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First published on Friday, Jul 18, 2025 and last modified on Friday, Jul 18, 2025 by François Chaplais.
Universitat Pompeu Fabra, Barcelona, Spain Email
Institut de Mathématiques de Toulouse, Université Paul Sabatier, Toulouse, France Email
Laboratoire Modal’X, Université Paris Nanterre, Nanterre, France Email
Abstract
We investigate the problem of estimating the drift parameter from \( N\) independent copies of the solution of a stochastic differential equation driven by a multiplicative fractional Brownian noise with Hurst parameter \( H\in (1/3,1)\) . Building on a least-squares-type object involving the Skorokhod integral, a key challenge consists in approximating this unobservable quantity with a computable fixed-point estimator, which requires addressing the correction induced by replacing the Skorokhod integral with its pathwise counterpart. To this end, a crucial technical contribution of this work is the reformulation of the Malliavin derivative of the process in a way that does not depend explicitly on the driving noise, enabling control of the approximation error in the multiplicative setting. For the case \( H\in (1/3,1/2]\) , we further exploit results on two-dimensional Young integrals to manage the more intricate correction term that appears. As a result, we establish the well-posedness of a fixed-point estimator for any \( H\in (1/3,1)\) , together with both an asymptotic confidence interval and a non-asymptotic risk bound. Finally, a numerical study illustrates the good practical performance of the proposed estimator.
1 Introduction
In this work, we consider the stochastic differential equation (SDE)
(1)
where \( T > 0\) is fixed, \( x_0\in\mathbb R\) , and \( B\) denotes a fractional Brownian motion (fBm) with Hurst parameter \( H\in (1/3,1)\) . The map \( (b,\sigma)\) belongs to \( C^{[1/H] + 1}(\mathbb R;\mathbb R^2)\) and has bounded derivatives. The integral with respect to \( B\) is understood pathwise, in the sense of Young for \( H\in (1/2,1)\) , and in the sense of rough paths for \( H\in (1/3,1/2]\) . Under these conditions, Equation (1) admits a unique solution \( X\) .
Our objective is to estimate the drift parameter \( \theta_0\) based on continuous observations of \( N\) i.i.d. copies of the equation above, in the asymptotic regime where \( T\) is fixed and the number of observed SDEs \( N\) tends to infinity.
Stochastic differential equations are widely employed to model real-world phenomena thanks to their versatility. Classical applications are found in finance [1], biology [2], neurosciences [3], and economics [4], as well as in physics [5] and mechanics [6]. In pharmacology, SDEs are commonly used to model variability in biomedical experiments, as reviewed in [7, 8, 9, 10, 11].
Traditionally, inference procedures were designed for a single SDE, either over a fixed time interval or in the limit as \( T\rightarrow\infty\) . However, with the advent of big data, statisticians increasingly have access to large amounts of data generated by different individuals. This motivates an asymptotic regime where one observes many independent copies over a finite time horizon, each copy corresponding to a different individual. Such a framework has found applications in biology and the social sciences [12, 13, 14, 15], in data-science (notably for neural networks) [16, 17, 18, 19], in optimization [20, 21, 22], and in MCMC methods [23]. For two detailed motivating examples of i.i.d. SDEs driven by standard Brownian motion, we refer to Section 2 of [24].
In recent years, statistical inference from copies of diffusion processes driven by classical Brownian motion - and in particular, estimation of the drift function - has been extensively investigated. See for instance [25, 26, 27] for the nonparametric setting, and [28] for the parametric one. The literature is even broader when dependence among copies is allowed, giving rise to observations from diffusions with correlated driving signals (see [29]) or interacting particle systems. In this last setting, drift parameter estimation has been thoroughly studied, see [30, 31, 32, 33] for nonparametric approaches and [34, 35] for parametric ones.
In the case of independent copies, extensions to privacy-constrained estimation procedures have been considered in [36], while replacing Brownian motion with L
specialChar{39}evy processes is studied in [37]. All of this confirms that statistical inference from i.i.d. copies of SDEs is a very active area of research of broad interest to the statistics community.
In the classical SDE framework, replacing standard Brownian motion with fractional Brownian motion has proven highly effective for modeling certain real-world phenomena. Indeed, fractional Brownian motion has emerged as a powerful tool for capturing processes driven by long-range dependence and self-similar noise. It has found applications across diverse fields, including finance [38, 39, 40, 41, 42], geophysics [43, 44], traffic modeling [45, 46, 47], and medical research [48]. These developments have naturally stimulated interest in statistical inference for models driven by fBm.
For example, systems of \( N\) (dependent) SDEs have recently been employed to model turbulent kinetic energy [49]. However, empirical evidence suggests that non-Markovian driving noise may offer a more accurate description of such phenomena [50, 51], thus motivating the replacement of standard Brownian motion with fractional Brownian motion. Moreover, [52] studies the growth of \( N\) axon cells in vertebrate brains using the fractional modeling approach, highlighting the relevance and interest of this topic within the scientific community.
It is important to note that statistical inference procedures must be adapted when replacing standard Brownian motion with fractional Brownian motion. In the classical Brownian case, estimation techniques rely crucially on the Markov and semimartingale properties of the process - properties that fractional Brownian motion lacks.
In the context of drift parameter estimation for SDEs driven by fractional Brownian motion, the earliest methods were developed for one long-term observation of the stationary ergodic solution to Equation (1), whose existence and uniqueness are guaranteed by a dissipativity condition on the drift function. In this framework, the most common estimation approaches in the literature are the maximum likelihood estimator (MLE) and the least squares estimator (LSE). Notably, for Brownian motion, these two estimators coincide, while for fractional Brownian motion this equivalence no longer holds (see [53, 54, 55, 56]).
