (5.2) (see Proposition 5.14 below). Otherwise, simple sufficient conditions are available (see e.g. Proposition 1 in [START_REF] Pardoux | On the Poisson equation and diffusion approximation[END_REF]). Lastly, we strengthen assumptions on b and σ to deal altogether with finite or infinite boundaries and keep a general, simple and clear framework. We also need a moment assumption for ⇡ and that σ be bounded.

[A5] Let I =[l, r] \ R.

(i) Assume that b 2 C 1 (I), b 0 bounded on I, σ 2 2 C 2 (I), (σ 2 ) 00 bounded on I. (ii) For all x 2 I, σ 2 (x)  σ 2

1 .

[A6] E(⌘ 8 ) < 1.

The following property will be useful.

Lemma 5.1 Under Assumptions [A1] -[A3] and [A5] -[A6]

, for all t, s such that |t -s|1, for 1  i  4, E((X t -X s ) 2i )  c|t -s| i .

Proof. Using the strict stationarity, we only need to study E((X t -X 0 ) 2i ) for t  1. By the Minkowski inequality,

(X t -X 0 ) 2i  2 2i-1 [( Z t 0 b(X s )ds) 2i +( Z t 0 σ(X s )dW s ) 2i ].
For the drift term, we use the Hölder inequality, [A5] and the strict stationarity to get:

E  ( Z t 0 b(X s )ds) 2i  t 2i-1 Z t 0 E(b 2i (X s ))ds  t 2i C(1 + E(⌘ 2i )),
with C a constant. For the diffusion term, we use the Burkholder-Davis-Gundy inequality and obtain:

E  ( Z t 0 σ(X s )dW s ) 2i  C E( Z t 0 σ 2 (X s )ds) i  Ct i σ 2i 1
with C a constant. This gives the result.

Observations and asymptotic framework

We assume that the sample path X t is observed at n +1discrete instants with sampling interval ∆. The asymptotic framework that we consider is the context of high-frequency data: the sampling interval ∆=∆ n tends to 0 as n tends to infinity. Moreover, we assume that the total length n∆ n of the time interval where observations are taken tends to infinity. This is a classical framework for ergodic diffusion models: it allows us to estimate simultaneously the drift b and the diffusion coefficient σ and to enlighten us about the different rates of estimation of these two coefficients. For the penalized non-parametric method developed here, the asymptotic framework has to fulfill the following strengthened condition.

[A7] As n tends to infinity, ∆=∆ n ! 0 and n∆ n / ln 2 n ! +1.

To simplify notations, we will drop the subscript and simply write ∆ for the sampling interval.

Estimation method

General description

We aim at estimating functions b and σ 2 of model (5.1) on a compact subset A of the state space (l, r). For simplicity, and without loss of generality, we assume from now on that A =[0, 1],

and we set b A = b1 A ,σ A = σ1 A .

(5.4)

The estimation method is inspired by what is done for regression models (see e.g. [START_REF] Comte | Adaptive estimation of the spectrum of a stationary Gaussian sequence[END_REF]Rozenholc (2002, 2004)). Suppose we have observations (x i ,y i ), i =1,...,nsuch that

y i = f (x i )+noise, (5.5)
where f is unknown. We consider a regression contrast of the form

t ! γ n (t)= 1 n n X i=1 (y i -t(x i )) 2 .
The aim is to build estimators for f by minimizing the least-square criterion γ n (t). For that purpose, we consider a collection of finite dimensional linear subspaces of L 2 ([0, 1]) and compute for each space an associated least-squares estimator. Afterwards, a data-driven procedure chooses from the resulting collection of estimators the "best" one, in a sense to be precised, through a penalization device. For adapting the method to discretized diffusion processes, we have to find a regression equation analogous to (5.5), i.e. find for f = b, σ 2 the appropriate couple (x i ,y i ) and the adequate regression equation. Of course, starting with a diffusion model, we do not find an exact regression equation but only a regression-type equation, one for the drift and another one for the diffusion coefficient. Hence, estimators for b and for σ 2 are built from two distinct constructions and the method does not require one to estimate the stationary density ⇡.

Spaces of approximation

Let us describe now some possible collections of spaces of approximation. We focus on two specific collections, the collection of dyadic regular piecewise polynomial spaces, denoted hereafter by [DP], and the collection of general piecewise polynomials, denoted by [GP], which is more complex. As for numerical implementation, algorithms for both collections are available and have been implemented on several examples.

In what follows, several norms for [0, 1]-supported functions are needed and the following notations will be used:

ktk =( R 1 0 t 2 (x)dx) 1/2 , ktk ⇡ =( R 1 0 t 2 (x)⇡(x)dx) 1/2 , ktk 1 =sup x2[0,1] |t(x)|.
(5.6) By our assumptions, the stationary density ⇡ is bounded from below and above on every compact subset of (l, r). Hence, let ⇡ 0 ,⇡ 1 denote two positive real numbers such that,

8x 2 [0, 1], 0 <⇡ 0  ⇡(x)  ⇡ 1 .
(5.7)

Thus, for [0, 1]-supported functions, the norms k.k and k.k ⇡ are equivalent.

Dyadic regular piecewise polynomials

Let r ≥ 0, p ≥ 0 be integers. On each subinterval I j =[ ( j -1)/2 p ,j/2 p ), j =1,...,2 p , consider r +1polynomials of degree `, ' j,`( x), `=0, 1,...r and set ' j,`( x)=0outside I j . The space S m , m =( p, r), is defined as generated by the D m =2 p (r + 1) functions (' j,`) . A function t in S m may be written as

t(x)= 2 p X j=1 r X `=0
t j,`'j,`( x).

The collection of spaces (S m ,m 2M n ) is such that, for r max a fixed integer,

M n = {m =(p, r),p 2 N,r 2{0, 1,...,r max }, 2 p (r max + 1)  N n }.
(5.8) In other words, D m  N n with N n  n. The maximal dimension N n will be subject to additional constraints. 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 `,`≥ 0), where the degree of Q `is equal to `, generating L 2 ([-1, 1]) (see Abramowitz and Stegun (1972), p.774). They satisfy

|Q `(x)|1, 8x 2 [-1, 1], Q `(1) = 1 and R 1 -1 Q 2 `(u)du =2 /(2`+ 1). Then we set P `(x)=( 2 `+ 1) 1/2 Q `(2x -1), to get an orthonormal basis of L 2 ([0, 1]). Finally, ' j,`( x)=2 p/2 P `(2 p x -j + 1)1 I j (x),j =1,...,2 p ,`=0, 1,...,r.
The space S m has dimension D m =2 p (r + 1) and its orthonormal basis described above satisfies

2 p X j=1 r X `=0 ' 2 j,` 1  D m (r + 1).
Hence, using notations (5.6), for all t 2 S m , ktk 1  (r + 1)

1/2 D 1/2 m ktk.
Collection [DP] is simple in the sense that one dimension D m is associated with a single space S m . In particular, since N n  n, the following holds:

Σ= X m2M n exp (-D m )= r max X r=0 X p:2 p (r+1)N n exp (-2 p (r + 1)) < 1. (5.9)
Finally, let us sum up the two key properties that are fulfilled by this collection. In Birgé and Massart (1998, p.337, Lemma 1), it is proved that property (5.10) is equivalent to:

There exists Φ 0 > 0, k X λ2Λ m ' 2 λ k 1  Φ 2 0 D m .
(5.11)

There are other collections of spaces satisfying the above two properties and for which our proofs apply (for instance, the trigonometric spaces, or the dyadic wavelet generated spaces).

