Nicolas Marie
email: nmarie@parisnanterre.fr

Amélie Rosier
email: amelie.rosier@esme.fr

NADARAYA-WATSON ESTIMATOR FOR I.I.D. PATHS OF DIFFUSION PROCESSES

Keywords: Diffusion processes, Nonparametric drift estimation, Nadaraya-Watson estimator, PCO method, Cross validation

This paper deals with a nonparametric Nadaraya-Watson estimator b of the drift function computed from independent continuous observations of a diffusion process. Risk bounds on b and its discrete-time approximation are established. The paper also deals with extensions of the PCO and leave-one-out cross validation bandwidth selection methods for b. Finally, some numerical experiments are provided. MSC2020 subject classifications. 62G05 ; 62M05.

Let I : (x, w) → I(x, w) be the Itô map for Equation (1) and, for N ∈ N * copies W 1 , . . . , W N of W , consider X i = I(x 0 , W i ) for every i ∈ {1, . . . , N }. The estimation of the drift function b from continuous-time and discrete-time observations of (X 1 , . . .

Introduction

Consider the stochastic differential equation ( 1)

X t = x 0 + t 0 b(X s )ds + t 0 σ(X s )dW s ,
where b, σ : R → R are two continuous functions and W = (W t ) t∈[0,T ] is a Brownian motion.

Since the 1980's, the statistical inference for stochastic differential equations (SDE) has been widely investigated by many authors in the parametric and in the nonparametric frameworks. Classically (see Hoffmann [START_REF] Hoffmann | Adaptive Estimation in Diffusion Processes[END_REF], Kessler [START_REF] Kessler | Simple and Explicit Estimating Functions for a Discretely Observed Diffusion Process[END_REF], Kutoyants [START_REF] Kutoyants | Statistical Inference for Ergodic Diffusion Processes[END_REF], Dalalyan [START_REF] Dalalyan | Sharp Adaptive Estimation of the Trend Coefficient for Ergodic Diffusion[END_REF], Comte et al. [START_REF] Comte | Penalized Nonparametric Mean Square Estimation of the Coefficients of Diffusion Processes[END_REF], etc.), the estimators of the drift function are computed from one path of the solution to Equation [START_REF] Cohen | On the Stability and Accuracy of Least Squares Approximations[END_REF] and converge when T goes to infinity. The existence and the uniqueness of the stationary solution to Equation (1) are then required, and obtained thanks to restrictive conditions on b.

for any t 0 > 0. Clearly, f is a density function:

∞ -∞ f (x)dx = 1 T -t 0 T t0 ∞ -∞ p t (x 0 , x)dxdt = 1.
Let K : R → R be a kernel (i.e. an integrable function such that K = 1) and consider K h (x) := h -1 K(h -1 x) with h ∈ (0, 1]. In the spirit of Comte and Genon-Catalot [START_REF] Comte | Nonparametric Drift Estimation for i.i.d. Paths of Stochastic Differential Equations[END_REF][START_REF] Comte | Drift Estimation on Non Compact Support for Diffusion Models[END_REF], our paper deals first with the continuous-time Nadaraya-Watson estimator [START_REF] Comte | Estimation non-paramétrique. 2e édition[END_REF] b N,h (x) := bf N,h (x)

f N,h (x)
of the drift function b, where

(3) f N,h (x) := 1 N (T -t 0 ) N i=1 T t0 K h (X i t -x)dt
is an estimator of f and

(4) bf N,h (x) := 1 N (T -t 0 ) N i=1 T t0 K h (X i t -x)dX i t
is an estimator of bf . From independent copies of X continuously observed on [0, T ], b N,h is a natural extension of the Nadaraya-Watson estimator already well-studied in the regression framework (see Comte [START_REF] Comte | Estimation non-paramétrique. 2e édition[END_REF], Chapter 4 or Györfi et al. [START_REF] Györfi | A Distribution-Free Theory of Nonparametric Regression[END_REF], Chapter 5). The paper also deals with a discrete-time approximate of the previous Nadaraya-Watson estimator:

(5) b n,N,h (x) := bf n,N,h (x)

f n,N,h (x) , where [START_REF] Comte | Penalized Nonparametric Mean Square Estimation of the Coefficients of Diffusion Processes[END_REF] f n,N,h (x) := 1 nN

N i=1 n-1 j=0 K h (X i tj -x)
is an estimator of f , [START_REF] Comte | Nonparametric Estimation for Stochastic Differential Equations with Random Effects[END_REF] bf n,N,h (x

) := 1 N (T -t 0 ) N i=1 n-1 j=0 K h (X i tj -x)(X i tj+1 -X i tj )
is an estimator of bf , and (t 0 , t 1 , . . . , t n ) is the dissection of [t 0 , T ] such that t j = t 0 + (T -t 0 )j/n for every j ∈ {1, . . . , n}. Finally, our paper deals with a risk bound on the adaptive double bandwidths Nadaraya-Watson's estimator b N, h, h (x) := bf N, h (x)

f N, h (x) ,
where h (resp. h ) is selected via a penalized comparison to overfitting (PCO) type criterion for its numerator (resp. denominator). However, in the nonparametric regression framework, it is established in Comte and Marie [START_REF] Comte | On a Nadaraya-Watson Estimator with Two Bandwidths[END_REF] that the leave-one-out cross-validation (looCV) bandwidth selection method for Nadaraya-Watson's estimator is numerically more satisfactory than two alternative procedures based on Goldenshluger-Lepski's method and on the PCO method. For this reason, an extension of the looCV method to b n,N,h is also provided, with numerical experiments, even if it seems difficult to establish a risk bound on the associated adaptive estimator.

• For every p 1, L p (R, dx) is equipped with its usual norm . p such that

ϕ p := ∞ -∞ ϕ(x) p dx 1/p
; ∀ϕ ∈ L p (R, dx).

• H 2 is the space of the processes (Y t ) t∈[0,T ] , adapted to the filtration generated by W , such that

T 0 E(Y 2 t )dt < ∞.
• For a given kernel δ, the usual scalar product on L 2 (R, δ(x)dx) is denoted by ., . 2,δ and the associated norm by . 2,δ .

Preliminaries: regularity of the density and estimates

This section deals with the existence and the regularity of the density p t (x 0 , .) of X t for every t ∈ (0, T ], with the Kusuoka-Stroock bounds on (t, x) → p t (x 0 , x) and its derivatives, and then with a Nikol'skii type condition fulfilled by f . In the sequel, in order to ensure the existence and the uniqueness of the (strong) solution to Equation [START_REF] Cohen | On the Stability and Accuracy of Least Squares Approximations[END_REF], b and σ fulfill the following regularity assumption.

Assumption 2.1. The functions b and σ are Lipschitz continuous. Now, assume that the solution X to Equation (1) fulfills the following assumption. Assumption 2.2. There exists β ∈ N * such that, for any t ∈ (0, T ], the distribution of X t has a β times continuously derivable density p t (x 0 , .). Moreover, for every x ∈ R, where all the constants are positive, depend on T , but not on t and x.

0 < p t (x 0 , x) c 2.2,
At Section 4, the following assumption on X is also required.

Assumption 2.3. For any x ∈ R, the function t ∈ (0, T ] → p t (x 0 , x) is continuously derivable. Moreover,

|∂ t p t (x 0 , x)| c 2.3,3 t q3 exp -m 2.3,3 (x -x 0 ) 2 t ; ∀t ∈ (0, T ],
where c 2.3,3 , m 2.3,3 and q 3 are three positive constants depending on T but not on t and x.

Let us provide some examples of diffusion processes categories satisfying Assumptions 2.2 and/or 2.3.

Examples:

(1) Assume that the functions b and σ belong to C ∞ b (R), and that there exists α > 0 such that [START_REF] Comte | On a Nadaraya-Watson Estimator with Two Bandwidths[END_REF] |σ(x)| > α ; ∀x ∈ R.

Then, by Kusuoka and Stroock [START_REF] Kusuoka | Applications of the Malliavin Calculus, Part II[END_REF], Corollary 3.25, X fulfills Assumptions 2.2 and 2.3. (2) Assume that b is Lipschitz continuous (but not bounded) and that σ ∈ C 1 b (R). Assume also that σ satisfies the non-degeneracy condition [START_REF] Comte | On a Nadaraya-Watson Estimator with Two Bandwidths[END_REF] and that σ is Hölder continuous. Then, by Menozzi et al. [START_REF] Menozzi | Density and Gradient Estimates for Non Degenerate Brownian SDEs with Unbounded Measurable Drift[END_REF], Theorem 1.2, X fulfills Assumption 2.2 with β = 1 (but not necessarily Assumption 2.3). Note that the conditions required to apply Menozzi et al. [START_REF] Menozzi | Density and Gradient Estimates for Non Degenerate Brownian SDEs with Unbounded Measurable Drift[END_REF], Theorem 1.2 are fulfilled by the so-called Ornstein-Uhlenbeck process, that is the solution to the Langevin equation: [START_REF] Dalalyan | Sharp Adaptive Estimation of the Trend Coefficient for Ergodic Diffusion[END_REF] X t = x 0 -θ t 0 X s ds + σW t ; t ∈ R + , where θ, σ > 0 and x 0 ∈ R + . In this special case, since it is well-known that the solution to Equation ( 9) is a Gaussian process such that E(X t ) = x 0 e -θt and var(X t ) =

σ 2 2θ (1 -e -2θt ) ; ∀t ∈ [0, T ],
one can show that X also fulfills Assumption 2.3.

Remark 2.4. Under Assumptions 2.1 and 2.2, for any p 1 and any continuous function

ϕ : R → R having polynomial growth, t ∈ [0, T ] → E(|ϕ(X t )| p ) is bounded. Indeed, for any t ∈ [0, T ], E(|ϕ(X t )| p ) c 1 (1 + E(|X t | pq )) = c 1 ∞ -∞ (1 + |x| pq )p t (x 0 , x)dx c 1 c 2.2,1 ∞ -∞ (1 + |t 1/2 x + x 0 | pq )e -m2.2,1x 2 dx c 2 (1 ∨ T pq/2 )
where

c 2 = c 1 c 2.2,1 ∞ -∞ [1 + (|x| + |x 0 |) pq ]e -m2.2,1x 2 dx
and the constants c 1 , q > 0 only depend on ϕ. Moreover,

ϕ p,f := ∞ -∞ |ϕ(x)| p f (x)dx = 1 T -t 0 T t0 E(|ϕ(X t )| p )dt c 2 (1 ∨ T pq/2 ).
Then, ϕ ∈ L p (R, f (x)dx) and ϕ p,f is bounded by a constant which doesn't depend on t 0 . In particular, the remark applies to b and σ with q = 1 by Assumption 2.1.

Finally, let us show that f fulfills a Nikol'skii type condition.

Corollary 2.5. Under Assumption 2.1, f (x) > 0 for every x ∈ R. Moreover, under Assumptions 2.1 and 2.2, there exists c 2.5 > 0, depending on T but not on t 0 , such that for every ∈ {0, . . . , β -1} and

θ ∈ R, ∞ -∞ [f ( ) (x + θ) -f ( ) (x)] 2 dx c 2.5 t 2q2( +1) 0 (θ 2 + |θ| 3 ).
Remark 2.6. Assumption 2.2, Assumption 2.3 and Corollary 2.5 are crucial in the sequel, but t 0 has to be chosen carefully to get reasonable risk bounds on the estimators b N,h and b n,N,h . Indeed, the behavior of the Kusuoka-Stroock bounds on (t, x) → p t (x 0 , x) and its derivatives is singular at point (0, x 0 ). This is due to the fact that the distribution of X at time 0 is a Dirac measure while that it has a smooth density with respect to Lebesgue's measure for every t ∈ (0, T ]. Moreover, since X is not a stationary process in general, the Kusuoka-Stroock bounds on (t, x) → p t (x 0 , x) and its derivatives explode when T → ∞.

The same difficulty appears with the estimators studied in Comte and Genon-Catalot [START_REF] Comte | Nonparametric Drift Estimation for i.i.d. Paths of Stochastic Differential Equations[END_REF] and in Della Maestra and Hoffmann [START_REF] Maestra | Nonparametric Estimation for Interacting Particle Systems: McKean-Vlasov Models[END_REF]. So, it is recommended to take T as small as possible in practice. In the sequel, only the dependence in t 0 is tracked in the risk bounds derived from Assumption 2.2, Assumption 2.3 and Corollary 2.5 because it is specific to our approach. Finally, these risk bounds only depend on t 0 through a multiplicative constant of order 1/ min{t α 0 , T -t 0 } (α > 0). So, to take t 0 ∈ [1, T -1] when T > 1 gives constants not depending on t 0 .

Risk bound on the continuous-time Nadaraya-Watson estimator

This section deals with risk bounds on f N,h , bf N,h , and then on the Nadaraya-Watson estimator b N,h .

In the sequel, the kernel K fulfills the following usual assumptions. Assumption 3.1. The kernel K is symmetric, continuous and belongs to L 2 (R, dx).

Assumption 3.2. There exists

υ ∈ N * such that ∞ -∞ |z υ+1 K(z)|dz < ∞ and ∞ -∞ z K(z)dz = 0 ; ∀ ∈ {1, . . . , υ}.
About the construction of kernels fulfilling both Assumptions 3.1 and 3.2, the reader can refer to Comte [START_REF] Comte | Estimation non-paramétrique. 2e édition[END_REF], Proposition 2.10. The following proposition provides a risk bound on f N,h (see [START_REF] Comte | Regression Function Estimation on Non Compact Support as a Partly Inverse Problem[END_REF]). Proposition 3.3. Under Assumptions 2.1, 2.2, 3.1 and 3.2 with υ = β,

E( f N,h -f 2 2 ) c 3.3 (t 0 )h 2β + K 2 2 N h with c 3.3 (t 0 ) = c 2.5 |(β -2)!| 2 t 2q2(β) 0 ∞ -∞ |z| β (1 + |z| 1/2 )|K(z)|dz 2 .
Note that thanks to Proposition 3.3, the bias-variance tradeoff is reached by (the risk bound on) f N,h when h is of order N -1/(2β+1) , leading to a rate of order N -2β/(2β+1) . Moreover, by Remark 2.6, to take t 0 1 when T > 1 gives

E( f N,h -f 2 2 ) c 3.3 h 2β + K 2 2 N h with c 3.3 = c 2.5 |(β -2)!| 2 ∞ -∞ |z| β (1 + |z| 1/2 )|K(z)|dz 2 .
Note also that in the risk bound on f N,h of Proposition 3.3, only the control of the bias term depends on T , trough the constant c 2.5 , depending itself on the constants c 2.2,2 ( ), ∈ {1, . . . , β}, involved in the Kusuoka-Strook bounds (see Assumption 2.2). Indeed, except in the special case of the Ornstein-Uhlenbeck process which is stationary, for all the examples of diffusion processes fulfilling Assumption 2.2 (see Kusuoka and Stroock [23] and Menozzi et al. [START_REF] Menozzi | Density and Gradient Estimates for Non Degenerate Brownian SDEs with Unbounded Measurable Drift[END_REF]), the constants c 2.2,2 ( ), ∈ {1, . . . , β}, depend on T . The variance term doesn't depend on time at all.

