Émeline Schmisser
email: emeline.schmisser@univ-lille.fr

Non parametric estimation of the jump coefficient of a diffusion with jumps

Keywords: jump diffusions, nonparametric estimation, model selection. Subject classification: 62G05, 62M05

Non parametric estimation of the jump coefficient of a diffusion with jumps

Émeline Schmisser

Introduction

We consider the jump diffusion given by the stochastic differential equation dX t = b(X t )dt + σ(X t )dW t + ξ(X t -)dL t , X 0 = η

(1)

with η a random variable, (W t ) t≥0 a Brownian motion independent of η and (L t ) t≥0 a pure jump centred Lévy process independent of (W t , η). It can be written

L t = t 0 R z(µ(dz, dt) -ν(dz)dt)
where µ is a Poisson measure of intensity ν(dz)dt. We do not assume that the jumps are of finite intensity, only that L t is centred and has moments of any order.

The diffusion is observed at discrete times t = 0, ∆, . . . , n∆ under the asymptotic framework:

n∆ → ∞ and ∆ → 0. Our aim is to construct adaptive non-parametric estimator of the jump coefficient ξ 4 on a compact set A.

Jump diffusions are used to modelize dynamical systems where the noise is discontinuous or too intensive to be modeled by a Brownian motion, for instance polymerazation phenomenons [START_REF] Berestycki | Exchangeable fragmentation-coalescence processes and their equilibrium measures[END_REF]), telephone noise [START_REF] Protter | The Euler scheme for Lévy driven stochastic differential equations[END_REF]), or finance (see Aït-Sahalia and Jacod 1 (2009) or [START_REF] Protter | The Euler scheme for Lévy driven stochastic differential equations[END_REF] for instance). The estimation of the drift function b, the diffusion coefficient σ and the second moment, σ 2 +ξ 2 , is well known. Fort the drift, [START_REF] Shimizu | Estimation of parameters for diffusion processes with jumps from discrete observations[END_REF] construct parametric maximum-likelihood estimators in the stationary case. [START_REF] Mancini | Threshold estimation of Markov models with jumps and interest rate modeling[END_REF] use local time to construct kernel point-wise estimators. [START_REF] Schmisser | Non-parametric adaptive estimation of the drift for a jump diffusion process[END_REF] bound the empirical risk of an adaptive estimator. The diffusion coefficient is even more studied. Indeed, for instance, [START_REF] Shimizu | Some remarks on estimation of diffusion coefficients for jump-diffusions from finite samples[END_REF] construct maximam-likelihood estimators of the coefficients σ 2 and ξ 2 .

Their estimators converge with rate √ n and √ n∆ respectively. [START_REF] Mancini | Threshold estimation of Markov models with jumps and interest rate modeling[END_REF] (for the finite intensity case) and [START_REF] Song | Threshold reweighted Nadaraya-Watson estimation of jump-diffusion models[END_REF] (for compound Poisson processes) estimate the diffusion coefficient σ 2 , using local time and kernel estimators. [START_REF] Hanif | Reweighted Nadaraya-Watson estimation of jump-diffusion models[END_REF] provide local polynomials estimators of the second moment σ 2 + ξ 2 , also in the finite intensity case. [START_REF] Schmisser | Non parametric estimation of the diffusion coefficients of a diffusion with jumps[END_REF] construct a robust and adaptive estimator of σ 2 and of the second moment, even for infinite intensity Lévy processes. There exist fewer results for the estimation of the jump coefficient ξ 2 . The parametric estimator of [START_REF] Shimizu | Some remarks on estimation of diffusion coefficients for jump-diffusions from finite samples[END_REF] converges with rate √ n∆. For nonparametric estimation, [START_REF] Park | Nonparametric estimation of jump diffusion models[END_REF] construct estimators for the three coefficients, using local times and kernel estimator. The rate of convergence of the kernel estimator of ξ 2 is (n∆) -2/5 + ∆ 1/2 .

There are two possibilities to construct an estimator of the jump component. Either compute estimators ĝ and σ2 of the second moment g(x) = σ 2 (x) + ξ 2 (x) and of the diffusion coefficient, σ 2 (x), and deduce an estimator of ξ 2 by substraction, or equivalently, like [START_REF] Park | Nonparametric estimation of jump diffusion models[END_REF], use the square of the increments when they are big enough (when a jump is suspected): indeed,

(X (k+1)∆ -X k∆ ) 2 ∆ ½ |X (k+1)∆ -X k∆ |≥∆ 1/2-ε = ξ 2 (X k∆ ) + centred terms + small terms.
The problem of this method is that the small jumps will not be kept. It is not very important if the Lévy process is a compound Poisson process, but if the jumps have infinite activity or infinite intensity, the small terms will not be so small.

In this article, we use the second possibility and consider an estimator based on the fourth moments of the increments:

T k∆ = (X (k+1)∆ -X k∆ ) 4 ∆ = ξ 4 (X k∆ ) + centred terms + small terms .

The Brownian terms are small because of the power 4, and it only remains the jumps. We introduce a sequence of increasing subspaces S m of L 2 (A) and construct a sequence of estimators ξ4 m by minimizing on each S m the contrast function γ n (t) = 1 n n k=1 (T k∆ -t(X k∆ )) 2 . Its L 2 -risk is bounded by

E ξ4 -ξ 4 2 L 2 (A) ξ 4 -ξ 4 m 2 L 2 (A) bias term + Ξ D m n∆ variance term + ∆
where D m is the dimension of the subspace S m and Ξ a constant depending on ξ and ν. The bias term decreases when the dimension D m increases, whereas the variance term increases. To find a good bias-variance compromise, we introduce a penalty, pen(m), proportional to the variance term. As the penalty constant, Ξ, is unknown, we replace it by an estimator Ξ1 and the penalty by pen(m). To choose the dimension, we minimize the quantity γ n ( ξ4 m ) + pen(m) and deduce the "best" dimension m. Finally, we prove that the risk of robust estimator, ξ m, satisfies an oracle inequality.

The paper is divided as follows: the model and its assumptions are stated in Section 2. In Section 3, we construct the estimator and bound its risk. Section 4 is devoted to the simulations and the proofs are gathered in Section 5.

In all the paper, C means a constant that does not depend on n, ∆ or m, but may be different from one line to another.

Model and assumptions

We consider the stochastic differential equation given by (1) and assume the following assumptions:

Assumption A1.

a. The functions b, σ and ξ are Lipschitz.

b. The drift function b is elastic:

∃M > 0, ∃C > 0 , ∀x, |x| ≥ M , xb(x) ≤ -Cx 2 .
c. The diffusion and jump functions σ and ξ are bounded: there exists σ

0 , σ 1 , ξ 0 , ξ 1 ∈ (R + * ) 4 such that ∀x, 0 < ξ 0 ≤ ξ(x) ≤ ξ 1 and 0 < σ 0 ≤ σ(x) ≤ σ 1 .
d. The Lévy measure has moments of any order: ∀k ≥ 2, R |z| k ν(dz) < +∞.

We introduce the notation

I k := R z k ν(dz). e. I 4 = R z 4 ν(dz) = 1.
Item a ensures that SDE (1) has a unique strong solution (see Applebaum (2004, Theorem 6.2.9).

According to Masuda (2007, Theorem 2.2 and Lemma 2.4), under Assumptions 1a-d, the process (X t ) t≥0 has a unique stationary distribution, is ergodic and exponentially β-mixing. Assumption e ensures that the model is identifiable: indeed, if we replace ν(.) by cν(c.) and ξ by ξ/c, we obtain the same process (X t ) t≥0 . If this assumption is not fulfilled, our estimator will estimate ξ 4 (X k∆ )I 4 .

We can know assume:

Assumption A2.

a. The process (X t ) t≥0 is stationary.

b. Its stationary measure has a density π which is bounded on any compact set.

c. the density π is bounded from below and above on the compact of estimation A:

∃π 0 , π 1 ; ∀x ∈ A, 0 < π 0 ≤ π(x) ≤ π 1 .
By Masuda (2007, Theorem 2.2), it implies that X t has also moments of any order. Let us consider the σ-algebra

F t = σ (η, (W s ) 0≤s≤t , (L s ) 0≤s≤t ).
To bound the risk the adaptive estimator, we apply a Talagrand's inequality. To this end, we need independent and bounded random variables.