General piecewise polynomials

A more general family can be described, the collection of general piecewise polynomials spaces denoted by [GP]. We first build the largest space S n of the collection whose dimension is denoted as above by N n (N n  n). For this, we fix an integer r max and let d max be an integer such that d max (r max +1) = N n . The space S n is linearly spanned by piecewise polynomials of degree r max on the regular subdivision of [0, 1] with step 1/d max . Any other space S m of the collection is described by a multi-index m =( d, j 1 ,...,j d-1 ,r 1 ,...,r d ) where d is the number of intervals of the partition,

j 0 := 0 <j 1 < ••• < j d-1 <j d := 1 are integers such that j i 2{1,...,d max -1} for i =1,...d- 1.
The latter integers define the knots j i /d max of the subdivision. Lastly r i  r max is the degree of the polynomial on the interval [j i-1 /d max ,j i /d max ), for i =1,...,d. A function t in S m can thus be described as

t(x)= d X i=1 P i (x)1 [j i-1 /d max ,j i /d max ) (x),
with P i a polynomial of degree r i . The dimension of S m is still denoted by D m and is equal to

P d i=1 (r i + 1) for all the d max -1 d-1
choices of the knots (j 1 ,...,j d-1 ). Note that the P i 's can still be decomposed by using the Legendre basis rescaled on the intervals [j i-1 /d max ,j i /d max ). The collection [GP] of models (S m ) m2M n is described by the set of indexes

M n = {m =(d, j 1 ,...,j d-1 ,r 1 ,...,r d ), 1  d  d max , j i 2{1,...,d max -1},r i 2{0,...,r max }} .
It is important to note that now, for all m 2M n , for all t 2 S m , since S m ⇢ S n , ktk 1  p (r max + 1)N n ktk.

(5.12)

Hence, the norm connection property still holds but only on the maximal space and no more on each individual space of the collection as for collection [DP]. This comes from the fact that regular partitions are involved in all spaces of [DP] and only in the maximal space of [GP]. Obviously, collection [GP] has higher complexity than [DP]. The complexity of a collection is usually evaluated through a set of weights (L m ) that must satisfy P m2M n e -L m D m < 1. For [DP], L m =1suits (see (5.9)). For [GP], we have to look at

X m2M n e -L m D m = d max X d=1 X 1j 1 <•••<j d-1 <d max X 0r 1 ,...,r d r max e -L m P d i=1 (r i +1) .
From the equality above, we deduce that the choice

L m D m = D m +ln ✓ d max -1 d -1 ◆ + d
X m2M n e -L m D m  d max X d=1 ✓ d max -1 d -1 ◆ (r max + 1) d e -dL n  d max -1 X d=0 ✓ d max -1 d ◆ [(r max + 1)e -L n ] d+1  (r max + 1) ⇥ 1+(r max + 1)e -L n ⇤ d max -1  (r max + 1) exp(d max (r max + 1)e -L n )  (r max + 1) exp(N n e -L n ).
Thus L m = L n =ln(N n ) ensures that the series is bounded. (For more details on these collections, see e.g. [START_REF] Comte | A new algorithm for fixed design regression and denoising[END_REF] or [START_REF] Baraud | Model selection for (auto)regression with dependent data[END_REF].

Drift estimation

Drift estimators: Statements of the results

The regression-type equation for the drift is as follows:

Y k∆ := X (k+1)∆ -X k∆ ∆ = b(X k∆ )+Z k∆ + R k∆ (5.14)
where

Z k∆ = 1 ∆ Z (k+1)∆ k∆ σ(X s )dW s ,R k∆ = 1 ∆ Z (k+1)∆ k∆ (b(X s )-b(X k∆ ))ds.
(5.15) The couple (X k∆ ,Y k∆ ) stands for the data (x k ,y k ). The term Z k∆ is a martingale increment (with respect to the filtration F k∆ = σ(X s ,s  k∆)) and plays the role of the noise term. The term R k∆ is a remainder due to the discretization. Now, we consider a collection which may be either [DP] or [GP]. For S m a space of the collection and for t 2 S m , we set

γ n (t)= 1 n n X k=1 [Y k∆ -t(X k∆ )] 2 .
(5.16)

We define an estimator bm of b A belonging to S m as any solution of: bm = arg min t2S m γ n (t).

(5.17)

Note that, with this definition, only the random R n -vector ( bm (X ∆ ),..., bm (X n∆ )) 0 is uniquely defined. Indeed, let Π m denote the orthogonal projection (with respect to the inner product of R n ) onto the subspace

{(t(X ∆ ),..., t(X n∆ )) 0 ,t 2 S m } of R n . Then ( bm (X ∆ ),..., bm (X n∆ )) 0 =Π m Y where Y =( Y ∆ ,...,Y n∆ ) 0 . Any function t in S m such that t(X k∆ )= bm (X k∆ ), k =1,...,n
, is a solution of (5.17). This is the reason why we adopt a specific definition of the risk of an estimator. Consider the following empirical norm for a function t:

ktk 2 n = 1 n n X k=1 t 2 (X k∆ ).
(5.18)

The risk of bm is defined as the expectation of the empirical norm:

E(k bm -b A k 2 n ).
Note that, for a deterministic function t, E(ktk

2 n )=ktk 2 ⇡ = R t 2 (x)⇡(x)dx.
The following proposition provides an upper bound for the risk of an estimator bm with fixed m. Let b m denote the orthogonal projection of b on S m .

Proposition 5.2 Assume that [A1] -[A7] hold. Consider a space S m in col- lection [DP] or [GP] with maximal dimension satisfying N n = o(n∆/ ln 2 (n)).
Then the estimator bm of b is such that (see (5.4))

E(k bm -b A k 2 n )  7⇡ 1 kb m -b A k 2 + K σ 2 1 D m n∆ + K 0 ∆+ K 00 n∆ , (5.19)
where K, K 0 and K 00 are some positive constants.

As usual, there appears to be a squared bias term kb mb A k 2 and a variance term of order D m /(n∆), plus additional terms due to the discretization. It is standard for diffusion models in high-frequency data to assume that ∆=o(1/(n∆)) so that the two last terms in (5.19) are negligible with respect to the variance term. It remains to select the dimension D m that leads to the best compromise between the squared bias term and the variance term.

Consider the case [DP].

To compare the result of Proposition 5.2 with the optimal non-parametric rates exhibited by [START_REF] Hoffmann | Adaptive estimation in diffusion processes[END_REF], let us assume that b A belongs to a ball of some Besov space, b A 2B ↵,2,1 ([0, 1]), and that r +1 ≥ ↵. DeVore and Lorentz (1993, p.359) or Lemma 12 in [START_REF] Barron | Risk bounds for model selection via penalization[END_REF]). Thus, choosing D m =(n∆) 1/(2↵+1) , we obtain

Then, for kb A k ↵,2,1  L, it is known that kb A -b m k 2  C(↵, L)D -2↵ m (see
E(k bm -b A k 2 n )  C(↵, L)(n∆) -2↵/(2↵+1) + K 0 ∆+ K 00 n∆ .
The first term (n∆) -2↵/(2↵+1) is exactly the optimal non-parametric rate (see [START_REF] Hoffmann | Adaptive estimation in diffusion processes[END_REF]). Under the condition ∆=o(1/(n∆)), the last two terms in (5.19) are negligible with respect to (n∆) -2↵/(2↵+1) .

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 knowing its regularity ↵. This selection is done by defining m = arg min

m2M n h γ n ( bm )+pen(m) i , (5.20)
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(k b m -b A k 2 n )  C inf m2M n ✓ kb A -b m k 2 + σ 2 1 D m n∆ ◆ + K 0 ∆+ K 00 n∆ ,
with C a constant, which should not be too large. We almost reach this aim.

Theorem 5.3 Assume that [A1] -[A7] hold and consider the nested collection of models [DP] with L m =1or the collection [GP] with L m given by (5.13), both with maximal dimension

N n = o(n∆/ ln 2 (n)). Let pen(m) ≥ σ 2 1 (1 + L m )D m n∆ , (5.21)
where  is a universal constant. Then the estimator b m of b with m defined in (5.20) is such that

E(k b m -b A k 2 n )  C inf m2M n kb m -b A k 2 +pen(m) + K 0 ∆+ K 00 n∆ . (5.22)
Inequality (5.22) shows that the adaptive estimator automatically realizes the bias-variance compromise. Nevertheless, some comments need to be made. It is possible to choose the equality in (5.21) but this is not what is done in practice. It is better to have additional terms to avoid underpenalization. The constant  in the penalty is numerical and must be calibrated for the problem.

Another important point is that σ 2 1 is unknown. In practice, it is replaced by a rough estimator.