In the sequel, L 2 (R, f (x)dx) is equipped with the f -weighted norm . 2,f defined at the end of the introduction section. Let us recall that by Remark 2.4, for every ϕ ∈ L 2 (R, f (x)dx), ϕ 2,f is bounded by a constant which doesn't depend on t 0 .

The following proposition provides a risk bound on bf N,h (see (4)).

Proposition 3.4. Under Assumptions 2.1 and 3.1,

E( bf N,h -bf 2 2 ) (bf ) h -bf 2 2 + c 3.4 (t 0 ) N h with (bf ) h := K h * (bf ) and c 3.4 (t 0 ) = 2 K 2 2 b 2 2,f + 1 T -t 0 σ 2 2,f .
Assume that bf is γ ∈ N * times continuously derivable and that there exists

ϕ ∈ L 1 (R, |z| γ-1 K(z)dz) such that, for every θ ∈ R and h ∈ (0, 1], (10)
∞ -∞ [(bf ) (γ-1) (x + hθ) -(bf ) (γ-1) (x)] 2 dx ϕ(θ)h 2 .
If in addition K fulfills Assumption 3.2 with υ = γ, then (bf ) h -bf 2 2 is of order h 2γ , and by Proposition 3.4, the bias-variance tradeoff is reached by bf N,h when h is of order N -1/(2γ+1) , leading to the rate N -2γ/(2γ+1) . Moreover, by Remark 2.6, to take t 0 T -1 when T > 1 gives

E( bf N,h -bf 2 2 ) (bf ) h -bf 2 2 + c 3.4 N h with c 3.4 = 2 K 2 2 ( b 2 2,f + σ 2 2,f ).
Note also that the variance term in this risk bound doesn't depend on T .

N,h,h (x) := bf N,h (x) f N,h (x) 1 f N,h (x)>m/2 with h, h > 0,
and assume that f (x) > m > 0 for every x ∈ [A, B] (m ∈ (0, 1] and A, B ∈ R such that A < B). Under Assumptions 2.1, 2.2, 3.1 and 3.2 with υ = β,

E( b N,h,h -b 2 f,A,B ) c 3.5 m 2 (bf ) h -bf 2 2 + c 3.4 (t 0 ) N h + 2 b 2 2,f c 3.3 (t 0 )(h ) 2β + K 2 2 N h with c 3.5 := 8( f ∞ ∨ b 2 f ∞ ) and ϕ f,A,B := ϕ1 [A,B] 2,f for every ϕ ∈ L 2 (R, f (x)dx).
Proposition 3.5 says that the risk of b N,h,h can be controlled by the sum of those of bf N,h and f N,h up to a multiplicative constant. Now, if K fulfills Assumption 3.2 with υ = β ∨ γ, and if bf satisfies Condition [START_REF] Delattre | Parametric Inference for Discrete Observations of Diffusion Processes with Mixed Effects[END_REF], then the risk bound on b N,h,h is of order h 2γ + (h ) 2β + 1/(N h) + 1/(N h ), and the bias-variance tradeoff is reached when h (resp. h ) is of order N -1/(2γ+1) (resp. N -1/(2β+1) ), leading to the rate

N -2[( γ 2γ+1 )∧( β 2β+1 )] = N -2(β∧γ) 2(β∧γ)+1 ,
which is of same order than the rate of the nonadaptive version of the estimator of Della Maestra and Hoffmann [START_REF] Maestra | Nonparametric Estimation for Interacting Particle Systems: McKean-Vlasov Models[END_REF] (see their Theorem 15). Note also that to consider the 2bNW estimator is crucial to extend the PCO method to our framework in the spirit of Comte and Marie [START_REF] Comte | On a Nadaraya-Watson Estimator with Two Bandwidths[END_REF] (see Subsection 5.1). However, by taking h = h of order N -1/(2(β∧γ)+1) , the bias-variance tradeoff is reached by the 1 bandwidth (truncated) Nadaraya-Watson estimator with the same rate. Finally, if h = h and σ is bounded, then the variance term in the risk bound of Proposition 3.5 is comparable to the variance term in the risk bound obtained by Comte and Genon-Catalot in [START_REF] Comte | Nonparametric Drift Estimation for i.i.d. Paths of Stochastic Differential Equations[END_REF] for their least squares projection estimator (see [START_REF] Comte | Nonparametric Drift Estimation for i.i.d. Paths of Stochastic Differential Equations[END_REF], Propositions 2.1 and 2.2). Indeed, for a d-dimensional projection space, the variance term in the risk bound of Comte and Genon-Catalot [START_REF] Comte | Nonparametric Drift Estimation for i.i.d. Paths of Stochastic Differential Equations[END_REF], Proposition 2.1 is of order d/N which is comparable to 1/(N h). The rate of convergence of their least squares projection estimator depends on the regularity space associated to the projection basis but, as in the nonparametric regression framework, not on the regularity of f . The limitation of our Proposition 3.5 is that m is unknown in general and must be replaced by an estimator as well. Most of the time, as stated in Comte [START_REF] Comte | Estimation non-paramétrique. 2e édition[END_REF], Chapter 4, the minimum of an estimator of f is taken to choose m in practice:

m N,h = min{ f N,h (x) ; x ∈ [A, B]}
for instance. A more naive way to solve this difficulty in practice is to take

m = m N = cN -ε 2 • 2(β∧γ) 2(β∧γ)+1 ----→ N →∞ 0,
where c > 0 is a fixed constant and ε ∈ (0, 1) is chosen as close as possible to 0. Under Assumption 2.2, by Corollary 2.5,

∃N 0 ∈ N : ∀N > N 0 , ∀x ∈ [A, B], f (x) > m N .
So, by Proposition 3.5, when h (resp. h ) is of order N -1/(2γ+1) (resp. N -1/(2β+1) ), b N,h,h converges with the slightly degraded rate

N -(1-ε) 2(β∧γ) 2(β∧γ)+1 .
This last comment remains true for Proposition 4.5 and Corollary 5.3.

Risk bound on the discrete-time approximate Nadaraya-Watson estimator

This section deals with risk bounds on f n,N,h , bf n,N,h , and then on the approximate Nadaraya-Watson estimator b n,N,h .

In the sequel, in addition to Assumptions 3.1 and 3.2, K fulfills the following one.

Assumption 4.1. The kernel K is two times continuously derivable on R and K , K ∈ L 2 (R, dx).

Compactly supported kernels belonging to C 2 (R) or Gaussian kernels fulfill Assumption 4.1. The following proposition provides a risk bound on f n,N,h (see [START_REF] Comte | Penalized Nonparametric Mean Square Estimation of the Coefficients of Diffusion Processes[END_REF]

E( f n,N,h -f 2 2 ) c 4.2 min{t 2q2(β) 0 , t 2q3 0 } h 2β + 1 N h + 1 n 2 + 1 N nh 3 .
Assume that β = 1 (extreme case) and h is of order N -1/3 . As mentioned at Section 3, under this condition, the bias-variance tradeoff is reached by the continuous-time estimator of f . Then, the approximation error of f n,N,h is of order 1/n, which is the order of the variance of the Brownian motion increments along the dissection (t 0 , t 1 , . . . , t n ) of [t 0 , T ]. For this reason, the risk bound established in Proposition 4.2 is satisfactory. Moreover, by Remark 2.6, to take t 0 1 when T > 1 gives

E( f n,N,h -f 2 2 ) c 4.2 h 2β + 1 N h + 1 n 2 + 1 N nh 3 .
The following proposition provides a risk bound on bf n,N,h (see [START_REF] Comte | Nonparametric Estimation for Stochastic Differential Equations with Random Effects[END_REF]). Proposition 4.3. Consider ε ∈ (0, 1). Under Assumptions 2.1, 2.2, 2.3, 3.1 and 4.1, if

1 nh 2-ε 1,
the kernel K belongs to L 4 (R, dx) and z → zK (z) belongs to L 2 (R, dx), then there exist a constant c 4.3 > 0, not depending on ε, h, N , n and t 0 , and a constant c 4.3 (ε) > 0, depending on ε but not on h, N , n and t 0 , such that

E( bf n,N,h -bf 2 2 ) c 4.3 min{t 1/2 0 , t 2q3 0 , T -t 0 } (bf ) h -bf 2 2 + 1 N h + 1 n + c 4.3 (ε) min{1, t (1-ε)/2 0 } • 1 N nh 3+ε .
Remark 4.4. Note that if b and σ are bounded, Proposition 4.3 can be improved. Precisely, with ε = 0 and without the additional conditions K ∈ L 4 (R, dx) and z → zK (z) belongs to L 2 (R, dx), the risk bound on bf n,N,h is of same order than in Proposition 4.2 (see Remark A.3 for details).

Assume that bf fulfills Condition [START_REF] Delattre | Parametric Inference for Discrete Observations of Diffusion Processes with Mixed Effects[END_REF] with γ = 1 (extreme case), and that h is of order N -1/3 . Then, for ε > 0 as close as possible to 0, the approximation error of bf n,N,h is of order N ε/3 /n. If in addition b and σ are bounded, thanks to Remark 4.4, with ε = 0 and without the additional conditions K ∈ L 4 (R, dx) and z → zK (z) belongs to L 2 (R, dx), then the approximation error of bf n,N,h is of order 1/n as the error of f n,N,h . Moreover, by Remark 2.6, to take

t 0 ∈ [1, T -1] when T > 1 gives E( bf n,N,h -bf 2 2 ) c 4.3 (bf ) h -bf 2 2 + 1 N h + 1 n + c 4.3 (ε)
N nh 3+ε . Finally, Propositions 4.2 and 4.3 allow to provide a risk bound on a truncated version the approximate Nadaraya-Watson estimator b n,N,h (see [START_REF] Comte | Drift Estimation on Non Compact Support for Diffusion Models[END_REF]). Proposition 4.5. Consider ε > 0, m ∈ (0, 1], and assume that f (x) > m > 0 for every x ∈ [A, B] (A, B ∈ R such that A < B). Under the assumptions of Proposition 4.3 and, in addition, Assumptions 2.3 and 4.1, there exist a constant c 4.5 > 0, not depending on ε, A, B, h, N , n and t 0 , and a constant c 4.5 (ε) > 0, depending on ε but not on A, B, h, N , n and t 0 , such that

E( b n,N,h -b 2 f,A,B ) c 3.5 m 2 min{1, t (1-ε)/2 0 , t 1/2 0 , t 2q2(β) 0 , t 2q3 0 , T -t 0 } × c 4.5 (bf ) h -bf 2 2 + h 2β + 1 N h + 1 n + c 4.5 (ε) N nh 3+ε with b n,N,h (.) := b n,N,h (.)1 f n,N,h (.)>m/2 .
The proof of Proposition 4.5 given Propositions 4.2 and 4.3 is almost the same than the proof of Proposition 3.5 given Propositions 3.3 and 3.4. Of course one can establish a risk bound on the discrete-time approximate 2bNW estimator, but to focus on the 1 bandwidth estimator is clearer and sufficient to introduce the looCV selection method based on discrete-time observations of X 1 , . . . , X N at Subsection 5.2. Now, assume that bf satisfies Condition [START_REF] Delattre | Parametric Inference for Discrete Observations of Diffusion Processes with Mixed Effects[END_REF] with γ = β, and that K fulfills Assumption 3.2 with υ = β. Then, (bf ) h -bf 2 2 is of order h 2β . For the sake of simplicity, assume also that b is bounded, and then let's take ε = 0 in Proposition 4.5. First, note that the minimization problem

min h∈(0,∞) h 2β + 1 N h + 1 n + 1 N nh 3
has unfortunately no explicit solutions. However, let us provide an upper-bound on the rate of our discrete-time estimator. Since (nh 2 ) -1

1, Proposition 4.5 says that the risk of b n,N,h is at most of order h 2β + 1/(N h) + 1/n. So, the optimal bandwidth for this bound is of order N -1/(2β+1) , leading to the rate

N -2β 2β+1 + 1 n .
Moreover, by taking a bandwidth of order N -1/(2β+1) such that (nh 2 ) -1 1, N is at most of order n (2β+1)/2 . So, clearly, the more f and bf are regular, the more N can be chosen freely with respect to n, and if N is of order n (2β+1)/2 , then the risk of b n,N,h is at most of order 1/n. Finally, note that if β = 1, for a bandwidth of order N -1/3 such that (nh 2 ) -1 1, then

1/n h 2 ∝ N -2/3 ,
and the rate of b n,N,h is of order N -2/3 (the optimal rate).

Bandwidth selection and numerical experiments

This section deals with extensions of the PCO (see Lacour et al. [START_REF] Lacour | Estimator Selection: New Method with Applications to Kernel Density Estimation[END_REF]) and looCV methods to the Nadaraya-Watson estimator studied in this paper (see Subsections 5.2 and 5.1). Subsection 5.3 deals with some numerical experiments on the looCV based adaptive Nadaraya-Watson estimator which is, as explained in Comte and Marie [START_REF] Comte | On a Nadaraya-Watson Estimator with Two Bandwidths[END_REF] in the nonparametric regression framework, numerically more satisfactory than the PCO based one. However, and this is its main advantage, the PCO based adaptive Nadaraya-Watson estimator offers theoretical guarantees: an oracle inequality is established in Subsection 5.1. Note also that the PCO method is easier to implement and numerically faster than the Goldenshluger-Lepski method which has been extended by Della Maestra and Hoffmann in [START_REF] Maestra | Nonparametric Estimation for Interacting Particle Systems: McKean-Vlasov Models[END_REF] for their estimator of the drift function in McKean-Vlasov models.

An extension of the Penalized Comparison to

Overfitting method. Let H N (resp. H N ) be a finite subset of [h 0 , 1] (resp. [h 0 , 1]), where h 0 > 0 and (N h 3 0 ) -1 1 (resp. h 0 > 0 and (N h 0 ) -1 1
). Consider an additional kernel δ, [START_REF] Delattre | Maximum Likelihood Estimation for Stochastic Differential Equations with Random Effects[END_REF] h ∈ arg min

h∈H N { bf N,h -bf N,h0 2
2,δ + pen(h)} with (12) pen(h) := 2 (T -t 0 ) 2 N 2 N i=1 T t0 K h (X i s -•)dX i s , T t0 K h0 (X i s -•)dX i s 2,δ ; ∀h ∈ H N , and (13)
h ∈ arg min

h∈H N { f N,h -f N,h 0 2 2 + pen (h)} with pen (h) := 2 (T -t 0 ) 2 N 2 N i=1 T t0 K h (X i s -•)ds, T t0 K h 0 (X i s -•)ds 2 ; ∀h ∈ H N .
This subsection deals with risk bounds on the adaptive estimators bf N, h (.) (see ( 3)), f N, h (.) (see ( 4)) and

b N, h, h (x) = bf N, h (x) f N, h (x) 1 f N, h (x)>m/2 ; x ∈ [A, B]
with the notations of Proposition 3.5. In the sequel, K, δ and σ fulfill the following technical assumption.

Assumption 5.1. The kernels K and δ are continuously derivable on R, the derivative of K belongs to L2 (R, dx), δ is positive and its derivative is bounded, and σ is bounded.