The literature is broadly divided between these two estimation strategies. For instance, MLE-based approaches can be found in [57], [55] and [56], while the LSE has been used in works such as [53], [58] and [54].
More recently, estimators computed from multiple copies of \( X\) driven by fBm, observed over short time intervals, have also been investigated. For example, [59] analyzes a maximum likelihood estimator, although without theoretical guarantees, while [60] develops a procedure based on the LSE, proposing a fixed-point approach to estimate \( \theta_0\) when \( H\in (1/2,1)\) and with constant \( \sigma\) . Then, [61] further extends this methodology to interacting particle systems driven by additive fractional noise.
In this work, we follow this second line of research, choosing to base our estimation procedure on an unobservable quantity, \( \widehat\theta_N\) , inspired by a least-squares-type estimator. However, this quantity cannot be considered as a genuine estimator, because it depends on the data through a Skorokhod integral, rendering it unobservable. Consequently, borrowing the playful terminology introduced in [61], we will refer to this quantity as a fakestimator in the sequel. It can be expressed as
where \( N\in\mathbb N^*\) , \( X^i =\mathcal I(x_0,B^i)\) for every \( i\in\{1,\dots,N\}\) , \( B^1,\dots,B^N\) are independent copies of \( B\) , \( \mathcal I\) denotes the solution map for Equation (1), and the stochastic integral is taken in the sense of Skorokhod.
Moving from this fakestimator to a computable estimator involves replacing the Skorokhod integral with a corrected pathwise one. In the case \( H > 1/2\) , one can exploit the classical relationship between Skorokhod and pathwise integrals, where the Malliavin derivative appears in the correction term. However, since we consider the multiplicative setting, the Malliavin derivative depends on the fBm explicitly in a complicated way, making it very challenging to manage, and consequently to control the error arising when transitioning from the fakestimator to a fixed-point estimator. To overcome this difficulty, our strategy relies on a refined argument that allows us to reformulate the Malliavin derivative of the process so that it can be expressed in a form that does not explicitly depend on \( B\) (see Proposition 1 for details).
Moreover, we also study the case \( H\in (1/3,1/2]\) , for which this change of integral is even more challenging. In this regime, one can first employ an elegant result from [62] to move from the Skorokhod integral to the pathwise one, obtaining a correction term involving the Malliavin derivative trough a 2D Young integral. However, this correction term is difficult to evaluate. Crucially, to obtain the main results for \( H\in (1/3,1/2]\) , we rely on our Proposition 2, which allows us to handle the rectangular integrals and recover an expression similar to the case \( H > 1/2\) . We believe this result is not only essential for our analysis, but is also of independent interest as it could be applied in other contexts of statistical inference for fBm-driven processes with \( H\in (1/3,1/2]\) .
Thanks to this result, we can move from the Skorokhod integral to the pathwise integral with a computable correction term depending on \( \theta_0\) , the parameter we wish to estimate. This justifies introducing a fixed-point estimator, denoted by \( \overline\theta_N\) .
Let us remark that the case \( H\in (1/3,1/2]\) is particularly challenging, as confirmed by the scarcity of literature in this context. The same applies to statistical inference with multiplicative fractional noise. Indeed, among all the previously cited works on statistical inference for fractional-noise-driven systems, only [54] addresses the multiplicative case, in the asymptotic regime with \( N = 1\) and \( T\rightarrow\infty\) .
Note that, by the change-of-variable formula for the pathwise integral,
where \( Y =\Sigma\circ X\) and \( \Sigma\) is a primitive function of \( 1/\sigma\) . Of course, for \( H \in (1/2,1)\) , one could apply the results of [60] to the fixed-point estimator computed from the transformed data \( \Sigma\circ X^1,\dots,\Sigma\circ X^N\) . However, although \( \sigma\) is assumed to be known, there is no reason that \( \Sigma\) can be calculated explicitly, which is unsatisfactory from a statistical perspective. For this reason, even for \( H \in (1/2,1)\) , the theoretical results in the present paper are established directly on \( \overline\theta_N\) .
Moreover, note that for an additive fractional noise with Hurst parameter \( H \in (1/3,1/2]\) , the fixed-point estimation strategy was already introduced in [60], but without any theoretical guarantees. In contrast, thanks to the probabilistic tools developed in this work, we are able to show that, for any \( H \in (1/3, 1]\) , the fixed-point estimator is well-defined, and we can establish both an asymptotic confidence interval and a non-asymptotic risk bound for \( \overline\theta_N\) . We emphasize that this constitutes a major step forward in the statistical inference for processes driven by fractional Brownian motion, as it simultaneously addresses (and solves!) two major challenges: the multiplicative noise setting and the regime \( H < 1/2\) .
The results provided in this paper not only offer valuable insights for tackling related challenges - for instance, nonparametric estimation of the drift or parametric inference under high-frequency observations in the presence of multiplicative noise and/or with \( H < 1/2\) - but also validate their practical relevance. Indeed, we conclude our analysis with a numerical study demonstrating that our theoretical fixed-point estimator also performs well in practice.
The outline of the paper is as follows. Section 2 introduces the probabilistic tools mentioned above. All theoretical guarantees for our fixed-point estimator are established in Section 3, with Section 3.1 devoted to the case \( H \in (1/2,1)\) , and Section 3.2 addressing the case \( H \in (1/3,1/2]\) . Section 4 presents some basic numerical experiments, while Section 5 collects the proofs of our main results.
2 Probabilistic preliminaries
As mentioned in the introduction, our estimation procedure for \( \theta_0\) relies on tools from Malliavin calculus as well as Young/rough differential equations theory. For background on these topics, we refer the reader to [63] and [64] respectively.