Proof of Proposition 5.2

Let us set (see (5.15)), for any function t(.),

⌫ n (t)= 1 n n X k=1 t(X k∆ )Z k∆ ,R n (t)= 1 n n X k=1 t(X k∆ )R k∆ .
(5.23)

Using (5.14) -(5.16) -(5.18), we have:

γ n (t) -γ n (b)=kt -bk 2 n +2⌫ n (b -t)+2R n (b -t). Recall that b m denotes 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 k bm -bk 2 n kb m -bk 2 n +2⌫ n ( bm -b m )+2R n ( bm -b m ).
The functions bm and b m being A-supported, we can cancel the terms kb1 A c k 2 n that appear in both sides of the inequality. This yields

k bm -b A k 2 n kb m -b A k 2 n +2⌫ n ( bm -b m )+2R n ( bm -b m ). Recall notations (5.6) and let B ⇡ m = {t 2 S m , ktk ⇡ =1}. We use the standard inequality 2xy  ✓ 2 x 2 + y 2 /✓ 2 which holds for all ✓ 6 =0 with ✓ 2 =8: 2⌫ n ( bm -b m )  2k bm -b m k ⇡ sup t2B π m |⌫ n (t)| 1 8 k bm -b m k 2 ⇡ +8 sup t2B π m [⌫ n (t)] 2 .
Similarly,

2R n ( bm -b m )  2k bm -b m k n 1 n n X k=1 R 2 k∆ ! 1/2  1 8 k bm -b m k 2 n +8 1 n n X k=1 R 2 k∆ .
Because the L 2 ⇡ -norm, k.k ⇡ , and the empirical norm (5.18) 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 (5.8))

Ω n = 8 < : !/ ktk 2 n ktk 2 ⇡ -1  1 2 , 8t 2 [ m,m 0 2M n (S m + S m 0 )/{0} 9 = ; . (5.24) On Ω n , k bm -b m k 2 ⇡  2k bm -b m k 2 n , and k bm -b m k 2 n  2(k bm -b A k 2 n + kb m -b A k 2 n ).
Hence, some elementary computations yield:

1 4 k bm -b A k 2 n 1 Ω n  7 4 kb m -b A k 2 n +8 sup t2B π m [⌫ n (t)] 2 + 8 n n X k=1 R 2 k∆ .
Now, using [A5] and Lemma 5.1, we get

E(R 2 k∆ )  1 ∆ E Z (k+1)∆ k∆ (b(X s ) -b(X k∆ )) 2 ds  c 0 ∆.
Consequently,

E(k bm -b A k 2 n 1 Ω n )  7kb m -b A k 2 ⇡ +32 E sup t2B π m [⌫ n (t)] 2 ! +32c 0 ∆. (5.25)
Consider a basis of S m , say { λ ,λ 2 J m }, which is orthonormal with respect to L 2 ⇡ , and with

|J m | = D m . For t 2 B ⇡ m , [⌫ n (t)] 2  X λ2J m [⌫ 2 n ( λ )].
Using the martingale property of (5.23) and the bound of σ 2 (.), we get:

E[⌫ 2 n ( λ )] = 1 n 2 ∆ 2 n X k=1 E ( 2 λ (X k∆ ) Z (k+1)∆ k∆ σ 2 (X s )ds )  σ 2 1 n∆ Z 2 λ (x)⇡(x)dx = σ 2 1 n∆ .
Therefore,

E sup t2B π m [⌫ n (t)] 2 !  σ 2 1 n∆ D m .
Gathering bounds, and using the upper bound ⇡ 1 defined in (5.7), we get

E(k bm -b A k 2 n 1 Ω n )  7⇡ 1 kb m -b A k 2 + 32 σ 2 1 D m n∆ + 32c 0 ∆. Now, it remains to deal with Ω c n . Since k bm -b A k 2 n k bm -bk 2 n , it is enough to check that E(k bm -bk 2 n 1 Ω c n )  c/n. Write Y k∆ = b(X k∆ )+" k∆ with " k∆ = Z k∆ + R k∆ . 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∆ )) 0 ,t 2 S m } of R n . We have ( bm (X ∆ ),..., bm (X n∆ )) 0 =Π m Y where Y =( Y ∆ ,..., Y n∆ ) 0 .
Using the same notation for the function t and the vector (t(X ∆ ),..., t(X n∆ )) 0 , we see that

kb -bm k 2 n = kb -Π m bk 2 n + kΠ m "k 2 n kbk 2 n + n -1 n X k=1 " 2 k∆ .
Therefore,

E ⇣ kb -bm k 2 n 1 Ω c n ⌘  E kbk 2 n 1 Ω c n + 1 n n X k=1 E " 2 k∆ 1 Ω c n  ⇣ E 1/2 (b 4 (X 0 )) + E 1/2 (" 4 ∆ ) ⌘ P 1/2 (Ω c n ). Using [A5], we have E(b 4 (X 0 ))  c(1 + E(X 4 0 )) = K.
With the Burholder-Davis-Gundy inequality, we find

E(" 4 ∆ )  2 3 ⇢ 1 ∆ Z ∆ 0 E[(b(X s ) -b(X ∆ )) 4 ]ds + 36 ∆ 3 E ✓Z ∆ 0 σ 4 (X s )ds ◆ .

Under [A1] -[A3],

[A5] -[A6] and using Lemma 5.1, we obtain E(" 4 ∆ )  C(1 + σ 4 1 /∆ 2 ):=C 0 /∆ 2 . The next lemma enables us to complete the proof.

Lemma 5.4 Let Ω n be defined by (5.24). Then, if

N n  O(n∆ n / ln 2 (n)) P(Ω c n )  c n 4 .
(5.26)

The proof of this technical lemma is given in [START_REF] Comte | Penalized nonparametric mean square estimation of the coefficients of diffusion processes[END_REF] and relies on inequalities proved in Baraud, Comte, and Viennet (2001a). We stress the fact that it is for this lemma that we need the exponential β-mixing assumption for (X t ). It is also for this lemma that we have constraints on the maximal dimension N n . Now, we gather all terms and use (5.26) to get (5.19). 2

Proof of Theorem 5.3

The proof relies on the following Bernstein-type inequality:

Lemma 5.5 Under the assumptions of Theorem 5.3, for any positive numbers ✏ and v, we have (see (5.23)):

P ⌫ n (t) ≥ ✏, ktk 2 n  v 2  exp ✓ - n∆✏ 2 2σ 2 1 v 2 ◆ .
Proof of Lemma 5.5. Consider the process:

H n u = H u = n X k=1 1 [k∆,(k+1)∆[ (u)t(X k∆ )σ(X u ) which satisfies H 2 u  σ 2 1 ktk 2 1 for all u ≥ 0. Then, denoting by M s = R s 0 H u dW u , we get that M (n+1)∆ = n X k=1 t(X k∆ ) Z (k+1)∆ k∆ σ(X s )dW s = n∆⌫ n (t)
and that

hM i (n+1)∆ = n X k=1 t 2 (X k∆ ) Z (k+1)∆ k∆ σ 2 (X s )ds. Moreover, hM i s = R s 0 H 2 u du  nσ 2 1 ∆ktk 2 n , 8s ≥ 0, so that (M s )
and exp(λM sλ 2 hM i 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, hM i s  d)  P ⇣ e λM s -λ 2 2 hM i s ≥ e λc-λ 2 2 d ⌘  e -(λc-λ 2 2 d) .
Therefore,

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

P ⌫ n (t) ≥ ✏, ktk 2 n  v 2 = P(M (n+1)∆ ≥ n∆✏, hM i (n+1)∆  nv 2 σ 2 1 ∆)  exp ✓ - (n∆✏) 2 2nv 2 σ 2 1 ∆ ◆ =exp ✓ - n✏ 2 ∆ 2v 2 σ 2 1 ◆ . 2
Now we turn to the proof of Theorem 5.3. As in the proof of Proposition 5.2, we have to split

k b m -b A k 2 n = k b m -b A k 2 n 1 Ω n + k b m -b A k 2 n 1 Ω c n .
For the study on Ω c n , the end of the proof of Proposition 5.2 can be used. It remains to look at what happens on Ω n . Let us introduce the notation

G m (m 0 )= sup t2S m +S m 0 ,ktk π =1 |⌫ n (t)|.
From the definition of b m, we have, 8m 2M n , γ n ( b m)+pen( m)  γ n (b m )+ pen(m). We proceed as in the proof of Proposition 5.2 with the additional penalty terms (see (5.25)) and obtain

E(k b m -b A k 2 n 1 Ω n )  7⇡ 1 kb m -b A k 2 + 4pen(m) +32E [G m (m)] 2 1 Ω n -4E(pen( m)) + 32c 0 ∆.
The main problem here is to control the supremum of ⌫ n (t) on a random ball (which depends on the random m). This will be done using Lemma 5.5 and Proposition 5.6 below, proceeding first as follows. We plug in a function p(m, m 0 ), which will in turn fix the penalty:

[G m (m)] 2 1 Ω n { ([G m (m)] 2 -p(m, m))1 Ω n } + + p(m, m)  X m 0 2M n {([G m (m 0 )] 2 -p(m, m 0 ))1 Ω n } + + p(m, m).
The penalty pen(.) is chosen such that 8p(m, m 0 )  pen(m)+pen(m 0 ). More precisely, the next proposition determines the choice of p(m, m 0 ).