Moreover, recall that under Assumptions 2.1 and 2.2, b 2 and σ 2 belong to

L 1 (R, f (x)dx) (see Remark 2.4).
Theorem 5.2. Under Assumptions 2.1, 2.2, 3.1 and 5.1, (1) There exist two deterministic constants c 5.2,1 , c 5.2,2 > 0, not depending on N , such that for every ϑ ∈ (0, 1) and λ > 0, with probability larger than 1

-c 5.2,1 |H N |e -λ , bf N, h -bf 2 2,δ (1 + ϑ) min h∈H N bf N,h -bf 2 2,δ + c 5.2,2 ϑ (bf ) h0 -bf 2 2,δ + (1 + λ) 3 N .
(2) There exist two deterministic constants c 5.2,1 , c 5.2,2 > 0, not depending on N , such that for every ϑ ∈ (0, 1) and λ > 0, with probability larger than 1

-c 5.2,1 |H N |e -λ , f N, h -f 2 2 (1 + ϑ) min h ∈H N f N,h -f 2 2 + c 5.2,2 ϑ f h 0 -f 2 2 + (1 + λ) 3 N . Corollary 5.3. Under Assumptions 2.1, 2.2, 3.1 and 5.1, if f (x), δ(x) > m > 0 for every x ∈ [A, B] (m ∈ (0, 1] and A, B ∈ R such that A < B)
, then there exists a deterministic constant c 5.3 > 0, not depending on N , A and B, such that for every ϑ ∈ (0, 1),

E( b N, h, h -b 2 f,A,B ) 2c 3.5 (1 ∨ δ ∞ ) m 3 (1 + ϑ) min (h,h )∈H N ×H N {E( bf N,h -bf 2 2 ) + E( f N,h -f 2 2 )} + c 5.3 ϑ (bf ) h0 -bf 2 2 + f h 0 -f 2 2 + 1 N .
Corollary 5.3 says that the risk of the adaptive estimator b N, h, h is controlled by the sum of the minimal risks of bf N,h and f N,h ; (h, h ) ∈ H N , up to a multiplicative constant and a negligible additive term.

Remark 5.4. The condition (N h 3 0 ) -1 1 on the bandwidths collection H N is quite uncomfortable but not that much because if bf satisfies Condition [START_REF] Delattre | Parametric Inference for Discrete Observations of Diffusion Processes with Mixed Effects[END_REF] with γ = β 2, then the (unknown) bandwidth h * of order N -1/(2β+1) such that our estimator of bf reaches the bias-variance tradeoff (see Section 3) possibly belongs to H N . Indeed, there exists an unknown constant c * > 0 such that h * = c * N -1/(2β+1) , and then

1 N (h * ) 3 = (c * ) -3 N
for N large enough. Moreover, the proof of Proposition 5.2 remains true by replacing the condition

(N h 3 0 ) -1 1 by (N h 3 0 ) -1 m with m > 0. So, even for β = 1, (N (h * ) 3 ) -1 (c * ) -3
and then h * possibly belongs to H N when m is large enough.

Remark 5.5. A nice choice for δ is the standard normal density:

δ(x) := e -x 2 /2 √ 2π ; ∀x ∈ R.
First, δ obviously fulfills Assumption 5.1. Moreover, δ ∞ 1. Finally, by assuming that

f (x) > m 1 for every x ∈ [A, B] (m 1 ∈ (0, 1] and A, B ∈ R such that A < B), since δ is continuous and positive on R (supp(δ) = R), necessarily there exists m 2 > 0 such that δ(x) > m 2 for every x ∈ [A, B]. So, f (x), δ(x) > m = m 1 ∧ m 2 > 0 for every x ∈ [A, B].
E( b N, h, h -b 2 f,A,B ) 2c 3.5 m 3 (1 + ϑ) min (h,h )∈H N ×H N {E( bf N,h -bf 2 2 ) + E( f N,h -f 2 2 )} + c 5.3 ϑ (bf ) h0 -bf 2 2 + f h 0 -f 2 2 + 1 N
for every ϑ ∈ (0, 1).

5.

2. An extension of the leave-one-out cross-validation method. First of all, note that the estimator b n,N,h (see [START_REF] Comte | Drift Estimation on Non Compact Support for Diffusion Models[END_REF]) can be written the following way:

b n,N,h (x) = N i=1 n-1 j=0 ω i j (x)(X i tj+1 -X t i j )
with

ω i j (x) := K h (X i tj -x) N k=1 n-1 =0 K h (X k t -x)(t +1 -t ) ; ∀(j, i) ∈ {0, . . . , n -1} × {1, . . . , N }, satisfying N i=1 n-1 j=0 ω i j (x)(t j+1 -t j ) = 1.
This nice (weighted) representation of b n,N,h (x) allows us to consider the following extension of the well-known looCV criterion in our framework:

CV(h) := N i=1   n-1 j=0 b -i n,N,h (X i tj ) 2 (t j+1 -t j ) -2 n-1 j=0 b -i n,N,h (X i tj )(X i tj+1 -X i tj )   with b -i n,N,h (x) := k∈{1,...,N }\{i} n-1 j=0 ω k j (x)(X k tj+1 -X k tj ) ; ∀i ∈ {1, . . . , N }.
Let us explain heuristically this extension of the looCV criterion. By assuming that dX t = Y t dt, Equation (1) leads to the regression model

Y tj = b(X tj ) + ε tj with tj 0 ε s ds = tj 0 σ(X s )dW s .
Then, a natural extension of the looCV criterion is 5)) estimator is computed on 4 datasets generated by SDEs with various types of vector fields. In each case, the bandwidth of the NW estimator is selected via the looCV method introduced at Subsection 5.2. On the one hand, two models with the same linear drift function are considered, but with an additive noise for the first one and a multiplicative noise for the second one:

CV * (h) := N i=1 n-1 j=0 (Y i tj -b -i n,N,h (X i tj )) 2 (t j+1 -t j ) ≈ CV(h) + N i=1 n-1 j=0 (Y i tj ) 2 (t j+1 -t j ) because Y tj (t j+1 -t j ) ≈ X tj+1 -X
1. The so-called Langevin equation, that is

X t = x 0 - t 0 X s ds + 0.1 • W t .
2. The hyperbolic diffusion process, that is

X t = x 0 - t 0 X s ds + 0.1 t 0 1 + X 2 s dW s .
On the other hand, two models having the same non-linear drift function involving sin(.) are considered, but here again with an additive noise for the first one and a multiplicative noise for the second one: 3. The third model is defined by

X t = x 0 - t 0 (X s + sin(4X s ))ds + 0.1 • W t .
4. The fourth model is defined by

X t = x 0 - t 0 (X s + sin(4X s ))ds + 0.1 t 0 (2 + cos(X s ))dW s .
The models and the estimator are implemented by taking N = 200, n = 50, T = 5, x 0 = 2, t 0 = 1 and K the Gaussian kernel z → (2π) -1/2 e -z 2 /2 . For Models 1 and 2, the estimator of the drift function is computed for the bandwidths set

H 1 := {0.02k ; k = 1, . . . , 10},
and for Models 3 and 4, it is computed for the bandwidths set

H 2 := {0.01k ; k = 1, . . . , 10}.
Each set of bandwidths has been chosen after testing different values of h, to see with which ones the estimation performs better. To choose smaller values in the second set of bandwidths allows to check that the looCV method does not systematically select the smallest bandwidth for Models 3 and 4.

For each of the previous models, on Figures 1, 2, 3 and 4 respectively, the true drift function (in red) and the looCV adaptive NW estimator (in blue) are plotted on the left-hand side, and the beam of proposals is plotted in green on the right-hand side. On Figures 1 and2, one can see that the drift function is well estimated by the looCV adaptive NW estimator, with a MSE equal to 2.95 • 10 -4 for the Langevin equation and to 8.31 • 10 -4 for the hyperbolic diffusion process. As presumed, the multiplicative noise in Model 2 slightly degrades the MSE. Note that when the bandwidth is too small, the estimation degrades, but the looCV method selects a higher value of h which performs better on the estimation. This means that, as expected, the looCV method selects a reasonable approximation of the bandwidth for which the NW estimator reaches the bias-variance tradeoff. On Figures 3 and4, one can see that the drift function of Models 3 and 4 is still well estimated by our looCV adaptive NW estimator. However, note that there is a significant degradation of the MSE, which is equal to 2.89 • 10 -3 for Model 3 and to 9.26 • 10 -3 for Model 4. This is probably related to the nonlinearity of the drift function to estimate. Once again, to consider a multiplicative noise in Model 4 degrades the estimation quality with respect to Model 3. As for Models 1 and 2, note that the looCV does not systematically select the smallest bandwidth. For Model 1, at levels n = 10, 20, . . . , 100, Figure 5 shows the evolution of the MSE of the looCV adaptive NW estimator as a function of N . For this study, the value of N ranges from 20 to 200. Figure 5 shows that the MSE of our adaptive estimator remains low regardless to the value of (n, N ) (from 4.50 • 10 -5 to 9.01 • 10 -3 ), decreases when N increases (for each n), and decreases when n increases (for a fixed N ). This is consistent with the risk bounds of Section 4. Note also that for N 70, there is no significant gain to take n larger than 30. For Model 3, Figure 6 shows the evolution of the MSE of the looCV adaptive NW estimator as a function of N and leads to the same conclusions than for Model 1.

Note anyway that due to the nonlinearity of b, the MSE of our adaptive estimator reaches higher values (from 2.67 • 10 -4 to 4.49 • 10 -2 ) than for Model 1. Again, there is no significant gain to take n larger than 30, and above all larger than 70.

Finally, for each model, Table 1 gathers the mean MSE of 100 looCV NW estimations of the drift function as well as the mean MSE of the corresponding 100 oracle estimations. The mean MSEs are globally low, but significantly higher for the models with a nonlinear drift function (Models 3 and 4) than for the models with a linear one (Models 1 and 2). Moreover, for each drift function, the mean MSE is slightly degraded for the models with a multiplicative noise (Models 2 and 4) with respect to the models with an additive one (Models 1 and 3). Note also that for each model, the mean MSE of the looCV estimations is close to the mean MSE of the corresponding oracle estimations. This means that our looCV method performs well in practice. Remark 5.6. Note that to take t 0 1 (here t 0 = 1) is recommended even in numerical experiments. Indeed, for instance, the mean MSE of 10 looCV estimations for Model 1 is significantly lower with t 0 = 1 (2.49 • 10 -4 ) than with t 0 = 0 (3.82 • 10 -3 ).

Concluding remarks

In this paper, first, a risk bound on our continuous-time Nadaraya-Watson estimator of b has been established. This bound is satisfactory because it leads to a rate of same order than in the classic nonparametric regression framework (see Comte [START_REF] Comte | Estimation non-paramétrique. 2e édition[END_REF], Chapter 4), and of same order than in Della Maestra Table 1. Mean MSEs of 100 looCV adaptive NW estimations compared to the oracle estimations. and Hoffmann [START_REF] Maestra | Nonparametric Estimation for Interacting Particle Systems: McKean-Vlasov Models[END_REF] for their estimator of the drift function in McKean-Vlasov models. Then, a risk bound on a discrete-time approximate estimator of b has been established too. The bound is satisfactory when b and σ are bounded, but a bit degraded when b is unbounded. To improve this bound will be the subject of future investigations.

In a second part, two bandwidth selection methods are provided. The first one is an extension of the PCO method to the 2bNW estimator of b in the spirit of Comte and Marie [START_REF] Comte | On a Nadaraya-Watson Estimator with Two Bandwidths[END_REF]. An oracle inequality is established but under the condition (N h 3 ) -1 1 (instead of (N h) -1 1) on the bandwidths collection. Unfortunately, it seems difficult to bypass this condition because of some constants involved in Bernstein's inequality and in the concentration inequality for U-statistics of Giné and Nickl [START_REF] Giné | Mathematical Foundations of Infinite-Dimensional Statistical Models[END_REF] (see Subsection A.7), but as explained at Remark 5.4 this condition is not so bad. The second bandwidth selection method is an extension of the looCV procedure for the discrete-time approximate estimator written has a convex combination. As in the nonparametric regression framework, this method is numerically satisfactory but it seems difficult to establish a theoretical risk bound on the associated adaptive estimator.

Finally, the estimation of b has been only investigated in the case of one-dimensional diffusion processes because of its simplicity, but by following the same ideas than Halconruy and Marie used in the nonparametric regression framework in [START_REF] Halconruy | Kernel Selection in Nonparametric Regression[END_REF], the major part of the results of the present paper should be extendable to multidimensional diffusion processes.

for every x ∈ R. Consider ∈ {0, . . . , β -1} and θ ∈ R + . Thanks to the bound on (t, x) → ∂ +1

x p t (x 0 , x) given in Assumption 2.2,

f ( ) (• + θ) -f ( ) 2 2 = ∞ -∞ [f ( ) (x + x 0 + θ) -f ( ) (x + x 0 )] 2 dx 1 T -t 0 T t0 ∞ -∞ (∂ 2 p t (x 0 , x + x 0 + θ) -∂ 2 p t (x 0 , x + x 0 )) 2 dxdt θ 2 T -t 0 T t0 ∞ -∞ sup z∈[x,x+θ] |∂ +1 2 p t (x 0 , z + x 0 )| 2 dxdt c 2.2,2 ( + 1) 2 θ 2 T -t 0 T t0 1 t 2q2( +1) ∞ -∞ sup z∈[x,x+θ] exp -2m 2.2,2 ( + 1) z 2 t dxdt = c 2.2,2 ( + 1) 2 θ 2 T -t 0 T t0 1 t 2q2( +1) × -θ -∞ exp -2m 2.2,2 ( + 1) (x + θ) 2 t dx + θ + ∞ 0 exp -2m 2.2,2 ( + 1) x 2 t dx dt 1 t 2q2( +1) 0 c 1 θ 2 + θ 3 max k∈{0,...,β-1} c 2.2,2 (k + 1) 2 with c 1 = 2 max k∈{0,...,β-1} c 2.2,2 (k + 1) 2 ∞ 0 exp -2m 2.2,2 (k + 1) x 2 T dx ,
and the same way,

f ( ) (• -θ) -f ( ) 2 2 θ 2 T -t 0 T t0 ∞ -∞ sup z∈[x-θ,x] |∂ +1 2 p t (x 0 , z + x 0 )| 2 dxdt c 2.2,2 ( + 1) 2 θ 2 T -t 0 T t0 1 t 2q2( +1) × 0 -∞ exp -2m 2.2,2 ( + 1) x 2 t dx + θ + ∞ θ exp -2m 2.2,2 ( + 1) (x -θ) 2 t dx dt 1 t 2q2( +1) 0 c 1 θ 2 + θ 3 max k∈{0,...,β-1} c 2.2,2 (k + 1) 2 .
This concludes the proof.