This section builds on these classical results to develop several probabilistic statements required for our statistical analysis in Section 3. In particular, as already emphasized, a crucial step for our methodology is to reformulate the Malliavin derivative of \( X_t\) (\( t\in (0,T]\) ) in a form that does not explicitly depend on \( B\) , and to handle 2D Young integrals arising in the analysis.
In the sequel, we adopt the following standard non-degeneracy assumption on \( \sigma\) :
Assumption 1
The function \( \sigma\) is bounded and \( \inf_{\mathbb R}|\sigma| > 0\) .
We are now ready to state the following proposition, which provides an expression for the Malliavin derivative of \( X_t\) that is independent of \( B\) . This result will play a central role in establishing our theoretical guarantees on \( \overline\theta_N\) , and applies both for \( H\in (1/3,1/2]\) and \( H > 1/2\) .
Proposition 1
Under Assumption 1, for every \( s,t\in [0,T]\) ,
This expression for the Malliavin derivative is crucial in transitioning from the fakestimator to the fixed-point estimator.
We begin by considering the case \( H \in (1/2,1)\) , where the relationship between the Skorokhod and pathwise integrals is well established (see Proposition 5.2.3 in [63]). Let us define
(2)
Then, combining this relationship with Proposition 1 yields
(3)
where \( \alpha_H = H(2H - 1)\) .
The situation is more challenging in the case \( H\in (1/3,1/2]\) . Here, it is again necessary to establish a link between the Skorokhod and pathwise integrals. In this context, Theorem 3.1 in [62] is instrumental, as it provides an explicit relationship between the two, involving a correction term expressed through a two-dimensional integral.
In order to handle this correction term and to establish statistical guarantees for the fixed-point estimator, the following proposition will play a crucial role.
Proposition 2
Consider
Assume that \( H\in (1/3,1/2]\) , and let \( x : [0,T]^2\rightarrow\mathbb R\) be a function such that
(4)
and
(5)
where \( \alpha\in (1 - 2H,H)\) and \( \mathfrak c_x\) is a positive constant. Then, the 2D Young integral of \( x\) with respect to the covariance function \( R\) of the fBm is well-defined, and
We have stated Proposition 2 here because we believe it deserves independent attention, as it may be applied in several contexts beyond our specific drift parameter estimation problem. In the following proposition, we instead detail its application within our framework. In particular, it provides an explicit expression for the Skorokhod integral of \( b\circ X\) with respect to \( X\) in the case \( H\in (1/3,1/2]\) .
Proposition 3
Assume that \( H\in (1/3,1/2]\) . Under Assumption 1, if \( b\) is bounded, then
(6)
and
(7)
Remark 1
Let us add a few remarks on Proposition 3.
- Since \( \sigma\) , \( 1/\sigma\) , and the derivatives of \( b\) and \( \sigma\) are already assumed to be bounded, if \( b\) is itself bounded, then so are \( \varphi\) and \( \psi\) . This additional boundedness condition on \( b\) , which is not needed when \( H \in (1/2,1)\) , is the price to pay in order to apply Theorem 3.1 of Song and Tindel as well as Proposition 2 in the proof of Proposition 3 (see Section 5).
It is worth noting that, after application of Proposition 2, the relationships between Skorokhod and pathwise integrals in the two cases \( H > 1/2\) and \( H\in (1/3,1/2]\) , summarized in Equations (3) and (7) respectively, appear quite similar. However, there is a crucial difference: in Equation (7), the right-hand side cannot be simplified in such a way that the second integral cancels the \( -1\) appearing in the third term, as happens in Equation (3). This is because the map
\[ (s,t)\in [0,T]^2\longmapsto \varphi(X_t)\exp\left(\theta_0\int_{s}^{t}\psi(X_u)du\right)\mathbf 1_{[0,t)}(s) \]does not satisfy the assumptions of Proposition 2.
3 The fixed-point estimator of the drift parameter
By the (standard) law of large numbers, and observing that the Skorokhod integral is centered, we obtain
so that \( \widehat\theta_N\) is a consistent estimator of \( \theta_0\) . However, as already mentioned, \( \widehat\theta_N\) cannot be computed directly from the data. Thanks to the relationships provided in Equations (3) and (7), we can replace the Skorokhod integral appearing in the fakestimator by the corrected pathwise integral, thereby moving to a computable approximation of \( \widehat\theta_N\) .
The goal of this section is to provide theoretical guarantees for such a computable approximation of \( \widehat\theta_N\) , defined as the fixed-point of a suitably chosen random functional. Due to Remark 1.(2), the cases \( H\in (1/2,1)\) and \( H\in (1/3,1/2]\) need to be treated separately.
In the sequel, assume that \( X^1,\dots,X^N\) have been observed on \( [0,T_0]\) (\( T_0 > 0\) ), and then on \( [0,T]\) for any \( T\in (0,T_0)\) . Assume also that for every \( t\in (0,T]\) , the probability distribution of \( X_t\) has a density \( f_t\) with respect to the Lebesgue measure on \( (\mathbb R,\mathcal B(\mathbb R))\) such that \( s\mapsto f_s(x)\) (\( x\in\mathbb R\) ) belongs to \( \mathbb L^1([0,T],dt)\) . Under Assumption 1, this condition on the distribution of \( X_t\) (\( t\in (0,T]\) ) is satisfied when \( \sigma\) is constant by [65], Theorem 1.3, and when \( \sigma\) is not constant but \( b\) is bounded by [66], Theorem 1.5. So, one can consider the density function \( f\) defined by
Notation. The usual norm on \( \mathbb L^2(\mathbb R,f(x)dx)\) is denoted by \( \|.\|_f\) .