Proposition 5.6 Under the assumptions of Theorem 5.3, there exists a numer-

ical constant  1 such that, for p(m, m 0 )= 1 σ 2 1 (D m +(1+L m 0 )D m 0 )/(n∆), we have E{([G m (m 0 )] 2 -p(m, m 0 ))1 Ω n } +  cσ 2 1 e -L m 0 D m 0 n∆ .
Proof of Proposition 5.6. The result of Proposition 5.6 follows from Lemma 5.5 applying the L 2 ⇡ -chaining technique used in [START_REF] Baraud | Model selection for (auto)regression with dependent data[END_REF] (see Proposition 6.1, p.42, and Section 7, pp. 44-47, Lemma 7.1, with s 2 = σ 2 1 /∆). 2 Recall that the weights L m are such that Σ= P m 0 2M n e -L m 0 D m 0 < 1. Thus, the result of Theorem 5.3 follows from Proposition 5.6 with pen(m) ≥ σ 2

1 (1 + L m )D m /(n∆) and  =8 1 . 2

Bound for the L 2 -risk

In Theorem 5.3, the risk of b m is not measured as a standard L 2 -risk. In this paragragh, we prove that a simple truncation of b m allows to study an integrated loss over a compact set instead of our empirical loss.

Proposition 5.7 Let b⇤ = ⇢ b if k bkk n 0 else, where b = b m and k n = O(n). Then, E(k b⇤ -b A k 2 )  C inf m2M n kb m -b A k 2 +pen(m) + K 0 ∆+ K 00 n∆ . Proof. Recall that k b⇤ -b A k 2  (1/⇡ 0 )k b⇤ -b A k 2 ⇡ . Then, we decompose the L 2 ⇡ -norm into: k b⇤ -b A k 2 ⇡ = k b⇤ -b A k 2 ⇡ 1 k bkk n 1 Ω n + k b⇤ -b A k 2 ⇡ 1 k bk>k n 1 Ω n + k b⇤ -b A k 2 ⇡ 1 Ω c n = T 1 + T 2 + T 3 .
First, it is easy to see that

E(T 3 )  2(⇡ 1 k 2 n + kb A k 2 ⇡ )P(Ω c n ),
and with k n = O(n), as we know that P(Ω c n )  c/n 4 , we get a negligible term of order 1/n 2 .

Next, T 1 can be studied as above, except that some constants are increased:

T 1 k b⇤ -b A k 2 ⇡ 1 Ω n  2(k b -b m k 2 ⇡ + kb m -b A k 2 ⇡ )1 Ω n  4k b -b m k 2 n 1 Ω n +2kb m -b A k 2 ⇡ 1 Ω n  8k b -b A k 2 n 1 Ω n +8kb m -b A k 2 n 1 Ω n +2kb m -b A k 2 ⇡ 1 Ω n
and we can use the bound obtained in Theorem 5.3 to get that, for all m 2M n ,

E(T 1 )  C(kb m -b A k 2 +pen(m)) + K∆+ K 0 n∆ . Lastly, T 2 = kb A k 2 ⇡ 1 k bk>k n 1 Ω n . On Ω n , k bk 2  1 ⇡ 0 k bk 2 ⇡  3 2⇡ 0 k bk 2 n  3 ⇡ 0 (kb A -bk 2 n + kb A k 2 n )
and the study of this term leads to the bound

kb A -bk 2 n kb A k 2 n + 1 n n X k=1 " 2 k∆ with E(" 2 k∆ )  c/∆. It follows that, with Markov's inequality, E(T 2 ) k b A k 2 ⇡ P({k bk≥k n }\Ω n ) k b A k 2 ⇡ P(6kbk 2 n ≥ ⇡ 0 k 2 n )+P( 3 n n X k=1 " 2 k∆ ≥ ⇡ 0 k 2 n ) ! k b A k 2 ⇡ ✓ 4kb A k 2 ⇡ ⇡ 0 k 2 n + c ∆ 3 ⇡ 0 k 2 n ◆ = o( 1 n ), since k n = O(n).
Gathering all terms gives that Theorem 5.3 extends to E(k b⇤b A k 2 ).

Diffusion coefficient estimation

Diffusion coefficient estimator: Statement of the results

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. We set

U k∆ = (X (k+1)∆ -X k∆ ) 2 ∆ .
The couple of data is now (U k∆ ,X k∆ ). The regression-type equation is as follows: (5.27) where

U k∆ = σ 2 (X k∆ )+V k∆ + ⌧ k∆ ,
V k∆ = V (1) k∆ + V (2) k∆ + V (3) k∆ with V (1) k∆ = 1 ∆ 2 4 ( Z (k+1)∆ k∆ σ(X s )dW s ) 2 - Z (k+1)∆ k∆ σ 2 (X s )ds 3 5 , V (2) k∆ = 1 ∆ Z (k+1)∆ k∆ ((k + 1)∆ -s)(σ 2 ) 0 (X s )σ(X s )dW s , V (3) k∆ =2 b(X k∆ ) Z (k+1)∆ k∆ σ(X s )dW s , ⌧ k∆ = ⌧ (1) k∆ + ⌧ (2) k∆ + ⌧ (3) k∆ with ⌧ (1) k∆ = 1 ∆ Z (k+1)∆ k∆ b(X s )ds ! 2 , ⌧ (2) k∆ = 2 ∆ Z (k+1)∆ k∆ (b(X s ) -b(X k∆ ))ds Z (k+1)∆ k∆ σ(X s )dW s , ⌧ (3) k∆ = 1 ∆ Z (k+1)∆ k∆ [(k + 1)∆ -s] (X s )ds, and
= σ 2 2 (σ 2 ) 00 + b(σ 2 ) 0 = Lσ 2 , (5.28)
where Lf = σ 2 2 f 00 + bf 0 is the infinitesimal generator of (5.1). The above relations are obtained by applying the Itô and the Fubini formulae. The term V k∆ is a sum of martingale increments whose variances have different orders. The term V

(1) k∆ plays the role of the main noise. The term ⌧ k∆ is a remainder due to the discretization and to the presence of the drift. The scheme is similar to what is done for the drift. To estimate σ 2 on A =[0, 1], we define

σ2 m = arg min t2S m γn (t), with γn (t)= 1 n n X k=1 [U k∆ -t(X k∆ )] 2 . ( 5
E(kσ 2 m -σ 2 A k 2 n )  7⇡ 1 kσ 2 m -σ 2 A k 2 + K σ 4 1 D m n + K 0 ∆ 2 + K 00 n , (5.30)
where K, K 0 , K 00 are some positive constants.

Let us make some comments on the rates of convergence for estimators built with [DP]. If σ 2 A belongs to a ball of some Besov space, say

σ 2 A 2 B ↵,2,1 ([0, 1]), and kσ 2 A k ↵,2,1  L, with r +1 ≥ ↵, then kσ 2 A -σ 2 m k 2  C(↵, L)D -2↵
m . Therefore, if we choose D m = n 1/(2↵+1) , we obtain

E(kσ 2 m -σ 2 A k 2 n )  C(↵, L)n -2↵/(2↵+1) + K 0 ∆ 2 + K 00 n .
The first term n -2↵/(2↵+1) is the optimal non-parametric rate proved by [START_REF] Hoffmann | Adaptive estimation in diffusion processes[END_REF]. Moreover, under the standard condition ∆ 2 = o(1/n), the last two terms are O(1/n), i.e. negligible with respect to n -2↵/(2↵+1) .