A.2. Proof of Proposition 3.3. First of all, the bias of f N,h (x) is denoted by b(x) and its variance by v(x). Moreover, let us recall the bias-variance decomposition of the L 2 -risk of f N,h :

E( f N,h -f 2 2 ) = ∞ -∞ b(x) 2 dx + ∞ -∞ v(x)dx.
On the one hand, let us find a suitable bound on the integrated variance of f N,h . Since X 1 , . . . , X N are i.i.d. copies of X, and thanks to Jensen's inequality,

v(x) = var 1 N (T -t 0 ) N i=1 T t0 K h (X i t -x)dt = 1 N (T -t 0 ) 2 var T t0 K h (X t -x)dt 1 N E   T t0 K h (X t -x) dt T -t 0 2   1 N (T -t 0 ) T t0 E(K h (X t -x) 2 )dt = 1 N ∞ -∞ K h (z -x) 2 f (z)dz. Thus, since K is symmetric, ∞ -∞ v(x)dx 1 N ∞ -∞ f (z) ∞ -∞ K h (z -x) 2 dxdz = 1 N h ∞ -∞ f (z)dz ∞ -∞ K(x) 2 dx = K 2 2 N h .
On the other hand, let us find a suitable bound on the integrated squared-bias of f N,h (x). Since X 1 , . . . , X N are i.i.d. copies of X,

b(x) = 1 T -t 0 T t0 E(K h (X t -x))dt -f (x) = 1 h ∞ -∞ K z -x h f (z)dz -f (x) = ∞ -∞ K(z)(f (hz + x) -f (x))dz.
First, assume that β = 1. By Assumption 3.2, the generalized Minkowski inequality and Corollary 2.5,

∞ -∞ b(x) 2 dx ∞ -∞ ∞ -∞ K(z)(f (hz + x) -f (x))dz 2 dx ∞ -∞ K(z) ∞ -∞ (f (hz + x) -f (x)) 2 dx 1/2 dz 2 c 1 (t 0 )h 2 with c 1 (t 0 ) = c 2.5 t 2q2(1) 0 ∞ -∞ |z|(1 + |z| 1/2 )|K(z)|dz 2 .
Now, assume that β 2. By the Taylor formula with integral remainder, for every z ∈ R,

f (hz + x) -f (x) = 1 β 3 β-2 =1 (hz) ! f ( ) (x) + (hz) β-1 (β -2)! 1 0 (1 -τ ) β-2 f (β-1) (τ hz + x)dτ.
Then, by Assumption 3.2, the generalized Minkowski inequality (two times) and Corollary 2.5,

∞ -∞ b(x) 2 dx = ∞ -∞ ∞ -∞ K(z)(f (hz + x) -f (x))dz 2 dx = h 2(β-1) |(β -2)!| 2 ∞ -∞ ∞ -∞ z β-1 K(z) 1 0 (1 -τ ) β-2 [f (β-1) (τ hz + x) -f (β-1) (x)]dτ dz 2 dx h 2(β-1) |(β -2)!| 2 × ∞ -∞ |z| β-1 |K(z)| 1 0 (1 -τ ) β-2 ∞ -∞ [f (β-1) (τ hz + x) -f (β-1) (x)] 2 dx 1/2 dτ dz 2 c 2.5 h 2β |(β -2)!| 2 t 2q2(β) 0 ∞ -∞ |z| β-1 |K(z)| 1 0 (1 -τ ) β-2 [τ |z| + (τ |z|) 3/2 ]dτ dz 2 c 2 (t 0 )h 2β with c 2 (t 0 ) = c 2.5 |(β -2)!| 2 t 2q2(β) 0 ∞ -∞ |z| β (1 + |z| 1/2 )|K(z)|dz 2 .
This concludes the proof.

A.3. Proof of Proposition 3.4. First of all,

E( bf N,h -bf 2 2 ) = ∞ -∞ b(x) 2 dx + ∞ -∞ v(x)dx
where b(x) (resp. v(x)) is the bias (resp. the variance) term of bf N,h (x) for any x ∈ R. On the one hand, let us find a suitable bound on the integrated variance of bf N,h . Since X 1 , . . . , X N are i.i.d. copies of X,

v(x) = var 1 N (T -t 0 ) N i=1 T t0 K h (X i t -x)dX i t 1 N (T -t 0 ) 2 E   T t0 K h (X t -x)dX t 2   2 N E   T t0 K h (X t -x)b(X t ) dt T -t 0 2 + 1 (T -t 0 ) 2 T t0 K h (X t -x)σ(X t )dW t 2   .
In the right-hand side of the previous inequality, Jensen's inequality on the first term and the isometry property for Itô's integral on the second one give

v(x) 2 N (T -t 0 ) T t0 E[K h (X t -x) 2 b(X t ) 2 ]dt + 2 N (T -t 0 ) 2 T t0 E[K h (X t -x) 2 σ(X t ) 2 ]dt = 2 N ∞ -∞ K h (z -x) 2 b(z) 2 f (z)dz + 2 N (T -t 0 ) ∞ -∞ K h (z -x) 2 σ(z) 2 f (z)dz. Moreover, K is symmetric and K ∈ L 2 (R, dx) by Assumption 3.1, and b, σ ∈ L 2 (R, f (x)dx) by Remark 2.4. Then, ∞ -∞ v(x)dx 2 N R 2 K h (z -x) 2 b(z) 2 f (z)dzdx + 2 N (T -t 0 ) R 2 K h (z -x) 2 σ(z) 2 f (z)dzdx = 2 N h ∞ -∞ b(z) 2 f (z) ∞ -∞ K(x) 2 dxdz + 2 N (T -t 0 )h ∞ -∞ σ(z) 2 f (z) ∞ -∞ K(x) 2 dxdz 2 K 2 2 N h ∞ -∞ b(z) 2 f (z)dz + 1 T -t 0 ∞ -∞ σ(z) 2 f (z)dz .
On the other hand, let us find a suitable bound on the integrated squared-bias of bf N,h (x). Again, since X 1 , . . . , X N are i.i.d. copies of X, and since Itô's integral restricted to H 2 is a martingale-valued map,

b(x) = E 1 N (T -t 0 ) N i=1 T t0 K h (X i t -x)dX i t -b(x)f (x) = 1 T -t 0 E T t0 K h (X t -x)dX t -b(x)f (x) = 1 T -t 0 E T t0 K h (X t -x)b(X t )dt + E T t0 K h (X t -x)σ(X t )dW t -b(x)f (x) = 1 T -t 0 T t0 E(K h (X t -x)b(X t ))dt -b(x)f (x) = ∞ -∞ K h (z -x)b(z)f (z)dz -b(x)f (x). Then, b(x) 2 = ((bf ) h -bf )(x) 2 with (bf ) h = K h * (bf ).
Therefore, since f is bounded and b belongs to

L 2 (R, f (x)dx) by Remark 2.4, ∞ -∞ b(x) 2 dx = bf -(bf ) h 2 2 .
This concludes the proof.

N,h,h -b = bf N,h -bf f N,h + 1 f N,h - 1 f bf 1 f N,h (.)>m/2 -b1 f N,h (.) m/2 .
Then,

b N,h,h -b 2 f,A,B 2 bf N,h -bf f N,h + 1 f N,h - 1 f bf 1 f N,h (.)>m/2 2 f,A,B + 2 b1 f N,h (.) m/2 2 f,A,B . Moreover, for any x ∈ [A, B], since f (x) > m, for every ω ∈ { f N,h (.) m/2}, |f (x) -f N,h (x, ω)| f (x) -f N,h (x, ω) > m - m 2 = m 2 .
Thus,

b N,h,h -b 2 f,A,B 8 m 2 bf N,h -bf 2 2,f + 8 m 2 (f -f N,h )b 2 f,A,B + 2 b1 |f (.)-f N,h (.)|>m/2 2 f,A,B 8 m 2 ∞ -∞ ( bf N,h -bf )(x) 2 f (x)dx + 8 m 2 B A (f (x) -f N,h (x)) 2 b(x) 2 f (x)dx +2 B A b(x) 2 f (x)1 |f (x)-f N,h (x)|>m/2 dx.
Since f has a sub-Gaussian tail by Assumption 2.2, and since b has at most linear growth because it is

Lipschitz continuous from R into itself (see Assumption 2.1), b 2 f is bounded on R. So, b N,h,h -b 2 f,A,B 8 f ∞ m 2 bf N,h -bf 2 2 + 8 b 2 f ∞ m 2 f N,h -f 2 2 + 2 b 2 f ∞ ∞ -∞ 1 |f (x)-f N,h (x)|>m/2 dx.
Therefore, thanks to Markov's inequality,

E( b N,h,h -b 2 f,A,B ) 8 f ∞ m 2 E( bf N,h -bf 2 2 ) + 8 b 2 f ∞ m 2 E( f N,h -f 2 2 ) + 8 b 2 f ∞ m 2 ∞ -∞ E(|f (x) -f N,h (x)| 2 )dx 8( f ∞ ∨ b 2 f ∞ ) m 2 [E( bf N,h -bf 2 2 ) + 2E( f N,h -f 2 2 )].
Propositions 3.4 and 3.3 allow to conclude.

A.5. Proof of Proposition 4.2. First of all, note that

E( f n,N,h -f 2 2 ) 2E( f N,h -f 2 2 ) + 2E( f N,h -f n,N,h 2 2
)

2 c 3.3 (t 0 )h 2β + 1 N h + ∞ -∞ E( f N,h (x) -f n,N,h (x)) 2 dx + ∞ -∞ var( f N,h (x) -f n,N,h (x))dx
by Proposition 3.3, and note also that

f N,h (x) -f n,N,h (x) = 1 N (T -t 0 ) N i=1 n-1 j=0 tj+1 tj (K h,x (X i t ) -K h,x (X i tj ))dt with K h,x (.) := K h (• -x).
On the one hand, for every s, u ∈ [t 0 , T ] such that s u, by Itô's formula, Jensen's inequality, the isometry property for Itô's integral and Remark 2.4,

∞ -∞ E[(K h,x (X u ) -K h,x (X s )) 2 ]dx = ∞ -∞ E u s K h,x (X t )dX t + 1 2 u s K h,x (X t )d X t 2 dx c 1 ∞ -∞ E u s K h,x (X t )b(X t )dt 2 +E u s K h,x (X t )σ(X t ) 2 dt 2 +E u s K h,x (X t )σ(X t )dW t 2 dx c 1 (u -s) u s E b(X t ) 2 ∞ -∞ K h,x (X t ) 2 dx dt +(u -s) u s E σ(X t ) 4 ∞ -∞ K h,x (X t ) 2 dx dt + u s E σ(X t ) 2 ∞ -∞ K h,x (X t ) 2 dx dt c 2 (u -s) 2 h 3 + (u -s) 2 h 5 + u -s h 3
where c 1 and c 2 are two positive constants not depending on s, u, h, N , n and t 0 . Then,

∞ -∞ var( f n,N,h (x) -f N,h (x))dx = 1 N (T -t 0 ) 2 ∞ -∞ var   n-1 j=0 tj+1 tj (K h,x (X t ) -K h,x (X tj ))dt   dx 1 N (T -t 0 ) n-1 j=0 tj+1 tj ∞ -∞ E[(K h,x (X t ) -K h,x (X tj )) 2 ]dxdt c 3 N nh 3
where the constant c 3 > 0 is not depending on h, N , n and t 0 . On the other hand, by Assumption 2.3,

|E( f N,h (x) -f n,N,h (x))| 1 T -t 0 n-1 j=0 tj+1 tj |E(K h,x (X t )) -E(K h,x (X tj ))|dt 1 T -t 0 n-1 j=0 tj+1 tj ∞ -∞ |K h (z -x)| • |p t (x 0 , z) -p tj (x 0 , z)|dzdt 1 T -t 0 n-1 j=0 tj+1 tj (t -t j )dt × ∞ -∞ |K h (z -x)| sup u∈[t0,T ] |∂ u p u (x 0 , z)|dz c 2.3,3 T -t 0 nt q3 0 ∞ -∞ |K(z)| exp -m 2.3,3 (hz + x -x 0 ) 2 T dz.
Then, by Jensen's inequality,

∞ -∞ E( f N,h (x) -f n,N,h (x)) 2 dx c 5 n 2 t 2q3 0 ∞ -∞ |K(z)| ∞ -∞ exp -2m 2.3,3 (hz + x -x 0 ) 2 T dxdz = c 6 n 2 t 2q3 0 where c 6 = c 5 K 1 ∞ -∞ exp -2m 2.3,3 (x -x 0 ) 2 T dx
and the constant c 5 > 0 is not depending on h, N , n and t 0 . This concludes the proof.

A.6. Proof of Proposition 4.3. The proof of Proposition 4.3 relies on the two following technical lemmas.

Lemma A.1. Consider a symmetric and continuous function ϕ 1 : R → R such that ϕ 1 : z → zϕ 1 (z) belongs to L 2 (R, dx). Consider also ϕ 2 , ψ ∈ C 0 (R) having polynomial growth. Under Assumptions 2.1 and 2.2, for every p > 0, there exists a constant c A.1 (p) > 0, not depending on ϕ 1 and t 0 , such that for every s, t

∈ [t 0 , T ] satisfying s < t, ∞ -∞ E t s ϕ 1 (x -X u )ϕ 2 (X u )dW u 2 ψ(X t ) 2 dx c A.1 (p)(t -s) ϕ 1 2 2 + ϕ 1 2 2 + 1 t 1/(2p) 0 ∞ -∞ ϕ 1 (z) 2p dz 1/p .
Lemma A.2. Consider ϕ ∈ C 0 (R). Under Assumptions 2.1 and 2.2, for every s, t

∈ [t 0 , T ] such that s < t, ∞ -∞ E(K h,x (X s )ϕ(X s , X t )) 2 dx c 2.2,1 K 2 1 t 1/2 0 E[ϕ(X s , X t ) 2 ].
The proof of Lemma A.1 (resp. Lemma A.2) is postponed to Subsubection A.6.1 (resp. Subsubsection A.6.2).

First of all, note that

E( bf n,N,h -bf 2 2 ) 2E( bf N,h -bf 2 2 ) + 2E( bf N,h -bf n,N,h 2
2 ) 2 (bf ) h -bf 2 2 + c 3.4 (t 0 ) N h + ∞ -∞ E( bf N,h (x) -bf n,N,h (x)) 2 dx + ∞ -∞ var( bf N,h (x) -bf n,N,h (x))dx =: 2 (bf ) h -bf 2 2 + c 3.4 (t 0 ) N h + B n,N,h + V n,N,h
by Proposition 3.4, and note also that

bf N,h (x) -bf n,N,h (x) = 1 N (T -t 0 ) N i=1 n-1 j=0 tj+1 tj (K h,x (X i t ) -K h,x (X i tj ))dX i t .
The proof is dissected in two steps. The term V n,N,h is controlled in the first step, and then B n,N,h is controlled in the second one.