As we have an exponentially β-mixing process, we can construct independent random variables thanks to the Berbee's coupling lemma. But to work with nearly bounded variables, we need the following assumption:

Assumption A3.

a. The jumps are sub-exponential:

∃M, C, λ, ∀z, |z| ≥ M ν([-z, z] c ) ≤ Ce -λ|z| .
b. The random variables (X ∆ , . . . , X n∆ ) have exponential moments: there exists C, K such that:

E e CX∆ K c. ∃η, η > 1, such that ∆ η = O(n -1 ).
d. The Blumenthal-Getoor index is strictly less than 2: there exists β < 2,

1 -1 z β ν(dz) < ∞.
This is not a very strong assumption, as

1 -1 z 2 ν(dz) is already finite.
The following proposition helps us to control the increments of the process. It is proved for instance in Applebaum (2004, Theorems 4.4.21 and 4.4.23)).

Proposition 1. According to the Burkholder-Davis-Gundy inequality, for any p, there exists a constant C p such that:

E sup s∈[t,t+h] t s σ(X u )dW u p |F t ≤ C p E   t+h t σ 2 (X u )du p/2 |F t   and E sup s∈[t,t+h] t s ξ(X u -)dL u p |F t ≤ C p E   t+h t ξ 2 (X u )du p/2 |F t   I p/2 2 + C p E t+h t |ξ p (X u )| du |F t I p .
We deduce the following lemma:

Lemma 2. For any integer p:

∀u > 0, E sup 0≤s≤∆ (X s+u ) 2p |F t 1 + |X u | 2p and ∀u ≥ 0; E sup 0≤s≤t (X t+u -X u ) 2p |F t ≤ C 2p tξ 2p 1 I 2p + t p σ 2p 1 + ξ 2p 1 I p 2 .
Then, as b, σ 2 and ξ 2 are Lipshitz:

E |B k∆ | 2p ∆ 2p , E J 2p k∆ ∆ p and E (J k∆ ) 2p = ∆ξ 2p (X k∆ )I 2p + C∆ 3/2 .
The bound of J k∆ is sometimes not precise enough. By the Lévy-Khintchine formula, E e iuL∆ = e -∆ R (e iuz -1-uz)ν (dz) . We obtain:

Lemma 3. We have the following moments:

E L 2 k∆ = I 2 ∆ E L 4 k∆ = I 4 ∆ + 3I 2 2 ∆ 2 E L 6 k∆ = I 6 ∆ + (15I 2 I 4 + 10I 2 3 )∆ 2 + 15I 3 2 ∆ 3 E L 8 ∆ = I 8 ∆ + (35I 2 4 + 56I 3 I 5 + 28I 2 I 6 )∆ 2 + (210I 2 2 I 4 + 280I 2 I 2 3 )∆ 3 + 105I 4 2 ∆ 4 .

Estimator

Our aim is to construct an adaptive nonparametric estimator of the jump function ξ on a compact set A. To do so, the first idea is to consider the random variable

(X (k+1)∆ -X k∆ ) 2 ∆ ≃ σ 2 (X k∆ ) + ξ 2 (X k∆ ) + centred terms + small terms.
Schmisser (2019) construct an adaptive estimator ĝ m of σ 2 + ξ 2 . Its L 2 -risk is bounded by C(n∆) -2αg /(2αg +1) + C ′ ∆ where α g is the regularity of g (more precisely, if g belongs to the

Besov space B α 2,∞ ). By cutting off the jumps thanks to a threshold, they obtain an estimator of

σ 2 : (X k+1 -X k∆ ) ∆ ½ |X (k+1)∆ -X k∆ |≤c∆ 1/2 ln(n) = σ 2 (X k∆ ) + centred terms + small terms.
The risk of the adaptive estimator σ2 m is bounded by

Cn -2ασ/(2ασ +1) + C ′ ∆ 1-β/2
, where α σ is the regularity of σ 2 and β the Blumenthal-Getoor index.

Then we can deduce ξ 2 by substraction. Let us set ξ2 = ĝ m -σ2 m. Its risk is bounded by:

E ξ2 -ξ 2 2 L 2 (A) (n∆) 2αg /(2αg +1) + n -2ασ/(2ασ +1) + ∆ 1-β/2 .
The problem is, the estimators obtained by subtracting terms are not very powerful. In particular, the rate of convergence does not only depends on the regularity of ξ 2 , but on the regularity of σ 2 and σ 2 + ξ 2 . Moreover, when Blumenthal-Getoor index β is strictly positive, the remainder term is quite big.

That is why we consider the fourth moments of the increments:

T k∆ = (X (k+1)∆ -X k∆ ) 4 ∆ .
The Brownian terms are small because of the power 4, and it only remains the jumps.

Decomposition of T k∆

Let us set:

B k∆ = (k+1)∆ k∆ b(X s )ds, Z k∆ = (k+1)∆ k∆ σ(X s )dW s , J k∆ = (k+1)∆ k∆ ξ(X s -)dL s .
Then

T k∆ = (B k∆ + Z k∆ + J k∆ ) 4 ∆ .
The drift and diffusion terms B k∆ and Z k∆ are quite small, the jump term J k∆ is a good approximation of ∆ξ 4 (X k∆ ). We can write T k∆ = ξ 4 (X k∆ ) + E k∆ + F k∆ where

∆E k∆ := (B k∆ + Z k∆ + J k∆ ) 4 -(Z k∆ + J k∆ ) 4 + E (Z k∆ + J k∆ ) 4 |F k∆ -ξ 4 (X k∆ )I 4 ∆F k∆ := (Z k∆ + J k∆ ) 4 -E (Z k∆ + J k∆ ) 4 |F k∆ .
The terms E k∆ are small, whereas F k∆ are centred:

Lemma 4.

•

E E 2 k∆ ∆, E E 4 k∆ ∆ and E E 8 k∆ ∆. • E (F k∆ |F k∆ ) = 0, E F 4 k∆ 1/∆ 3 and Var (F k∆ |F k∆ ) = ξ 8 (X k∆ )I 8 ∆ + C √ ∆ .

Subspaces of approximation

To construct an adaptive estimator of ξ 4 , we consider a sequence of increasing subspaces S 0 ⊆ . . . ⊆ S m , . . . such that ∪ m S m is dense in L 2 (K). We minimize a contrast function γ n (t) on each S m and then choose the best estimator by introducing a penalty function (see for instance [START_REF] Barron | Risk bounds for model selection via penalization[END_REF]). The rate of convergence of our estimator will depend on the regularity of the coefficient ξ,

i.e. its modulus of smoothness.

Definition (Modulus of smoothness). The modulus of continuity of a function f is defined by

ω(f, t) = sup |x-y|≤t |f (x) -f (y)|. If f is Lipschitz, the modulus of continuity is proportional to t. If ω(f, t) = o(t), then f is constant:
the modulus of continuity cannot measure higher smoothness.

We define the modulus of smoothness by

ω r (f, t) p = sup 0<h≤t ∆ r h (f, .) L p where ∆ r h (f, x) = r k=0 (-1) k r k f (x + kh). If f ∈ C r , then for 1 ≤ p ≤ ∞: ω r (f, t) p ≤ t r ω(f (r) , t) p .
Definition (Besov space). The Besov space B α 2,∞ is the set of functions:

B α 2,∞ = {f ∈ L 2 , sup t>0 t -α ω r (f, t) 2 < ∞}
where r = ⌊α + 1⌋. The norm on a Besov space is defined by:

f B α 2,∞ := sup t>0 t -α ω r (f, t) 2 + f L 2 .
For more details see [START_REF] Devore | Constructive approximation, volume 303 of Grundlehren der Mathematischen Wissenschaften[END_REF].

Assumption A4. We consider a series of vectorial subspaces of L 2 (A), (S m ) m∈Mn satisfying the assumptions a. The subspaces S m have finite dimension D m and are increasing: ∀m, S m ⊆ S m+1 .

b. The . L 2 and . ∞ norms are connected:

∃φ 1 , ∀m, ∀t ∈ S m , t 2 ∞ ≤ φ 1 D m t 2 L 2 with t 2 L 2 = A t 2 (x)dx and t ∞ = sup x∈A |t(x)|. This implies that, for an orthonormal basis ϕ λ of S m , Dm λ=1 ϕ 2 λ ∞ ≤ φ 1 D m .
c. There exists a constant φ 2 such that for any m ∈ N, there exists an orthonormal basis

(ψ λ ) λ∈Λ of S m such that ∀λ, card(λ ′ , ψ λ ψ λ ′ ∞ = 0) ≤ φ 2 . d. For any function t ∈ B α 2,∞ , ∃c, ∀m, t -t m 2 L 2 ≤ cD -2α m
where t m is the orthogonal projection L 2 of t on S m .

