1. Introduction
Consider the stochastic differential equation
(1) 
where , is a fractional Brownian motion of Hurst index , is a continuous map and .
In this work, we assume that we observe i.i.d. paths of the solution of Equation (1). Our aim is to propose and study a nonparametric estimator of the drift function
based on these observations. This problem is related to functional data analysis, and more specifically, there are various recent contributions about i.i.d. parametric models of (non fractional) stochastic differential equations with mixed effects (see, e.g., Ditlevsen and De Gaetano
[23], Overgaard et al. [42], Picchini, De Gaetano and Ditlevsen [43], Picchini and Ditlevsen [44], Comte, GenonCatalot and Samson [13], Delattre and Lavielle [19], Delattre, GenonCatalot and Samson [16], Dion and GenonCatalot [22], Delattre, GenonCatalot and Larédo [17]). Also, i.i.d. samples of stochastic differential equations have been recently considered in the framework of multiclass classification of diffusions (see Denis, Dion and Martinez [20]). The need of flexibility to deal with the information contained in functional data analysis make it interesting to use a nonparametric approach.Along the last two decades, many authors studied statistical inference from observations drawn from stochastic differential equations driven by fractional Brownian motion, considering the observation of one path, either in continuous time, or in discrete time with fixed or small step.
Most references on the estimation of the trend component in Equation (1) deal with parametric estimators. Let us start by papers considering continuous time observations. In Kleptsyna and Le Breton [27] and Hu and Nualart [29], estimators of the trend component in Langevin’s equation are studied. Kleptsyna and Le Breton [27] provide a maximum likelihood estimator, where the stochastic integral with respect to the solution of Equation (1) returns to an Itô integral. In [49], Tudor and Viens extend this estimator to equations with a drift function depending linearly on the unknown parameter. Hu and Nualart [29] provide a least squares estimator, where the stochastic integral with respect to the solution of Equation (1) is taken in the sense of Skorokhod. In [30], Hu, Nualart and Zhou extend this estimator to equations with a drift function depending linearly on the unknown parameter. Finally, in [36], Marie and Raynaud de Fitte extend this estimator to nonhomogeneous semilinear equations with almost periodic coefficients.
Now, considering discrete time observations, still in parametric context, Tindel and Neuenkirch [39] study a least squarestype estimator defined by an objective function, tailormaid with respect to the main result of Tudor and Viens [50] on the rate of convergence of the quadratic variation of the fractional Brownian motion. In [46], Panloup, Tindel and Varvenne extend the results of [39] under much more flexible conditions. In [8], Chronopoulou and Tindel provide a likelihood based numerical procedure to estimate a parameter involved in both the drift and the volatility functions in a stochastic differential equation with multiplicative fractional noise.
Concerning nonparametric methods for the estimation of the function in Equation (1), there are only few references. Saussereau [47] and Comte and Marie [14] study the consistency of NadarayaWatson type estimators of the drift function in Equation (1). In [37], Mishra and Prakasa Rao established the consistency and a rate of convergence of a nonparametric estimator of the whole trend of the solution to Equation (1) extending that of Kutoyants [31]. Marie [35] deals with the same estimator but for reflected fractional SDE. For nonparametric kernelbased estimators in Itô’s calculus framework, the reader is referred to Kutoyants [31] and [32].
The present paper deals with nonparametric projection estimators of , computed from independent continuous time observations of the solution of Equation (1) on . Let us mention that it became usual that such functional data are available and can be processed thanks to the improvements of computers. The question of nonparametric drift estimation in stochastic differential equations from such data has been studied in Comte and GenonCatalot [12] who consider an Itô’s calculus framework. Here, we propose to extend their functional least squares strategy to fractional SDE in Malliavin’s calculus framework. Almost all the references cited above on the statistical inference for fractional SDE are based on longtime behavior properties of the solution which are often difficult to check in practice, but not required here.
The estimator studied in this paper is introduced in Sections 3.1 and 3.2. Section 3.3 presents the main risk bound results of the paper and Sections 3.4 and 3.5 provide examples of function bases well adapted in our situation. We can in these frameworks obtain convergence results and rates.
Before that, Section 2 deals with some preliminaries on stochastic integration with respect to the fractional Brownian motion. More precisely, the Skorokhod integral with respect to the solution of Equation (1) is required for the definition of the projection estimators studied in Sections 3.1 and 3.2. However, it is difficult in practice to compute Skorokhod’s integral when . For this reason, Section 4 proposes an approximated and calculable estimator requiring an observed path of the solution of Equation (1) for two close but different values of the initial condition. Clearly, such a requirement is not possible in any context, but we have in mind the pharmacokinetics application field and explain why it is meaningful in this context.
Lastly, concluding remarks are gathered in Section 5 while most proofs are postponed in Section 6.
Notations.
The vector space of Lipschitz continuous maps from
into itself is denoted by and equipped with the usual Lipschitz seminorm . Now, consider . The Euclidean norm on is denoted by ,and
Note that . Finally, for every , the vector space of infinitely continuously differentiable maps such that and all its partial derivatives have polynomial growth is denoted by .
2. Stochastic integrals with respect to the fractional Brownian motion