Step 1. First of all, by Jensen's inequality,

V n,N,h = 1 N (T -t 0 ) 2 ∞ -∞ var   n-1 j=0 tj+1 tj (K h,x (X t ) -K h,x (X tj ))dX t   dx 2 N (T -t 0 ) ∞ -∞   n-1 j=0 tj+1 tj E[(K h,x (X t ) -K h,x (X tj )) 2 b(X t ) 2 ]dt   dx + V σ n,N,h with V σ n,N,h := 2 N (T -t 0 ) 2 ∞ -∞ E      n-1 j=0 tj+1 tj (K h,x (X t ) -K h,x (X tj ))σ(X t )dW t   2    dx.
In order to control V n,N,h as in the proof of Proposition 4.2, a preliminary bound on V σ n,N,h has to be established via the isometry property of Itô's integral:

V σ n,N,h = 2 N (T -t 0 ) 2 ∞ -∞ E      T t0   n-1 j=0 (K h,x (X t ) -K h,x (X tj ))σ(X t )1 [tj ,tj+1] (t)   dW t   2    dx = 2 N (T -t 0 ) 2 ∞ -∞ T t0 E      n-1 j=0 (K h,x (X t ) -K h,x (X tj ))σ(X t )1 [tj ,tj+1] (t)   2    dtdx = 2 N (T -t 0 ) ∞ -∞   n-1 j=0 tj+1 tj E[(K h,x (X t ) -K h,x (X tj )) 2 σ(X t ) 2 ]dt   dx. Then, V n,N,h 2 N (T -t 0 ) n-1 j=0 tj+1 tj ∞ -∞ E[(K h,x (X t ) -K h,x (X tj )) 2 b(X t ) 2 ]dxdt + 2 N (T -t 0 ) n-1 j=0 tj+1 tj ∞ -∞ E[(K h,x (X t ) -K h,x (X tj )) 2 σ(X t ) 2 ]dxdt. For ϕ = b or ϕ = σ, by Itô's formula, n-1 j=0 tj+1 tj ∞ -∞ E[(K h,x (X t ) -K h,x (X tj )) 2 ϕ(X t ) 2 ]dxdt c 1 n-1 j=0 tj+1 tj ∞ -∞   E   t tj K h,x (X u )b(X u )du 2 ϕ(X t ) 2   +E   t tj K h,x (X u )σ(X u ) 2 du 2 ϕ(X t ) 2   +E   t tj K h,x (X u )σ(X u )dW u 2 ϕ(X t ) 2     dxdt
where the constant c 1 > 0 is not depending on ϕ, h, N , n and t 0 . Moreover, for every j ∈ {0, . . . , n -1} and t ∈ [t j , t j+1 ], by Lemma A.

1 with p = 1/(1 -ε), ∞ -∞ E   t tj K h,x (X u )σ(X u )dW u 2 ϕ(X t ) 2   dx c A.1 (p)(t -t j ) ∞ -∞ K h (z) 2 dz +c A.1 (p)(t -t j ) ∞ -∞ z 2 K h (z) 2 dz + c A.1 (p) t 1/(2p) 0 (t -t j ) ∞ -∞ K h (z) 2p dz 1/p c 2 (ε)(t -t j ) 1 + 1 h 3 + t -(1-ε)/2 0 h 3+ε
where the constant c 2 (ε) > 0 depends on ε, but not on ϕ, j, t, h, N , n and t 0 . Thus, by Jensen's inequality and Remark 2.4,

n-1 j=0 tj+1 tj ∞ -∞ E[(K h,x (X t ) -K h,x (X tj )) 2 ϕ(X t ) 2 ]dxdt c 3 (ε) n-1 j=0 tj+1 tj (t -t j ) t tj E ϕ(X t ) 2 b(X u ) 2 ∞ -∞ K h,x (X u ) 2 dx du +(t -t j ) t tj E ϕ(X t ) 2 σ(X u ) 4 ∞ -∞ K h,x (X u ) 2 dx du + (t -t j ) 1 + 1 h 3 + t -(1-ε)/2 0 h 3+ε dt c 4 (ε) min{1, t (1-ε)/2 0 } (T -t 0 ) 3 1 n 2 h 3 + 1 n 2 h 5 + 1 nh 3+ε
where c 3 (ε) and c 4 (ε) are two positive constants depending on ε, but not on ϕ, h, N , n and t 0 . Therefore,

V n,N,h c 5 (ε) min{1, t (1-ε)/2 0 } • 1 N nh 3+ε
where the constant c 5 (ε) > 0 depends on ε, but not on h, N , n and t 0 .

Step 2. First of all, since Itô's integral restricted to H 2 is a martingale-valued map, since K h,x is a kernel, by Lemma A.2, by Assumptions 2.2 and 2.3, and since b is Lipschitz continuous,

∞ -∞ E( bf N,h (x) -bf n,N,h (x)) 2 dx = ∞ -∞   1 T -t 0 n-1 j=0 tj+1 tj E((K h,x (X t ) -K h,x (X tj ))b(X t ))dt + 1 T -t 0 n-1 j=0 E tj+1 tj (K h,x (X t ) -K h,x (X tj ))σ(X t )dW t   2 dx 2 T -t 0 n-1 j=0 tj+1 tj ∞ -∞ E(K h,x (X t )b(X t ) -K h,x (X tj )b(X tj )) 2 dxdt + 2 T -t 0 n-1 j=0 tj+1 tj ∞ -∞ E(|K h,x (X tj )| • |b(X t ) -b(X tj )|) 2 dxdt 2 K 1 T -t 0 n-1 j=0 tj+1 tj ∞ -∞ ∞ -∞ |K h,x (z)|dx b(z) 2 (p t (x 0 , z) -p tj (x 0 , z)) 2 dzdt + 2c 2.2,1 K 2 1 t 1/2 0 (T -t 0 ) n-1 j=0 tj+1 tj E[(b(X t ) -b(X tj )) 2 ]dt 4 K 2 1 T -t 0 n-1 j=0 tj+1 tj (t -t j ) 2 dt ∞ -∞ b(z) 2 sup u∈[t0,T ] |∂ u p u (x 0 , z)| 2 dz + 2c 2.2,1 K 2 1 t 1/2 0 (T -t 0 ) b 2 ∞ n-1 j=0 tj+1 tj E[(X t -X tj ) 2 ]dt c 6   1 t 2q3 0 n 2 + 1 t 1/2 0 (T -t 0 ) n-1 j=0 tj+1 tj E[(X t -X tj ) 2 ]dt  
where the constant c 6 > 0 is not depending on h, N , n and t 0 . Moreover, for any j ∈ {0, . . . , n -1} and t ∈ [t j , t j+1 ],

X t -X tj = t tj b(X u )du + t tj σ(X u )dW u
and then, by Jensen's inequality, the isometry property of Itô's integral and Remark 2.4,

E[(X t -X tj ) 2 ] (t -t j ) t tj E(b(X u ) 2 )du + t tj E(σ(X u ) 2 )du (t -t j ) 2 sup u∈[t0,T ] E(b(X u ) 2 ) + (t -t j ) sup u∈[t0,T ] E(σ(X u ) 2 ) c 7 (t -t j )
where the constant c 7 > 0 is not depending on j, t, h, N , n and t 0 . Therefore,

∞ -∞ E( bf N,h (x) -bf n,N,h (x)) 2 dx c 8 min{t 1/2 0 , t 2q3 0 } 1 n 2 + 1 n
where the constant c 8 > 0 is not depending on n, N , h and t 0 .

= σ, n-1 j=0 tj+1 tj ∞ -∞ E[(K h,x (X t ) -K h,x (X tj )) 2 ϕ(X t ) 2 ]dxdt c 1 n-1 j=0 tj+1 tj ∞ -∞   E   t tj K h,x (X u )b(X u )du 2 ϕ(X t ) 2   +E   t tj K h,x (X u )σ(X u ) 2 du 2 ϕ(X t ) 2   +E   t tj K h,x (X u )σ(X u )dW u 2 ϕ(X t ) 2     dxdt c 1 ϕ 2 ∞ n-1 j=0 tj+1 tj ∞ -∞   E   t tj K h,x (X u )b(X u )du 2   +E   t tj K h,x (X u )σ(X u ) 2 du 2   + E   t tj K h,x (X u )σ(X u )dW u 2     dxdt.
So, in this special case, the bound on V n,N,h is established by using the exact same arguments than in the proof of Proposition 4.2. In particular, one can take ε = 0, the additional conditions K ∈ L 4 (R, dx) and z → zK(z) belongs to L 2 (R, dx) are not required, and the bound on V n,N,h is of order 1/(N nh 3 ) and doesn't depend on t 0 . When ϕ = b or ϕ = σ is not bounded, since ϕ(X t ) is not σ(W u )-measurable for every u ∈ [t j , t) (j ∈ {0, . . . , n -1}), the Hölder inequality has to be used to get a suitable bound on

∞ -∞ E   t tj K h,x (X u )σ(X u )dW u 2 ϕ(X t ) 2   dx
(see the proof of Lemma A.1), and for this reason the variance term in the bound of Proposition 4.3 is of order 1/(N nh 3+ε ) instead of 1/(N nh 3 ) as when b and σ are bounded.

A.6.1. Proof of Lemma A.1. Consider ϕ(x, z) := ϕ 1 (x -z)ϕ 2 (z) for every z ∈ R, q > 0 such that 1/p + 1/q = 1, and s, t ∈ [0, T ] such that s < t. First of all, by the isometry property of Itô's integral, Burkholder-Davis-Gundy's inequality, Hölder's inequality, Markov's inequality, Remark 2.4, and the generalized Minkowski inequality,

E t s ϕ(x, X u )dW u 2 ψ(X t ) 2 x 2 E t s ϕ(x, X u )dW u 2 1 ψ(Xt) 2 x 2 +E t s ϕ(x, X u )dW u 2p 1/p E(ψ(X t ) 4q ) 1/(2q) P(ψ(X t ) 2 > x 2 ) 1/(2q) x 2 t s E[ϕ(x, X u ) 2 ]du + c 1 (p)E t s ϕ(x, X u ) 2 du p 1/p E(ψ(X t ) 4q ) 1/(2q) × E(ψ(X t ) 2 ) 1/(2q) x 1/q 1 [-1,1] (x) + E(ψ(X t ) 4q ) 1/(2q) x 2 1 R\[-1,1] (x) x 2 t s E[ϕ(x, X u ) 2 ]du + c 2 (p) t s E[ϕ(x, X u ) 2p ] 1/p du 1 x 1/q 1 [-1,1] (x) + 1 x 2 1 R\[-1,1] (x)
where c 1 (p) and c 2 (p) are two positive constants depending on p, but not on x, s, t, ϕ and t 0 . On the one hand, since ϕ 2 has polynomial growth, by Remark 2.4, for every u ∈ [s, t],

∞ -∞ x 2 E[ϕ(x, X u ) 2 ]dx = ∞ -∞ ∞ -∞ x 2 ϕ 1 (x -z) 2 ϕ 2 (z) 2 p u (x 0 , z)dxdz = ∞ -∞ ∞ -∞ (x + z) 2 ϕ 1 (x) 2 ϕ 2 (z) 2 p u (x 0 , z)dxdz 2 ∞ -∞ x 2 ϕ 1 (x) 2 dx ∞ -∞ ϕ 2 (z) 2 p u (x 0 , z)dz +2 ∞ -∞ ϕ 1 (x) 2 dx ∞ -∞ z 2 ϕ 2 (z) 2 p u (x 0 , z)dz c 3 ( ϕ 1 2 2 + ϕ 1 2 2 )
where the constant c 3 > 0 is not depending on u, ϕ 1 and t 0 . Then,

∞ -∞ x 2 t s E[ϕ(x, X u ) 2 ]dudx c 3 (t -s)( ϕ 1 2 2 + ϕ 1 2 2 ).
On the other hand, since ϕ 2 has polynomial growth, by Assumption 2.2, for every x ∈ R,

t s E[ϕ(x, X u ) 2p ] 1/p du = t s ∞ -∞ ϕ 1 (z) 2p ϕ 2 (z + x) 2p p u (x 0 , z + x)dz 1/p du c 4 (p) t 1/(2p) 0 (t -s) ∞ -∞ ϕ 1 (z) 2p dz 1/p
where the constant c 4 (p) > 0 depends on p, but not on x, s, t, ϕ 1 and t 0 . Then,

∞ -∞ t s E[ϕ(x, X u ) 2p ] 1/p du 1 x 1/q 1 [-1,1] (x) + 1 x 2 1 R\[-1,1] (x) dx c 5 (p) t 1/(2p) 0 (t -s) ∞ -∞ ϕ 1 (z) 2p dz 1/p with c 5 (p) = c 4 (p) 1 -1 dx x 1/q + R\[-1,1] dx x 2 < ∞.
A.6.2. Proof of Lemma A.2. Consider s, t ∈ [t 0 , T ] such that s < t, and let p s,t (resp. p t|s ) be the density of (X s , X t ) (resp. the conditional density of X t with respect to X s ). Moreover, for the sake of readability, p s (x 0 , .) is denoted by p s (.) in this proof. By Assumption 2.2,

E(K h,x (X s )ϕ(X s , X t )) 2 = K 2 1 ∞ -∞ K h,x (y) K h,x 1 ∞ -∞ ϕ(y, z)p t|s (z|y)dz p s (y)dy 2 K 1 ∞ -∞ |K h,x (y)| ∞ -∞ ϕ(y, z)p t|s (z|y)dz 2 p s (y) 2 dy K 1 sup s∈[t0,T ] sup y∈R p s (y) ∞ -∞ |K h,x (y)| ∞ -∞
ϕ(y, z) 2 p t|s (z|y)dz p s (y)dy

c 2.2,1 K 1 t 1/2 0 ∞ -∞ |K h,x (y)| ∞ -∞ ϕ(y, z) 2 p s,t (y, z)dzdy. Therefore, ∞ -∞ E(K h,x (X s )ϕ(X s , X t )) 2 dx c 2.2,1 K 1 t 1/2 0 ∞ -∞ ∞ -∞ |K h,x (y)|dx ∞ -∞
ϕ(y, z) 2 p s,t (y, z)dz dy

= c 2.2,1 K 2 1 t 1/2 0 ∞ -∞ ∞ -∞
ϕ(y, z) 2 p s,t (y, z)dzdy

= c 2.2,1 K 2 1 t 1/2 0 E[ϕ(X s , X t ) 2 ]. A.
1 T -t 0 T t0 K h (X t -x)dX t = Φ h (X, x) ; ∀x ∈ R, ∀h > 0, where (x, h, ϕ) → Φ h (ϕ, x) is the map from R × (0, ∞) × C 0 ([t 0 , T ]; R) into R defined by Φ h (ϕ, x) := 1 T -t 0 K ϕ(T ) -x h -K ϕ(t 0 ) -x h - 1 2h 2 T t0 K ϕ(t) -x h σ(ϕ(t)) 2 dt
for every x ∈ R, h > 0 and ϕ ∈ C 0 ([t 0 , T ]; R). Moreover, (1) For every x ∈ R, h > 0 and ϕ ∈ C 0 ([t 0 , T ]; R),

|Φ h (ϕ, x)| 2 K ∞ T -t 0 + σ 2 ∞ K ∞ 2h 2 .
(2) For every h > 0 and ϕ ∈ C 0 ([t 0 , T ]; R),

Φ h (ϕ, .) 2 2,δ 6 K 2 ∞ (T -t 0 ) 2 + δ ∞ σ 4 ∞ K 2 2 h 3 .
(3) There exists a deterministic constant c A.4,1 > 0 such that, for every h, h > 0,

E( Φ h (X 1 , .), Φ h (X 2 , .) 2 2,δ ) c A.4,1 m(h ) with m(h ) = E( Φ h (X, .) 2 2,δ ). (4
) There exists a deterministic constant c A.4,2 > 0 such that, for every h > 0 and ϕ ∈ L 2 (R, dx),

