# Evolving privacy: drift parameter estimation for discretely observed i.i.d.
diffusion processes under LDP

Chiara Amorino Arnaud Gloter Hélène Halconruy  Université du Luxembourg,
L-4364 Esch-Sur-Alzette, Luxembourg. The author gratefully acknowledges
financial support of ERC Consolidator Grant 815703 “STAMFORD: Statistical
Methods for High Dimensional Diffusions”.Laboratoire de Mathématiques et
Modélisation d’Evry, CNRS, Univ Evry, Université Paris-Saclay, 91037, Evry,
France.Télécom SudParis, Institut polytechnique de Paris, 91120, Palaiseau,
France.

###### Abstract

The problem of estimating a parameter in the drift coefficient is addressed
for $N$ discretely observed independent and identically distributed stochastic
differential equations (SDEs). This is done considering additional
constraints, wherein only public data can be published and used for inference.
The concept of local differential privacy (LDP) is formally introduced for a
system of stochastic differential equations. The objective is to estimate the
drift parameter by proposing a contrast function based on a pseudo-likelihood
approach. A suitably scaled Laplace noise is incorporated to meet the privacy
requirements. Our key findings encompass the derivation of explicit conditions
tied to the privacy level. Under these conditions, we establish the
consistency and asymptotic normality of the associated estimator. Notably, the
convergence rate is intricately linked to the privacy level, and in some
situations may be completely different from the case where privacy constraints
are ignored. Our results hold true as the discretization step approaches zero
and the number of processes $N$ tends to infinity.

Keywords: local differential privacy, parameter drift estimation, high
frequency data, convergence rate, privacy for processes

AMS 2010 subject classifications: 62F12, 62E20, 62M05, 60G07, 60H10

###### Contents

  1. 1 Introduction
  2. 2 Mathematical framework
    1. 2.1 Setting and assumptions
    2. 2.2 Problem formulation
  3. 3 Statistical procedure and main results
    1. 3.1 Privatization mechanism and estimation procedure
    2. 3.2 Main results: consistency and asymptotic normality
    3. 3.3 Effective privacy
  4. 4 Concluding remarks
  5. 5 Preliminary results
    1. 5.1 Splines, some tools
    2. 5.2 Technical results
  6. 6 Proof of main results
    1. 6.1 Consistency
    2. 6.2 Asymptotic normality
      1. 6.2.1 Proof of Proposition 4
      2. 6.2.2 Proof of Proposition 5
      3. 6.2.3 Proof of Corollary 1
      4. 6.2.4 Proof of Proposition 6
  7. 7 Proof of preliminary results
    1. 7.1 Proof of Lemma 2
    2. 7.2 Proof of Proposition 2
    3. 7.3 Proof of Lemma 5
    4. 7.4 Proof of Lemma 7
    5. 7.5 Proof of Lemma 8
    6. 7.6 Proof of Lemma 9

##  1 Introduction

In recent years, the rapid accumulation of large-scale data, including medical
records, cellphone location information, and internet browsing history,
underscores the critical need for a nuanced understanding of the tradeoffs
between privacy and the utility derived from collected data. Traditional
privacy-preserving mechanisms like permutation or basic anonymization have
proven inadequate, leading to notable privacy breaches with genomic and movie
rating data (see [37]). These incidents, such as Netflix canceling a
competition, highlight the urgency of finding a balance between utility and
safeguarding sensitive information.

To address these challenges, researchers in statistics, databases, and
computer science started studying differential privacy as a formalization of
disclosure risk limitation (for example in [20, 22, 21]). Differential privacy
has evolved from a theoretical paradigm to a widely deployed technology in
various applications over the last decade [18, 23, 25]. It addresses the need
for protecting individual data while allowing statistical analysis of
aggregate databases. This is achieved through a trusted curator holding
individual data, and the protection is ensured by injecting noise into
released information. The challenge is to optimize statistical performance
while preserving privacy in a remote access query system.
The first attempt to design a private methodology traces back to Dwork et
al.’s work [21], that formalized global differential privacy. Research in this
field now encompasses central or global privacy models, where a curator
collects and privatizes data before releasing only the output, and local
privacy models, involving randomized data collection. Major technology
companies like Apple [5, 48] and Google [1, 23] have adopted local
differential privacy protections, reflecting the broader impact of privacy
measures in billions of devices. Understanding the fundamental limitations and
possibilities of learning with local privacy notions is crucial in this
context.

Historically, methods for locally private analysis were primarily focused on
estimating parameters within a binomial distribution [50]. However, recent
advancements in research have introduced mechanisms that extend to a broader
array of statistical challenges. These encompass hypothesis testing [8, 33],
M-estimators [6], robustness [35], change point analysis [9], mean and median
estimation [20], and nonparametric estimation [12, 11, 31], among others. With
the escalating importance of data protection, striking the right balance
between statistical utility and privacy becomes crucial. It is imperative to
safeguard data against privacy breaches while still facilitating the
extraction of valuable insights. Consequently, determining the optimal
equilibrium between these two considerations has acquired heightened
significance.

To the best of our understanding, the exploration of statistical inference
under privacy constraints has traditionally centered on $N$ random variables.
The consideration of variables having a historical context, involving $N$
stochastic processes across a time horizon $[0,T]$, has been noticeably absent
from prior investigations. This paper aims to address this gap by examining
the drift parameter estimation from i.i.d. paths of diffusion processes while
adhering to local differential privacy constraints.

Note that the literature on statistical estimation for stochastic differential
equations is extensive, owing to the model’s versatility and its applicability
across various domains. Examples abound, spanning biology [44], neurology
[28], finance [29], and economics [7]. Classical applications extend to
physics [38] and mechanics [32]. The framework mentioned can also be employed
to model the inherent variability of biomedical experiments, with a specific
emphasis on pharmacology (refer to [49, 13, 41, 42] and [19] for an overview
of stochastic differential equation estimation for pharmacodynamic models).
This is why focusing on diffusion processes becomes interesting when dealing
with sensitive data that requires privacy guarantees. However, a significant
challenge arises when applying privacy in this context due to the dependency
structure of the process. Indeed, it becomes possible to recover information
about the private process at a particular time instant through observations
about its past or future.

In this work we consider $N$ i.i.d. diffusion processes
$(X_{t}^{i})_{t\in[0,T]}$ for $i\in\\{1,\dots,N\\}$, with a fixed time horizon
$T>0$. They follow the dynamics

$X_{t}^{\theta,i}=X_{0}^{\theta,i}+\int_{0}^{t}b(\theta,X_{s}^{\theta,i})ds+\int_{0}^{t}{\color[rgb]{0,0,0}\sigma}(X_{s}^{\theta,i})dW_{s}^{i},$

where the processes $(W^{i}_{t})_{t\in[0,T]}$ are independent standard
Brownian motions. We aim to estimate the parameter $\theta^{\star}\in\Theta$
based on the discrete observations
$(X_{t_{j}}^{\theta,i})_{i=1,\ldots,N,j=1,\ldots,n}$ under $\alpha$ local
differential privacy constraints, for $N,n\rightarrow\infty$. In our context,
the discrete observations of the private processes
$(X_{t}^{1})_{t},\ldots,(X_{t}^{N})_{t}$ are not directly available. Instead,
our estimator relies on a public sample derived from the original discrete
observations, subject to $\alpha$ local differential privacy. To achieve this,
we begin by adapting the concept of local differential privacy to a context
that accommodates the presence of processes, leveraging the definition of
componentwise local differential privacy (CLDP) proposed in [2] (refer to
Section 2.2 for comprehensive details). In our interpretation, the $N$
processes represent $N$ independent individuals evolving over time. Each
individual contributes $n$ observations, subject to componentwise local
differential privacy, ensuring that the information at different time points
is privatized in a different way. This approach enables us to make public each
observation corresponding to the same individual separately, using a distinct
privacy channel with a privacy level $\alpha_{j}$, for $j\in\\{1,...,n\\}$.
This strategy is advantageous as it allows for tailoring the level of privacy
protection based on the specific context, recognizing that different
observations corresponding to various moments in an individual’s life may
warrant varying degrees of privacy safeguards.