Assumptions A4 a, b and d are quite standard (see for instance [START_REF] Comte | Penalized nonparametric mean square estimation of the coefficients of diffusion processes[END_REF]). Assumption A4 c is an additional assumption, which is not satisfied by the trigonometric basis, for instance.

However, it is fulfilled by subspaces generated by piecewise polynomials of degree r, spline functions of degree r, or compactly supported wavelets.

Estimator with fixed m

We then define the contrast function

γ n (t) = 1 n n k=1 (T k∆ -t(X k∆ )) 2 (2)
and for any m ∈ M n = {m, D 2 m ≤ (n∆)/ ln(n)}, we consider the estimator

ξ4 m = arg min t∈Sm γ n (t). (3)
The easiest bound is obtained for the empirical risk:

R n (t) := E t -ξ 4 2 n where t 2 n = 1 n n k=1 t 2 (X k∆ ).
Proposition 5. Under Assumptions A1-A4, for any m such as

D m ∈ M n : R n ( ξ4 m ) ≤ ξ 4 -ξ 4 m,π 2 π + 12Ξ D m n∆ + C∆ + C ′ n
where ξ 4 m,π is the orthogonal projection of ξ 4 on S m (for the π-norm), t 2 π = A t 2 (x)π(x)dx and

Ξ = min ξ 8 1 I 8 , φ 1 π 0 E π ξ 8 (X ∆ ) I 8 .
The empirical risk is not very intuitive: indeed, the empirical norm is random, and the stationary measure π is unknown. Nevertheless, the L 2 -risk can also be bounded, even if the bound is less sharp.

Corollary 6. Under Assumptions A1-A4, for any m such as

D m ≤ D n = n∆/ ln(n): E ξ4 m -ξ 4 2 π ≤ 5 ξ 4 -ξ 4 m,π 2 π + 48Ξ D m n∆ + 4C∆.
Moreover,

E ξ4 m -ξ 4 2 L 2 (A) ≤ 5 π 1 π 0 ξ 4 -ξ 4 m 2 L 2 (A) + 48 Ξ π 0 D m n∆ + 4C π 0 ∆.
where ξ 4 m is the orthogonal projection (for the L 2 -norm) of ξ 4 on S m .

Adaptive estimator

We obtain a sequence of estimators of ξ 4 , and have to choose the best. If ξ 4 belongs to the Besov space B α 2,∞ , then the bias term

ξ 4 -ξ 4 m 2 L 2 is smaller than D -2α
m and the risk is optimal for 1+2α) . The optimal estimator satisfies:

D m = (n∆) -1/(
R n ( ξ4 mopt ) (n∆) -2α/(2α+1) + ∆.
As the regularity of ξ 4 is unknown, we have to choose the best estimator automatically. To this aim, we introduce the penalty function pen(m) = κΞD m /(n∆) and we choose the dimension m:

m = arg min m∈Mn {γ n ( ξ4 m ) + pen(m).}
Then we have the following theorem:

Theorem 7. Under Assumptions A1-A3, R n ( ξ4 m) inf m∈Mn ξ 4 -ξ 4 m,π 2 π + Ξ D m n∆ + 1 n∆ + ∆.
We obtain the bound of the L 2 -risk:

Corollary 8. E ξ4 m -ξ 4 2 L 2 (A) inf m∈Mn ξ 4 -ξ 4 m 2 L 2 (A) + Ξ D m n∆ + 1 n∆ + ∆.

Robust estimator

The constant Ξ = min(ξ

8 1 I 8 , φ1 π0 E π ξ 8 (X ∆ ) I 8 ) + ∆ 1/4
is unknown and has to be estimated. Let us

first estimate h(x) = ξ 8 (x)I 8 . Let us note V k∆ := (X (k+1)∆ -X k∆ ) 8 ∆ = ξ 8 (X k∆ )I 8 + G k∆ + H k∆ ,
where G k∆ is small and H k∆ is centred:

∆G k∆ = (B k∆ + Z k∆ + J k∆ ) 8 -(Z k∆ + J k∆ ) 8 + E (Z k∆ + J k∆ ) 8 |F k∆ -ξ 8 (X k∆ )I 8 ∆H k∆ = (Z k∆ + J k∆ ) 8 -E (Z k∆ + J k∆ ) 8 |F k∆ .
The following lemma bounds the moments of G and H:

Lemma 9. E G 2 k∆ |F k∆ ∆, E G 4 k∆ |F k∆ ∆, E ( H k∆ ) |F k∆ ) = 0, E H 2 k∆ |F k∆ 1 ∆ , E H 4 k∆ |F k∆ 1 ∆ 3 .
We consider the contrast function γ 2,n (t) = 1 n n k=1 (V k∆ -t(X k∆ )) 2 and the estimator ĥm = arg min t∈Sm γ 2,n (t). This estimator is consistent:

Lemma 10. E h -ĥm 2 n h -h m 2 L 2 + D m n∆ + ∆. Let us set ĥ1 = sup x∈A | ĥln(n) (x)|.
Let us now estimate the second term, φ 1 /π 0 E π ξ 8 (X ∆ ) I 8 . We consider the rectangular kernel

K(x) := ½ |x|≤1/2 and set πh (x) = 1 n n k=1 K h (X i -x) with K h (x) = 1 h K h (x/h).
We take a grid (x 1 , . . . , x ln 2 (n) ) of equally spaced points of A, and compute the minimum of π√ n∆ on this grid:

π0 := min 1≤j≤ln 2 (n) π√ n∆ (x j ).
Finally, we obtain an estimator of the constant Ξ:

Ξ := min ĥ1 , φ 1 π0 Vn + ∆ 1/4 + 1/ ln(n).
We can prove that Ξ is a good estimator of Ξ:

Lemma 11. Under Assumptions A1-3, we have that

P | Ξ -Ξ| ≥ C/ ln(n) + ∆ 1/4 n -4 .
We then are able to construct a robust estimator: let us set

m = arg inf m∈Mn γ n ( ξm ) + κ Ξ D m n∆ .
We deduce the convergence of the robust estimator:

Corollary 12. The robust estimator satisfies the oracle inequality: Under Assumptions A1-A3,

R n ( ξ4 m) inf m∈Mn ξ 4 -ξ 4 m 2 π + Ξ D m n∆ + 1 n∆ + ∆.
Moreover,

E ξ m -ξ 4 A 2 L 2 (A) inf m∈Mn ξ 4 -ξ 4 m 2 L 2 (A) + Ξ D m n∆ + 1 n∆ + ∆.

Comparison with the substraction estimator

The risk of the estimator obtained by substraction, ξ2 = g -σ2 , is

E ξ2 -ξ 2 A 2 L 2 (A) (n∆) - 2αg 2αg )+1 + n -2ασ /(2ασ+1) + ∆ 1-β/2
where α g is the regularity of g and α σ 2 the regularity of σ 2 and β the Blumentoor-Getoo index. See [START_REF] Schmisser | Non parametric estimation of the diffusion coefficients of a diffusion with jumps[END_REF] for the proofs. In [START_REF] Park | Nonparametric estimation of jump diffusion models[END_REF], the Lévy process has finite intensity, and all the coefficients are C 2 . The pointwise risk of the estimator of ξ 2 is bounded by C(n∆) -4/5 +C ′ ∆ for ∆ ≤ (n∆) -r . The remainder term, ∆, is smaller than ∆ 1-β/2 , but r depends on the Blumenthal-Getoor index β and on the regularity of the coefficients. Moreover, this bound is not valid for an infinite intensity Lévy process.

In order to compare those bounds to the risk of our estimator ξ m, we can remark that

ξ2 m -ξ 2 A 2 L 2 (A) ≤ ξ4 m -ξ 4 A 2 L 2 (A) min A ξ 2 (x) .
Therefore, we get that

E ξ2 m -ξ 2 A 2 n (n∆) -2α/(2α+1) + ∆.
If σ is less regular than ξ or if β is not null, then our estimator ξ4 m, based on the fourth moment, has a better convergence rate than ξ2 .

Simulations

We consider the linear subspaces S m generated by the spline functions of degree r (that is, piecewise polynomials of degree r and C r-1 . We compute the series of estimators ξ4 m,r for r between 0 and 4 and m between 0 and max( n∆/ ln(n), 7) (for m = 7, we already have D m = 128, for m bigger, the computations become too slow). To obtain the adaptive estimator, we minimise first with regard to m, then to r.