E( Φ h (X, .), ϕ 2 2,δ ) c A.4,2 ϕ 2 2,δ . (5
) There exists a deterministic constant c A.4,3 > 0 such that, for every h, h ∈ H N ,

| Φ h (X, .), (bf ) h 2,δ | c A.4,3 a.s. Lemma A.5. Consider (14) U h,h (N ) := i =j Φ h (X i , .) -(bf ) h , Φ h (X j , .) -(bf ) h 2,δ ; ∀h, h ∈ H N .
Under Assumptions 2.1, 2.2, 3.1 and 5.1, there exists a deterministic constant c A.5 > 0, not depending on N , such that for every θ ∈ (0, 1) and λ > 0, with probability larger than 1 -5.4|H N |e -λ ,

sup h∈H N |U h,h0 (N )| N 2 - θm(h) N c A.5 (1 + λ) 3 θN and sup h∈H N |U h,h (N )| N 2 - θm(h) N c A.5 (1 + λ) 3 θN . Lemma A.6. Consider V h (N ) := 1 N N i=1 Φ h (X i , .) -(bf ) h 2 2,δ ; ∀h ∈ H N .
Under Assumptions 2.1, 2.2, 3.1 and 5.1, there exists a deterministic constant c A.6 > 0, not depending on N , such that for every θ ∈ (0, 1) and λ > 0, with probability larger than 1 -

2|H N |e -λ , sup h∈H N 1 N |V h (N ) -m(h)| - θm(h) N c A.6 (1 + λ) θN . Lemma A.7. Consider (15) W h,h (N ) := bf N,h -(bf ) h , (bf ) h -bf 2,δ ; ∀h, h ∈ H N .
Under Assumptions 2.1, 2.2, 3.1 and 5.1, there exists a deterministic constant c A.7 > 0, not depending on N , such that for every θ ∈ (0, 1) and λ > 0, with probability larger than 1 -

2|H N |e -λ , sup h∈H N {|W h,h0 (N )| -θ (bf ) h0 -bf 2 2,δ } c A.7 (1 + λ) 2 θN , sup h∈H N {|W h0,h (N )| -θ (bf ) h -bf 2 2,δ } c A.7 (1 + λ) 2 θN and sup h∈H N {|W h,h (N )| -θ (bf ) h -bf 2 2,δ } c A.7 (1 + λ) 2 θN . A.7.1.
Steps of the proof. The proof of Theorem 5.2.( 1) is dissected in four steps.

Step 1. This first step provides a suitable decomposition of bf N, h -bf 2 2,δ . First,

bf N, h -bf 2 2,δ = bf N, h -bf N,h0 2
2,δ + bf N,h0 -bf 2 2,δ -2 bf N,h0 -bf N, h , bf N,h0 -bf 2,δ .
Then, by [START_REF] Delattre | Maximum Likelihood Estimation for Stochastic Differential Equations with Random Effects[END_REF] and the definition of pen(.) (see [START_REF] Delattre | Coupling the SAEM Algorithm and the Extended Kalman Filter for Maximum Likelihood Estimation in Mixed-Effects Diffusion Models[END_REF]), for any

h ∈ H N , bf N, h -bf 2 2,δ bf N,h -bf N,h0 2
2,δ + pen(h) -pen( h) + bf N,h0 -bf 2 2,δ -2 bf N,h0 -bf N, h , bf N,h0 -bf 2,δ bf N,h -bf 2 2,δ + pen(h) -pen( h) + bf N,h0 -bf 2 2,δ -2 bf N,h -bf N, h , bf N,h0 -bf 2,δ = bf N,h -bf 2 2,δ -ψ N (h) + ψ N ( h) (16)
where

ψ N (h) := 2 bf N,h -bf, bf N,h0 -bf 2,δ -pen(h).
Let's complete the decomposition of bf N, h -bf 2 2,δ by writing

ψ N (h) = 2(ψ 1,N (h) + ψ 2,N (h) + ψ 3,N (h)),
where

ψ 1,N (h) := 1 (T -t 0 ) 2 N 2 N i=1 T t0 K h (X i s -•)dX i s , T t0 K h0 (X i s -•)dX i s 2,δ + U h,h0 (N ) N 2 - 1 2 pen(h) = U h,h0 (N ) N 2 , ψ 2,N (h) := - 1 N 2   N i=1 1 T -t 0 T t0 K h0 (X i s -•)dX i s , (bf ) h 2,δ + + N i=1 1 T -t 0 T t0 K h (X i s -•)dX i s , (bf ) h0 2,δ   + 1 N (bf ) h0 , (bf ) h 2,δ and
ψ 3,N (h) := W h,h0 (N ) + W h0,h (N ) + (bf ) h -bf, (bf ) h0 -bf 2,δ .
Step 2. This step deals with bounds on E(ψ j,N (h)) and E(ψ j,N ( h)) for j = 1, 2, 3.

• By Lemma A.5, for any λ > 0 and θ ∈ (0, 1), with probability larger than

1 -5.4|H N |e -λ , |ψ 1,N (h)| θm(h) N + c A.5 (1 + λ) 3 θN and |ψ 1,N ( h)| θm( h) N + c A.5 (1 + λ) 3 θN .
• On the one hand, for any h, h ∈ H N , consider

Ψ 2,N (h, h ) := 1 N N i=1 Φ h (X i , .), (bf ) h 2,δ . By Lemma A.4, |Ψ 2,N (h, h )| 1 N N i=1 ∞ -∞ Φ h (X i , x)(bf ) h (x)δ(x)dx c A.4,3 a.s.
On the other hand,

| (bf ) h , (bf ) h0 2,δ | δ ∞ K h * (bf ) ∞ K h0 * (bf ) 1 δ ∞ K 2 1 bf ∞ bf 1 .
Then, there exists a deterministic constant c 1 > 0, not depending on N and h, such that

|ψ 2,N (h)| c 1 N and |ψ 2,N ( h)| sup h ∈H N |ψ 2,N (h )| c 1 N a.s.
• By Lemma A.7 and Cauchy-Schwarz's inequality, with probability larger that 1

-|H N |e -λ , |ψ 3,N (h)| θ 4 ( (bf ) h -bf 2 2,δ + (bf ) h0 -bf 2 2,δ ) + 8c A.7 (1 + λ) 2 θN + 1 θ (bf ) h0 -bf 2 2,δ + 8c A.7 (1 + λ) 2 θN .
Step 3. Let us establish that there exist two deterministic constants c 2 , c 2 > 0, not depending on N and θ, such that with probability larger than 1 -

c 2 |H N |e -λ , sup h∈H N bf N,h -bf 2 2,δ -(1 + θ) (bf ) h -bf 2 2,δ + m(h) N c 2 (1 + λ) 3 θN and sup h∈H N (bf ) h -bf 2 2,δ + m(h) N - 1 1 -θ bf N,h -bf 2 2,δ c 2 (1 + λ) 3 θ(1 -θ)N .
On the one hand, note that bf [START_REF] Dion | Bidimensional Random Effect Estimation in Mixed Stochastic Differential Model[END_REF]). Moreover, for any h ∈ H N , [START_REF] Giné | Mathematical Foundations of Infinite-Dimensional Statistical Models[END_REF] bf

N,h -bf 2 2,δ -(1 + θ) (bf ) h -bf 2 2,δ + m(h) N can be written bf N,h -(bf ) h 2 2,δ - (1 + θ)m(h) N + 2W h (N ) -θ (bf ) h -bf 2 2,δ , where W h (N ) := W h,h (N ) (see
N,h -(bf ) h 2 2,δ = U h (N ) N 2 + V h (N ) N with U h (N ) = U h,h (N ) (see (14)). So, with probability larger than 1 -c 2 |H N |e -λ , sup h∈H N bf N,h -(bf ) h 2 2,δ - m(h) N - θm(h) N 2(c A.5 + c A.6 )(1 + λ) 3 θN
by Lemmas A.5 and A.6, and then

sup h∈H N bf N,h -bf 2 2,δ -(1 + θ) (bf ) h -bf 2 2,δ + m(h) N c 2 (1 + λ) 3 θN
by Lemma A.7. On the other hand, for any h ∈ H N ,

(bf ) h -bf 2 2,δ = bf N,h -bf 2 2,δ -bf N,h -(bf ) h 2 2,δ -W h (N ). Then, (1 -θ) (bf ) h -bf 2 2,δ + m(h) N -bf N,h -bf 2 2,δ |W h (N )| -θ (bf ) h -bf 2 2,δ + Λ h (N ) - θm(h) N where Λ h (N ) := bf N,h -(bf ) h 2 2,δ - m(h) N .
By Equality [START_REF] Giné | Mathematical Foundations of Infinite-Dimensional Statistical Models[END_REF],

Λ h (N ) = U h (N ) N 2 + V h (N ) N - m(h) N .
By Lemmas A.6 and A.5, there exist two deterministic constants c 3 , c 3 > 0, not depending N and θ, such that with probability larger than

1 -c 3 |H N |e -λ , sup h∈H N Λ h (N ) -θ m(h) N c 3 (1 + λ) 3 θN .
By Lemma A.7, with probability larger than

1 -2|H N |e -λ , sup h∈H N {|W h (N )| -θ (bf ) h -bf 2 2,δ } c A.7 (1 + λ) 2 θN .
Therefore, with probability larger than

1 -c 2 |H N |e -λ , sup h∈H N (bf ) h -bf 2 2,δ + m(h) N - 1 1 -θ bf N,h -bf 2 2,δ c 2 (1 + λ) 3 θ(1 -θ)N .
Step 4. By step 2, there exist two deterministic constants c 4 , c 4 > 0, not depending on N , θ, h and h 0 , such that with probability larger than

1 -c 4 |H N |e -λ , |ψ N (h)| θ (bf ) h -bf 2 2,δ + m(h) N + θ 2 + 2 θ (bf ) h0 -bf 2 2,δ + c 4 (1 + λ) 3 θN and |ψ N ( h)| θ (bf ) h -bf 2 2,δ + + c 6 θ (bf ) h0 -bf 2 2,δ + c 6 θ(1 -θ) • (1 + λ) 3 N .
This concludes the proof.

A.7.2. Proof of Lemma A.4. First of all, for any x ∈ R and h > 0, by Itô's formula,

K X T -x h = K X t0 -x h + T t0 K h (X t -x)dX t + 1 2h 2 T t0 K X t -x h d X t . So, T t0 K h (X t -x)dX t = K X T -x h -K X t0 -x h - 1 2h 2 T t0 K X t -x h σ(X t ) 2 dt = (T -t 0 )Φ h (X, x).
Lemma A.4.( 1) is a straightforward consequence of the previous equality and Lemma A.4.( 2) is easy to establish: for every h > 0 and ϕ ∈ C 0 ([t 0 , T ]; R),

(T -t 0 ) 2 Φ h (ϕ, .) 2 2,δ 2 ∞ -∞ K ϕ(T ) -x h 2 δ(x)dx + 4 ∞ -∞ K ϕ(t 0 ) -x h 2 δ(x)dx + 1 h 4 ∞ -∞ T t0 K ϕ(t) -x h σ(ϕ(t)) 2 dt 2 δ(x)dx 6 K 2 ∞ + T -t 0 h 4 T t0 σ(ϕ(t)) 4 ∞ -∞ K ϕ(t) -x h 2 δ(x)dxdt 6 K 2 ∞ + (T -t 0 ) 2 δ ∞ σ 4 ∞ K 2 2 h 3 .
Let us prove Lemma A.4. [START_REF] Comte | Regression Function Estimation on Non Compact Support as a Partly Inverse Problem[END_REF][START_REF] Comte | Nonparametric Drift Estimation for i.i.d. Paths of Stochastic Differential Equations[END_REF][START_REF] Comte | Drift Estimation on Non Compact Support for Diffusion Models[END_REF]. First, for any h, h > 0,

E( Φ h (X 1 , .), Φ h (X 2 , .) 2 2,δ ) = 1 (T -t 0 ) 4 E   ∞ -∞ T t0 K h (X 1 t -x)dX 1 t T t0 K h (X 2 t -x)dX 2 t δ(x)dx 2   2 (T -t 0 ) 4 (E(A 2 h,h ) + E(B 2 h,h )) with A h,h := ∞ -∞ T t0 K h (X 1 t -x)σ(X 1 t )dW 1 t T t0 K h (X 2 t -x)dX 2 t δ(x)dx and B h,h := ∞ -∞ T t0 K h (X 1 t -x)b(X 1 t )dt T t0 K h (X 2 t -x)dX 2 t δ(x)dx.
Bound on E(A 2 h,h ). Since (X 1 , W 1 ) and X 2 are independent,

E(A 2 h,h ) = ∞ -∞ ∞ -∞ E T t0 K h (X 1 t -x)σ(X 1 t )dW 1 t T t0 K h (X 1 t -y)σ(X 1 t )dW 1 t ×E T t0 K h (X 2 t -x)dX 2 t T t0 K h (X 2 t -y)dX 2 t δ(x)δ(y)dxdy.
On the one hand, for every x, y ∈ R, by the isometry property of Itô's integral and the definition of f ,

E T t0 K h (X 1 t -x)σ(X 1 t )dW 1 t T t0 K h (X 1 t -y)σ(X 1 t )dW 1 t = T t0 E(K h (X 1 t -x)K h (X 1 t -y)σ(X 1 t ) 2 )dt = (T -t 0 ) ∞ -∞ K h (z -x)K h (z -y)σ(z) 2 f (z)dz.
Then,

E(A 2 h,h ) = (T -t 0 ) ∞ -∞ ∞ -∞ ∞ -∞ K h (z -x)K h (z -y)σ(z) 2 f (z) ×E T t0 K h (X 2 t -x)dX 2 t T t0 K h (X 2 t -y)dX 2 t δ(x)δ(y)dxdydz = (T -t 0 ) ∞ -∞ σ(z) 2 f (z)E   ∞ -∞ K h (z -x)δ(x) T t0 K h (X 2 t -x)dX 2 t dx 2   dz.
On the other hand, for every z ∈ R, x → |K h (z -x)|/ K 1 is a density function. Then, by Jensen's inequality,

E(A 2 h,h ) (T -t 0 ) K 1 ∞ -∞ σ(z) 2 f (z) ∞ -∞ |K h (z -x)|δ(x) 2 E   T t0 K h (X 2 t -x)dX 2 t 2   dxdz (T -t 0 ) σ 2 f ∞ K 2 1 δ ∞ ∞ -∞ δ(x)E   T t0 K h (X 2 t -x)dX 2 t 2   dx (T -t 0 ) 3 σ 2 f ∞ K 2 1 δ ∞ m(h ).
Bound on E(B 2 h,h ). Since x → |K h (X t (ω) -x)|/ K 1 is a density function for every (t, ω) ∈ [t 0 , T ] × Ω, by Jensen's inequality,