The rationale behind this choice lies in the recognition that disclosing
information about the distant past may have different implications than
disclosing information about the present. By treating each observation
independently, we can tailor the level of privacy protection based on the
specific circumstances.

In our study, we examine a privacy mechanism where the data holder can observe
two consecutive realizations of $X^{i}$ each time. This scenario models
situations such as when a patient’s vitals are taken, and the doctor (data
holder) has access not only to the current data but also to the data from the
previous check-up (control data).
Given the correlated nature of the observations, it becomes crucial to operate
within a framework that acknowledges this dependency. A pertinent question
arises: Is it possible to extract sensitive information about the present by
exploiting the fact that some extra information is carried by another
observation from the past?
To answer this question, it is important to understand the constraints on the
privacy levels $\alpha_{1},...,\alpha_{n}$ necessary to obtain well-performing
estimators. Specifically, we will explore how the behavior of
$\alpha_{1},...,\alpha_{n}$ as functions of $N$ and $n$ leads to two distinct
asymptotic regimes, which we will refer to as the "significant contribution of
privacy" and "negligible contribution of privacy".

Without the presence of privacy constraints, a natural approach to estimating
unknown parameters from the continuous observation of a SDE would be to use
maximum likelihood estimation. However, the likelihood function based on the
discrete sample is not tractable, as it depends on the transition densities of
the process, which are not explicitly known. To overcome this difficulty,
several methods have been developed for high-frequency estimation of
discretely observed classical SDEs. A widely-used method involves considering
a pseudo-likelihood function, often based on the high-frequency approximation
of the process dynamics using the Euler scheme, as seen in [24, 30, 52].
This statistical analysis relies on the minimization of a contrast function,
akin to methods proposed for classical SDEs, extended to Lévy-driven SDEs [47,
3] and interacting particle systems [46, 4].

Even in our context, with the presence of local differential privacy
constraints, it seems natural to leverage the minimization of a contrast
function technique, incorporating this quantity into the definition of the
privacy mechanism employed in our estimation procedure. Furthermore,
introducing centered Laplace-distributed noise to bounded random variables is
known to ensure ${\alpha}$-differential privacy (see [2], [20], [31]). This
motivates our choice of the anonymization procedure. Specifically, we adopt a
Laplace mechanism to construct the public counterpart of the raw samples, as
illustrated in (3) below. Some technical challenges arise from such a
definition, mainly due to the fact that Laplace random variables are defined
only for some values of $\theta$ on a grid, whose size $L_{n}$ goes to
$\infty$ for $n\rightarrow\infty$. Consequently, our definition of the public
sample $(Z^{i}_{j}(\theta))_{i=1,\dots,N,\,j=1,\dots,n}$ holds true only on
the grid as well. However, to define our estimator, we need to extend the
definition of the public sample to $\theta$ belonging to the whole parameter
space $\Theta$. To address this, we rely on a spline approximation method,
detailed in Section 5.1. Then, with access to the public data
$Z_{j}^{i}(\theta)$ for $i=1,\dots,N$ and $j=1,\dots,n$, the statistician can
propose an estimator obtained by minimizing the contrast function, which is
the spline approximation of the private version of the contrast function in
the case of classical SDEs (see Section 3 for details).

The main result of the paper is the consistency and asymptotic normality of
the resulting estimator, demonstrated using a central limit theorem for
martingale difference triangular arrays.

Let us introduce

$r_{n,N}:=\frac{L_{n}^{2}\log(n)}{\sqrt{\bar{\alpha}^{2}}},$ (1)

where $\bar{\alpha}^{2}$ is the harmonic mean over different privacy levels,
defined as $1/\bar{\alpha}^{2}:=n^{-1}\sum_{j=1}^{n}1/\alpha_{j}^{2}$ and that
may be dependent on $N$. We establish the consistency of the proposed
estimator under the assumption that

$\frac{1}{L_{n}}\sqrt{\frac{\log(L_{n})}{N}}r_{n,N}\rightarrow
0\quad\text{as}\;n,N,L_{n}\rightarrow\infty.$

This requirement implies that the case of perfect privacy ($\bar{\alpha}=0$)
is not allowed and informs us about the price to pay for the privacy guarantee
in obtaining reasonable statistical results.

Moreover, as anticipated earlier, we prove the asymptotic normality of our
estimator under the two regimes delineated by different values of
$\bar{\alpha}$. In particular, we find that if the privacy parameters
$\alpha_{1},...,\alpha_{n}$ are large enough to guarantee
$r_{n,N}\sqrt{\log(L_{n})}\rightarrow 0$, with some technical conditions, we
essentially obtain the same result as in the case without privacy:

$\sqrt{N}(\widehat{\theta}_{n}^{N}-\theta^{\star})\xrightarrow{\mathcal{L}}{\mathcal{N}}\bigg{(}0,2\Big{(}\int_{0}^{T}{\mathbb{E}}\bigg{[}\Big{(}\frac{\partial_{\theta}b(\theta^{\star},X_{s})}{{\color[rgb]{0,0,0}\sigma}(X_{s})}\Big{)}^{2}\bigg{]}ds\Big{)}^{-1}\bigg{)}=:{\mathcal{N}}\big{(}0,2(\Sigma_{0})^{-1}\big{)}\quad\mbox{as
}n,N\rightarrow\infty.$

Indeed, the convergence above asserts the asymptotic Gaussianity of our
estimator with a convergence rate and a variance that resemble the classic
scenario without any privacy constraints.

When the contribution of privacy is instead the dominant one (i.e.
$r_{n,N}\rightarrow\infty$, with $r_{n,N}$ defined by (1)), we still recover
the asymptotic normality of our estimator, but with a different convergence
rate, depending on the average amount of privacy $\bar{\alpha}^{2}$. In this
context, some extra challenges appear, leading us to replace the previous grid
with a random one, depending on a uniform random variable $S$ (see Equation
(13)). Then, subject to some technical conditions we are able to prove the
following:

$\frac{\sqrt{N\bar{\alpha}^{2}}}{4(a+1)L_{n}^{2}\log(n)\sqrt{T}}(\widehat{\theta}_{n}^{N}-\theta^{\star})\xrightarrow{\mathcal{L}}\sqrt{\widehat{v}(S)}(\Sigma_{0})^{-1}\,{\mathcal{N}},$

where ${\mathcal{N}}$ is a gaussian $\mathcal{N}(0,1)$ random variable
independent of $S$, $a$ and $\widehat{v}$ are respectively a tuning parameter
and an auxiliary function properly defined in (14); both depending on the
spline functions under consideration.

It is noteworthy that our results distinctly reveal $L_{n}^{2}\log(n)$ as the
threshold for $\bar{\alpha}$, thereby delineating whether the newly introduced
term arising from the privacy constraints is a primary contributing factor.
The significance of this threshold becomes apparent through our findings.
Theorem 2 indicates that the case of ’negligible’ privacy corresponds to the
constraint $r_{n,N}\sqrt{\log(L_{n})}\rightarrow 0$, while Theorem 3
establishes the condition $r_{n,N}\rightarrow\infty$ for achieving
’significant’ privacy. In Corollary 1, we investigate the scenario where
$r_{n,N}$ converges to a constant. In this case, we are able to demonstrate
the convergence in law of our estimator to the sum of the two Gaussian random
variables obtained in Theorems 3 and 2, respectively.
This emphasizes that the threshold demarcating the two asymptotic regimes of
’significant’ and ’negligible’ privacy is dictated by the asymptotic behavior
of $r_{n,N}$. The additional $\log(L_{n})$ in the condition outlined in
Theorem 2 is introduced for technical reasons associated with the grid,
representing a non-optimal condition necessary to mitigate the impact of
privacy constraints. Additionally, we provide a discussion about effective
privacy.