3.1 The case \( H\in (1/2,1)\)
First, let us define the following quantities:
and
(8)
where the last equality follows from the change-of-variable formula for the Young integral. By Equality (3), we can rewrite \( \widehat\theta_N - I_N\) in the following way:
(9)
Consider the random functional
From Equality (9), and since \( \widehat\theta_N\) is a consistent estimator of \( \theta_0\) ,
which motivates introducing the estimator \( \overline\theta_N = I_N + R_N\) , where \( R_N\) is (when it exists and is unique) a fixed-point of the map \( \Theta_N\) .
Proposition 4
Assume that \( b'\) , \( \varphi\) and \( \psi\) are nonpositive. Under Assumption 1, if
(10)
where \( \mathfrak c\) is a deterministic constant arbitrarily chosen in \( (0,1)\) ,
then \( \Theta_N\) is a contraction from \( \mathbb R_+\) into itself. Therefore, \( R_N\) exists and is unique.
Remark 2
Note that, in particular, the conditions of Proposition 4 on \( (\varphi,\psi)\) imply that it is bounded. Indeed,
leading to
Since \( b'\) and \( \sigma\) are bounded, the ratio \( \sigma'b/\sigma\) is also bounded, and therefore so are \( (\varphi,\psi)\) .
Example 1
Let us provide examples of drift and volatility functions satisfying the condition \( {\rm (A)}\) :
- If \( \sigma\) is constant, then \( (\varphi,\psi) = (\sigma^2b',b')\) satisfies \( {\rm (A)}\) if and only if \( b'\leqslant 0\) (as in [60], Proposition 7).
Assume that \( b(x) = -x\) , which is quite common. Then,
\[ ({\rm A})\Longleftrightarrow -1\leqslant\frac{\sigma'(x)}{\sigma(x)}x\leqslant 1. \]For instance,
If \( \sigma(x) =\pi +\arctan(x)\) , then \( \sigma'(x) = (1 + x^2)^{-1}\) , leading to
\[ \frac{\sigma'(x)}{\sigma(x)}x = \frac{x}{(1 + x^2)(\pi +\arctan(x))}\in [-1,1]. \]If \( \sigma(x) = 1 + e^{-x^2}\) , then \( \sigma'(x) = -2xe^{-x^2}\) , leading to
\[ \frac{\sigma'(x)}{\sigma(x)}x =\frac{-2x^2e^{-x^2}}{1 + e^{-x^2}}\in [-1,1]. \]
The previous examples illustrate that, although the conditions imposed on the coefficients for our analysis may appear rather restrictive, there is nonetheless a broad class of drift and volatility functions satisfying them.
Let us define \( \overline\theta_{N}^{\mathfrak c} =\overline\theta_N\mathbf 1_{\Delta_N}\) , where
(11)
Thanks to Proposition 4, we know that the fixed-point \( R_N\) of \( \Theta_N\) exists and is unique on \( \Delta_N\) . To ensure that \( \overline\theta_{N}^{\mathfrak c}\) inherits the desirable properties of \( \widehat\theta_N\) , it is necessary to control the probability of the complement event \( \mathbb P(\Delta_N^c)\) .
Proposition 5
Assume that \( b'\) , \( \varphi\) and \( \psi\) are nonpositive, \( \theta_0 > 0\) and that
(12)
Under Assumption 1, there exists a constant \( \mathfrak c_{5} > 0\) , not depending on \( N\) , such that
Now, the following proposition provides an asymptotic confidence interval for the computable fixed-point estimator \( \overline\theta_{N}^{\mathfrak c}\) .
Proposition 6
Assume that \( b'\) , \( \varphi\) and \( \psi\) are nonpositive, and that \( \theta_0 > 0\) . Under Assumption 1, if \( T\) satisfies the condition (12) with \( \mathfrak c = 1/2\) , then
for every \( \alpha\in (0,1)\) , where \( u_{\cdot} =\phi^{-1}(\cdot)\) , \( \phi\) is the standard normal distribution function, and
Finally, the following proposition provides a non-asymptotic risk bound on the truncated estimator
(13)
Proposition 7
Assume that \( b'\) , \( \varphi\) and \( \psi\) are nonpositive, and that \( \theta_0 > 0\) . Under Assumption 1, if \( T\) satisfies the condition (12), then there exists a constant \( \mathfrak c_{7} > 0\) , not depending on \( N\) , such that
In this way, in the case \( H > 1/2\) , we have established both an asymptotic confidence interval and a non-asymptotic risk bound for the fixed-point estimator. Let us now turn to the case \( H\in (1/3,1/2]\) .
3.2 The case \( H\in (1/3,1/2]\)
First, as in the case \( H > 1/2\) , we start by defining the functional underlying our fixed-point estimator. Note that, by applying the change-of-variable formula for the rough integral (instead of the Young integral), the quantity \( I_N\) retains the same form (8) as in the case \( H > 1/2\) . However, the connection between the Skorokhod and pathwise integrals, previously expressed in Equality (3), is now replaced by the relation in Equality (7). This leads us to introduce a modified version of the random functional \( \Theta_N\) from the previous section:
Again, since \( \widehat\theta_N\) is a consistent estimator of \( \theta_0\) , we have
which justifies considering the estimator \( \overline\theta_N = I_N + R_N\) of \( \theta_0\) , where \( R_N\) is the fixed-point (when it exists and is unique) of the map \( \widetilde\Theta_N\) .
Proposition 8
Assume that \( b\) is bounded, and that \( b'\) , \( \varphi\) and \( \psi\) are nonpositive. Under Assumption 1, if \( M_N/D_N\) satisfies the condition (10), then \( \widetilde\Theta_N\) is a contraction from \( \mathbb R_+\) into itself. Therefore, \( R_N\) exists and is unique.