E(B 2 h,h ) = E   T t0 ∞ -∞ K h (X 1 t -x)b(X 1 t )δ(x) T t0 K h (X 2 s -x)dX 2 s dxdt 2   (T -t 0 ) K 1 T t0 ∞ -∞ E(|K h (X 1 t -x)|b(X 1 t ) 2 )δ(x) 2 E   T t0 K h (X 2 s -x)dX 2 s 2   dxdt = (T -t 0 ) 2 K 1 ∞ -∞ ∞ -∞ |K h (z -x)|b(z) 2 f (z)dz δ(x) 2 E   T t0 K h (X 2 s -x)dX 2 s 2   dx (T -t 0 ) 2 b 2 f ∞ K 2 1 δ ∞ ∞ -∞ δ(x)E   T t0 K h (X 2 s -x)dX 2 s 2   dx (T -t 0 ) 4 b 2 f ∞ K 2 1 δ ∞ m(h ).
Now, for any h > 0 and ϕ ∈ L 2 (R, dx),

E( Φ h (X, .), ϕ 2 2,δ ) = 1 (T -t 0 ) 2 E   ∞ -∞ ϕ(x)δ(x) T t0 K h (X t -x)dX t dx 2   2 (T -t 0 ) 2 (E(C 2 h ) + E(D 2 h ))
with

C h := ∞ -∞ ϕ(x)δ(x) T t0
K h (X t -x)σ(X t )dW t dx and

D h := ∞ -∞ ϕ(x)δ(x) T t0 K h (X t -x)b(X t )dtdx.
Bound on E(C 2 h ). By the isometry property of Itô's integral and the definition of f ,

E(C 2 h ) = ∞ -∞ ∞ -∞ ϕ(x)ϕ(y)δ(x)δ(y) T t0 E(K h (X t -x)K h (X t -y)σ(X t ) 2 )dtdxdy = (T -t 0 ) ∞ -∞ ∞ -∞ ϕ(x)ϕ(y)δ(x)δ(y) ∞ -∞ K h (z -x)K h (z -y)σ(z) 2 f (z)dzdxdy = (T -t 0 ) ∞ -∞ (K h * (ϕδ))(z) 2 σ(z) 2 f (z)dz (T -t 0 ) σ 2 f ∞ K 2 1 δ ∞ ϕ 2 2,δ .
Bound on E(D 2 h ). By the definition of f ,

E(D 2 h ) = E   T t0 b(X t ) ∞ -∞ K h (X t -x)ϕ(x)δ(x)dxdt 2   (T -t 0 ) T t0 E(b(X t ) 2 (K h * (ϕδ))(X t ) 2 )dt (T -t 0 ) 2 ∞ -∞ (K h * (ϕδ))(z) 2 b(z) 2 f (z)dz (T -t 0 ) 2 b 2 f ∞ K 2 1 δ ∞ ϕ 2 2,δ .
Finally, since X is a semi-martingale and since the map (t, ω, x) → K h (X t (ω) -x)(bf ) h (x)δ(x) is measurable and bounded for any h, h ∈ H N , by the stochastic Fubini theorem and Itô's formula,

(T -t 0 ) Φ h (X, .), (bf ) h 2,δ = (T -t 0 ) ∞ -∞ Φ h (X, x)(bf ) h (x)δ(x)dx = T t0 ∞ -∞
K h (X t -x)(bf ) h (x)δ(x)dxdX t a.s.

= T t0 [K h * ((bf ) h δ)](X t )dX t = Ψ h,h (X T ) -Ψ h,h (X t0 ) - 1 2 T t0 ψ h,h (X t )σ(X t ) 2 dt
where ψ h,h := K h * ((bf ) h δ), and Ψ h,h := K(./h) * ((bf ) h δ) is a primitive function of ψ h,h . On the one hand,

ψ h,h = K h * ((bf ) h δ) = K h * ((bf ) h δ ) + K h * ((K h * (bf ) )δ) = K h * ((bf ) h δ ) + K h * ((K h * (bf ))δ) + K h * ((K h * (b f ))δ).
Then, since bf , bf and b f are bounded under Assumption 2.2,

ψ h,h ∞ K h 1 K h * (bf ) ∞ δ ∞ + K h 1 K h * (bf ) ∞ δ ∞ + K h 1 K h * (b f )) ∞ δ ∞ K 2 1 bf ∞ δ ∞ + K 2 1 bf ∞ δ ∞ + K 2 1 b f ∞ δ ∞ < ∞. On the other hand, Ψ h,h ∞ K(./h) ∞ (K h * (bf ))δ 1 K ∞ δ ∞ K 1 bf 1 < ∞.
This concludes the proof because On the one hand, since E(g h,h (ϕ, X)) = 0 for every ϕ ∈ E, by Giné and Nickl [START_REF] Giné | Mathematical Foundations of Infinite-Dimensional Statistical Models[END_REF], Theorem 3.4.8, there exists a universal constant m 1 such that for any λ > 0, with probability larger than 1 -5.4e -λ ,

(T -t 0 )| Φ h (X,
|U h,h (N )| N 2 m N 2 (c h,h (N )λ 1/2 + d h,h (N )λ + b h,h (N )λ 3/2 + a h,h (N )λ 2 )
where the constants a h,h (N ), b h,h (N ), c h,h (N ) and d h,h (N ) are defined and controlled later. First, note that [START_REF] Györfi | A Distribution-Free Theory of Nonparametric Regression[END_REF] U h,h (N ) = i =j (g h,h (X i , X j ) -g h,h (X i ) -g h ,h (X j ) + E(g h,h (X i , X j )))

where, for every η, η ∈ H N and ϕ 1 , ϕ 2 , ψ ∈ E, g η,η (ϕ 1 , ϕ 2 ) := Φ η (ϕ 1 , .), Φ η (ϕ 2 , .) 2,δ and g η,η (ψ) := Φ η (ψ, .), (bf ) η 2,δ = E(g η,η (ψ, X)).

Let us now control a h,h (N ), b h,h (N ), c h,h (N ) and d h,h (N ):

4 6 K 2 ∞ (T -t 0 ) 2 + δ ∞ σ 4 ∞ K 2 2 h 3 1/2 6 K 2 ∞ (T -t 0 ) 2 + δ ∞ σ 4 ∞ K 2 2 (h ) 3 1/2 c 1 h 3 0 with c 1 = 4 6 K 2 ∞ (T -t 0 ) 2 + δ ∞ σ 4 ∞ K 2 2 .
So, since (N

h 3 0 ) -1 1, a h,h (N )λ 2 N 2 c 1 λ 2 N 2 h 3 0 c 1 λ 2 N .
• • The constant c h,h (N ). Consider c h,h (N ) 2 := N 2 E(g h,h (X 1 , X 2 ) 2 ).

By [START_REF] Györfi | A Distribution-Free Theory of Nonparametric Regression[END_REF] and Lemma A.4,

c h,h (N ) 2 16N 2 E( Φ h (X 1 , .), Φ h (X 2 , .) 2 2,δ ) c 3 m(h )N 2 with c 3 = 16c A.4,1 .
So, as previously, c h,h (N )λ 1/2 N 2 θm(h ) 3mN + 3c 3 mλ θN .

• The constant d h,h (N ). Consider

Finally, Propositions 3 .Proposition 3 . 5 .

 335 3 and 3.4 allow to provide a risk bound on a truncated version of the Nadaraya-Watson estimator b N,h (see (2)). Consider the 2 bandwidths (truncated) Nadaraya-Watson (2bNW) estimator b

Figure 1 .

 1 Figure 1. LooCV NW estimation for Model 1 (Langevin equation), h = 0.04.

Figure 2 .Figure 3 .

 23 Figure 2. LooCV NW estimation for Model 2 (hyperbolic diffusion process), h = 0.04.

Figure 4 .Figure 5 .

 45 Figure 4. LooCV NW estimation for Model 4, h = 0.06.

Figure 6 .

 6 Figure 6. MSE of the looCV estimator with respect to N and n for Model 3.

A. 4 .

 4 Proof of Proposition 3.5. First of all, b

Remark A. 3 .

 3 Assume that b and σ are bounded. Then, in Step 1, for ϕ = b or ϕ

- 1 i=1E

 1 X i )b j (X j )g h,h (X i , X j ) (a i (X i ) 2 ) 1 and N j=2 E(b j (X j ) 2 ) 1    .

  ).

	Proposition 4.2. Under Assumptions 2.1, 2.2, 2.3, 3.1, 3.2 with υ = β, and 4.1, if
	1 nh 2	1,
	then there exists a constant c 4.2 > 0, not depending on h, N , n and t 0 , such that

  Therefore, under Assumptions 2.1, 2.2, 3.1 and 5.1, by Corollary 5.3,

  tj thanks to the assumption dX t = Y t dt. Of course CV * (h) is not satisfactory because the last term of its previous decomposition doesn't exist, but since this term doesn't depend on h, to minimize CV * (.) is almost equivalent to minimize CV(.) which only involves quantities existing without the condition dX t = Y t dt.5.3. Numerical experiments.Some numerical experiments on our estimation method are presented in this subsection. The discrete-time approximate Nadaraya-Watson (NW) (see (

  [START_REF] Comte | Nonparametric Estimation for Stochastic Differential Equations with Random Effects[END_REF]. Proof of Theorem 5.2. Throughout this subsection, K is a primitive function of the kernel K. The proof of Theorem 5.2.(1) relies on the following technical lemmas proved at the end of this subsection. The proof of Theorem 5.2.(2) is left to the reader because it is similar but simpler than the proof of Theorem 5.2.(1) detailed in this subsection.

	Lemma A.4. Under Assumptions 2.1, 2.2, 3.1 and 5.1,

  h,h (ϕ 1 , ϕ 2 ) := Φ h (ϕ 1 , .) -(bf ) h , Φ h (ϕ 2 , .) -(bf ) h 2,δ .

.), (bf ) h 2,δ | 2 Ψ h,h ∞ + T -t 0 2 σ 2 ∞ ψ h,h ∞ a.s. A.7.3. Proof of Lemma A.5. For any h, h ∈ H N , U h,h (N ) = i =j g h,h (X i , X j )

with, for every ϕ 1 , ϕ 2 ∈ E = C 0 ([0, T ]; R), g

•

  The constant a h,h (N ). Considera h,h (N ) := sup ϕ1,ϕ2∈E |g h,h (ϕ 1 , ϕ 2 )|. Φ h (ϕ 1 , .), Φ h (ϕ 2 , .) 2,δ | 4 sup ϕ1∈E Φ h (ϕ 1 , .) 2,δ sup

	By (18), Cauchy-Schwarz's inequality and Lemma A.4,
	a h,h (N )	4 sup
		ϕ1,ϕ2∈E

| ϕ2∈E Φ h (ϕ 2 , .) 2,δ

  The constant b h,h (N ). Consider b h,h (N ) 2 := N sup

	So, for any θ ∈ (0, 1), since (N h 3 0 ) -1 1,
	b h,h (N )λ 3/2 N 2	2	θ 3m	1/2 m(h ) 1/2 N h 3/2 ×	3m θ	1/2 c 1/2 2 λ 3/2 N 1/2
		θm(h ) 3mN 2 h 3 +	3c 2 mλ 3 θN	θm(h ) 3mN	+	3c 2 mλ 3 θN	.

ϕ∈E E(g h,h (ϕ, X) 2 ).

By (18), Cauchy-Schwarz's inequality and Lemma

A.4, b h,h (N ) 2 16N sup ϕ∈E E( Φ h (ϕ, .), Φ h (X, .) 2 2,δ ) 16N E( Φ h (X, .) 2 2,δ ) sup ϕ∈E Φ h (ϕ, .) 2 2,δ c 2 m(h )N h 3 with c 2 = 4c 1 .

2β+1(1-β)

Acknowledgments. Many thanks to Fabienne Comte and Valentine Genon-Catalot for their constructive comments about this work.

Appendix A. Proofs A.1. Proof of Corollary 2.5. First of all, since p t (x 0 , x) > 0 for every (t, x) ∈ (0, T ] × R,

p t (x 0 , x)dt > 0 By [START_REF] Györfi | A Distribution-Free Theory of Nonparametric Regression[END_REF], Cauchy-Schwarz's inequality, Jensen's inequality and Lemma A.4,

Therefore, there exists a deterministic constant c 4 > 0, not depending on N , h and h , such that with probability larger than 1 -5.4e -λ ,

In conclusion, with probability larger than

A.7.4. Proof of Lemma A.6. First, the two following results are used several times in the sequel:

and

Moreover, by Inequality [START_REF] Halconruy | Kernel Selection in Nonparametric Regression[END_REF] and Lemma A.4,

and, by Inequality [START_REF] Halconruy | Kernel Selection in Nonparametric Regression[END_REF], Equality [START_REF] Hoffmann | Adaptive Estimation in Diffusion Processes[END_REF] and Lemma A.4,

Then, for any θ ∈ (0, 1), since (N h 3 0 ) -1 1, with probability larger than 1 -2e -λ ,

So, with probability larger than

Therefore, by Equality [START_REF] Hoffmann | Adaptive Estimation in Diffusion Processes[END_REF], with probability larger that 1 -

with, for every ϕ ∈ E, g h,h (ϕ) := Φ h (ϕ, .), (bf ) h -bf 2,δ .

By Bernstein's inequality, for any λ > 0, with probability larger than 1 -2e -λ ,

Moreover, by Lemma A.4,

Then, for any θ ∈ (0, 1), with probability larger than 1 -2e -λ ,

So, with probability larger than 1 -2|H N |e -λ ,

A.8. Proof of Corollary 5.3. On the one hand, as in the proof of Proposition 3.5 and since δ(x) > m for every x ∈ [A, B],

]. On the other hand, by Theorem 5.2 and union bounds,

Therefore,


Frédéric Baret

Benoit De Solan

Samuel Thomas

Philippe Burger

Shouyang Liu

Alexis Comar

Christophe Rousset

Robin Vanhove

Romain Regnier

Jerome Terreni

Laurent Combe

Phenomobile: a fully automatic robot for high-throughput field phenotyping of a large range of crops with active measurements

Keywords: Field phenotyping, Remote Sensing, Robot, Unmanned Ground Vehicle

High-throughput phenotyping data acquisition in the field is highly required to provide breeders with a set of accurate and heritable traits for the evaluation of the genotypes performances. An entirely automatic robot was developed, called Phenomobile, able to sample with a 100-200 plots/hour throughput using a 12m long telescopic arm moving from 1m to 4m height. The vehicle is mounted on caterpillars to operate under difficult soil conditions. The measurement heads include LiDARS, high resolution RGB cameras and multispectral cameras used in active mode with flashes. The phenomobile is driven by RTK-GPS with centimetric accuracy. Sensors are automatically triggered according to a predefined mission, and the data stored under standard format for easy processing. Fist results and conclusions are given in this paper.

Introduction

Acquisition of phenotyping data in field conditions is mandatory to support breeders with pertinent information on the performances of the genotypes. Although UAVs allow to characterize the genotypes with a number of traits, their application is often limited to some of the traits requiring either very high resolution imagery, high power consumption or heavy payload sensors. Further, UAV observations are mainly completed in passive mode, making the measurements sensitive to the illumination conditions prevailing during the data acquisition. Finally, flying UAVs might be limited by local regulations. Although associated with a lower throughput, Unmanned Ground Vehicles (UGV) is a versatile solution that carries a number of sensors operating in active mode from very low distance to the canopy allowing to record very high spatial resolution images. This paper presents a UGV called Phenomobile, that was specifically developed to sample a range of crops including tall crops such as sunflower or maize and crops that fills rapidly the interval left between two consecutive plots such as rapeseed or peas.