The paper unfolds as follows. In Section 2, we introduce the model and its
underlying assumptions. Notably, we formulate the definition of local
differential privacy tailored to stochastic processes in Section 2.2. Section
3 is dedicated to presenting the contrast function pivotal for our estimator’s
definition, alongside articulating our main results, whose strengths and
weaknesses we discuss in Section 4. Turning to Section 5, we furnish essential
tools essential for proving our main results. This includes an exploration of
spline functions in Section 5.1, followed by technical results outlined in
Section 5.2. In summary, the proofs of our main and technical results find
their place in Sections 6 and 7, respectively.

##  2 Mathematical framework

###  2.1 Setting and assumptions

Let $T>0$. Let $W^{1},\dots,W^{N}$ be $N\in{\mathbb{N}}^{*}=\\{1,2,3,...\\}$
independent standard Brownian motions. For any $i\in\\{1,\dots,N\\}$, we
consider the diffusion process $(X_{t}^{i})_{t\in[0,T]}$ defined by

$X_{t}^{\theta,i}=X_{0}^{\theta,i}+\int_{0}^{t}b(\theta,X_{s}^{\theta,i})ds+\int_{0}^{t}{\color[rgb]{0,0,0}\sigma}(X_{s}^{\theta,i})dW_{s}^{i},$
(2)

where $b:\Theta\times{\mathbb{R}}\rightarrow{\mathbb{R}}$,
${\color[rgb]{0,0,0}\sigma}:{\mathbb{R}}\rightarrow{\mathbb{R}}$ and
$\theta\in\Theta$. We fix $\Theta:=[0,1]$ to simplify the notation. Assume
that for any $i\in\\{1,\dots,N\\}$, the processes $(W^{i}_{t})_{t\in[0,T]}$
are independent of the initial value
$(X_{0}^{\theta,1},\dots,X_{0}^{\theta,N})$. We also assume that
$(X_{0}^{\theta,i})_{i}$ are i.i.d with $X_{0}^{\theta,i}\in\cap_{p\geq
1}L^{p}$.
As anticipated in the introduction, we aim at estimating the parameter
$\theta^{\star}\in\overset{\circ}{\Theta}=(0,1)$ given the observations
$(X_{t_{j}}^{\theta,i})_{i=1,\dots,N,\,j=1,\dots,n}$ subject to local
differential privacy constraints, see next section for a formal definition and
a rigorous introduction of the problem.
We introduce the discretization step as $\Delta_{n}:=T/n$, so that
$t_{j,n}=jT/n=j\Delta_{n}$. The asymptotic framework here considered is such
that both $N,n\rightarrow\infty$ while the time horizon $T$ is fixed.
Moreover, $N$ goes to $\infty$ as a polynomial of $n$, i.e. there exists $r>0$
such that $N=O(n^{r})$. In the sequel we will write simply $X_{t_{j}}^{i}$
when the process is computed in the true value of the parameter
$\theta^{\star}$, that is $X_{t_{j}}^{\theta^{\star},i}$.

Let us consider the assumptions :

######  Assumption 1.

For all $\theta\in\Theta$ the function $b(\theta,\cdot)$ is bounded. Moreover,
$b(\theta,\cdot)$ and ${\color[rgb]{0,0,0}\sigma}$ are globally Lipschitz, i.e
there exists $c>0$ such that, for all $x,y\in{\mathbb{R}}$,

$|b(\theta,x)-b(\theta,y)|+|{\color[rgb]{0,0,0}\sigma}(x)-{\color[rgb]{0,0,0}\sigma}(y)|\leq
c|x-y|.$

######  Assumption 2.

We assume that the diffusion coefficient is bounded away from $0$: for some
${\color[rgb]{0,0,0}\sigma_{\min}}>0$,

${\color[rgb]{0,0,0}\sigma_{\min}}^{2}\leq{\color[rgb]{0,0,0}\sigma^{2}}(x).\\\
$

Under assumptions 1 and 2, Equation (2) has a unique strong solution
$(X_{t}^{\theta,i})_{t\in[0,T]}$ taking its values in
$({\mathbb{R}},{\mathcal{B}}({\mathbb{R}}))$.

######  Assumption 3.

For all $k\in\mathbb{N}=\\{0,1,2,...\\},$ the function
$\frac{\partial^{k}}{\partial\theta^{k}}b$ is bounded uniformly in $\theta$:
$\sup_{x,\theta}|\frac{\partial^{k}}{\partial\theta^{k}}b(\theta,x)|<\infty$.

######  Assumption 4.

[Identifiability]
For all $\theta\in\Theta$ such that $\theta\neq\theta^{\star}$, it is

$\int_{0}^{T}{\mathbb{E}}\bigg{[}\frac{\big{(}b(\theta,X_{s})-b(\theta^{\star},X_{s})\big{)}^{2}}{{\color[rgb]{0,0,0}\sigma^{2}}(X_{s})}\bigg{]}ds>0.$

One can easily check that Assumption 4 is equivalent to ask that, for any
$\theta\in\Theta$ such that $\theta\neq\theta^{\star}$, it is
$b(\theta,\cdot)\neq b(\theta^{\star},\cdot)$. This will be crucial in order
to prove the consistency of the estimator we will propose.
Observe that, in the sequel, we will often shorten the notation
$\partial_{\theta}^{k}g$ for $\frac{\partial^{k}}{\partial\theta^{k}}g$, for
any derivable function $g$.

######  Assumption 5.

[Invertibility]
Define the asymptotic Fisher information as
$I(\theta):=\int_{0}^{T}{\mathbb{E}}\big{[}(\frac{\partial_{\theta}b(\theta,X_{s})}{{\color[rgb]{0,0,0}\sigma}(X_{s})})^{2}\big{]}ds$.
We assume that, for any $\theta\in\Theta$, $I(\theta)>0$.

We will see that Assumption 5 will be essential to prove the asymptotic
normality of our estimator.

###  2.2 Problem formulation

Let $({X}_{t}^{1}),\ldots,({X}_{t}^{N})$ be solutions of the stochastic
differential equation (2). Since they are driven by independent Brownian
motions, we can regard them as $N$ independent realizations of the same
diffusion process. It is important to note that, for classical SDEs, it is
generally not feasible to estimate the drift over a finite time horizon.
However, in this context, the number of copies $N$ will serve a similar
purpose as the time horizon $T$ does in parameter estimation for classical
SDEs. As $N$ tends to infinity, the sample size will effectively increase,
allowing for our estimation procedure. The $N$ copies follow the law of the
same stochastic process $(X_{t})_{t\in[0,T]}$, and we observe each of them in
$n+1$ different instants of time $0=t_{0}\leq t_{1}\leq\dots\leq t_{n}=T$.
They can represent the information coming from $N$ different individuals, that
evolves in time.

In our approach, we adopt componentwise local differential privacy as
introduced in [2]. It means that we do not release the public data pertaining
to each individual based on the whole vector of its private data, but instead
release data relying on componentwise observation of this private vector. As
shown in [2], this constraint usually makes harder to infer the joint law of
the private data, but it might be more suitable in practice when dealing with
temporal series.

Let us now formalize the framework discussed earlier. We introduce
$\bm{X}^{i}:=(X^{i}_{t_{0}},\dots,X^{i}_{t_{n}})$ for any
$i\in\\{1,\dots,N\\}$. The process of privatizing the raw samples
$(\bm{X}^{i})_{i=1,\ldots,N}$ and transforming them into the public set of
samples $(\bm{Z}^{i})_{i=1,\ldots,N}$ is captured by a conditional
distribution, known as privacy mechanism or channel distribution. We make the
assumption that each component of a disclosed observation, denoted by
$Z^{i}_{j}$, is privatized independently and belongs to a specific space
$\mathcal{Z}^{i}$, which may vary for each $i$. This implies that the
observation $\bm{Z}^{i}$ belongs to the product space
$\bm{\mathcal{Z}}:=\prod_{j=1}^{n}\mathcal{Z}_{j}$. Additionally, we assume
that the spaces $\mathcal{Z}_{j}$ are separable complete metric spaces. Their
corresponding Borel sigma-fields define measurable spaces
$(\mathcal{Z}_{j},\Xi_{\mathcal{Z}_{j}})$ for all $j\in\\{0,\dots,n\\}$.