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 m2M n ⇥ γn (σ 2 m )+g pen(m) ⇤ .
(5.31)

We denote by σ2 m the resulting estimator and we need to determine the penalty g pen as for b. For simplicity, we use the same notation m in (5.31) as in (5.20) although they are different. We can prove the following theorem.

Theorem 5.9 Assume that [A1]-[A7] hold. Consider collection [DP] with L m =1or [GP] with L m given by (5.13) both with maximal dimension

N n  n∆/ ln 2 (n). Let g pen(m) ≥ σ 4 1 (1 + L m )D m n ,
where  is a universal constant. Then, the estimator σ2 m of σ 2 with m defined by (5.31) is such that

E(kσ 2 m -σ 2 A k 2 n )  C inf m2M n kσ 2 m -σ 2 A k 2 + g pen(m) + K 0 ∆ 2 + K 00 n .
Analogous comments as those given for the drift can be made.

Proof of Proposition 5.8

Let us set ⌫n (t)=⌫ ( 1)

n (t)+⌫ (2) n (t)+⌫ (3) n (t) (5.32) with ⌫(i) n (t)= 1 n n X k=1 t(X k∆ )V (i) k∆ , (5.33)
and

⌧n (t)= 1 n n X k=1 t(X k∆ )⌧ k∆ .
We begin with some lemmas. The first one concerns the remainder term.

Lemma 5.10 We have (see (5.27))

E( 1 n n X k=1 ⌧ 2 k∆ )  K∆ 2 .
(5.34)

Proof of Lemma 5.10. We prove that E[(⌧

(i) k∆ ) 2 ]  K i ∆ 2 for i =1, 2, 3. Using [A5] and Lemma 5.1, E[(⌧ (1) k∆ ) 2 ]  E Z (k+1)∆ k∆ b 2 (X s )ds ! 2  ∆E Z (k+1)∆ k∆ b 4 (X s )ds !  ∆ 2 E(b 4 (X 0 ))  c∆ 2 , E[(⌧ (2) k∆ ) 2 ]  1 ∆ 2 0 @ E Z (k+1)∆ k∆ (b(X s ) -b(X k∆ ))ds ! 4 ⇥ E Z (k+1)∆ k∆ σ(X s )dW s ! 4 1 A 1/2
Using [A5], Lemma 5.1 and the Burkholder-Davis-Gundy inequality, we get

E[(⌧ (2) k∆ ) 2 ]  c 0 ∆ 2 . Lastly, [A5] implies that | (x)|K(1 + x 2 ) (see (5.28)), hence E[(⌧ (3) k∆ ) 2 ]  1 ∆ E Z (k+1)∆ k∆ ((k + 1)∆ -s) 2 2 (X s )ds !  E( 2 (X 0 )) ∆ 2 3  c 00 ∆ 2 .
Therefore (5.34) is proved. 2

Now, we deal with the noise terms and show that i =1gives the main term. In the statement below, K, K 0 denote constants which may vary from line to line.

Lemma 5.11 1. For S m in collection [DP] or [GP],

E sup t2S m ,ktk π =1 (⌫ (1) n (t)) 2 !  K D m n σ 4 1 .
2. Recall that S n denotes the maximal space for both collections. For i =2, 3,

E sup t2S n ,ktk π =1 (⌫ (i) n (t)) 2 !  K ∆N n n  K 0 ∆ 2 .
(5.35)

Proof of Lemma 5.11. To study ⌫( 1) n (t), we consider, as for the drift case, an orthonormal basis

( λ ,λ 2 J m ) of S m with respect to L 2 ⇡ . So, E sup t2S m ,ktk π =1 (⌫ (1) n (t)) 2 !  X λ2J m E((⌫ (1) n ( λ )) 2 ).
Then, we use the fact that V

(1)

k∆ is a martingale increment and obtain:

E((⌫ (1) n ( λ )) 2 )= 1 n 2 n X k=1 E( 2 λ (X k∆ )E([V (1) k∆ ] 2 |F k∆ )).
Then,

E((V

(1)

k∆ ) 2 |F k∆ )  2 ∆ 2 " E(( Z (k+1)∆ k∆ σ(X s )dW s ) 4 |F k∆ ) +E(( Z (k+1)∆ k∆ σ 2 (X s )ds) 2 |F k∆ ) # .
Using the Burkholder-Davis-Gundy inequality, we obtain:

E((V (1) k∆ ) 2 |F k∆ )  Cσ 4
1 . This gives the first part.

For the second part, note that the maximal space S n is equipped with an orthonormal basis (' λ ,λ 2L n ) with respect to L 2 which satisfies for both collections (see (5.11) -(5.12))

k X λ2L n ' 2 λ k 1  Φ 2 0 N n , with Φ 2 0 = r max +1. For i =2, 3, E sup t2S n ,ktk1 (⌫ (i) n (t)) 2 ) !  X λ2L n E((⌫ (i) n (' λ )) 2 ).
Since the martingale increments (V

(i)
k∆ ) are uncorrelated, we have:

E((⌫ (i) n (' λ )) 2 )= 1 n 2 n X k=1 E ⇣ ' 2 λ (X k∆ )(V (i) k∆ ) 2 ⌘ .
Therefore, interchanging sums in λ and k, we get:

X λ2L n E((⌫ (i) n (' λ )) 2 )  Φ 2 0 N n n 1 n n X k=1 E((V (i) k∆ ) 2 ).
(5.36)

Now,

E((V

(2)

k∆ ) 2 )= 1 ∆ 2 E[((σ 2 ) 0 (X 0 )σ(X 0 )) 2 ] Z (k+1)∆ k∆ ((k + 1)∆ -s) 2 ds  C∆(1 + (E(X 0 )) 4 ) and E((V (3) k∆ ) 2 )=4 E(b 2 (X k∆ ) Z (k+1)∆ k∆ σ 2 (X s )ds)  4 E(b 4 (X k∆ ))E[( Z (k+1)∆ k∆ σ 2 (X s )ds) 2 ] ! 1/2  4 E(b 4 (X 0 ))E(σ 4 (X 0 )) 1/2 ∆  C∆(1 + E(X 4 0 )). Since ktk 2 k tk 2
⇡ /⇡ 0 , we join the above bounds and (5.36) and obtain the first inequality in (5.35). Since N n  n∆/ ln 2 n, N n ∆/n  ∆ 2 / ln 2 n. This gives the second inequality. 2 Now, we can prove Proposition 5.8. As for the drift, the starting point is:

γn (t) -γn (σ 2 )=kσ 2 -tk 2 n +2⌫ n (σ 2 -t)+2⌧ n (σ 2 -t).
Introducing the orthogonal projection σ 2 m of σ 2 on S m , we have: γn (σ 2 m )γn (σ 2 )  γn (σ 2 m )γn (σ 2 ). After some computations analogous to those done for the drift study, we are led to the following inequality which holds on Ω n (see (5.24)):

1 4 kσ 2 m -σ 2 A k 2 n  7 4 kσ 2 m -σ 2 A k 2 n +8 sup t2B π m (0,1) ⌫2 n (t)+ 8 n n X k=1 ⌧ 2 k∆ ,
where B ⇡ m (0, 1) = {t 2 S m , ktk ⇡ =1 }. Now we apply Lemma 5.10 and Lemma 5.11. This yields the first three terms of the right-hand-side of (5.30). The study on Ω c n is the same as for b with the regression model

U k∆ = σ 2 (X k∆ )+⇠ k∆ , where ⇠ k∆ = V k∆ + ⌧ k∆ . By standard inequalities, E(⇠ 4 ∆ )  K{∆ 4 E(b 8 (X 0 )) + E(σ 8 (X 0 ))}. Hence, E(⇠ 4
∆ ) is bounded. Moreover, using Lemma 5.4, P(Ω c n )  c/n 2 . 2 5.6.3 Proof of Theorem 5.9

This proof follows the same lines as the proof of Theorem 5.3. We start with a Bernstein-type inequality.