As in the case \( H > 1/2\) , let us consider \( \overline\theta_{N}^{\mathfrak c} =\overline\theta_N\mathbf 1_{\Delta_N}\) , where the event \( \Delta_N\) remains defined by Equality (11). Again, to transfer suitable properties from \( \widehat\theta_N\) to \( \overline\theta_{N}^{\mathfrak c}\) , it is necessary to control \( \mathbb P(\Delta_{N}^{c})\) . This control is already established in Proposition 5, and a brief review of its proof confirms that the result does not depend on \( H\) nor on the considered functional (\( \Theta_N\) or \( \widetilde\Theta_N\) ).
Now, we proceed to the following proposition providing an asymptotic confidence interval for the computable fixed-point estimator \( \overline\theta_{N}^{\mathfrak c}\) .
Proposition 9
Assume that \( b\) is bounded, \( b'\) , \( \varphi\) and \( \psi\) are nonpositive, and that \( \theta_0\leqslant\theta_{\max}\) with a known \( \theta_{\max} > 0\) . Under Assumption 1, if \( T\) satisfies the condition (12) with \( \mathfrak c = 1/2\) , then
where
Remark 3
Comparing the confidence interval in Proposition 6 with that of Proposition 9, one can notice a difference in the random variables appearing in the widths of the intervals: \( Y_N\) is now replaced by \( \mathfrak Y_N\) . This discrepancy arises because, in the case \( H > 1/2\) , when analyzing the variance of the Skorokhod integral, we could invoke Theorem 3.11.1 in [67], which allows us to propose a Lebesgue integral bounded by \( Y_N\) and converging in probability to the variance of the Skorokhod integral.
For \( H\in (1/3,1/2]\) , this theorem is no longer applicable. Instead, we handle the variance of the Skorokhod integral by directly using the relationship between the Skorokhod and pathwise integrals, as established through our probabilistic tools in Equality (7).
Finally, the following proposition provides a non-asymptotic risk bound on the truncated estimator \( \overline\theta_{N}^{\mathfrak c,\mathfrak d}\) defined as in (13).
Proposition 10
Assume that \( b\) is bounded, and that \( b'\) , \( \varphi\) and \( \psi\) are nonpositive. Under Assumption 1, if \( T\) satisfies the condition (12), then there exists a constant \( \mathfrak c_{10} > 0\) , not depending on \( N\) , such that
An experienced reader might have anticipated that the proofs of Propositions 7 and 10 rely on the risk bounds for the fakestimator, together with a control on the approximation error incurred when moving from the fakestimator to the fixed-point estimator. The first part does not depend on \( H\) , while the second uses the bound on \( \mathbb P(\Delta_{N}^{c})\) which, as discussed above, is also independent of \( H\) . It then follows directly that the proof of Proposition 10 proceeds along the same lines as that of Proposition 7, and is therefore omitted.
4 Numerical experiments
In this section, some numerical experiments on the computable approximation of the least squares estimator of \( \theta_0\) are presented for the three following models:
- (A) \( dX_t = -\theta_0X_tdt + dB_t\) with \( \theta_0 = 1\) ,
- (B) \( dX_t = -\theta_0X_tdt +\sigma_1(X_t)dB_t\) with \( \theta_0 = 1\) and \( \sigma_1(x) = 1 + e^{-x^2}\) ,
- (C) \( dX_t = -\theta_0X_tdt +\sigma_2(X_t)dB_t\) with \( \theta_0 = 1\) and \( \sigma_2(x) =\pi +\arctan(x)\) .
For each model, with \( H = 0.7\) and \( H = 0.9\) , \( \overline\theta_N\) is computed from \( N = 1,\dots,50\) paths of the process \( X\) . This experiment is repeated \( 100\) times. The means and the standard deviations of the error \( |\overline\theta_{50} -\theta_0|\) are stored in Table 1.
| \( H\) | Mean error | Error StD. | |
| Model (A) | 0.7 | 0.0489 | 0.0336 |
| 0.9 | 0.0186 | 0.0139 | |
| Modal (B) | 0.7 | 0.0510 | 0.0500 |
| 0.9 | 0.0294 | 0.0214 | |
| Model (C) | 0.7 | 0.1597 | 0.1443 |
| 0.9 | 0.0823 | 0.0922 | |
As expected, both the mean error and the standard deviation of the estimator \( \overline{\theta}_{50}\) are larger for the models driven by multiplicative noise (Models (B) and (C)) compared to the fractional Ornstein-Uhlenbeck process (Model (A)). However, the mean error of \( \overline\theta_{50}\) remains small for both Model (A) and Model (B): lower than \( 5.1\cdot 10^{-2}\) . Since - for these two models - the error standard deviation of \( \overline\theta_{50}\) is also small (\( < 5\cdot 10^{-2}\) ), on average, the error of \( \overline\theta_{50}\) for one repetition of the experiment should be near of its mean error. Finally, note also that for the three models ((A), (B) and (C)), the mean error of \( \overline\theta_{50}\) is higher when \( H = 0.7\) than when \( H = 0.9\) ; probably because \( H\) controls the H
specialChar{34}older exponent of the paths of the fractional Brownian motion.
In Figures 1, 4 and 7, for \( N = 1,\dots,50\) , \( \overline\theta_N\) and the bounds of the \( 95\%\) -asymptotic confidence interval (ACI) in Proposition 6 are plotted for one of the 100 datasets generated from Models (A), (B) and (C) respectively. These figures illustrate both that \( \overline\theta_N\) is consistent and its rate of convergence.
5 Proofs
This section is dedicated to the proofs of our main results. We begin by proving the statements presented in Section 2, which introduces the key probabilistic tools. Subsequently, we proceed to prove the main results concerning the fixed-point estimator, as formulated in Section 3.