Let us now discuss the properties and structure of the privacy mechanism in
our framework. For simplicity, the privacy mechanism is designed to be non
interactive between the $N$ individuals. This gives rise to the the following
conditional independence structure : for $j\in\\{1,\dots,n\\}$,

$\\{{X}^{i}_{t_{j}},{X}^{i}_{t_{j-1}}\\}\rightarrow Z^{i}_{j},\qquad
Z^{i}_{j}\perp\\!\\!\\!\perp
X^{k}_{t_{l}}\mid\\{{X}^{i}_{t_{j}},{X}^{i}_{t_{j-1}}\\}\,\text{ for }k\neq
i,\,\forall l\in\\{0,\dots,n\\}.$

More precisely, for $j=1,\dots,n$ and $i=1,\dots,N$, given
$X^{i}_{t_{j}}=x^{i}_{j}\in{\mathbb{R}}$ and
$X^{i}_{t_{j-1}}=x^{i}_{j-1}\in{\mathbb{R}}$, the public output
$Z^{i}_{j}\in\mathcal{Z}_{j}$ is drawn as

${}Z^{i}_{j}\sim
Q_{j}(\cdot|X^{i}_{t_{j}}=x^{i}_{j},X^{i}_{t_{j-1}}=x^{i}_{j-1})$ (3)

for Markov kernels
$Q_{j}:\Xi_{\mathcal{Z}_{j}}\times({\mathbb{R}}\times{\mathbb{R}})\rightarrow[0,1]$.
The notation
$(\bm{\mathcal{Z}},\Xi_{\bm{\mathcal{Z}}})=(\prod_{j=1}^{n}\mathcal{Z}_{j},\otimes_{j=1}^{n}\Xi_{\mathcal{Z}_{j}})$
refers to the measurable space of non-private (or public) data. The space of
public data, denoted by $\bm{\mathcal{Z}}$, can be quite general, as it is
selected by the statistician based on a specific privatization mechanism.
Nonetheless, in the parameter estimation discussed below, it will be valued in
$\bm{\mathcal{Z}}=\mathbb{R}^{d_{Z}\times n}$ for some dimension $d_{Z}$ given
in Section 3.

This privatization structure highlights a major difference compared to the
definition of componentwise local differential privacy (CLDP) introduced in
[2]. In our case, the variable $Z^{i}_{j}$ is derived not only from the
private value of $X^{i}_{t_{j}}$, but also from the previous observation at
time $t_{j-1}$. This allows for incorporating additional temporal information
in the privatization process. Such privacy mechanism is in practice more
flexible than a usual privatization based on the whole vector $\bm{X}^{i}$. It
enables for instance a physician accessing to the record of just two
consecutive private values of an individual to output the public data.
However, it excludes the situation where the private data can be accessed only
for a single date before releasing the public value. This would be the case if
the collected private data is immediately destroyed after the public view is
sent. However, by [2], we know that this situation increases the error of
estimation for the joint law of the diffusion process $X$, and in turn
certainly would deteriorate the estimation of the drift parameter. For this
reason, we exclude it from our analysis.

Let us mention that a more general concept of interactive privacy mechanism
could be considered. In that case, the output $Z_{j}^{i}$ would be constructed
on the basis of some private data and all the public data already available.
Although it is typically easier to work with non-interactive algorithms, as
they yield independent and identically distributed privatized samples, there
are situations where it is advantageous for the channel’s output to depend on
previous computations. Stochastic approximation schemes, for instance, require
this kind of dependency (see [40]). In our framework the temporal aspect of
the private data $X_{j}^{i}$ is along the index $j$, whereas the usual
situation of interactive mechanism is to construct $\bm{Z}^{i}$ inductively on
the index $i$ corresponding to the individuals. Thus, to cope with real
situation the definition of interactive mechanism should be modified
accordingly in our case. For this reason, we do not pursue the discussion here
about interactive mechanism.
The notion of privacy can be quantified using the concept of local
differential privacy. Let $\bm{\alpha}=(\alpha_{1},\ldots,\alpha_{n})$ be a
given parameter, where $\alpha_{j}\geq 0$ for each $j\in\\{1,\ldots,n\\}$. We
say that the random variable $Z^{i}_{j}$ is an $\alpha_{j}$-differentially
locally privatized view of $X^{i}_{t_{j}}$ and $X^{i}_{t_{j-1}}$ if, for all
$x_{j},x_{j}^{\prime}\in{\mathbb{R}}$, and
$x_{j-1},x_{j-1}^{\prime}\in{\mathbb{R}}$, the following condition holds:

${}\sup_{A\in\Xi_{\mathcal{Z}_{j}}}\frac{Q_{j}(A|X^{i}_{t_{j}}=x_{j},X^{i}_{t_{j-1}}=x_{j-1})}{Q_{j}(A|X^{i}_{t_{j}}=x^{\prime}_{j},X^{i}_{t_{j-1}}=x^{\prime}_{j-1})}\leq\exp(\alpha_{j}).$
(4)

We define the privacy mechanism $\bm{Q}=(Q_{1},\ldots,Q_{n})$ to be
$\bm{\alpha}$-differentially locally private if each variable $Z^{i}_{j}$
satisfies the condition of being an $\alpha_{j}$-differentially locally
privatized view of $X^{i}_{t_{j}}$ and $X^{i}_{t_{j-1}}$.
The parameter $\alpha_{j}$ serves as a measure of the level of privacy
guaranteed to the variables $X^{i}_{t_{j}}$ and $X^{i}_{t_{j-1}}$. By setting
$\alpha_{j}=0$, we ensure perfect privacy, meaning that it is impossible to
recover these variables from the perspective of $Z^{i}_{j}$. On the other
hand, as $\alpha_{j}$ tends to infinity, the privacy restrictions become less
stringent.

In the upcoming section, we will introduce a privacy mechanism that serves as
the foundation for our estimation procedure. Firstly, we will establish its
adherence to the $\bm{\alpha}$-local differential privacy condition, as
outlined in Equation (4). Subsequently, we will demonstrate that the estimator
we propose, based solely on observations of the privatized views $Z^{i}_{j}$
for $i=1,\dots,N$ and $j=1,\dots,n$, is consistent and asymptotically normal,
with the convergence rate depending on the privacy level $\bm{\alpha}$, in the
case of significant privacy.

##  3 Statistical procedure and main results

###  3.1 Privatization mechanism and estimation procedure

We assume that the functions $b$ and $\sigma$ are known and we aim at
estimating the unknown parameter $\theta^{\star}$ under $\bm{\alpha}$-local
differential privacy, as properly formalized in previous section. Hence, we
need to define a public sample $(Z_{j}^{i})_{1\leq i\leq N,1\leq j\leq n}$
obtained from observations $(X_{t_{j}}^{\theta,i})_{1\leq i\leq N,0\leq j\leq
n}$ of the initial data via a privatization mechanism that satisfies the
condition in (4).

Consider the classical scenario in statistics, where the goal is to estimate
the parameter ${\color[rgb]{0,0,0}\theta^{\star}}$ based on continuous
observations of the stochastic differential equation given in (2). It is well-
known that the maximum likelihood estimator (MLE) performs optimally in this
case, being consistent and asymptotically Gaussian with an optimal variance.
However, when privacy is not a concern and only discrete observations of the
equation are available, the transition density (and hence the likelihood) of
the process is no longer accessible. To address this challenge, a commonly
used approach in the literature is to employ a contrast function that serves
as a substitute for the likelihood. This contrast function approximates the
likelihood based on the Euler approximation scheme. Specifically, for any
$\theta\in\Theta$, it takes the form :

$\sum_{i=1}^{N}\sum_{j=1}^{n}\frac{(X_{t_{j}}^{i}-X_{t_{j-1}}^{i}-\Delta_{n}b(\theta,X_{t_{j-1}}^{i}))^{2}}{{\color[rgb]{0,0,0}\sigma}^{2}(X_{t_{j-1}}^{i})}.$

The proposed estimator, denoted as $\hat{\theta}_{n}^{N}$, minimizes the above
quantity over the parameter set $\Theta$. It can be verified that minimizing
the aforementioned quantity is equivalent to maximizing the following
expression:

$\sum_{i=1}^{N}\sum_{j=1}^{n}\frac{2b(\theta,X_{t_{j-1}}^{i})(X_{t_{j}}^{i}-X_{t_{j-1}}^{i})-\Delta_{n}b^{2}(\theta,X_{t_{j-1}}^{i})}{{\color[rgb]{0,0,0}\sigma}^{2}(X_{t_{j-1}}^{i})}=:\sum_{i=1}^{N}\sum_{j=1}^{n}f(\theta;X_{t_{j-1}}^{i},X_{t_{j}}^{i}).$
(5)

Hence, it is natural to incorporate the above quantity into the definition of
the privacy mechanism employed in our estimation procedure.
Furthermore, as said in the introduction, it is widely recognized that
introducing centered Laplace-distributed noise to bounded random variables
ensures $\bm{\alpha}$-differential privacy. Laplace random variables will
therefore play a role in defining the privacy mechanism.