Description of the phenomobile

The phenomobile is made of three main components: (1) the vehicle, (2) the measurement head and (3) the data acquisition system.

a. The vehicle

The platform weigh eight tons (Figure 1). It is equipped with four caterpillars allowing to operate even under difficult soil conditions with limited damage. The platform can rotate 360° along a vertical axis above the caterpillars. It supports the diesel engine with 8 hours autonomy that runs both the hydraulic and electric units. The 12 m telescopic boom can raise the measurement head from 1 to 4 m height. A cockpit allows an operator to ensure the security when the regulations do not authorize a full autonomous driving. The vehicle is running on the alleys, making stops on specific positions to sample a group of 1 to 12 microplots on both sides of the alley. The throughput is in between 100-200 plots per hour depending on the plot size.

b. The measurement head

The measurement head (Figure 2) is made of two units: one looking vertically, the other looking at a given incidence angle from the side. The two units are positioned to sample approximately the same crop volume. Each unit is equipped with LiDARS, high resolution RGB cameras and multispectral cameras. The cameras are operated in active mode using powerful flashes, making the measurements fully independent from the illumination conditions.

c. The data acquisition system

The acquisition system is run by a PC using the ROS operating system dedicated to robotics. The acquisition system triggers the measurements that are then recorded in HDF5 format along with all the metainformation required to ensure FAIR principles in data management, including position and time stamp. The system is driven by phenoIHM (Figure 3), the application that defines and runs the mission.

Figure 3. One typical page of the "phenoIHM" interface used to define a script for a given microplot.

A mission corresponds to a measurement session, i.e. the sampling of a group of plots completed at a given date/stage along the growth cycle. It is therefore necessary to define the vehicle and measurement head used, with all the settings of the sensors. The trajectory (path) of the vehicle is then defined, with the several stops required to sample the plots considered. Then the script that describes the set of measurements to make on a given plot is defined by the start and stop positions (beginning and end of the plot) and all the positions where images are shot (Figure 3). Several tools help to define the mission by automatically repeating a script for a large number of plots. Once a mission is defined, it can be run automatically up to its completion. Warnings and a log file allows to control the quality of the measurements both in real time and after the mission ended.

Sample results and conclusion

The phenomobile started to operate in 2017 in Toulouse and two more copies are currently operating in Clermont and Montpellier. First results indicate a very good accuracy (Figure 3) due to the high spatial resolution available and the independency from the illumination conditions. Further, a very high repeatability (Figure 3) is also achieved allowing to get high values of heritability (Figure 4) and very good time consistency. It is concluded that the phenomobile is a very efficient system for a detailed monitoring of structural traits. It is further possible to easily add sensors to access new traits, particularly those benefiting from active measurements difficult to make from a UAV platform.

Figure 1 .

 1 Figure 1. The phenomobile with caterpillars showing the 12m long boom supporting the measurement head.

Figure 2 .

 2 Figure 2. The measurement head with its vertical and side units. Each unit includes LiDARS (in blue), RGB and multispectral cameras operated with flashes. The white box on the right hosts the acquisition system.

Figure 3 .

 3 Figure 3. Comparison of phenomobile derived GAI trait as compared to reference ground measurements. Data from Greoux experiment on wheat conducted in 2018.

Figure 4 .

 4 Figure 4. Heritability levels reached for several structural traits (Average Leaf Angle (ALA), Fraction of light intercepted (FIPAR), Green area index (GAI) and green fractions at 0° (GF0) and 45°(GF45)) derived from the Phenomobile over a wheat experiment in 2018 in Gréoux with well-watered (red) and water-stress modalities.

Acknowledgements. This study was supported by "Programme d'investissement d'Avenir" PHENOME


Laurent Barbiéro
email: barbiero@civil.iisc.ernet.in

Vincent Valles

Claude Cheverry

Reply to J.W. Van Hoorn "Some observations with respect to sodicity hazard of irrigation waters"

In the comments "Some observations with respect to sodicity hazard of irrigation waters", Dr. Van Hoorn opened a constructive discussion about the assessment of sodicity hazard of soils subjected to irrigation in arid or semi arid areas. The following are three concepts usually used for the afore mentioned assessment, which are reminded in his comment on our article:

.

. Indeed, the sodicity hazard is low, low-moderate, or very high according to the SAR, the SAR adj , or the RSC respectively. Therefore, the donors face with a difficult choice in developing new irrigation schemes in Niger valley. Dr. Van Hoorn concludes that the RSC is the most suitable indicator for the assessment of sodicity hazard of Niger irrigation water. With the second point Dr. Van Hoorn is considering if the process of magnesium precipitation should be taken into account for the assessment of sodicity hazard. Both questions will be answered successively.

The SAR :

, where the evolution of both SAR and Electrical Conductivity of three types of water during a pan evaporation experiment are plotted. The

data were taken from [START_REF] Valles | Etude et modélisation des transferts d'eau et de sel dans un sol argileux. Application au calcul de doses d'irrigation[END_REF] in the Medjerdah valley in Tunisia, [START_REF] Gonzalez Barrios | Eau d'irrigation et salinité des sols en zone aride mexicaine[END_REF] in the Comarca Lagunera in Mexico, and [START_REF] Barbiéro | Les sols alcalinisés sur socle dans la vallée du fleuve Niger. Origine de l'alcalinisation et évolution des sols sous irrigation[END_REF] in Niger. On the Tunisian site, the sodicity hazard remains low during the evaporation. For the Mexican water, it increases slowly and reaches the second class (moderate sodicity hazard) with the higher concentration factors. Finally, in Niger the sodicity hazard increases strongly and reaches dangerous levels even while at the moderate concentration factors. Thus, the three irrigation waters present the same soil sodization hazard but evolve in three different ways, hence, emphasizing the limitation of the SAR concept.

The adjusted SAR :

The concept of Adjusted SAR, used for carbonate waters, is a first step towards a dynamic view of sodicity hazard assessment during the evaporation and concentration, although it remains inadequate. Considering the precipitation of a mineral AB according to the reaction :

A + B ←→ AB The equilibrium between the solute A, B and the mineral AB is governed by the equation :

(A) . (B) / (AB) = K Where K is the constant of solubility and ( ) denote the activities. The activity of the crystal AB is conventionally equal to 1, the relation is :

(A) . (B) = K Taking the Log of this equation, we have : Log (A) + Log (B) = Log (K) Therefore, the equilibrium is represented by a negative unit-slope line in the diagram Log(A) versus Log(B) (Fig. 2). For a diluted solution, the product (A).(B) is below K. When the solution concentrates, the equilibrium is reached and the product (A).(B) becomes equal to K. As the solution keeps concentrating, it remains at equilibrium with respect to the mineral but the activities of A and B cannot increase simultaneously. If A increases, B decreases and vice versa. Two cases may occur : if (A)>(B), the activity of A keeps increasing and the activity of B decreases. If (B)>(A), the activities evolve in the opposite direction. The above mentioned theoretical case applied to the precipitation of the calcite, allows the identification of both saline ((Ca) > (Alk.)) and alkaline ((Alk.) > (Ca)) evolution ways.

The use of the SAR adj instead of the SAR (Suarez, 1981, Ayers and[START_REF] Ayers | Water Quality for Agriculture[END_REF] amounts to considering the solution at equilibrium with respect to the calcite instead of the bulk solution, i.e. shifting from point 1 to point 2 (or 1' to 2') in Fig. 2. However, the adjustment of the SAR does not take into account the evolution way during the concentration process, and yet again the sodicity hazard remains underestimated using this indicator in the case of solutions evolving in alkaline way. The adjustment of the SAR is a correct adjustment but it cannot be used as quantitative (multiplicative) factor for the assessment of the alkalisation hazard.

The RSC :

The concept of RSC is much closer to that of Residual Alkalinity, used in our article (Barbiero et al., 2001) and commented by Dr. Van Hoorn. RSC corresponds to RA calcite + magnesite , i.e. Residual Alkalinity after the complete precipitation of Ca and Mg as carbonate salt. When the precipitation of Ca and Mg carbonates alone occurs without any other precipitation involving transfer of protons, the RSC can be used as a quantitative tool for assessment of sodicity hazard. However, Ca and Mg carbonates are usually not the only precipitate in natural waters therefore the RSC concept has its own limitation.

Taking into account the precipitation of Mg:

The precipitation of Mg as carbonate, not as magnesite but as Mg-calcite, has been frequently observed. In the alkaline environment of Chad [START_REF] Cheverry | Contribution à l'étude pédologique des polders du lac Tchad. Dynamique des sels en milieu continental subaride dans les sédiments argileux et organiques[END_REF] or Niger [START_REF] Barbiéro | Les sols alcalinisés sur socle dans la vallée du fleuve Niger. Origine de l'alcalinisation et évolution des sols sous irrigation[END_REF], the proportion of Mg ranges from 5 to 8 %, and Ca from 92 to 95 %. Such quantities are generally too low to explain the control of Mg in the solution. Moreover, the precipitation of Mg as silicate (stevensite, sepiolite, Mg-montmorillonite, palygorskite…) has been detected principally in alkaline environments [START_REF] Tardy | Néoformation d'une argile magnésienne dans les dépressions interdunaires du lac Tchad. Application aux domaines de stabilité des phyllosilicates alumineux, magnésiens et ferrifères[END_REF][START_REF] Gac | Geochemical behaviour of silica and magnesium during the evaporation of waters in Chad[END_REF][START_REF] Valles | Geochemistry of water in the Kouroumari region, Mali. Al, Si and Mg in water concentrated by evaporation: development of a model[END_REF]. However, the exact mineralogy of Mg-silicates is difficult to identify because of the low quantity of clay involved in the precipitation in comparison to the quantity of clay already present in the soil profile. Mg-clay formations involve transfer of protons, therefore they have an influence on the alkalinity, and should be logically included into the balance made on the Residual Alkalinity as developed in the referred article [START_REF] Barbiéro | Residual alkalinity as tracer to estimate the changes induced by a forage cultivation in a non-saline irrigated sodic soil[END_REF]. The calculation requires knowing the exact stoichiometry of the Mg-silicate produced.

In the specific case of the Niger site, we took into consideration the precipitations occurring in the soil and not in the Niger river during evaporation for the assessment of water quality. The behaviour of Ca, Mg and alkalinity in the soil solution during concentration is plotted on Fig. 3. The control of Ca (evolving in a 1/10 ratio) is much stronger than that of Mg (evolving in a 1/2 ratio) and the quantities of Mg involved (in both carbonate and silicate precipitations) are much lower than the alkalinity of the solution, and therefore do not significantly influence the control of alkalinity. The behaviour of Mg in the soil solution and the quantities involved do not justify the introduction of Mg precipitation in the assessment of salinity hazard of irrigation water. On the other hand, although fluoride amounts are not controlled during the concentration of Niger water, fluorite precipitation interferes significantly in the balance of alkalinity in the soil solution during evaporation and was therefore taken into account in the article, using RA calcite+fluorite .

Final considerations

Qualitative and quantitative assessment of salinity and/or sodicity hazard must be developed using several residual alkalinities that have to be adapted to each site according to the precipitations occurring in the soil [START_REF] Valles | Ivariant criteria for irrigation water quality assessment in arid and semi-arid regions[END_REF] (Table 1).

In the Medjerdah valley, the irrigation water belongs to a saline calcic family. A neutral salinization will occur if irrigation is poorly managed ; a leaching fraction must be applied in order to eliminate the salt brought at each irrigation. The agricultural problems are osmotic and toxicities with respect to Na, Cl, B… The calcite + gypsum residual alkalinity, defined as RA calcite+gypsum = (Alk.) -(Ca) + (SO4) can be used for quantification of salinity hazard.

The Mexican water belongs to a sulphate alkaline family. This water evolves in an alkaline way, but during evaporation the equilibrium with respect to the sodium sulphates is reached. The sodium amounts are controlled and remain moderate. No sodization of the complex will be observed in the soil. In this case, the agricultural problems are mainly osmotic. Two independent residual alkalinities can be used as tracer for the assessement of salinity hazard, the calcite residual alkalinity RA calcite = (Alk.) -(Ca) and a thenardite residual alkalinity that could be defined as RA thenardite = (Na) -(SO4) [START_REF] Droubi | Generalized residual alkalinity concept application to prediction of the chemical evolution of natural waters by evaporation[END_REF].

The water of the Niger River belongs to a sodic alkaline family, which can induce a sodization of the soil complex, even for moderate concentration factors. Physical problems are expected, namely asphyxia, loss of structure, low permeability and hardsetting after drying [START_REF] Sumner | Sodic soils ", new perspectives[END_REF]. RA calcite usually presents a conservative behaviour during concentration and can be used as a conservative tracer. When Mg amounts are rather high and when Mg is involved in precipitation which consumes alkalinity, the RA calcite+magnesite (or RSC) is a suitable tracer of the concentration process. However, in the case of Niger site, it is necessary to adapt the RA, using RA calcite+fluorite = (Alk.) -(Ca) + (F) to take into account the concomitant precipitation of both calcite and fluorite.

As a conclusion, from the three concepts usually used for the assessment of sodicity hazard (SAR, SAR adjusted and RSC), the RSC is the best one because it does not underestimate the hazard of carbonate irrigation water. The RSC concept is included in the RA concept, and our article described the conservative behaviour of the RA concept [START_REF] Barbiéro | Residual alkalinity as tracer to estimate the changes induced by a forage cultivation in a non-saline irrigated sodic soil[END_REF] which was already applied in quantitative studies [START_REF] Ribolzi | Contrôle géochimique des eaux par la formation de calcite en milieu méditerranéen et en milieu tropical. Arguments d'équilibre et argument de bilan[END_REF][START_REF] Ribolzi | Comparison of hydrograph deconvolutions using residual alkalinity, chloride and oxygen 18 as hydrochemical tracers[END_REF], Barbiero et al., 1995). Therefore, RA should support the development of new reliable criteria for assessment of irrigation waters in arid or semi arid regions.

Figure captions

Fig. 1 :

 1 Figure captionsFig.1: Advance of SAR and EC during a pan evaporation experiment of three irrigation waters from Mexico, Tunisia and Niger.

Fig. 2 :

 2 Fig. 2: Theoretical advance of a solution reaching equilibrium with respect to a mineral AB, in a Log(A) versus Log(B) diagram.

Table 1 :

 1 Subdivision of irrigation water of arid and semi-arid area into 5 families (modified from[START_REF] Valles | Ivariant criteria for irrigation water quality assessment in arid and semi-arid regions[END_REF].

	Alk < 0		Alk > 0
	Acid	Alk. -Ca < 0	Alk. -Ca > 0
	group	Saline group	Alkaline group
		Alk. -Ca + SO 4	Alk. -Ca + SO 4	Na -SO 4 < 0	Na -SO 4 > 0
		< 0	> 0
	(1)	(2)	(3)	(4)	(5)
	Acid	Calcic	Sulfate	Sulfate	Carbonate
	family	saline	saline	alkaline	alkaline
		family	family	family	family