We choose κ = 8 for the simulations. We consider 3 different models. For each model, we let n and ∆ vary and realize 200 simulations for each value of (n, ∆). We compute the robust estimator ξ4 m,r , the selected dimension D m,r , the estimated penalty constant Ξ and the L 2 -error

risk = N -1 j=0 ξ4 m,r (x j ) -ξ 4 (x j ) 2 (x j+1 -x j ) ≃ A ξ4 m,r (x) -ξ 4 (x) 2 dx
for (x 0 , x 1 , . . . , x N ) a regular subdivision of A. We also compute the empirical error for each ξm,r .

Then we deduce the dimension (m opt , r opt ) that minimizes the empirical error (denoted by err min ).

To compare this oracle estimator to the robust one, we compute an oracle or which is the mean of the logarithmic ratio between the two errors. To compare our estimator to the one obtained by substraction, we also compute the error risk1 = ξ2 m,r -ξ 2 A 2 L 2 and the risk of the estimator obtained by substraction: risk2

:= ξ2 m,r -ξ 2 A 2 L 2 .
We add the empirical risks of the estimators of the diffusion coefficient σ 2 and the second moment g(x) = σ 2 (x) + ξ 2 (x).

For each model, we provide two tables. The first one show some result for the estimator ξ4 m,r :

the mean and standard deviation of the risk of the robust estimator ξ4 m,r (in fact, the standard deviation of these quantity, to have a norm), a criteria of comparison between the risks of the robust estimator and the 'oracle' estimator; or = mean(ln(risk/err)), the mean and the standard deviation of the estimated penalty constant Ξ and the mean and the standard deviation of the selected dimension D m,r .

On the second table, we write the mean and standard deviations of risk1 and risk2 (more precisely, the square roots of those quantities) in order to compare these two estimators. We also add the means and standard deviation for the risk of the estimators ĝ m and σ2 m.

Models

In the first two models, L t is simply a compound Poisson Process:

t 0 ξ(X s -)dL s = Nt k=1 ξ(X t - k )ζ k
where N t is a Poisson Process of intensity λ, t 1 , . . . , t Nt the times of the jumps, and (ζ k ) k≥0 are independent, identically distributed of binomial law:

P (ζ k = ±1) = 1/2.
Model 1: Ornstein-Uhlenbeck Let us consider the process given by the equation:

dX t = -2X t dt + dW t + dL t
with binomial jumps: P(ζ = 1) = P(ζ = -1) = 0.5 and the intensity of the Poisson process is 1.

We have that I 2 = 1, I 4 = 1, I 6 = 1, I 8 = 1. We find Model 2: non-constant coefficient

dX t = -2X t dt + 2 + 0.5 sin(piX t )dW t + X 2 t -+ 3 X 2 t -+ 1 dL t
with binomial jumps: P (z = 1) = P (z = -1) = 0.5.

Model 3: nearly stable processes -0.4 -0.2 0.0 0.2 0.4 0.0 0.5 1.0 1.5 2.0

dX t = -2X t + 3 + X 2 t 1 + X 2 t dW t + 2 + 0.5 sin(X t -)dL t
--estimator ξ2 m . . . estimator ĝ m -σ2 m. n = 10 5 , ∆ = 10 -2 . Risk of ξ4 m,r n ∆ risk Ξ D m,
(dz) = 2-β 2|z| 1+β ½ |z|≤1 .
β is the Blumenthal-Getoor index of the process. We take β = 1/2 and we get that I 2 = 1, I 4 = 3/7, I 6 = 3/11 and I 8 = 3/15.

Results

The risk of the adaptive estimator ξ4 m,r decreases when n increases (for fixed ∆). When the period of observation (n∆) is fixed, the risk seems to be either stable, either decreasing with ∆, especially when ∆ is not small enough. When n∆ ≤ 100, our estimator nearly always select the smallest dimension, whereas for n∆ ≥ 10 3 , the algorithm can select more complex models. Our estimator is truly adaptive, and the oracle seems to be quite small.

For the estimated penalty constant Ξ, it can be noted than when n increases, the variance decreases as well: Ξ converges to a constant depending on ∆. When ∆ decreases, this constant decreases as well. This is because

Vn → E (V ∆ ) ≃ ∆ -1 E (J ∆ + Z k∆ ) 8 ≥ E ξ 8 (X ∆ ) I 8 .
This is not a drawback, as the penalty should be the variance of the centred terms, more precisely

Ξ ∆ = ∆ min Var (F k∆ |F k∆ ) ∞ ; φ 1 π 0 Var π (F ∆ ) .
Ξ is an estimator of this quantity, that converges when ∆ → 0 and n∆ → ∞: by Proposition 5, Ξ ∆ = Ξ + C∆ 1/2 . In the simple case where the jump and diffusion coefficients are constants, then

Ξ ∆ = ∆ -1 Var (F ) k∆ = ∆ -1 Var (ξL k∆ + σZ k∆ ) 4 = ∆ -1 ξ 8 E L 8 k∆ + 28σ2ξ 6 E L 6 Z 2 -ξ8(E L 4 ) 2 + O(∆ 2 ) = ξ 8 I8 + ∆ ξ 8 (34I 2 4 + 28I 2 I 6 ) + σ 2 ξ 6 28I6 + O(∆ 2 ).
When ∆ is not small enough, the term in ∆ is not negligeable, neither is the term in O(∆ 2 ).

Our estimator ξ4 m is better than ξ2 as soon as the path of discretisation ∆ is small enough (∆ ≤ 10 -2 ), even for compound Poisson processes.

Proof

Proof of Lemma 4 and 9

According to Proposition 1, we have that

E B 2p k∆ |F k∆ ∆ 2p , E Z 2p k∆ |F k∆ ∆ p , and E J 2p k∆ |F k∆ ∆I 2p (4)
Let us set

Bk∆ = b(X k∆ )∆ Bk∆ = B k∆ -Bk∆ Zk∆ = σ(X k∆ )(W (k+1)∆ -W k∆ ), Zk∆ = Z k∆ -Zk∆ , Jk∆ = ξ(X k∆ )(L (k+1)∆ -L k∆ ), Jk∆ = J k∆ -Jk∆ .
dX t = -2X t dt + X 2 t -+3 X 2 t -+1 dW t + 2 + 0.5 sin(πX t -)dL t ν(z) = 1.5 2|z| 1.5 ½ |z|≤1
-1.0 -0.5 0.0 0.5 1.0 1.0 1.5 2.0 2.5 3.0 (5) Indeed, by Hölder inequality and Lemma 2, as the drift function b is Lipschitz,

--estimator ξ2 m . . . estimator ĝ m -σ2 m. n = 10 5 , ∆ = 10 -2 . Estimation of ξ4 m,r . n ∆ risk Ξ D m,
E B2p k∆ |F k∆ = E   (k+1)∆ k∆ (b(X s ) -b(X k∆ ))ds 2p   |F k∆   ≤ ∆ 2p-1 (k+1)∆ k∆ E (b(X s ) -b(X k∆ )) 2p |F k∆ ds ∆ 2p-1 (k+1)∆ k∆ E |X s -X k∆ | 2p |F k∆ ds ∆ 2p+1 .
By Proposition 1 and the same arguments, we get:

E Z2p k∆ E (k+1)∆ k∆ (σ(X s ) -σ(X k∆ )) 2 ds p ∆ p-1 (k+1)∆ k∆ E (σ(X s ) -σ(X k∆ ) 2p ds ∆ p+1 and E J2p k∆ |F k∆ E (k+1)∆ k∆ (ξ(X s -) -ξ(X k∆ )) 2p ds |F k∆ I 2p + E (k+1)∆ k∆ (ξ(X s -) -ξ(X k∆ )) 2 ds p |F k∆ I p 2 ∆ 2 I 2p + ∆ 2p I 2p .
Bound of E k∆ . Let us decompose the bias term

E k∆ = (B k∆ + Z k∆ + J k∆ ) 4 -(Z k∆ + J k∆ ) 4 + E (Z k∆ + J k∆ ) 4 |F k∆ -ξ 4 (X k∆ I 4 ).
We can write E k∆ = E

(1)

k∆ + E

(2)

k∆ + E

(3)

k∆ + E (4)
k∆ where

∆E

(1)

k∆ = E (J k∆ + Z k∆ ) 4 -( Jk∆ + Zk∆ ) 4 |F k∆ ∆E (2) k∆ = E ( Jk∆ + Zk∆ ) 4 -J4 k∆ |F k∆ ∆E (3) k∆ = E J4 k∆ |F k∆ -∆ξ 4 (X k∆ )I 4 ∆E (4) k∆ = (J k∆ + Z k∆ + B k∆ ) 4 -(J k∆ + Z k∆ ) 4 .
The first three terms are conditional expectations, the last is random. For the first term, we factorize the difference of two squares:

∆E (1) k∆ = E ( Jk∆ + Zk∆ ) J k∆ + Jk∆ + Z k∆ + Zk∆ (J k∆ + Z k∆ ) 2 + ( Jk∆ + Zk∆ ) 2 |F k∆ Then, as E ( Jk∆ + Zk∆ ) 2
∆ 2 , we obtain by Cauchy-Schwarz:

∆E (1) k∆ ∆ 2 1/2 ∆ 1/4 ∆ 1/4 ≃ ∆ 3/2 .