We commence by establishing a grid of the parameter space $\Theta=[0,1]$ upon
which we will construct the Laplace random variables. Let us denote by
$\Xi=\\{0\leq\theta_{0}<\cdots<\theta_{L_{n}-1}\leq 1\\}$ this grid. It has
cardinality $L_{n}\in{\mathbb{N}}^{*}$. For $n\rightarrow\infty$, we assume
$L_{n}$ to go to $\infty$ with the restriction that there exists $r>0$ such
that $L_{n}=O(n^{r})$.

In order to define the privatized views of our data we need to introduce a
smooth version of the indicator function, that we denote as $\varphi$. It is
such that $\varphi(\xi)=0$ for $|\xi|\geq 2$, $\varphi(\xi)=1$ for $|\xi|\leq
1$ and, for $1<|\xi|<2$, $\varphi\in C^{\infty}({\mathbb{R}})$.
We fix $a\in\mathbb{N}^{*}$ and, for
$(i,j,k,\ell)\in\\{1,\dots,N\\}\times\\{1,\dots,n\\}\times\\{0,\dots,a\\}\times\\{0,\dots,L_{n}-1\\}$,
let us denote by $\mathcal{E}^{i,\ell,(k)}_{j}$ a random variable, such that
$(\mathcal{E}^{i,\ell,(k)}_{j})_{i,j,k,\ell}$ are independent variables with
law

$\mathcal{E}^{i,\ell,(k)}_{j}\sim{\mathcal{L}}(2\tau_{n}\,L_{n}\,(a+1)/\alpha_{j}),$
(6)

where $\tau_{n}:=\sqrt{\Delta_{n}}\log(n)$ and $L_{n}$ is the cardinality of
$\Xi$ as above. We assume that the variables $\mathcal{E}^{i,\ell,(k)}_{j}$
are independent from the data $X_{t_{j^{\prime}}}^{i^{\prime}}$ for
$i^{\prime}=1,\dots,N$ and $j^{\prime}=0,\dots,n$. Then, we set for any
$i\in\\{1,\dots,N\\}$, $j\in\\{1,\dots,n\\}$, $k\in\\{0,\dots,a\\}$,
$\ell\in\\{0,\dots,L_{n}-1\\}$

$\displaystyle Z_{j}^{i,(k)}(\theta_{\ell})$
$\displaystyle:=f^{(k)}(\theta_{\ell};X_{t_{j-1}}^{i},X_{t_{j}}^{i})\varphi\left(f^{(k)}(\theta_{\ell};X_{t_{j-1}}^{i},X_{t_{j}}^{i})/\tau_{n}\right)+\mathcal{E}_{j}^{i,\ell,(k)}$
(7)
$\displaystyle=f^{i,\ell,(k)}_{j}\times\varphi_{\tau_{n}}\left(f^{i,\ell,(k)}_{j}\right)+\mathcal{E}_{j}^{i,\ell,(k)},$

where

$f^{i,\ell,(k)}_{j}:=f^{(k)}(\theta_{\ell};X_{t_{j-1}}^{i},X_{t_{j}}^{i})=\frac{\partial^{k}}{\partial\theta^{k}}f(\theta_{\ell},X_{t_{j-1}}^{i},X_{t_{j}}^{i})$
(8)

and $\varphi_{\tau_{n}}(\cdot):=\varphi(\cdot/\tau_{n})$. Remark that the
dependence on $k$ in $\mathcal{E}_{j}^{i,\ell,(k)}$ does not stand for the
derivatives of the Laplace itself but it is a reminder of the fact we are
adding some noise to the derivatives of $f$. An analogous comment applies to
$Z_{j}^{i,(k)}$.

The public data is then defined by

$Z_{j}^{i}:=(Z_{j}^{i,(k)}(\theta))_{\theta\in\Xi,0\leq k\leq
a}\in\mathbb{R}^{L_{n}\times(a+1)}.$ (9)

It is easy to check that the local differential privacy control holds true, as
proven in the following lemma.

######  Lemma 1.

The public variables described in (9) are $\bm{\alpha}$-local differential
private views of the original $(X_{t_{j-1}}^{i},X_{t_{j}}^{i})$.

###### Proof.

Let us denote by $(z_{j}^{\theta,k})_{\theta\in\Xi,0\leq k\leq a}\mapsto
q_{j}\big{(}(z_{j}^{\theta,k})_{\theta\in\Xi,0\leq k\leq a}\mid
X_{t_{j}}^{i}=x_{j},X_{t_{j-1}}^{i}=x_{j-1}\big{)}$ the density of the public
data $Z_{j}^{i}=(Z_{j}^{i,(k)}(\theta))_{\theta\in\Xi,0\leq k\leq
a}\in\mathbb{R}^{L_{n}\times(a+1)}$ conditional to $X_{t_{j}}^{i}=x_{j}$,
$X_{t_{j-1}}^{i}=x_{j-1}$ for $j\in\\{1,\dots,n\\}$ and $i\in\\{1,\dots,N\\}$.
As $\mathcal{E}^{i,\ell,(k)}_{j}$ is distributed as a centered Laplace random
variable with scale parameter $2\tau_{n}L_{n}(a+1)/\alpha_{j}$, its density at
the point $x\in{\mathbb{R}}$ is given by
$\frac{1}{2\tau_{n}L_{n}(a+1)}\alpha_{j}\exp(-\frac{1}{2\tau_{n}L_{n}(a+1)}\alpha_{j}|x|)$.
Then, using the independence of the variables
$(Z_{j}^{i,(k)}(\theta))_{\theta\in\Xi,0\leq k\leq a}$ and denoting
$\varphi_{f^{(k)},\tau_{n}}^{\theta}(x,y):=\varphi\left(f^{(k)}(\theta;x,y)/\tau_{n}\right)$
for $\theta\in\Xi$, we have

$\displaystyle\frac{q_{j}((z_{j}^{\theta,k})_{\theta\in\Xi,0\leq k\leq a}\mid
X_{t_{j}}^{i}=x_{j},X_{t_{j-1}}^{i}=x_{j-1})}{q_{j}((z_{j}^{\theta,k})_{\theta\in\Xi,0\leq
k\leq a}\mid X_{t_{j}}^{i}=x_{j}^{\prime},X_{t_{j-1}}^{i}=x_{j-1}^{\prime})}$
$\displaystyle=\prod_{\theta\in\Xi}\prod_{k=0}^{a}\exp\bigg{[}\frac{\alpha_{j}}{2\tau_{n}L_{n}(a+1)}\big{(}z-f^{(k)}(\theta;x_{j},x_{j-1})\varphi_{f^{(k)},\tau_{n}}^{\theta}(x_{j-1},x_{j})\big{)}$
$\displaystyle\hskip
100.0pt-\frac{\alpha_{j}}{2\tau_{n}L_{n}(a+1)}\big{(}z-f^{(k)}(\theta;x_{j}^{\prime},x_{j-1}^{\prime})\varphi_{f^{(k)},\tau_{n}}^{\theta}(x_{j-1}^{\prime},x_{j}^{\prime})\big{)}\bigg{]}$
$\displaystyle=\prod_{\theta\in\Xi}\prod_{k=0}^{a}\exp\bigg{[}\frac{\alpha_{j}}{2\tau_{n}L_{n}(a+1)}\big{(}f^{(k)}(\theta;x_{j},x_{j-1})\varphi_{f^{(k)},\tau_{n}}^{\theta}(x_{j-1},x_{j})$
$\displaystyle\hskip
100.0pt-f^{(k)}(\theta;x_{j}^{\prime},x_{j-1}^{\prime})\varphi_{f^{(k)},\tau_{n}}^{\theta}(x_{j-1}^{\prime},x_{j}^{\prime})\big{)}\bigg{]}$
$\displaystyle\leq\prod_{\theta\in\Xi}\prod_{k=0}^{a}\exp\Big{(}\frac{\alpha_{j}}{L_{n}(a+1)}\Big{)}=\exp\big{(}\alpha_{j}\big{)},$