This section presents two different methods to define a stochastic integral with respect to the fractional Brownian motion. The first one is based on the pathwise properties of the fractional Brownian motion. Another stochastic integral with respect to the fractional Brownian motion is defined via the Malliavin divergence operator. This stochastic integral is called Skorokhod’s integral with respect to . If , which means that is a Brownian motion, the Skorokhod integral defined via the divergence operator coincides with Itô’s integral on its domain. This integral is appropriate for the estimation of the drift function in Equation (1), while the first one is used in section 4 to propose a calculable estimator.
2.1. The pathwise stochastic integral
This subsection deals with some definitions and basic properties of the pathwise stochastic integral with respect to the fractional Brownian motion of Hurst index greater than .
Definition 2.1.
Consider and two continuous functions from into . Consider a dissection of with and such that . The Riemann sum of with respect to on for the dissection is
Notation. With the notations of Definition 2.1, the mesh of the dissection is
The following theorem ensures the existence and the uniqueness of Young’s integral (see Friz and Victoir [25], Theorem 6.8).
Theorem 2.2.
Let (resp. ) be a Hölder (resp. Hölder) continuous map from into with such that . There exists a unique continuous map such that for every satisfying and any sequence of dissections of such that as ,
The map is the Young integral of with respect to and is denoted by
for every such that .
2.2. The Skorokhod integral
This subsection deals with some definitions and results on Malliavin calculus.
Consider the space of measurable functions from into , and the reproducing kernel Hilbert space
of , where is the inner product defined by
for every . Let be the isonormal Gaussian process defined by
which is the Wiener integral of with respect to .
Definition 2.3.
The key property of the operator is the following.
Proposition 2.4.
The map is closable from into . Its domain in , denoted by , is the closure of the smooth functionals space for the seminorm defined by
for every .
For a proof, see Nualart [41], Proposition 1.2.1.
Definition 2.5.
The adjoint of the Malliavin derivative is the divergence operator. The domain of is denoted by , and if and only if there exists a deterministic constant such that for every ,
For any process and every , if , then its Skorokhod integral with respect to is defined on by
and its Skorokhod integral with respect to is defined by
Note that since is the adjoint of the Malliavin derivative , the Skorokhod integral of with respect to on is a centered random variable. Indeed,
(2) 
Let be the space of the smooth functionals presented in Definition 2.3 and consider , the closure of
for the seminorm defined by
for every . The following proposition provides an isometry type property for the Skorokhod integral with respect to on , which is a subspace of by Nualart [41], Proposition 1.3.1. This result is useful for our purpose and is proved in Biagini et al. [3] (see Theorem 3.11.1).
Proposition 2.6.
For every ,
In the sequel, the function fulfills the following assumption.
Assumption 2.7.
The function belongs to and there exist such that
Under Assumption 2.7, the following result is a straightforward application of Proposition 2.6 to functionals of the solution of Equation (1).
The following theorem provides suitable controls of the moments of Skorokhod’s integral.
Theorem 2.9.
Under Assumption 2.7, for every , there exists a deterministic constant , only depending on , and , such that for every ,
where and
Note that if , then Theorem 2.9 has been already proved in Hu, Nualart and Zhou [30] (see Proposition 4.4.(2)).
Remark 2.10.
On the one hand, note that the control of the variance of Skorokhod’s integral provided in Theorem
2.9 is a straightforward consequence of Corollary 2.8. On the other hand, with the notations of Corollary 2.8, note that for , the solution of Equation (1) is adapted and thenThis reduces importantly the order of the variance of Skorokhod’s integral with respect to the case .
Lastly, the following proposition provides an expression and a bound for the density of the solution to Equation (1).
Proposition 2.11.
Under Assumption 2.7, for any
, the probability distribution of
has a smooth density with respect to Lebesgue’s measure such that for every ,where
and is the OrnsteinUhlenbeck operator. Moreover, for every ,
where
A straightforward consequence of Proposition 2.11 is that for any , the probability distribution of has a smooth density with respect to Lebesgue’s measure such that for every ,
and
(3) 
where
Since the paths of are Hölder continuous for any ,
where
which has a finite first order moment because and is Lipschitz continuous. Then, since ,
with
Therefore, by taking , Inequality (3) implies that for every , .
3. Projection estimators of the drift function
Under Assumption 2.7, is Lipschitz continuous on and its derivative is bounded. So, Equation (1) has a unique solution and the associated Itô map is continuously differentiable from into .
3.1. The objective function
Let be the density function defined by
where is the smooth density with respect to Lebesgue’s measure of the probability distribution of introduced in Proposition 2.11 for any . Consider also independent copies of , for every , and the objective function defined by
for every function .
Note that for any bounded function from into itself, thanks to Equality (2),
Then, the definition of gives
(4) 
Equality (4) shows that is the smallest for the nearest of . Therefore, minimizing its empirical version should provide a functional estimator near of .
Remark 3.1.
The pathwise stochastic integral with respect to , defined in Section 2.1, is not centered in general. For instance, if , then it coincides with Stratonovich’s integral. So this is not even the case for . This is the main reason why the objective function above is defined via Skorokhod’s integral.
3.2. Projection estimators
Consider and assume that (resp. ) is equipped with its usual inner product (resp. ). For any , consider also
where is an orthonormal family of . Moreover, assume that the functions , are bounded. So, .
Let
be the projection estimator of on . As in Comte and GenonCatalot [12], section 2.2,
where
with
and
Note that
and
where
for every bounded and measurable functions , and
By Equality (2), is centered, as expected for an error term in regression.
3.3. Risk of the projection estimators
Throughout this subsection, and the functions , fulfill the following assumption.
Assumption 3.2.
and, for ,

