Since the 1980’s, the statistical inference for stochastic differential equations (SDE) driven by a Brownian motion has been widely investigated by many authors in the parametric and in the nonparametric frameworks. Classically (see Kutoyants ), the estimators of the drift function are computed from one path of the solution to the SDE and converge when the time horizon goes to infinity. The existence and the uniqueness of the stationary solution to the SDE are then required, and obtained thanks to restrictive conditions on the drift function.
Since few years, a new type of parametric and nonparametric estimators is investigated ; those computed from multiple independent observations on of the SDE solution. Indeed, this functional data analysis problem is already studied in the parametric framework (see Ditlevsen and De Gaetano , Overgaard et al. , Picchini, De Gaetano and Ditlevsen , Picchini and Ditlevsen , Comte, Genon-Catalot and Samson , Delattre and Lavielle , Delattre, Genon-Catalot and Samson , Dion and Genon-Catalot , Delattre, Genon-Catalot and Larédo , etc.) and more recently in the nonparametric framework (see Comte and Genon-Catalot [4, 5], Della Maestra and Hoffmann , and Marie and Rosier ). In [4, 5], the authors extend to the diffusion processes framework the projection least squares estimators already well studied in the regression framework (see Cohen et al.  and Comte and Genon-Catalot ). Our paper deals with a nonparametric estimation problem close to this last one.
Consider the stochastic process , defined by
where is a centered, continuous and square integrable martingale vanishing at , and is an unknown function which belongs to . By assuming that the quadratic variation of is deterministic for every , our paper deals with the estimator of minimizing the objective function
on a -dimensional function space , where (resp. ) are independent copies of (resp. ) and . Precisely, risk bounds are established on and on the adaptive estimator , where
Now, consider the differential equation
where , is a fractional Brownian motion of Hurst parameter , the stochastic integral with respect to is taken pathwise (in Young’s sense), and , and are regular enough. An appropriate transformation (see Subsection 4.1) allows to rewrite Equation (2) as a model of type (1) driven by the Molchan martingale which quadratic variation is for every . Our paper also deals with a risk bound on an estimator of derived from .
Finally, let us consider a financial market model in which the prices of the risky asset are modeled by the following equation of type (2):
where and . This is a non-autonomous extension of the fractional Black-Scholes model defined in Hu et al. . An estimator of is derived from at Subsection 4.3.
Up to our knowledge, only Comte and Marie  deals with a nonparametric estimator of the drift function computed from multiple independent observations on of the solution to a fractional SDE.
At Section 2, a detailed definition of the projection least square estimator of is provided. Section 3 deals with risk bounds on and on the adaptive estimator . At Section 4, the results of Section 3 on the estimator of are applied to the estimation of in Equation (2) and then of in Equation (3). Finally, at Section 5, some numerical experiments on Model (1) are provided when is the Molchan martingale.
2. A projection least square estimator of the map
In the sequel, the quadratic variation of fulfills the following assumption.
The (nonnegative, increasing and continuous) process is a deterministic function.
For some results, fulfills the following stronger assumption.
There exists such that is continuous from into , and such that
2.1. The objective function
In order to define a least square projection estimator of , let us consider independent copies (resp. ) of (resp. ), and the objective function defined by
for every , where , and are continuous functions from into such that is an orthonormal family in .
Remark. Note that since is nonnegative, increasing and continuous, and since the ’s are continuous from into , the objective function is well-defined.
For any ,
Then, the more is close to , the more is small. For this reason, the estimator of minimizing is studied in this paper.
2.2. The projection least square estimator
3. Risk bound and model selection
In the sequel, the space is equipped with the scalar product defined by
for every . The associated norm is denoted by .
First, the following proposition provides a risk bound on for a fixed .
Under Assumption 2.1,
Note that Inequality (5
) says first that the bound on the variance of our least square estimator ofis of order , as in the usual nonparametric regression framework. Under Assumption 2.2, the following corollary provides a more understandable expression of the bound on the bias in Inequality (5).