where we have used the fact that
$f^{(k)}(\theta;X_{t_{j-1}}^{i},X_{t_{j}}^{i})\varphi_{f^{(k)},\tau_{n}}^{\theta}(X_{t_{j-1}}^{i},X_{t_{j}}^{i})$
is bounded by ${\tau_{n}}$ from the construction of
$\varphi_{f^{(k)},\tau_{n}}^{\theta}$, together with
$\mathrm{card}(\Xi)=L_{n}$. It gives the wanted result. ∎

In order to define our estimator and derive the convergence results, we must
smoothly extend the definition of
$Z_{j}^{i,(0)}(\theta_{\ell})=f(\theta_{\ell};X_{t_{j-1}}^{i},X_{t_{j}}^{i})\varphi\left(f(\theta_{\ell};X_{t_{j-1}}^{i},X_{t_{j}}^{i})/\tau_{n}\right)+\mathcal{E}_{j}^{i,\ell,{\color[rgb]{0,0,0}(0)}}$
from the grid $\Xi$ to the entire space $\Theta$. Indeed, recalling the
contrast function without privacy is as in (5), one can easily see it will
enable us to define a smooth contrast function by summing these extensions on
$i,j$.

The extension is achieved through the application of a spline approximation
method, which is comprehensively discussed in Section 5.1. More specifically,
starting with the initial definition of $Z_{j}^{i}(\theta)$, valid for all
$\theta\in\Xi$, we will construct its spline approximation of order $a$,
denoted as ${\color[rgb]{0,0,0}H_{\Xi}}Z_{j}^{i}(\theta)$. The extended
function is well-defined for any $\theta$ and coincides with
$Z_{j}^{i,(0)}(\theta)$ within the confines of $\Xi$. More formally, the
spline extension $H_{\Xi}$ is given by a linear operator

$H_{\Xi}:\left\\{\begin{aligned}
\mathbb{R}^{L_{n}\times(a+1)}&\to\mathcal{C}^{a}([\theta_{0},\theta_{L_{n}-1}])\\\
(g_{\ell}^{\color[rgb]{0,0,0}k})_{0\leq\ell\leq L_{n}-1,0\leq k\leq
a}&\mapsto(g(\theta))_{\theta\in[\theta_{0},\theta_{L_{n}-1}]}\end{aligned}\right.$
(10)

where the function $g$ is such that $g^{(k)}(\theta_{\ell})=g_{\ell}^{k}$ for
all $\theta_{\ell}\in\Xi$. A thorough elucidation of the process involved in
creating such a function can be found in Section 5.1.

The natural contrast function in this context, defined for $\theta\in\Xi$, is
$S_{n}^{N}(\theta):=\sum_{i=1}^{N}\sum_{j=1}^{n}Z_{j}^{i{\color[rgb]{0,0,0},(0)}}(\theta)$.
Its extension to $[\theta_{0},\theta_{L_{n}-1}]$ is given by

$S^{N,\text{pub}}_{n}(\theta):=\sum_{i=1}^{N}\sum_{j=1}^{n}H_{\color[rgb]{0,0,0}\Xi}\left(Z_{j}^{i}\right)(\theta)=\sum_{i=1}^{N}\sum_{j=1}^{n}H_{\color[rgb]{0,0,0}\Xi}\left((Z_{j}^{i,(k)}(\theta_{\ell}))_{\ell,k}\right)(\theta),\qquad\theta\in[\theta_{0},\theta_{L_{n}-1}].$
(11)

With access to public data $(Z_{j}^{i}(\theta))_{1\leq i\leq N,1\leq j\leq
n}$, the statistician considers the estimator

$\widehat{\theta}_{n}^{N}=\mathop{\rm
argmax}_{\theta\in{\color[rgb]{0,0,0}[\theta_{0},\theta_{L_{n}-1}]}}{\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}(\theta)}.$
(12)

To complete the construction of the private estimator it remains to choose the
grid $\Xi$. Let us start considering the deterministic uniform grid

$\Xi:=\bigg{\\{}\theta_{\ell}=\frac{\ell}{L_{n}},\quad\ell\in\\{0,\dots,L_{n}-1\\}\bigg{\\}}.$

We will see it will be crucial for the analysis of the variance of the
estimator to determine in certain occasions whether the true value of the
parameter, denoted as $\theta^{\star}$, resides within this grid. To address
this, it will be useful in the sequel to introduce a random grid in which the
probability of $\theta^{\star}$ coinciding with any point is zero. Let us
remark that the extension of the contrast function is from the grid $\Xi$ to
$[\theta_{0},\theta_{L_{n}-1}]=[0,1-1/L_{n}]$ rather than on the whole
parameter set $\Theta=[0,1]$. It is convenient to leave out some space on the
the rightmost part of parameter space as we will introduce a random shift of
size smaller than $1/L_{n}$ of the grid in this sequel, to create the
randomization. That is why we have assumed
$\theta^{\star}\in\overset{\circ}{\Theta}=(0,1)$.

###  3.2 Main results: consistency and asymptotic normality

It is feasible to demonstrate that the estimator in (12), introduced as a
natural extension of the Maximum Likelihood Estimator (MLE) in cases of
discrete observations and under local differential privacy constraints,
exhibits several desirable properties. Notably, it is consistent, as stated in
the following theorem and proved in Section 6.
Let us recall that $\bar{\alpha}^{2}$ is the harmonic mean over the different
privacy levels. It is such that
$1/{\bar{\alpha}^{2}}=n^{-1}\sum_{j=1}^{n}1/\alpha_{j}^{2}$.

######  Theorem 1 (Consistency).

Assume that A1\- A4 hold and that
$\frac{L_{n}\log(n)}{\sqrt{N}}\sqrt{\frac{\log(L_{n})}{\bar{\alpha}^{2}}}\rightarrow
0$ for $n,N,L_{n}\rightarrow\infty$. Then, the estimator
$\widehat{\theta}_{n}^{N}$ defined in (12) is consistent:

$\widehat{\theta}_{n}^{N}\xrightarrow{\mathbb{P}}\theta^{\star}.$

Furthermore, if $\bar{\alpha}^{2}$ is chosen to be sufficiently large to
ensure that the impact of privacy constraints is negligible, it becomes viable
to regain the asymptotic normality of the estimator with a convergence rate
and a variance reminiscent of the classical scenario where no privacy
constraints are imposed (see for example [4, 15, 16]). In this case the
privacy-related influence becomes negligible. This assertion is formally
established in the subsequent theorem whose proof is postponed to Section 6.
To establish this, we introduce the following assumptions regarding the spline
approximation and the discretization step.

######  Assumption 6.

[Condition spline approximation]

  1. 1.

Assume that $a>3$ and that $\frac{\sqrt{N}}{L_{n}^{a-2}}\rightarrow 0$.

  2. 2.

Assume that $a>3$ and that
$\frac{\sqrt{N\bar{\alpha}^{2}}}{L_{n}^{a}\log(n)}\rightarrow 0$.

######  Assumption 7.

[Condition discretization step]

  1. 1.

Assume that $\sqrt{N\Delta_{n}}\rightarrow 0$.

  2. 2.

Assume that
$\frac{\sqrt{N\Delta_{n}\bar{\alpha}^{2}}}{L_{n}^{2}\log(n)}\rightarrow 0$.