Lemma 5.12 Under the assumptions of Theorem 5.9,

P ⇣ ⌫(1) n (t)) ≥ ✏, ktk 2 n  v 2 ⌘  exp ✓ -Cn ✏ 2 /2 2σ 4 1 v 2 + ✏ktk 1 σ 2 1 v ◆ and P ⇣ ⌫(1) n (t) ≥ vσ 2 1 (2x) 1/2 + σ 2 1 ktk 1 x, ktk 2 n  v 2 ⌘  exp(-Cnx).
(5.37)

The proof that the first inequality implies the second one above is rather tricky and proved in [START_REF] Birgé | Minimum contrast estimators on sieves: exponential bounds and rates of convergence[END_REF]. Consequently, we just prove the first one.

Proof of Lemma 5.12. First we note that:

E ⇣ e ut(X n∆ )V (1) n∆ |F n∆ ⌘ =1 + +1 X p=2 u p p! E n (t(X n ∆)V (1) n∆ ) p |F n∆ o  1+ +1 X p=2 u p p! |t(X n∆ )| p E ⇣ |V (1) n∆ | p |F n∆ ⌘ .
Next we apply successively the Minkowski inequality and the Burkholder-Davis-Gundy inequality with best constant (Proposition 4.2 of [START_REF] Barlow | Semimartingale 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 =s u p st |M s | satisfies kM ⇤ k k  ck 1/2 khM i 1/2 k k , with c a universal constant. And we obtain:

E(|V (1) n∆ | p |F n∆ )  2 p-1 ∆ p 8 < : E 0 @ Z (n+1)∆ n∆ σ(X s )dW s 2p |F n∆ 1 A +E Z (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+ 1 X p=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 1 X p=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 ektk 1 )
. Now, let us set a = e(4σ 2 1 c 2 ) 2 and b =4σ 2 1 c 2 ektk 1 . 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 X k=1 ✓ ut(X k∆ )V (1) k∆ - au 2 t 2 (X k∆ ) 1 -bu ◆ )#  1.
Then, we deduce that

P n X k=1 t(X k∆ )V (1) k∆ ≥ n✏, ktk 2 n  v 2 !  e -nu✏ E
(

1 ktk 2 n v 2 exp u n X k=1 t(X k∆ )V (1) k∆ !)  e -nu✏ E  1 ktk 2 n v 2 exp ( n X 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 -nu✏ e (nau 2 v 2 )/(1-bu) E " exp ( n X k=1 (ut(X k∆ )V (1) k∆ - au 2 t 2 (X k∆ ) 1 -bu ) )#  e -nu✏ e (nau 2 v 2 )/(1-bu) .
The inequality holds for any u such that bu < 1. In particular, u = ✏/(2av 2 + ✏b) gives -u✏

+ av 2 u 2 /(1 -bu)=-(1/2)(✏ 2 /(2av 2 + ✏b) and therefore P n X k=1 t(X k∆ )V (1) k∆ ≥ n✏, ktk 2 n  v 2 !  exp ✓ -n ✏ 2 /2 2av 2 + ✏b ◆ . 2
We now finish the proof of Theorem 5.9. As for b m, we introduce the additional penalty terms and obtain that the risk satisfies

E(kσ 2 m -σ 2 A k 2 n 1 Ω n )  7⇡ 1 kσ 2 m -σ 2 A k 2 +4g pen(m) + 32E sup t2B π m, m (0,1) (⌫ n (t)) 2 1 Ω n ! -4E(g pen( m)) + K 0 ∆ 2 (5.38)
where B ⇡ m,m 0 (0, 1) = {t 2 S m + S m 0 , ktk ⇡ =1}. We use that

(⌫ n (t)) 2  2[(⌫ (1) n (t)) 2 +(⌫ (2) n (t)+⌫ (3) n (t)) 2 ]. By Lemma 5.11, since B ⇡ m,m 0 (0, 1) ⇢{t 2S n , ktk ⇡ =1}, E sup t2B π m, m(0,1) (⌫ (2) n (t)+⌫ (3) n (t)) 2 !  K∆ 2 .
There remains the main term to study

Gm (m 0 )= sup t2B π m,m 0 (0,1) |⌫ (1) n (t)|.
(5.39)

As for the drift, we write

E( G2 m (m))  E[( G2 m (m) -p(m, m))1 Ω n ] + + E(p(m, m))  X m 0 2M n E[( G2 m (m 0 ) -p(m, m 0 ))1 Ω n ] + + E(p(m, m)).
Now we have the following statement.

Proposition 5.13 Under the assumptions of Theorem 5.9, for

p(m, m 0 )=σ 4 1 D m + D m 0 (1 + L m 0 ) n + K∆ 2 ,
where  is a numerical constant, we have

E[( G2 m (m 0 ) -p(m, m 0 ))1 Ω n ] +  cσ 4 1 e -D m 0 L m 0 n .
The result of Proposition 5.13 is obtained from inequality (5.37) of Lemma 5.12 by a L 2 ⇡ -L 1 chaining technique. A description of this method, in a more general setting, is given in Propositions 2-4, pp. 282-287, in [START_REF] Comte | Adaptive estimation of the spectrum of a stationary Gaussian sequence[END_REF], Theorem 5 in [START_REF] Birgé | Minimum contrast estimators on sieves: exponential bounds and rates of convergence[END_REF] and Proposition 7, Theorem 8 and Theorem 9 in [START_REF] Barron | Risk bounds for model selection via penalization[END_REF]. For the sake of completeness and since the context is slightly different, we detail the proof in the Appendix, Section 5.9. Note that there is a difference between Propositions 5.6 and 5.13 which comes from the additional term ktk 1 appearing in Lemma 5.12.

Choosing g pen(m) ≥ σ 4 1 D m (1 + L m )/n with  = 16, we deduce from (5.38), Proposition 5.13 and

D m  N n  n∆/ ln 2 (n) that, E(kσ 2 m -σ 2 A k 2 n )  7⇡ 1 kσ 2 m -σ 2 A k 2 +8g pen(m)+cσ 4 1 X m 0 2M n e -D m 0 L m 0 n +K 0 ∆ 2 + E(kσ 2 m -σ 2 A k 2 n 1 Ω c n ). The bound for E(kσ 2 m -σ 2 k 2 n 1 Ω c n )
is the same as the one given in the end of the proof of Proposition 5.8. It is less than c/n. The result of Theorem 5.9 follows. 2

Examples and practical implementation

In this section, we consider classical examples of diffusions for which an exact simulation of sample paths is possible and for which the estimation method has been implemented with [GP]. For exact simulation of sample paths, when it is not directly possible, we have in view 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]. Models of Families 1 and 2 below can be simulated by the algorithm EA1. Among the assumptions, requiring that σ be bounded is rather stringent and not always satisfied in our examples. The other assumptions hold. More details may be found in [START_REF] Comte | Nonparametric estimation of a discretely observed integrated diffusion model[END_REF][START_REF] Comte | Penalized nonparametric mean square estimation of the coefficients of diffusion processes[END_REF]

Examples of diffusions Family 1

First, we consider (5.1) with

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

⇡(x) / 1 (1 + x 2 ) 1+✓/c 2 . If X 0 = ⌘ has distribution ⇡(x)dx, then, setting ⌫ =1+2 ✓/c 2 , ⌫ 1/2 ⌘ has Student distribution t(⌫)
. This distribution satisfies the moment condition [A6] for 2✓/c 2 > 7. See Figure 5.1 for the estimation of b and σ 2 in this case.

Then, we consider Assumptions [A1] -[A3] and [A5] hold for (⇠ t ) with ⇠ 0 = F 1 (X 0 ). Moreover, (⇠ t ) satisfies the conditions of Proposition 1 in [START_REF] Pardoux | On the Poisson equation and diffusion approximation[END_REF] implying that (⇠ t ) is exponentially β-mixing and has moments of any order. Hence, [A4] and [A6] hold. See Figure 5.2 for the estimation of b and σ 2 in this case.