is an orthonormal family of .

The functions , are bounded and belong to .

There exist such that
By Comte and GenonCatalot [12], Lemma 1, which remains true for without additional arguments,
is invertible under Assumption 3.2. In addition, we impose that
fulfill the following assumption.
Assumption 3.3.
There exists and such that and
The above condition is a generalization of the socalled stability condition introduced for standard regression by Cohen et al. [9, 10], also considered in Comte and GenonCatalot [12]. In order to ensure the existence and the stability of the estimator, is replaced by
where
The two following results provide controls of the empirical risk and of the weighted integrated risk of respectively.
Theorem 3.4.
Corollary 3.5.
Remark 3.6.
Note that
The risk decompositions given in Theorem 3.4 and Corollary 3.5 both involve the same types of terms:

The first one is equal or proportional to and is a squared bias term due to the projection strategy. It is decreasing when increases, because then the projection space grows.

The second one, , is a variance term. From the remark above, it is bounded by which is increasing with .

The last one is a residual negligible term, which is small when is large. Note that if the upperbound of is nonnegative, then explodes for large values of .
The order of the bias generally depends on the regularity of the function, and the order of the trace term is discussed below. Both quantities imply that a choice of ensuring a compromise between the bias and the variance is required, to obtain the consistency of the estimator and a rate.
Finally, let us provide a control for which allows comparison with non fractional results.
In the standard case, with and constant volatility function , it holds that
as established in Comte and GenonCatalot [12]. Here, for , becomes which is coherent. However, the additional term may have an important order in and substantially increase the variance. Thus, it will deteriorate the rate of the estimators. So, there is a discontinuity between the cases and , which is explained in Remark 2.10.
Now, for projection estimators, different bases can be considered. In the present setting, the bases have to be differentiable. We present two examples in the sequel.
3.4. Rates on FourierSobolev spaces for trigonometric basis
A first example is the compactly supported trigonometric basis. For , it is defined by
for every and . This basis satisfies, for odd and any ,
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