Moreover, E

(2) k∆ can be written ∆E

(2) k∆ = 4 j=1 C j 4 E Zj k∆ J4-j k∆ |F k∆ .
As Zk∆ and Jk∆ are centred and conditionnally independent, by (4), we obtain:

∆E (2) k∆ 4 j=1 ∆ j/2 (∆½ j<4 + ½ j=4 ) ∆ 3/2 .
Furthermore, by lemma 3, E L 4 k∆ = ∆I 4 + O(∆ 2 ) and therefore:

∆E (3) k∆ = ξ 4 (X k∆ )(E L 4 ∆ -∆I 4 ) ∆ 2 .
As ∆E

(4) k∆ = 4 j=1 C j 4 B j k∆ (Z k∆ + J k∆ ) 4-j , we get that E ∆E (4) k∆ 2p 4 j=1 E B 2jp k∆ (Z k∆ + J k∆ ) 8p-2pj . ( 6
)
By Cauchy-Schwartz inequality and ( 4), for 2 ≤ j ≤ 4,

E B 2jp k∆ (Z k∆ + J k∆ ) 8p-2pj ≤ E B 4jp k∆ E (Z k∆ + J k∆ ) 16p-4pj 1/2 ≤ ∆ 2jp ½ j=4 + ∆ 1/2 ½ j<4 ≤ ∆ 4p . (7)
Moreover, as B k∆ = ∆b(X k∆ ) + Bk∆ , by Lemma 4 and equation ( 5), for j = 1 we obtain:

E B 2p k∆ (Z k∆ + J k∆ ) 6p ∆ 2p E b(X k∆ ) 2p E (Z k∆ + J k∆ ) 6p |F k∆ + E B2p k∆ (Z k∆ + J k∆ ) 6p ∆ 2p+1 + E B4p k∆ E (Z k∆ + J k∆ ) 12p 1/2 ∆ 2p+1 + ∆ 2p+1/2 ∆ 1/2 = ∆ 2p+1 . Then E ∆E (4) k∆ 2p ∆ 2p+1 . As E E 2p k∆ E (1)
k∆ + E

(2)

k∆ + E (3) k∆ 2p + E E (4) k∆ 2p
, we get that

E E 2p k∆ ∆.
Bound of G k∆ We recall that

∆G k∆ = (B k∆ + Z k∆ + J k∆ )) 8 -(Z k∆ + J k∆ ) 8 + E (Z k∆ + J k∆ ) 8 |F k∆ -∆ξ 8 (X k∆ )I 8 .
As for the bound of E k∆ , we can write

G k∆ = G (1) k∆ + G (2) k∆ + G (3) k∆ + G (4)
k∆ with

∆G

(1)

k∆ := E (Z k∆ + J k∆ ) 8 |F k∆ -E ( Zk∆ + Jk∆ ) 8 |F k∆ ∆G (2) k∆ := E ( Zk∆ + Jk∆ ) 8 |F k∆ -E J8 k∆ |F k∆ ∆G (3) k∆ := E J8 k∆ |F k∆ -∆ξ 8 (X k∆ )I 8 ∆G (4) k∆ := (B k∆ + Z k∆ + J 8 k∆ ) -(Z k∆ + J k∆ ) 8 .
Let us note JZ := J k∆ + Z k∆ and JZ := Jk∆ + Zk∆ . Then by Cauchy-Schwarz,

∆G

(1)

k∆ = E ( Jk∆ + Jk∆ )(JZ + JZ) JZ 2 + JZ 2 JZ 4 + JZ 4 |F k∆ (∆ 2 ) 1/2 (∆) 1/4 (∆) 1/8 ∆ 1/8 ∆ 3/2 .
Moreover, as Zk∆ and Jk∆ are conditionnaly independant, we get:

∆G (2) k∆ = 8 j=1 C j 8 E Zj k∆ J8-j k∆ |F k∆ ∆ 3/2 .
And by Lemma 3:

∆G (3) k∆ = ξ 8 (X k∆ )(E L 8 ∆ -∆I 8 ) C∆ 2 .
Finally, by ( 5)

E ∆G (4) k∆ 2p |F k∆ 8 j=2 E B 2pj k∆ (Z k∆ + J k∆ ) 16p-2pj |F k∆ + B2p k∆ E (Z k∆ + J k∆ ) 14p |F k∆ + E B2p k∆ (Z k∆ + J k∆ ) 14p |F k∆ ∆ 2p+1 . Therefore E G 2 k∆ ∆ and E G 4 k∆ |F k∆ ∆.
Moments of F k∆ . The term

F k∆ = 1 ∆ (Z k∆ + J k∆ ) 4 -E (Z k∆ + J k∆ ) 4 |F k∆ is centred.
Moreover, by equation ( 4), for any p > 0,

E F 2p k∆ E J 8p k∆ + E Z 8p k∆ ∆ 2p 1 ∆ 2p-1 I 8p . Then E F 4 k∆ I16 ∆ 3 ∆ -3 . It remains to bound the quantity Var (F k∆ |F k∆ ) -ξ 8 (X k∆ )I8 ∆
. By equation ( 4),

∆ 2 Var (F k∆ |F k∆ ) = Var (Z k∆ + J k∆ ) 4 |F k∆ = E (Z k∆ + J k∆ ) 8 |F k∆ -E (Z k∆ + J k∆ ) 4 |F k∆ 2 = E (Z k∆ + J k∆ ) 8 |F k∆ + C∆ 2 . As E (Z k∆ + J k∆ ) 8 |F k∆ -∆ξ 8 (X k∆ )I 8 = G (1) k∆ + G (2) k∆ + G (3) k∆ ∆ 3/2 ,
we get:

Var (F k∆ |F k∆ ) - ξ(X k∆ ) 8 ∆ ∆ -1/2 .
Bound of H k∆ . We only need an upper bound for the moments of H k∆ . We have that

E H 2 k∆ |F k∆ ≤ 1 ∆ 2 E (J k∆ + Z k∆ ) 16 |F k∆ ∆ 8 + ∆ ∆ 2 = 1 ∆ and E H 4 k∆ |F k∆ ≤ 1 ∆ 4 E (J k∆ + Z k∆ ) 32 |F k∆ ∆ 16 + ∆ ∆ 4 = 1 ∆ 3 .

Proof of Proposition 5

We recall that the empirical norm is defined by

t 2 n = 1 n n k=1 t 2 (X k∆ ). Then for any t ∈ S m , γ n (t) = 1 n n k=1 T k∆ -ξ 4 (X k∆ ) + ξ 4 (X k∆ ) -t(X k∆ ) 2 = 1 n n k=1 (T k∆ -ξ 4 (X k∆ )) 2 + ξ 4 -t 2 n + 2 n n k=1 (T k∆ -ξ 4 (X k∆ ))(ξ 4 (X k∆ ) -t(X k∆ ).
Let us denote by Π m the orthogonal projection on S m for the . n -norm. As ξ4 m minimizes γ n (t),

γ n ( ξ4 m ) ≤ γ n (Π m ξ 4 ). Moreover, as T k∆ -ξ 4 (X k∆ ) = E k∆ + F k∆ , we get: ξ4 m -ξ 4 A 2 n ≤ Π m ξ 4 -ξ 4 A 2 n + 2 n n k=1 (E k∆ + F k∆ )( ξ4 m (X k∆ ) -Π m ξ 4 (X k∆ )).
As ξ4 m and Π m ξ 4 both have support in A. Moreover,

ξ4 m -ξ 4 A 2 n = ξ4 m -Π m ξ 4 2 n + Π m ξ 4 -ξ 4 A 2 n . (8)
Then we obtain:

ξ4 m -Π m ξ 4 2 n ≤ 2 n n k=1 (E k∆ + F k∆ )( ξ4 m (X k∆ ) -Π m ξ 4 (X k∆ )).
By geometric-arithmetic means inequality,

2 n n k=1 E k∆ ( ξ4 m (X k∆ ) -Π m ξ 4 (X k∆ )) ≤ 1 6 ξ4 m -Π m ξ 4 2 n + 6 n k=1 E 2 k∆ .
We introduce the unit ball for the π-norm