F 1 (x)= R x 0 1/(c(1 + x 2 ) 1/2 dx = arg sinh(x)/c. By the Itô formula, ⇠ t = F 1 (X t ) is solution of a stochastic differential equation with σ(⇠)=1and b(⇠)=-(✓/c + c/2) tanh(c⇠).
Since X t = F -1 1 (⇠ t ), this process is also β-mixing. It satisfies all assumptions except that σ 2 (x) is not bounded from above.

Family 2

For the second family of models, we consider a process

(⇠ t ) with diffusion coefficient σ(⇠)=1and drift b(⇠)=-✓ ⇠ (1 + c 2 ⇠ 2 ) 1/2 ,
(5.40)

(see [START_REF] Barndorff-Nielsen | Hyperbolic distributions and distributions on hyperbolae[END_REF]). The model is positive recurrent on R for ✓>0.

Its stationary distribution is a hyperbolic distribution given by ] hold for this model. For [A4], we apply Proposition 1 of [START_REF] Pardoux | On the Poisson equation and diffusion approximation[END_REF].

⇡(⇠)d⇠ / exp(-2 ✓ c 2 (1 + c 2 ⇠ 2 ) 1/2 ). Assumptions [A1] -[A3], [A5] -[A6
Next, we consider X t = F 2 (⇠ t ) = arg sinh(c⇠ t ) which satisfies a stochastic differential equation with coefficients:

b(x)=- ✓ ✓ + c 2 2 cosh(x) ◆ sinh(x) cosh 2 (x) ,σ (x)= c cosh(x)
.

The process (X t ) is exponentially β-mixing as (⇠ t ). The diffusion coefficient σ(x) has an upper bound. See Figure 5.3 for the estimation of b and σ 2 in this case.

To obtain a different shape for the diffusion coefficient, showing two bumps, we consider X t = G(⇠ t ) = arg sinh(⇠ t -5) + arg sinh(⇠ t + 5). The function 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 .
The drift is given by G 0 (G -1 (x))b(G -1 (x)) + 1 2 G 00 (G -1 (x)) with b given in (5.40). See Figure 5.4 for the estimation of b and σ 2 in this case.

Family 3

Consider Y t a stationary Ornstein-Uhlenbeck process given by dY t = -✓Y t dt+ cdW t with ✓>0 and Y 0 ; N (0,c 2 /(2✓)). The β-mixing coefficient of (Y t ) can be evaluated using the exact formula (5.2). This gives a direct proof of (5.3).

Proposition 5.14 The β-mixing coefficient of (Y t ) satisfies

β Y (t)  exp (-✓t) 2(1 -exp (-✓t))
.

Proof. We use the expansion of the transition density p t (x, y) of (Y t ) in terms of the sequence of eigenvalues and eigenfunctions of the infinitesimal generator Lf (y)= σ 2 2 f 00 (y) -✓yf 0 (y). For this, we refer e.g. to Karlin and Taylor (1981, p.333). To simplify notations, we assume that σ 2 /2✓ =1so that the stationary distribution of (Y t ) is ⇡(y)dy = N (0, 1). Let us now consider the n-th Hermite polynomial given, for n =0, 1,..., by:

H n (x)= (-1) n p n! exp (x 2 /2) d n dx n [exp (-x 2 /2)].
As defined above, this sequence is an orthonormal basis of L 2 (⇡) and satisfies, for all n ≥ 0, LH n = -n✓H n , i.e., H n is the eigenfunction associated with the eigenvalue -n✓ of L. This gives the following expansion:

p t (x, y)=⇡(y) +1 X n=0 exp (-n✓t)H n (x)H n (y).
Since H 0 (x)=1and the H n have L 2 (⇡)-norm equal to 1, we get Integrating w.r.t. ⇡(x)dx and repeating the same tool, we obtain:

β Y (t)  1 2 +1 X n=1 exp (-n✓t)= exp (-✓t) 2(1 -exp (-✓t))
.

The interest of this proof is that it can be mimicked for all models for which the infinitesimal generator has a discrete spectrum with explicit eigenfunctions and eigenvalues. Now, we consider X t = tanh(Y t ). By the Itô formula, we get that (X t ) has coefficients

b(x)=-(1 -x 2 )  c 2 x + ✓ 2 ln ✓ 1+x 1 -x ◆ ,σ (x)=c(1 -x 2 ).
Assumptions [A1] -[A6] are satisfied for (X t ).

Finally, we consider

dX t =  dc 2 4 -✓X t dt + c p X t dW t .
With d ≥ 2 integer, (X t ) has the distribution of P d i=1 Y 2 i,t where (Y i,t ) are i.i.d. Ornstein-Uhlenbeck processes as above. The process (X t ) satisfies all assumptions except that its diffusion coefficient is not bounded.

Calibrating the penalties

It is not easy to calibrate the penalties. The method is studied in full details in [START_REF] Comte | A new algorithm for fixed design regression and denoising[END_REF]. Implementation with [DP] is done on the above examples in [START_REF] Comte | Penalized nonparametric mean square estimation of the coefficients of diffusion processes[END_REF] and for [GP] in [START_REF] Comte | Nonparametric estimation of a discretely observed integrated diffusion model[END_REF]. We only give here a brief description.

For collection [GP], the drift penalty (i = 1) and the diffusion penalty (i = 2) are given by

2 ŝ2 i n 0 @ d -1+ln ✓ d max -1 d -1 ◆ +ln 2.5 (d)+ d X j=1
(r j +ln 2.5 (r j + 1))

1

A .

Moreover, d max =[ n∆/ ln 1.5 (n)], r max =5 . The constants  and  in both drift and diffusion penalties have been set equal to 2. The term ŝ2 1 replaces σ 2 1 /∆ for the estimation of b and ŝ2 2 replaces σ 4 1 for the estimation of σ 2 . Let us first explain how ŝ2 2 is obtained. We run once the estimation algorithm of σ 2 with a preliminary penalty where ŝ2

2 is taken equal to 2 max m (γ n (σ 2 m )). This gives a preliminary estimator σ2 0 . Now, we take ŝ2 equal to twice the 99.5%-quantile of σ2 0 . The use of the quantile here is to avoid extreme values. We get σ2 . We use this estimate and set ŝ2 1 = max 1kn (σ 2 (X k∆ ))/∆ for the penalty of b. In all the examples, 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).

Bibliographical remarks

Non-parametric estimation of the coefficients of diffusion processes has been widely investigated in the last decades. There are two first reference papers which are only devoted to drift estimation. One is by [START_REF] Banon | Nonparametric identification for diffusion processes[END_REF], who uses a spectral approach and a continuous time observation of the sample path. The other one is by [START_REF] Tuan | Nonparametric estimation of the drift coefficient in the diffusion equation[END_REF] who constructs and studies kernel estimators of the drift based on a continuous time observation of the sample path and also on a discrete observation of the sample path for an ergodic diffusion process.

More recently, several authors have considered drift estimation based on a continuous time observation of the sample path for ergodic models. Asymptotic results are given as the length of the observation time interval tends to infinity (Prakasa Rao (2010), [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]).

Discrete sampling of observations has also been investigated, 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 non-parametric 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 non-parametric 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 ∆ n =1 /n, so that the length of the observation time interval n∆ n =1is fixed, can only lead to estimating the diffusion coefficient consistently with no need of ergodicity assumptions. 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], [START_REF] Florens-Zmirou | On estimating the diffusion coefficient from discrete observations[END_REF]Genon-Catalot, Larédo, and[START_REF] Genon-Catalot | Nonparametric estimation of the diffusion coefficient by wavelet methods[END_REF]. 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, [START_REF] Hoffmann | Adaptive estimation in diffusion processes[END_REF] proposes non-parametric 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. Hence, Hoffmann's paper gives the benchmark for studying non-parametric estimation in this framework and assumptions. Nevertheless, Hoffmann's estimators are based on computations of some random times which make them difficult to implement.

Finally, let us mention that [START_REF] Bandi | Fully nonparametric estimation of scalar diffusion models[END_REF] also consider the same asymptotic framework but with nonstationary diffusion processes: they study kernel estimators using local time estimations and random normalization.