Note that Conditions A6.1 and A7.1 are essential for demonstrating asymptotic
normality in scenarios where the contribution of privacy is negligible. On the
other hand, A6.2 and A7.2 are required for cases where the contribution of
privacy is significant.
Recall that $r_{n,N}=\frac{L_{n}^{2}\log(n)}{\sqrt{\bar{\alpha}^{2}}}$, as
introduced in (1).
Remark that of A6.2 can be seen as
$\frac{\sqrt{N}}{L_{n}^{a-2}}\frac{1}{r_{n,N}}\rightarrow 0$ and, similarly,
the second point of A7 translates to
$\sqrt{N\Delta_{n}}\frac{1}{r_{n,N}}\rightarrow 0$. We highlight that, in case
of significant privacy, $r_{n,N}$ goes to $\infty$. It implies that Points 1
of both the Assumptions above would be enough to obtain the wanted result. We
decided to require the conditions gathered in A6.2 and A7.2 as they are
weaker.

######  Theorem 2 (Asymptotic normality with negligible contribution of
privacy).

Assume that A1\- A5, A6.1 and A7.1 hold. Assume moreover that
$\sqrt{\log(L_{n})}\,r_{n,N}\rightarrow 0$ and
$\frac{{\color[rgb]{0,0,0}L_{n}^{3}}\log(n)}{\sqrt{N}}\sqrt{\frac{\log(L_{n})}{\bar{\alpha}^{2}}}\rightarrow
0$ for $N,n,L_{n}\rightarrow\infty$. Then

$\sqrt{N}(\widehat{\theta}_{n}^{N}-\theta^{\star})\xrightarrow{\mathcal{L}}{\mathcal{N}}\bigg{(}0,2\Big{(}\int_{0}^{T}{\mathbb{E}}\bigg{[}\Big{(}\frac{\partial_{\theta}b(\theta^{\star},X_{s})}{{\color[rgb]{0,0,0}\sigma}(X_{s})}\Big{)}^{2}\bigg{]}ds\Big{)}^{-1}\bigg{)}=:{\mathcal{N}}\big{(}0,2(\Sigma_{0})^{-1}\big{)}=:\mathcal{Z}_{1}.$

One might question the efficiency of the proposed estimator, in the scenario
of ’negligible’ privacy contribution. A recent paper [17] establishes the
local asymptotic normality (LAN) property for drift estimation in
$d$-dimensional McKean-Vlasov models under continuous observations. If we
narrow down their result to dimension 1, with a single-dimensional parameter
$\theta$, the Fisher information derived by the authors can be expressed as
follows

$\int_{0}^{T}\int_{{\mathbb{R}}}\Big{(}\frac{\partial_{\theta}b}{{\color[rgb]{0,0,0}\sigma}}\Big{)}^{2}(\theta^{\star},{t},x,\bar{\mu}_{t})\bar{\mu}_{t}(dx)dt,$

where $\bar{\mu}_{t}=\bar{\mu}_{t}^{\theta^{\star}}$ is the law of the McKean-
Vlasov SDE under the true value $\theta^{\star}$ of the parameter. In the case
of i.i.d. diffusion processes under consideration, the above quantity
simplifies to
$\int_{0}^{T}{\mathbb{E}}\left[\left(\frac{\partial_{\theta}b(\theta^{\star},X_{s})}{{\color[rgb]{0,0,0}\sigma}(X_{s})}\right)^{2}\right]ds$.
It suggests that our estimator is asymptotically efficient when the privacy
contribution is ’negligible’.

We can establish the asymptotic normality of our proposed estimator in the
case where the contribution of the privacy is the dominant one as well.
However, as evident from Theorem 3 below, the issue of privacy significantly
affects the estimation of the parameter we intend to estimate. This is
demonstrated by the degradation of the convergence rate, which now becomes
dependent on the privacy levels $\alpha_{1},\dots,\alpha_{n}$. In this case,
it is important to observe that we need to ensure that the various levels of
privacy do not differ significantly from each other, as elaborated further
below.

######  Assumption 8.

[Privacy ratio] Assume that
$\frac{\max_{j}\alpha_{j}}{\min_{j}\alpha_{j}}=O(1)$.

Moreover, in this context we need to introduce a random grid, as anticipated
before.
Let us introduce a random variable, denoted as $S$, that is uniformly drawn in
the interval $(0,1)$. The concept is that the statistician initiates the
process by generating the uniform random variable $S$ and subsequently
calculates and assigns the value $\theta_{\ell}$ as
$\theta_{\ell}:=(\ell+S)/L_{n}$ for each $\ell$ in the range from $0$ to
$L_{n}-1$. This procedure leads to the establishment of the following random
grid:

${}\Xi_{S}:=\bigg{\\{}\theta_{\ell}=\frac{\ell+S}{L_{n}},\quad\ell\in\\{0,\dots,L_{n}-1\\}\bigg{\\}}.$
(13)

We define for $s\in[0,1]$,

$\widehat{v}(s){\color[rgb]{0,0,0}:=}(2a+1)^{2}\binom{2a}{a}^{2}s^{2a}(1-s)^{2a}.$
(14)

Let $\ell_{n}^{\star}\in\\{0,\dots,L_{n}-1\\}$ be such that
$\theta^{\star}\in[\theta_{\ell_{n}^{\star}},\theta_{\ell_{n}^{\star}+1})$. We
set

${}v_{n}(\theta^{\star}){\color[rgb]{0,0,0}:=}\hat{v}(L_{n}(\theta^{\star}-\theta_{\ell_{n}^{\star}})).$
(15)

Remark that $v_{n}(\theta^{\star})$ is zero if and only if $\theta^{\star}$
belongs to the grid $(\theta_{\ell})_{0\leq\ell\leq
L_{n}-1}=((\ell+S)/L_{n})_{0\leq\ell\leq L_{n}-1}$. As the shift variable $S$
has a continuous law, the probability that $\theta^{\star}$ lies in the grid
for some $n\geq 1$ is zero. Thus, we have that, almost surely,
$v_{n}(\theta^{\star})>0$ for all $n\geq 1$. Moreover, the law of
$v_{n}(\theta^{\star})$ is independent of $\theta^{\star}$ and $n$, as stated
in next lemma, whose proof can be found in Section 7.

######  Lemma 2.

For all $n\geq 1$, $v_{n}(\theta^{\star})$ has the same law as
$\widehat{v}(S)$.

This allows us to obtain the following asymptotic result.

######  Theorem 3 (Asymptotic normality with significant contribution of
privacy).

Assume that A1\- A5, A6.2, A7.2 and A8 hold. Assume moreover that
${\color[rgb]{0,0,0}L_{n}^{3}}\log(n)\sqrt{\frac{\log(L_{n})}{N\bar{\alpha}^{2}}}\rightarrow
0$ and $r_{n,N}\rightarrow\infty$ for $N,n,L_{n}\rightarrow\infty$. Then

$\frac{\sqrt{N\bar{\alpha}^{2}}}{4{\color[rgb]{0,0,0}(a+1)}L_{n}^{2}\log(n)\sqrt{T}}(\widehat{\theta}_{n}^{N}-\theta^{\star})\xrightarrow{\mathcal{L}}{\mathcal{N}}\big{(}0,\widehat{v}(S)(\Sigma_{0})^{-2}\big{)}=:\mathcal{Z}_{2}$

where $\mathcal{Z}_{2}$ has a mixed normal distribution, with mean $0$ and
conditional variance $\widehat{v}(S)(\Sigma_{0})^{-2}$. Moreover, this
convergence holds jointly with
$v_{n}(\theta^{\star})\xrightarrow{\mathcal{L}}\widehat{v}(S)$.

Theorems 2 and 3 provide insights into how the proposed estimator behaves with
varying contributions of privacy, particularly concerning the convergence of
$\sqrt{\log(L_{n})}r_{n,N}$ towards $0$ or those of $r_{n,N}$ towards
$\infty$. As already mentioned in the introduction, the asymptotic behaviour
of $r_{n,N}$ dictates the privacy regime. In the corollary below we explore
the scenario where privacy levels
$(\alpha_{1},\dots,{\color[rgb]{0,0,0}\alpha_{n}})$ satisfy $r_{n,N}\sim
1/c_{p}>0$. Its proof is presented in Section 6.