B m = {t ∈ S m , t 2 π ≤ 1} where t 2 π = A t 2 (x)π(x)dx and the contrast function ν n (t) = 1 n n k=1 F k∆ t(X k∆ ). Then by geometric-arithmetic means in- equality, 2 n n k=1 F k∆ ( ξ4 m (X k∆ ) -Π m ξ 4 (X k∆ )) ≤ ξ4 m -Π m ξ 4 2 π sup t∈Bm ν 2 n (t) 1/2 ≤ 1 6 ξ4 m -ξ 4 m 2 π + 6 sup t∈Bm ν 2 n (t).
Collecting terms, we get:

ξ4 m -Π m ξ 4 2 n ≤ 6 sup t∈Bm ν 2 n (t) + 1 6 ξ4 m -Π m ξ 4 2 π + 6 n k=1 E 2 k∆ + 1 6 ξ4 m -Π m ξ 4 2 n .
Then we introduce the space Ω n on which the norms . n et . π are equivalent:

Ω n = ω, ∀m ∈ M n , ∀t ∈ S m , t 2 n t 2 π -1 ≤ 1 2 .
Comte et al. (2007, Lemma 6.1) prove the following lemma for diffusion processes. However, it only relies on the boundedness of the stationary density π and of the mixing properties of the process

(X t ) t≥0 . Lemma 13. As i) (X t ) t≥0 is exponentially β-mixing,
ii) (X t ) t≥0 is stationary and its stationary density π is bounded from below and above on A, iii) the vectorial subspaces S m satisfy Assumption A4, then

P(Ω c n ) ≤ c/n 8 . On Ω n , ξ4 m -Π m ξ 4 2 π ≤ 2 ξ4 m -Π m ξ 4 2 n . Therefore ξ4 m -Π m ξ 4 2 n ½ Ωn ≤ 12 n n k=1 E 2 k∆ + 12 sup t∈Bm ν 2 n (t). (9)
The term sup t∈Bm ν 2 n (t) can be bounded in two ways. First, let us consider (ϕ λ ) λ∈Λm an orthonormal basis (for the . π -norm) of S m . As t ∈ B m , it can be written t = λ a λ ϕ λ where λ a 2 λ = 1. Then, as ν is a linear function

sup t∈Bm ν 2 n (t) = sup λ a 2 λ =1 ν 2 n λ a λ ϕ λ = sup λ a 2 λ =1 λ a λ ν n (ϕ λ ) 2 ≤ sup λ a 2 λ =1 λ a 2 λ λ ν 2 n (ϕ λ ) ≤ λ ν 2 n (ϕ λ ).
By Lemma 4, E ( F k∆ ) |F k∆ ) = 0 and therefore:

E ν 2 n (ϕ λ ) = E   1 n n k=1 F k∆ ϕ λ (X k∆ ) 2   = 1 n 2 n k=1 E ϕ λ (X k∆ )E F 2 k∆ |F k∆ + 2 n(n -1) j<k E (ϕ λ (X k∆ )ϕ λ (X j∆ )F j∆ E (F k∆ |F k∆ )) = 1 n 2 n k=1 E F 2 k∆ ϕ 2 λ (X k∆ ) .
By Lemma 4, we get that

E ν 2 n (ϕ λ ) ≤ E ϕ 2 λ (X k∆ ) ξ 8 (X k∆ )I 8 ∆ + C √ ∆ .
Using that E ϕ 2 λ (X k∆ ) = R ϕ 2 λ (x)π(x)dx = 1, and the boundness of ξ, we obtain

E ν 2 n (ϕ λ ) ≤ ξ 8 1 I 8 ∆ + C √ ∆ .
The second way of bounding sup t∈Bn ν 2 n (t) nearly use the same arguments. Let us remark that if

t ∈ B m , then t 2 L 2 ≤ 1 π0 t 2 π = 1 π0 . Then, if we consider (ψ λ ) λ∈Λm an orthonormal basis of S m for the L 2 -norm. Then sup t∈Bm ν 2 n (t) ≤ sup t∈Sm; t 2 L 2 ≤ 1 π 0 ν 2 n (t) ≤ 1 π 0 λ ν 2 n (ψ λ ).
By Assumption 4 b, λ∈Λm ψ 2 λ (X k∆ ) ≤ φ 1 D m , and as the process is stationary:

E sup t∈Bm ν 2 n (t) ≤ 1 π 0 1 n 2 n k=1 E F 2 k∆ λ∈Λm ψ 2 λ (X k∆ ) ≤ φ 1 π 0 D m E F 2 ∆ n .
Therefore:

E sup t∈Bm ν 2 n (t) ≤ Ξ D m n∆ . (10)
By Lemma 4, E E 2 k∆ ∆. By ( 9) and ( 10), we obtain:

E ξ4 m -Π m ξ 4 2 n ½ Ωn ≤ Ξ D m n∆ + c∆. (11)
As Π m ξ 4 is the orthogonal projection for the . n -norm,

Π m ξ 4 -ξ 4 A 4 n ≤ ξ 4 m -ξ 4 A 2 n . And as ξ m and ξ are deterministic, E ξ 4 m -ξ 4 A 2 n = E ξ 4 m -ξ 4 A 2 π .
Then by ( 8)

E ξ4 m -ξ 4 A 2 n ½ Ωn ≤ ξ 4 m -ξ 4 A 2 π + Ξ D m n∆ + c∆.
It remains to bound the risk on Ω c n . The function ξ4 m is the orthogonal projection (for the . n norm) of (T ∆ , . . . , T n∆ ) on the vectorial subspace {(t(X ∆ ), . . . , t(X n∆ )) , t ∈ S m }.

We have:

ξ4 m -ξ 4 A 2 n = Π m T -ξ 4 A 2 n = ξ 4 m -ξ 4 A 2 n + Π m E + Π m F 2 n ≤ ξ 4 A 2 n + E + F 2 n .
By stationarity and Cauchy-Schwarz:

E ξ4 m -ξ 4 A 2 n ½ Ω c n E ξ 4 A 2 n ½ Ω c n + E 1 n n k=1 E 2 k∆ + F 2 k∆ ½ Ω c n E ξ 4 A 4 n + E E 4 k∆ + F 4 k∆ P (Ω c n ) 1/2
.

We obtain:

E ξ4 m -ξ 4 A 2 n ½ Ω c n 1 ∆ 3/2 n 4 ≤ 1 n .

Proof of Corollary 6

As ξ 4 m,π is the orthogonal projection of ξ 4 on S m for the . π :

ξ4 m -ξ 4 2 π = ξ4 m -ξ 4 m,π 2 π + ξ 4 m,π -ξ 4 2 π }.
As ξ4 m and ξ 4 m,π belong to the subspace S m , on Ω n :

ξ4 m -ξ 4 m,π 2 π ½ Ωn ≤ 2 ξ4 m -ξ 4 m,π 2 n ≤ 4 ξ4 m -Π m ξ 4 2 n + 4 Π m ξ 4 -ξ 4 m,π 2
E ξ4 m -ξ 4 2 π ½ Ωn ≤ 5 ξ 4 -ξ 4 m,π 2 π + 48Ξ D m n∆ + 4C∆.
In addition,

E ξ4 m -ξ 4 m,π 2 π ½ Ω c n ≤ E Π m T 4 π + ξ 4 4 π 1/2 (P (Ω c n )) 1/2
1 n which give the bound for the . π -norm.

To bound the L 2 -risk, we simply remark that t

2 L 2 (A) ≤ 1 π0 t 2 π and that ξ 4 -ξ 4 π,m 2 π ≤ ξ 4 -ξ 4 m 2 π ≤ π 1 ξ 4 -ξ 4 m 2 L 2 (A) . Then E ξ2 m -ξ 4 A 2 L 2 (A) ≤ 5π 1 π 0 ξ 4 -ξ 4 m,π 2 L 2 + 48Ξ π 0 D m n∆ + 4C∆ π 0 + 4C ′ π 0 n∆ .

Proof of Theorem 7

For any m ∈ M n , γ n ( ξ4 m) + pen( m) ≤ γ n (Π m ξ 4 ) + pen(m).