5.1 Proof of Proposition 1
It is well-known (see, for instance, Theorem 2.2.1 in [63]) that for any \( s,t\in [0,T]\) with \( s < t\) ,
Then, by the change-of-variable formula for the Young integral (resp. rough integral) when \( H\in (1/2,1)\) (resp. \( H\in (1/3,1/2]\) ),
Moreover, once again by using the change-of-variable formula,
Therefore,
5.2 Proof of Proposition 2
First of all, let us define the rectangular increments of \( R\) :
Moreover, let \( \pi =\pi_n = (t_0,\dots,t_n)\) be a dissection of \( [0,T]\) such that
By [68], Section 6.4, the Riemann sum
converges to the 2D Young integral of \( x\mathbf 1_{\Delta_T}\) with respect to \( R\) when \( n\rightarrow\infty\) . So, it is sufficient to establish that
To that purpose, note that
where
The proof of Lemma 2 is dissected in three steps.
Step 1 (preliminaries). Let \( \partial_2R\) be the partial derivative of \( R\) with respect to its second variable. First, note that for every \( i,j\in\{1,\dots,n - 1\}\) such that \( i\neq j\) ,
(14)
and
(15)
Now, for every \( (s,t)\in\Delta_T\) ,
(16)
Finally, since \( z\mapsto z^{2H}\) is a \( 2H\) -H
specialChar{34}older continuous function from \( \mathbb R_+\) into itself,
(17)
Step 2 (control of \( I_{=}^{\pi}(x,R)\) ). By the condition (4),
Moreover, by Equality (15) and Inequality (17),
So, by the H
specialChar{34}older inequality with the conjugate exponents \( p = (1 +\varepsilon)/\alpha\) - where \( \varepsilon = \alpha + 2H - 1\) - and \( q = 1 + 1/(p - 1)\) ,
Therefore, since \( 2Hq - 1 =\alpha + 2H - 1 =\varepsilon\) and \( 1/p + 1/q = 1\) ,
Step 3 (control of \( I_{<}^{\pi}(x,R)\) ). By Equalities (14) and (16),
with
On the one hand, by the condition (4),
On the other hand, by the condition (5),
Therefore,
5.3 Proof of Proposition 3
In the sequel, the set of all the dissections of \( [0,T]\) is denoted by \( \mathfrak D_T\) . First of all, let us define the 1D and 2D \( p\) -variation norms (\( p\in [1,\infty)\) ). A continuous function \( x : [0,T]\rightarrow\mathbb R\) is of finite \( p\) -variation if and only if
and a continuous function \( \rho : [0,T]^2\rightarrow\mathbb R\) is of finite \( p\) -variation if and only if
The proof of Proposition 3 relies on Proposition 2 and on the following technical lemma.
Lemma 1
For every \( s,t\in [0,T]\) , consider
Under the assumptions of Proposition 3,
There exists a deterministic constant \( \mathfrak c_{1} > 0\) such that
\[ |L(s,t)|\leqslant \mathfrak c_{1}|t - s| \textrm{ \( ;\) }\forall (s,t)\in\Delta_T, \]and for every \( \alpha\in (0,H)\) ,
\[ |L(s,t) - L(u,v)|\leqslant \mathfrak c_{1}(1\vee\|X\|_{\alpha,T})(|t - v|^{\alpha} + |s - u|) \textrm{ \( ;\) }\forall (s,t),(u,v)\in\Delta_T, \]where \( \|X\|_{\alpha,T}\) is the \( \alpha\) -H
specialChar{34}older norm of \( X\) over \( [0,T]\) .- For every \( p > 1/H\) , the random variables \( \|\overline L(0,\cdot)\|_{p\textrm{-var},T}\) and \( \|\overline L\|_{p\textrm{-var},[0,T]^2}\) belong to \( \mathbb L^2(\Omega)\) .
The proof of Lemma 1 is postponed to Section 5.3.1. Note that Lemma 1.(1) allows to apply our Proposition 2 to the \( L(\omega)\) ’s, and that Lemma 1.(2) allows to apply [62], Theorem 3.1 to \( Y =\pi\circ X\) .
First, as established in [64], Section 7.3, the paths of \( Y\) are controlled by those of \( B\) . Precisely, for every \( (s,t)\in\Delta_T\) ,
where
is the Gubinelli derivative of \( Y\) at time \( s\) , and \( \mathfrak R\) is a stochastic process which paths are continuous and of finite \( p/2\) -variation for every \( p\in (1/H,3)\) . Moreover, recalling that \( \varphi\) and \( \psi\) are defined in (2), by Proposition 1, for every \( (s,t)\in\Delta_T\) ,
Let us introduce \( V(s) := R(s,s)\) (\( s\in [0,T]\) ). Thanks to Lemma 1.(2), by [62], Theorem 3.1,
Now, thanks to Lemma 1.(1), by Proposition 2,
Therefore,
leading to
5.3.1 Proof of Lemma 1
For every \( (s,t)\in\Delta_T\) , consider
First, for every \( (s,t)\in\Delta_T\) ,