5.9 Appendix. Proof of Proposition 5.13

The proof relies on the following Lemma (Lemma 9 in [START_REF] Barron | Risk bounds for model selection via penalization[END_REF]):

Lemma 5.15 Let µ be a positive measure on [0, 1]. Let ( λ ) λ2Λ be a finite orthonormal system in L 2 \ L 1 (µ) with |Λ| = D and S be the linear span of

{ λ }. Let r = 1 p D sup β6 =0 k P λ2Λ β λ λ k 1 |β| 1 .
For any positive δ, one can find a countable set T ⇢ S and a mapping p from S to T with the following properties:

• for any ball B with radius σ ≥ 5δ,

|T \B|(B 0 σ/δ) D with B 0 < 5.

•k up(u)k µ  δ for all u in S, and

sup u2p -1 (t)
kutk 1  rδ, for all t in T.

To use this lemma, the main difficulty is often to evaluate r in the different contexts. In our problem, the measure µ is ⇡. We consider a collection of models (S m ) m2M n which can be [DP] or [GP]. Recall that B ⇡ m,m 0 (0, 1) = {t 2 S m + S m 0 , ktk ⇡ =1 }. We have to compute r =r m,m 0 corresponding to S = S m + S m 0 . We denote by D(m, m 0 )=dim(S m + S m 0 ).

Collection [DP]

-S m + S m 0 = S max(m,m 0 ) , D(m, m 0 ) = max(D m ,D m 0 ), an orthonormal L 2 (⇡)-basis ( λ ) λ2Λ(m,m 0 ) can be built by orthonormalisation, on each sub-interval, of (' λ ) λ2Λ(m,m 0 ) . Then (5.42) where Ca(δ 0 )=ln(5/δ 0 )+δ 0 (4 ln(5/δ 0 ) + 6 ln(2) + 4). This leads to  (r max + 1) 3 N 2 n ⇡ 0 n 2 (1 + ln(n)) 2  2(r max + 1) 3 ∆ 2 /⇡ 0 .

+ 1+δ 0 X k≥1 2 -(k-1) ! ✓ L m 0 D m 0 nC + ⌧ nC ◆  a(δ 0 ) D(m, m 0 ) n +( 1+2δ 0 C )( L m 0 D m 0 n + ⌧ n ),
1 X k=0 ⌘ k ! 2  σ 4 1 2 " p 2 p 3x 0 ++r m,m 0 x 0 + 1 X k=1 δ k-1 p 15x k +3r m,m 0 x k # 2  σ 4 1 2 " p 6x 0 + 1 X k=1 δ k-1 p 15x k ! +r m,m 0 p 2x 0 +3 1 X k=1 δ k-1 x k !# 2  15σ 4 1 2 4 p x 0 + 1 X k=1 δ k-1 p x k ! 2 +r 2 m,m 0 x 0 + 1 X k=1 δ k-1 x k ! 2 3 5  15σ 4 1 " 1+ 1 X k=1 δ k-1 ! x 0 + 1 X k=1 δ k-1 x k ! +r 2 m,m 0 x 0 + 1 X k=1 δ k-1 x k ! 2 3
Thus, the bound for ( P ⌘ k ) 2 is proportional to

σ 4 1  D(m, m 0 ) n + L m 0 D m 0 n + 2(r max + 1) 3 ∆ 2 /⇡ 0 + ⌧ n +r 2 m,m 0 ⌧ 2 n 2 .
This term defines p(m, m 0 ) as given in Proposition 5.13.

We obtain, for K =(r max + 1) 3 /⇡ 0 ,

Figure 5 . 1

 51 Figure 5.1 First example: dX t = -θX t dt+,c p 1+X 2 t dW t , n =5 0 0 0 , ∆=1 /20, θ =2, c =1, dotted line: true function, full line: estimated function.

Figure 5 . 2

 52 Figure 5.2 Second example: dξ t = -(θ/c + c/2) tanh(cξ t )+dW t , n =5 0 0 0, ∆= 1/20, θ =6, c =2, dotted line: true function, full line: estimated function.

Figure 5 . 3

 53 Figure 5.3 Third example, dX t= -[θ + c 2 /(2 cosh(X t ))] (sinh(X t )/ cosh 2 (X t ))dt +(c/ cosh(X t ))dW t , n =5 0 0 0 , ∆=1 /20, θ =3 , c =2 ,dotted line: true function, full line: estimated function.

Figure 5 . 4

 54 Figure 5.4 Fourth example, the two-bumps diffusion coefficient X t = G(ξ t ), dξ t = -θξ t / p 1+c 2 ξ 2 t dt + dW t , G(x) = arg sinh(x -5) + arg sinh(x +5), n =5 0 0 0 , ∆=1/20, θ =1, c =10, dotted line: true function, full line: estimated function.

  kp t (x, y)dy -⇡(y)dyk TV n✓t)|H n (x)|.

  (m,m 0 ) x k ).Fix ⌧>0 and choose x 0 such thatCnx 0 = H 0 + L m 0 D m 0 + ⌧ and for k ≥ 1, x k such that Cnx k = H k-1 + H k + kD m 0 + L m 0 D m 0 + ⌧. If D m 0 ≥ 1, k-1) [(2k -1) ln(2) + 2 ln(5/δ 0 )+k]

5 .

 5 Now, fix δ 0  1/5 (say, δ 0 =1 /10) and use the bound (5.42). The bound for ( P +1 k=0 ⌘ k ) 2 is less than a quantity proportional to Now in the case of collection [DP], we have L m =1 , rm,m 0 is bounded uniformly with respect to m and m 0 and(D(m, m 0 )/n) 2  (N n /n) 2  ∆ 2 / ln 4 (n) with N n  n∆/ ln 2 (n).Thus the bound for ( collection [GP], we use that L m  c ln(n), r2 m,m 0  (r max + 1) 3 N n /(⇡ 0 D(m, m 0 )) and N n  n∆/ ln 2 (n) to obtain the bound r2 (r max + 1) 3 N n D(m, m 0 ) ⇡ 0 n 2 (1 + ln(n)) 2

✓

  D m + D m 0 (1 + L m 0 ) 2e -L m 0 D m 0 -⌧so that, reminding that Gm (m 0 ) is defined by (5.39), 0 e -L m 0 D m 0 σ 4 1 n which ends the proof. 2

  The collection (S m ) m2M n is composed of finite dimensional linear sub-spaces of L 2 ([0, 1]), indexed by a set M n depending on n. The space S m has dimension D m  N n  n, 8m 2M n , where N n designates a maximal dimension, and S m is equipped with an orthonormal basis (' λ ) λ2Λ m with |Λ m | = D m . ) Nesting condition: There exists a space denoted by S n , belonging to the collection, with 8m 2M n ,S m ⇢S n , with dimension denoted by N n .

	The following holds:
	(H 1 ) Norm connection: There exists Φ 0 > 0, such that,
	8m 2M n , 8t 2 S m , ktk 1  Φ 0 D 1/2 m ktk.	(5.10)
	(H 2

Thus here rm,m 0  (r max + 1) 3/2 / p ⇡ 0 .

Collection [GP]-Here we have rm,m 0  [(r max + 1) p N n ]/ p D(m, m 0 )⇡ 0 .

We now prove Proposition 5.13. We apply Lemma 5.15 to the linear space S m + S m 0 of dimension D(m, m 0 ) and norm connection measured by rm,m 0 bounded above. We consider δ k -nets, T k = T δ k \ B ⇡ m,m 0 (0, 1), with δ k = δ 0 2 -k with δ 0  1/5, to be chosen later and we set

(5.41) Given some point u 2 B ⇡ m,m 0 (0, 1), we can find a sequence

Thus we have the following decomposition that holds for any u 2 B ⇡ m,m 0 (0, 1),

Clearly

In the sequel we denote by P n (.) the measure P(. \ Ω n ), see (5.24), (actually only the inequality ktk 2 n  3 2 ktk 2 ⇡ holding for any t 2 S m + S m 0 is required). Let (⌘ k ) k≥0 be a sequence of positive numbers that will be chosen later on and

n is defined by (5.27) -(5.32) -(5.33). We have

Then using inequality (5.37) of Lemma 5.12 and (5.41), we straightforwardly infer that IP 1  exp(H 0 -Cnx 0 ) and IP 2  P k≥1 exp(H k-1 + H k -Cnx k )