As previously, we decompose the risk on Ω n and Ω c n . On Ω c n , the bound is the same as in the previous proof. On Ω n ,

ξ4 m -ξ 4 A 2 n ≤ Π m ξ 4 -ξ 4 A 2 n + pen(m) -pen( m) + 12 n k=1 E 2 k∆ + 12 sup t∈B m, m ν 2 n (t)
where B m,m ′ is the unit ball (for the . π -norm of the vectorial subspace S m + S ′ m . Let us set p(m, m ′ ) = (pen(m) + pen(m ′ ))/12. Then

ξ4 m -ξ 4 A 2 n ½ Ωn ≤ Π m ξ 4 -ξ 4 A 2 n + 2pen(m) + 12 n k=1 E 2 k∆ + 12( sup t∈B m, m ν 2 n (t) -p(m, m)). ( 12
)
It remains to bound

E sup t∈B m, m ν 2 n (t) -p(m, m) ≤ m∈Mn E sup t∈B m,m ′ ν 2 n (t) -p(m, m ′ ) .
The following lemma is deduced from the Berbee's coupling lemma and a Talagrand inequality. It is proved for instance in Schmisser (2019, Appendix A).

Lemma 14 (Talagrand inequality for β-mixing variables). Let us consider Z 1 , . . . , Z n some random variables exponentially β-mixing. Let us set b 0 ≥ 1/β where β is the β-mixing coefficient). We define q n := 2b 0 ln(n)/∆, p n = n/(2q n ). We have that β(q n ) ≤ ce -2βb0 ln(n) n -2 . Let us consider

I n (t) = 1 n n k=1 F t (Z k ) -E [F t (Z k )] .
If we can find a triplet (M 2 , V and H) such that:

∀i, sup t∈F Var   1 q n (i+1)qn k=iqn F t (Z k )   ≤ V q n , sup t∈B m,m ′ 1 q n (c+1)qn k=cqn F t (Z k ) ∞ ≤ M 2 and E sup t∈F |I n (t)| ≤ H √ n ,
then we have:

E sup t∈B m,m ′ |I 2 n (t) -6H 2 | + ≤ C V n exp -k 1 H 2 12V + C ′ M 2 2 p 2 n exp -k 2 √ p n H √ q n M 2 + 2 M 2 2 n 2
where C, C ′ , k 1 and k 2 are universal constants.

The random variables (F k∆ , X k∆ ) are exponentially β-mixing. Let us set F t (F k∆ , X k∆ ) = F k∆ t(X k∆ ). We have the following decomposition:

F k∆ J 4 k∆ + Z 4 k∆ ∆ .
The term Z k∆ is a Brownian motion term, it is nearly bounded. The small jumps (smaller than √ ∆) have the same comportement. The bigger jumps can also be bounded: as we have only subexponentials-jumps, the size of the jumps is nearly bounded. Then we count the jumps on a timeinterval of length 8 ln(n): the number of jumps will be smaller than ln 2 (n) with great probability.

The following lemma allows us to bound the terms of the sum, it is proved in [START_REF] Schmisser | Non parametric estimation of the diffusion coefficients of a diffusion with jumps[END_REF] Lemma 2):

Lemma 15. Under Assumptions A1-A3, we have:

• ∀ε > 0, ∀r > 0, P |B k∆ | ≥ ∆ 1-α n -r . • ∀r > 0, P |Z k∆ | ≥ rσ 1 ∆ 1/2 ln(n) ≤ 2n -r .
• ∀p > 0, ∀r > 0, for any c and c(p) large enough:

P |J k∆ | ≥ r 2 ∆ -α ln(n) n -r and P   1 q n qn j=1 J 2p k∆ ≥ (r + 1) 2 ∆ 1-α ln 2p (n)   n -r
Let us set, for any 0 < α < 1,

Ω B,α = ω, ∀k, ∀p ∈ N, p ≤ 8, |B k∆ | ≤ ∆ 1-ε ; |Z k∆ | ≤ 6σ 1 ∆ 1/2 ln(n); |J k∆ | ≤ 36∆ -α ln(n); 1 q n (c+1)qn k=cqn J 2p k∆ ≤ 49∆ 1-α ln 2p (n)    .
By Lemma 15:

P(Ω B,α ) 1 n 5 . ( 13
)
The bound of the risk of (Ω * ∩ Ω n ∩ Ω B,α ) c is exactly the same as on Ω c n . On Ω n ∩ Ω * ∩ Ω B,α , as t(X k∆ ) ≤ D 1/2 φ 1 /π 0 , and F k∆ (J 4 k∆ + Z 4 k∆ )/∆, we apply Talagrand's inequality (lemma 14)

to ν n (t) = 1 n n k=1 t(X k∆ )F k∆ with V = Ξ ∆ , H 2 = ΞD ∆ , M 2 = D 1/2 ln 4 (n) ∆ α φ 1 π 0 with D = dim(S m + S m ′ )
. By Lemma 14, we obtain:

E sup t∈B m,m ′ ν 2 n (t) -12H 2 + 1 n∆ exp -c D/∆ 1/∆) + D ln 8 (n) p 2 n ∆ 2α exp -c ′ √ p n D 1/2 / √ ∆ √ q n D 1/2 ln 4 (n)/∆ α + D ln 8 (n) n 2 ∆ 2α 1 n∆ exp(-c 1 D) + D ln 10 (n) n 2 ∆ 2+2α exp -c 2 √ n∆ 1+2α ln 5 (n) + D ln 8 (n) n 2 ∆ 2α .
We obtain that

E sup t∈B m, m ν 2 n (t) -p(m, m) ≤ Dm∈Mn E sup t∈B m,m ′ ν 2 n (t) -H 2 ≤ √ n∆ D=1 E sup t∈B m,m ′ ν 2 n (t) -H 2 1 n∆ + 1 n∆ 1+α e -n∆ 1+2α ln 5 (n) + ∆ 1-2α n .
As we can take α as small as we want, we get that

E sup t∈B m, m ν 2 n (t) -p(m, m) ≤ C ′ n∆ . (14)

Proof of Corollary 8

The proof is nearly the same as for the risk of the estimator with m fixed. We have that, for any m:

ξ4 m -ξ 4 A 2 π ≤ 2 ξ4 m -ξ 4 m,π 2 π + 2 ξ 4 m,π -ξ 4 A 2 π .
And on Ω n :

ξ4 m -ξ 4 m,π 2 π ½ Ωn ≤ 2 ξ4 m -ξ 4 m,π 2 n ≤ 4 ξ4 m -Π m ξ 4 2 n + 4 Π m ξ 4 -ξ 4 m,π 2 n ≤ 4 ξ4 m -ξ 4 A 2 n + 4 ξ 4 A -ξ 4 m,π 2 n . As Π m ξ 4 -ξ 4 A 2 n ≤ ξ 4 m,π -ξ 4 A 2 n
, by (12), we obtain:

ξ4 m -ξ 4 m,π 2 π ½ Ωn ≤ 8 ξ4 m -ξ 4 A 4 n + 8 ξ 4 A -ξ 4 m,π 2 n + 2 ξ 4 m,π -ξ 4 A 2 π ≤ 16 ξ 4 m,π -ξ 4 A 2 n + 2 ξ 4 m,π -ξ 4 A 2 π + 16pen(m) + 96 n k=1 E 2 k∆ + 96 sup t∈B m, m ν 2 n (t) -p(m, m) .
The risk on Ω c n is the same as for m fixed, so we get by ( 14):

E ξ4 m -ξ 4 A 2 π inf m∈Mn ξ 4 m,π -ξ 4 A 2 π + pen(m) + 1 n∆ + ∆ and E ξ4 m -ξ 4 A 2 L 2 (A) inf m∈Mn ξ 4 m,π -ξ 4 A 2 π + pen(m) + 1 n∆ + ∆

Proof of Lemma 10

As in the proof of Proposition 5, we decompose the risk on Ω n and Ω

E sup t∈Sm ν 2 n,2 (t) ≤ λ E ν 2 n,2 (ϕ λ ) = λ 1 n 2 n k=1 E H 2 k∆ ϕ 2 λ (X k∆ ) D m n∆ . As ĥm -h A 2 n = ĥm -Π m h 2 n + Π m h -h A 2 n , and E Π m h -h A 2 n ≤ E h m -h A 2 n = h m -h A 2 π , we get that E ĥm -h A 2 n ½ Ωn ≤ h m -h A 2 π + CD m n∆ + c∆.
And on Ω c n , we get that

E ĥm -h A 2 n E h A 4 n + E G 4 k∆ + H 4 k∆ P (Ω c n ) 1/2 1 ∆ 3/2 n 4 ≤ 1 n
which ends the proof.

Proof of lemma 11

Let us set h 1 = sup x∈A h(x) and ĥ1 = sup A ĥln(n) (x).