\[ \begin{eqnarray*} |L(s,t)| & = & |\varphi(X_t)\lambda(s,t)|\\ & \leqslant & \mathfrak c_1|t - s| ~ {\rm with}~ \mathfrak c_1 = |\theta_0|\cdot \|\varphi\|_{\infty}\|\psi\|_{\infty}e^{|\theta_0|\cdot\|\psi\|_{\infty}T}. \end{eqnarray*} \]Now, for any \( \alpha\in (0,H)\) and \( (s,t),(u,v)\in\Delta_T\) ,
\[ \begin{eqnarray*} |L(s,t) - L(u,v)| & = & |\varphi(X_t)\lambda(s,t) -\varphi(X_v)\lambda(u,v)|\\ & \leqslant & |\varphi(X_t)|\cdot |\lambda(s,t) -\lambda(u,v)| + |\lambda(u,v)|\cdot |\varphi(X_t) -\varphi(X_v)|\\ & \leqslant & \|\varphi\|_{\infty}|\lambda(s,t) -\lambda(u,v)| + \|\varphi'\|_{\infty}\|X\|_{\alpha,T} (1 + e^{|\theta_0|\cdot\|\psi\|_{\infty}T})|t - v|^{\alpha}. \end{eqnarray*} \]Moreover,
\[ \begin{eqnarray*} |\lambda(s,t) -\lambda(u,v)| & \leqslant & \mathfrak c_2 \left|\int_{s}^{t}\psi(X_r)dr -\int_{u}^{v}\psi(X_r)dr\right| ~ {\rm with}~ \mathfrak c_2 = |\theta_0|e^{|\theta_0|\cdot\|\psi\|_{\infty}T}\\ & = & \mathfrak c_2 \left|\int_{s}^{u}\psi(X_r)dr -\int_{t}^{v}\psi(X_r)dr\right| \leqslant \mathfrak c_2\|\psi\|_{\infty}(|s - u| + |t - v|). \end{eqnarray*} \]Then,
\[ \begin{equation} |L(s,t) - L(u,v)|\leqslant \mathfrak c_3(1\vee\|X\|_{\alpha,T})(|t - v|^{\alpha} + |s - u|) \end{equation} \](18)
with
\[ \mathfrak c_3 = (\mathfrak c_2\|\varphi\|_{\infty}\|\psi\|_{\infty})\vee [\|\varphi'\|_{\infty}(1 + e^{|\theta_0|\cdot\|\psi\|_{\infty}T}) + \mathfrak c_2\|\varphi\|_{\infty}\|\psi\|_{\infty}T^{1 -\alpha}]. \]Consider \( p > 1/H\) . First, by Inequality (18), for every \( s,t\in [0,T]\) ,
\[ \begin{eqnarray*} |\overline L(0,t) -\overline L(0,s)| & \leqslant & |L(0,t) - L(0,s)| + |\varphi(X_t) -\varphi(X_s)|\\ & \leqslant & \mathfrak c_4(1\vee\|X\|_{\alpha,T})|t - s|^{\alpha} ~ {\rm with}~ \mathfrak c_4 =\mathfrak c_3 +\|\varphi'\|_{\infty} ~ {\rm and}~ \alpha =\frac{1}{p}. \end{eqnarray*} \]Then,
\[ \begin{eqnarray*} \|\overline L(0,\cdot)\|_{p\textrm{-var},T}^{p} & = & \sup_{(t_j)\in\mathfrak D_T}\sum_j|\overline L(0,t_{j + 1}) -\overline L(0,t_j)|^p\\ & \leqslant & \mathfrak c_{4}^{p}(1\vee\|X\|_{\alpha,T})^p \sup_{(t_j)\in\mathfrak D_T}\sum_j|t_{j + 1} - t_j| \leqslant \mathfrak c_{4}^{p}T(1\vee\|X\|_{\alpha,T})^p. \end{eqnarray*} \]Now, using the definitions of \( \overline L\) , \( \overline\lambda\) and \( \lambda\) , for every \( s,t,u,v\in [0,T]\) such that \( s,u < t,v\) ,
\[ \begin{eqnarray*} & & \overline L(s,t) -\overline L(u,t) - (\overline L(s,v) -\overline L(u,v))\\ & & = \varphi(X_t)(\overline\lambda(s,t) -\overline\lambda(u,t)) - \varphi(X_v)(\overline\lambda(s,v) -\overline\lambda(u,v))\\ & & = \varphi(X_t)(\overline\lambda(s,u)\overline\lambda(u,t) -\overline\lambda(u,t)) - \varphi(X_v)(\overline\lambda(s,u)\overline\lambda(u,v) -\overline\lambda(u,v))\\ & & = (\overline\lambda(s,u) - 1)(\varphi(X_t)\overline\lambda(u,t) -\varphi(X_v)\overline\lambda(u,v))\\ & & = \lambda(s,u)(\varphi(X_t)(\overline\lambda(u,t) -\overline\lambda(u,v)) + \overline\lambda(u,v)(\varphi(X_t) -\varphi(X_v)))\\ & & = \lambda(s,u)\overline\lambda(u,v) (\varphi(X_t)\lambda(v,t) +\varphi(X_t) -\varphi(X_v)), \end{eqnarray*} \]leading to
\[ \begin{eqnarray*} & & |\overline L(s,t) -\overline L(u,t) - (\overline L(s,v) -\overline L(u,v))|\\ & & \leqslant \|\overline\lambda\|_{\infty}|\lambda(s,u)| (\|\varphi\|_{\infty}|\lambda(v,t)| +\|\varphi'\|_{\infty}\|X\|_{\alpha,T}|t - v|^{\alpha})\\ & & \leqslant \mathfrak c_5|s - u| (\mathfrak c_1|t - v| + \|\varphi'\|_{\infty}\|X\|_{\alpha,T}|t - v|^{\alpha}) ~ {\rm with}~ \mathfrak c_5 = |\theta_0|\cdot\|\psi\|_{\infty}e^{2|\theta_0|\cdot\|\psi\|_{\infty}T}\\ & & \leqslant \mathfrak c_6(1\vee\|X\|_{\alpha,T})|s - u|^{\alpha}|t - v|^{\alpha}\\ & & ~ \mathfrak c_6 =\mathfrak c_5T^{1 -\alpha}(\mathfrak c_1T^{1 -\alpha} +\|\varphi'\|_{\infty}) ~\textrm{and (again)}~ \alpha = p^{-1}. \end{eqnarray*} \]Then,