Under Assumption 2.2,
For instance, assume that , where
for every and satisfying . The basis of , orthonormal in , is obtained from
via the Gram-Schmidt process. Consider also the Sobolev space
and assume that there exists such that for every . Then, by DeVore and Lorentz , Theorem 2.3 p. 205, there exists a constant , not depending on , such that
Therefore, by Corollary 3.2,
Finally, consider , and
where is a constant to calibrate in practice via, for instance, the slope heuristic
slope heuristic. In the sequel, the ’s fulfill the following assumption.
For every , if , then .
The following theorem provides a risk bound on the adaptive estimator .
As in the usual nonparametric regression framework, since is of same order than the bound on the variance term of for every , Theorem 3.4 says that the risk of our adaptive estimator is controlled by the minimal risk of on up to a multiplicative constant not depending on .
4. Application to differential equations driven by the fractional Brownian motion
Throughout this section, is twice continuously differentiable with bounded derivatives, is -Hölder continuous with , and is continuous. Under these conditions on , and , Equation (2) has a unique solution which paths are -Hölder continuous from into for every (see Kubilius et al. , Theorem 1.42). The maps and are known and our purpose is to provide a nonparametric estimator of .
4.1. Auxiliary model
The model transformation used in the sequel has been introduced in Kleptsyna and Le Breton  in the parametric estimation framework. Consider the function space
let be the map defined by
and assume that . Consider also the Molchan martingale defined by
and the process defined by
for every . Then, Equation (2) leads to
for every and almost every . Note that the Molchan martingale fulfills Assumption 2.2 with for every .
4.2. An estimator of
Let us consider the function space
where is the Riemann-Liouville left-sided fractional integral of order . The reader can refer to Samko et al.  on fractional calculus.
In order to provide an estimator of with a closed-form expression, let us establish first the following technical proposition.
The map is one-to-one from into . Moreover, for every and almost every ,
By Proposition 4.1, if , then
). A simple vector subspace ofis provided at the end of this subsection.
The following proposition provides risk bounds on , , and on the adaptive estimator .
If the ’s belong to , then there exists a deterministic constant , not depending on , such that
If in addition the ’s fulfill Assumption 3.3, then there exists a deterministic constant , not depending on , such that
Proposition 4.2 says that the MISE of , , (resp. ) has at most the same bound than the MISE of (resp. ).
Finally, the following proposition provides a simple vector subspace of .
The function space
is a subset of .
Consider and such that is an orthonormal family of . In particular, note that are linearly independent. Moreover, assume that with for every and . The basis of , orthonormal in , is obtained from via the Gram-Schmidt process, and the ’s belong to . For every , there exist such that
where for every , and
So, by assuming that , the bound on the bias term of in the inequalities of Proposition 4.2 can be controlled the following way:
where, for every ,
If is the -dimensional trigonometric basis, and if there exists such that for every , then
4.3. Example: drift estimation in a non-autonomous fractional Black-Scholes model
Let us consider a financial market model in which the prices process of the risky asset is defined by
where and . This is a non-autonomous extension of the fractional Black-Scholes model defined in Hu et al. . Thanks to the change of variable formula for Young’s integral,
Then, is the solution to Equation (2) with , and
Consider independent copies of . For every and , consider also
If the volatility constant is known, thanks to Subsection 4.2, a nonparametric estimator of is given by
if the ’s belong to , then Proposition 4.2 provides risk bounds on , , and on the adaptive estimator .
5. Numerical experiments
Some numerical experiments on our estimation method of in Equation (1) are presented in this subsection when is the Molchan martingale:
with and the Brownian motion driving the Mandelbrot-Van Ness representation of the fractional Brownian motion . The estimation method investigated on the theoretical side at Section 3 is implemented here for the three following examples of functions :
These functions belong to as required. Indeed, one the one hand, is continuous on and
On the other hand, for every such that ,
Since for every , with , and since in our numerical experiments, is square-integrable with respect to