Bound of h 1 -ĥ1 . We have that

| ĥ1 -h 1 | ≤ ĥln(n) -h L ∞ (A) ≤ ĥln(n) -h ln(n) L ∞ (A) + h ln(n) -h L ∞ (A) .
By (DeVore and Lorentz, 1993, 13, p182) and (Barron et al., 1999, 4, Lemma 12), if h belongs to the Besov space B α 2,∞ (A), then

h m -h 2 L ∞ (A) ≤ CD 1-2α m . ( 15
)
As ξ is Lipschitz, h = ξ 8 I 8 belongs to B 1 2,∞ (A) and

h m -h A 2 L ∞ (A) D -1 m . It remains to bound ĥm -h m L ∞ (A)
. As ĥm and h m belongs to S m ,

ĥm -h m 2 L ∞ (A) ≤ φ 1 D m ĥm -h m 2 L 2 ≤ φ 1 D m π 0 ĥm -h m 2 π . On Ω n , ĥm -h m 2 π ≤ 2 ĥm -h m 2 n ≤ 2 ĥm -Π m h 2 n + 2 Π m h -h m 2 n Bound of Π m h -h m 2 n . As Π m h is the orthogonal projection of h on S m , Π m h -h m 2 n ≤ h m -h A 2 n . Its expectation is h m -h A 2 π D -2 m . By (15), h m -h 2 L ∞ (A) D -1 m . Moreover, Var   (j+1)qn jqn+1 (h m -h A ) 2 (X k∆ )   ≤ q 2 n E (h m -h A ) 4 (X ∆ ) ≤ q 2 n h m -h A 2 L ∞ h m -h A 2 π q 2 n D -3 m .
The following lemma is very useful to control the difference between a mean and its expectation.

Lemma 16 (Bennet inequality for β-mixing variables). Let us consider Z 1 , . . . , Z n some exponentially β-mixing random variables. Let us set b 0 ≥ 1/β where β is the β-mixing coefficient. We define q n := 2b 0 ln(n)/∆, p n = n/(2q n ). We have that β(q n ) ≤ ce -2βb0 ln(n) n -2 . Let us set Zn = 1 n n k=1 Z k . If there exists two constants V ′ and M ′ 2 such that, for any j,

Var   (j+1)qn k=jqn +1 Z k   ≤ q n V ′ and (j+1)qn k=jqn+1 Z k -E (Z k ) ≤ q n M ′ 2
then, for any x > 0,

P Zn -E Zn ≥ x ≤ 2 exp - nx 2 2(V ′ + xM ′ 2 q n )
.

By Bennett inequality with V ′ = q n D -3 m and M ′ 2 = CD -1 m , as D m = ln(n), H k∆ ϕ(X k∆ )   = q n Var (H ∆ ϕ(X ∆ )) q n ∆ .

P h m -h A 2 n -h m -h A
On Ω B,α , as ϕ |H k∆ | q n D m ln 8 (n).

And by Bennett's inequality (Lemma 16) with V ′ = 1/∆ and M ′ 2 = √ D m ln 8 (n), we obtain:

P |ν n (ϕ λ )| ≥ 1 √ D m ln 9 (n) P Ω c B,α + exp - n∆ D m ln 18 (n) n -5 + exp - n∆ ln 19 (n)
.

We have that G k∆ = G ∆ 2j-2jα-2 ∆ 8-j ln 16-2j (n) + ∆ 1-α ½ j<8 ln 16-2j (n) + ½ j=8 ∆ 1-α ln 16 (n).

Then P 1 n n k=1 G 2 k∆ ≥ ∆ + ∆ 1-α ln 16 (n) n -5 . We recall that

| ĥ1 -h 1 | D -1 m + D m h m -h A 2 n + D m ĥm -Π m h 2 n .
Collecting terms, as D m = ln(n), we obtain that

P | ĥ1 -h 1 | D -1 m + D m × D -2 m + D m D m D m ln 18 (n) + ∆ 1-α ln 16 (n) = P ĥ1 -h 1 | 1 ln(n) n -4 .
Bound of Vn -E ξ 8 (X ∆ ) I 8 . To bound this quantity, we use Bennett's inequality.

We have that

E Vn = E ξ 8 (X k∆ )I 8 + G k∆ + H k∆ = E ξ 8 (X k∆ ) I 8 + C ′ ∆ 1/2 .

Figure 1 :

 1 Figure 1: Model 1 dX t = -2X t dt + dW t + dL t with binomial jumps.

Figure 3 :

 3 Figure 3: Model 3:

  n and as Π m ξ 4 is the orthogonal projection of ξ 4 on S m for the empirical risk, Π m ξ 4 -ξ 4

.H

  Bound of ĥm -Π m hn We know that, on Ω n , ĥm -(ϕ λ ) λ∈Λm an orthonormal basis for the π-norm of S m , k∆ ϕ λ (X k∆ ) 2 and we use again Bennett's inequality. We recall that |H k∆ | ∆ -1 (Z 8 k∆ + J 8 k∆ ). For any t ∈ B m , this term is centred: E (ν n (ϕ λ )) = 0. In addition, as E ( H k∆ ) |F k∆ ) = 0, its variance is bounded by:

  k∆ | ∆ 1/2 . Moreover, on Ω B,α , by Lemma 15,

  Conditionnally to F k∆ , Bk∆ , Jk∆ and Zk∆ are independent. Moreover, we have thatE B2p k∆ |F k∆ ∆ 2p+1 , E Z2p k∆ |F k∆ ∆ p+1 , E J2p k∆ |F k∆ ∆ 2 I 2p .

									r
				mean	sd	or	mean	sd	mean	sd
		10 3 10 -1	3.91 2.83 0.0098	182	222	1	0
		10 4 10 -2	3.75 1.57 0.015	135	84	1.04 0.18
		10 5 10 -1	3.73 0.88 0.0081	117	45	1.58 0.55
		10 3 10 -2	1.41 2.24	0.23	23.0 61.9	1	0
		10 4 10 -2	0.86 0.66	0.67	17.3 18.4 1.03 0.17
		10 5 10 -2	0.61 0.35	0.20	15.1	8.1	2.11 0.97
		10 4 10 -3	1.07 1.33	0.29	12.8 27.9	1	0
		10 5 10 -3	0.68 0.45	1.16	10.2	17	1.14 0.35
			Comparison of the estimators ξ4 m,r and ξ2 .
	n	∆	risk1		risk2		risk g	risk σ 2
			mean	sd	mean	sd	mean	sd	mean	sd
		10 -1	1.60 1.01 1.33 0.77	0.97	0.58	1.99	0.34
		10 -1	1.58 0.61 1.30 0.47	0.94	0.36	1.99	0.21
		10 -1	1.66 0.58 1.28 0.26	0.86	0.32	1.99	0.13
		10 -2	0.59 0.86 0.93 0.97	0.84	0.72	0.82	0.37
		10 -2	0.36 0.42 0.83 0.49	0.65	0.23	0.78	0.23
		10 -2	0.59 0.36 0.71 0.31	0.24	0.28	0.76	0.11
		10 -3	0.48 0.62 0.84 0.69	0.84	0.70	0.11	0.11
		10 -3	0.33 0.50 0.64 0.30	0.65	0.30	0.057	0.038

  c n and get that, on Ω n ,

	ĥm -Π m h	2 n	½ Ωn ≤ 12 sup t∈Bm	ν 2 n,2 (t) +	12 n	n k=1	G 2 k∆
	where ν n,2 (t) = 1 n (ϕ λ ) 1≤λ≤Dm an orthonormal basis for the L 2 n k=1 t(X k∆ )H k∆ . By Lemma 9, we have that E G 2 k∆ π -norm of S m , we have	∆. Moreover, for

This work was supported by the Labex CEMPI (ANR-11-LABX-0007-

Moreover,

And by lemma 15, on Ω B,α :

Then, with V ′ = q n + 1/∆ and M ′ 2 = ln 8 (n)∆ -α , by Bennett's inequality, we get on Ω B,α :

) and ln(n) = o(1/∆), we get:

Bound of π 0 -π0 . We have that πh

The expectation of this term is

We have the following bounds:

Then by Bennett's inequality (lemma 16) with V ′ = q n /h ∝ ln(n) ∆h and M ′ 2 = 1/h, we get, for any

x, if h = 1/ √ n∆:

Moreover, as π is continuous, by [START_REF] Alexandre | Introduction à l'estimation non-paramétrique[END_REF], Proposition 1.2 with β = 1): (π h (x) -π(x)) 2 C ′ h. Let (x 1 , . . . , x ln 2 (n) ) be equally spaced points on A. Then

and therefore

which ends the proof.