\[ \begin{eqnarray*} \|\overline L\|_{p\textrm{-var},[0,T]^2}^{p} & = & \sup_{(s_i),(t_j)\in\mathfrak D_T}\sum_{i,j} |\overline L(t_{j + 1},s_i) -\overline L(t_j,s_i) - (\overline L(t_{j + 1},s_{i + 1}) -\overline L(t_j,s_{i + 1}))|^p\\ & \leqslant & \mathfrak c_{6}^{p}(1\vee\|X\|_{\alpha,T})^p \sup_{(s_i),(t_j)\in\mathfrak D_T} \sum_{i,j}|s_{i + 1} - s_i|\cdot |t_{j + 1} - t_j| \leqslant \mathfrak c_{6}^{p}T^2(1\vee\|X\|_{\alpha,T})^p. \end{eqnarray*} \]In conclusion, since \( b\) and \( \sigma\) are bounded, by [64], Proposition 8.3, and by [68], Theorem 15.33.(iii),
\[ \begin{eqnarray*} \mathbb E(\|\overline L(0,\cdot)\|_{p\textrm{-var},T}^{2}) & \leqslant & \mathfrak c_{4}^{2}T^{2\alpha}(1 +\mathbb E(\|X\|_{\alpha,T}^{2})) <\infty\\ & & ~ \mathbb E(\|\overline L\|_{p\textrm{-var},[0,T]^2}^{2}) \leqslant \mathfrak c_{6}^{2}T^{4\alpha}(1 +\mathbb E(\|X\|_{\alpha,T}^{2})) <\infty \Box \end{eqnarray*} \]
5.4 Proof of Proposition 4
Since \( \varphi\leqslant 0\) , \( \Theta_N\) is nonnegative and, in particular, \( \Theta_N(\mathbb R_+)\subset\mathbb R_+\) . Moreover, by (10), for every \( r,\overline r\in\mathbb R_+\) ,
So, \( \Theta_N\) is a contraction from \( \mathbb R_+\) into itself, and then \( R_N\) exists and is unique by Picard’s fixed-point theorem.\( \Box\)
5.5 Proof of Proposition 5
Note that
On the one hand, using Markov’s inequality and that \( X^1,\dots,X^N\) are i.i.d. processes, we obtain
(19)
On the other hand,
with
By the Bienaym
specialChar{39}e-Tchebychev inequality, and since \( X^1,\dots,X^N\) are i.i.d. processes,
Therefore,
5.6 Proof of Proposition 6
First, consider
where \( Z^1,\dots,Z^N\) are defined by
On the one hand, by the (usual) law of large numbers,
By the (usual) central limit theorem,
Then, since \( dX_{t}^{i} =\theta_0b(X_{t}^{i})dt +\sigma(X_{t}^{i})dB_{t}^{i}\) for every \( i\in\{1,\dots,N\}\) , and by Slutsky’s lemma,
On the other hand, let us introduce a random variable \( Y_{N}^{*}\) converging towards \( {\rm {var}}(Z) = \mathbb E(Z^2)\) . Precisely,
where, for every \( i\in\{1,\dots,N\}\) ,
By the law of large numbers, and by [67], Theorem 3.11.1,
So, by Slutsky’s lemma,
and then, for every \( x\in\mathbb R_+\) ,
Now, consider
Since \( \psi\leqslant 0\) and \( \theta_0 > 0\) , \( Y_{N}^{*}\leqslant Y_N\) , and then \( c_{N}^{*}\geqslant c_N\) . Moreover,
and since \( \mathfrak c\in (0,1)\) ,
(20)
So, by taking \( \mathfrak c = 1/2\) , for any \( x\in\mathbb R_+\) ,
Therefore,
Since, by Proposition 5,
for every \( x\in\mathbb R_+\) ,
For any \( \alpha\in (0,1)\) , observe that \( 4\phi(x) - 3 = 1 -\alpha\) if and only if \( \phi(x) = 1 -\alpha/4\) . Then,
5.7 Proof of Proposition 7
First, since \( dX_{t}^{i} =\theta_0b(X_{t}^{i})dt +\sigma(X_{t}^{i})dB_{t}^{i}\) for every \( i\in\{1,\dots,N\}\) ,
Let us establish a non-asymptotic risk bound on the auxiliary non-computable estimator \( \widehat\theta_{N}^{\mathfrak d} =\widehat\theta_N\mathbf 1_{D_N >\mathfrak d}\) of \( \theta_0\) . Observe that
Moreover, a computation along the same lines as in (19) guarantees that
So,
(21)
Now, Inequality (20) leads to
So,
Therefore, by Inequality (21) together with Proposition 5,
5.8 Proof of Proposition 8
First, since \( \alpha_H = H(2H - 1) < 0\) and \( \psi\leqslant 0\) , for every \( i\in\{1,\dots,N\}\) , \( t\in [0,T]\) and \( \theta\in\mathbb R_+\) ,
(22)
Then, since \( \varphi\leqslant 0\) and \( I_N\geqslant 0\) ,
Now, for any \( r,\overline r\in\mathbb R_+\) ,
So, exactly as in the proof of Proposition 4,
Therefore, \( \widetilde\Theta_N\) is a contraction from \( \mathbb R_+\) into itself, and then \( R_N\) exists and is unique by Picard’s fixed-point theorem.\( \Box\)
5.9 Proof of Proposition 9
First, as in the proof of Proposition 6,
Consider
By the law of large numbers,
So, by Slutsky’s lemma,
and then, for every \( x\in\mathbb R_+\) ,
Now, recall that \( \mathfrak Y_N\) was introduced in the statement of this proposition, and consider
For any \( i\in\{1,\dots,N\}\) , since \( \theta_0\leqslant\theta_{\max}\) ,
Moreover, by Inequality (22), and since \( \alpha_H < 0\) and \( \varphi,\psi\leqslant 0\) ,
So,
Therefore, as in the proof of Proposition 6,
In conclusion, by Proposition 5,
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