######  Corollary 1.

Assume that A1\- A5, A6.2, A7.2 and A8 hold. Assume moreover that
${\color[rgb]{0,0,0}L_{n}^{3}}\log(n)\sqrt{\frac{\log(L_{n})}{N\bar{\alpha}^{2}}}\rightarrow
0$ and there exists $c_{p}>0$ such that $r_{n,N}\rightarrow 1/c_{p}$ for
$N,n,L_{n}\rightarrow\infty$. Then

$\frac{\sqrt{N\bar{\alpha}^{2}}}{{\color[rgb]{0,0,0}L_{n}^{2}\log(n)}}(\widehat{\theta}_{n}^{N}-\theta^{\star})\xrightarrow{\mathcal{L}}c_{p}\mathcal{Z}_{1}+4(a+1)\sqrt{T}\mathcal{Z}_{2}={\mathcal{N}}\big{(}0,(2c_{p}^{2}\Sigma_{0}+{\color[rgb]{0,0,0}16(a+1)^{2}T}\widehat{v}(S))(\Sigma_{0})^{-2}\big{)}$

and this convergence holds jointly with
$v_{n}(\theta^{\star})\xrightarrow{\mathcal{L}}\widehat{v}(S)$.

Corollary 1 emphasizes the crucial roles played by both $\mathcal{Z}_{1}$ and
$\mathcal{Z}_{2}$ in our analysis. The convergence rate of the proposed
estimator towards the sum reveals that, in the scenario of negligible privacy,
the convergence is towards $\mathcal{Z}_{1}$, whereas in the case of a
significant contribution of privacy, the convergence is towards
$\mathcal{Z}_{2}$.

An intriguing question emerges: How can one thoughtfully choose the parameter
$L_{n}$ while ensuring its compliance with the aforementioned conditions? It
is noteworthy that $L_{n}=O(n^{r})$, and $r$ arbitrarily small is a feasible
choice. In this case, the choice of the order of the spline approximation $a$
becomes pivotal, requiring it to be sufficiently large to ensure that $ar$
satisfies the assumptions outlined in A6.

###  3.3 Effective privacy

When considering local differential privacy, a natural question arises
regarding the likelihood of recovering private data from the observation of
public data. Specifically, the definition of $\alpha_{j}$-LDP provided in (4)
informs us about how well the values of $X^{i}_{t_{j}}$ and $X^{i}_{t_{j-1}}$
are covered when the value of $Z^{i}_{j}$ is available. However, some
information about $X^{i}_{t_{j}}$ and $X^{i}_{t_{j-1}}$ could be conveyed by
$Z^{i}_{k}$, for $k\neq j$. Therefore, it is worth understanding how precisely
the values of $X^{i}_{t_{j}}$ and $X^{i}_{t_{j-1}}$ could be revealed by the
observations of $Z^{i}_{1},\dots,Z^{i}_{n}$, which are publicly available.
Similar questions have been explored, for example, in [51] and in Section 4.1
in [2]. In particular, it is well understood that if a vector is privatized
with an independent channel for each component and the components are
independent, then no information on a private component is carried by the
public views of the others.
The situation becomes more intricate if the components of the vector
$(X^{i}_{t_{0}},\dots,X^{i}_{t_{n}})$ are dependent, as the observation of
$Z^{i}_{1},\dots,Z^{i}_{n}$ imparts extra information on $X^{i}_{t_{j}}$ and
$X^{i}_{t_{j-1}}$. This is evident in the case under consideration, as the
observations $X^{i}_{t_{0}},\dots,X^{i}_{t_{n}}$ represent the evolution of
one individual, and they are therefore dependent.
For simplicity, assume that $\alpha_{j}=\alpha$ for all $j\in\\{1,\dots,n\\}$.
Let us denote by $q\big{(}(z_{h}^{\theta,k})_{\theta\in\Xi,0\leq k\leq a,0\leq
h\leq n}\mid X_{t_{j}}^{i}=x_{j},X_{t_{j-1}}^{i}=x_{j-1}\big{)}$ the density
of the public data $Z_{1}^{i}=(Z_{1}^{i,(k)}(\theta))_{\theta\in\Xi,0\leq
k\leq a}\in\mathbb{R}^{L_{n}\times(a+1)},\dots,\\\
Z_{n}^{i}=(Z_{n}^{i,(k)}(\theta))_{\theta\in\Xi,0\leq k\leq
a}\in\mathbb{R}^{L_{n}\times(a+1)}$ conditional to $X_{t_{j}}^{i}=x_{j}$,
$X_{t_{j-1}}^{i}=x_{j-1}$ for $j\in\\{1,\dots,n\\}$ and $i\in\\{1,\dots,N\\}$.
We apply Proposition 4.1 in [2] to the privacy mechanism introduced in Section
3.1 and obtain

$\frac{q((z_{h}^{\theta,k})_{\theta\in\Xi,0\leq k\leq a,0\leq h\leq n}\mid
X_{t_{j}}^{i}=x_{j},X_{t_{j-1}}^{i}=x_{j-1})}{q((z_{h}^{\theta,k})_{\theta\in\Xi,0\leq
k\leq a,0\leq h\leq n}\mid
X_{t_{j}}^{i}=x_{j}^{\prime},X_{t_{j-1}}^{i}=x_{j-1}^{\prime})}\leq\exp(\alpha+2(n-1)\alpha)=\exp((2n-1)\alpha),$
(16)

where in the application of Proposition 4.1 in [2] we use the crude bound
$\Delta_{\text{ind}}\leq 2$. This means that we neglect any possible
decorrelation arising in the different components of
$(X^{i}_{t_{j}})_{j=1,\dots,n}$. Equation (16) evaluates how the privacy of
$X^{i}_{t_{j}}$ and $X^{i}_{t_{j-1}}$ is deteriorated by the observation of
the side-channels $Z^{i}_{1},\dots,Z^{i}_{n}$.
It is important to note that, according to [51], if someone accesses some
private values $x_{j}$, $x_{j}^{\prime}$, $x_{j-1}$, and $x_{j-1}^{\prime}$,
it will be impossible for them to determine with a high level of certainty
which of the values corresponds to a specific public observation, denoted as
$(Z^{i}_{j})_{1\leq j\leq n}$. Any attempt to make a decision in this regard
would result in an error, with a probability of at least
$\frac{1}{1+e^{{\color[rgb]{0,0,0}(2n-1)\alpha}}}$, as $(2n-1)\alpha$ is the
privacy level appearing in (16). Hence, one would like the effective privacy
level $\alpha_{f}$ to be fixed, implying the choice
${\alpha}=:\frac{\alpha_{f}}{2n-1}$, where $f$ stands for ’fixed’.

Observe that, in such a case, the condition
$\frac{N^{-1/2}}{L_{n}}r_{n,N}\rightarrow 0$ needed for consistency translates
to
$\frac{1}{{\color[rgb]{0,0,0}\log(L_{n})}(L_{n}\log(n))^{2}}N\frac{\alpha_{f}^{2}}{n^{2}}\rightarrow\infty$.
If we stick to the policy where the grid size $L_{n}$ tends slowly to infinity
and neglect its contribution, as the one of the $\log$-terms, it implies
$N>>n^{2}$. One might wonder if that is compatible with the condition on the
discretization step that puts $N$ and $n$ in a relationship, as required for
the asymptotic normality of our estimator. The case of ’negligible’ privacy is
not compatible with the choice $\alpha=\frac{\alpha_{f}}{2n-1}$. Hence, we
turn to the study of the regime of ’significant’ privacy only. In such a case,
condition A7.2 is implied by $N\Delta_{n}\frac{\alpha_{f}^{2}}{n^{2}}\leq c$
for some $c>0$. Hence, we are asking $N\lesssim n^{3}$.

In summary, this section helps us understand that, to achieve an effective
level of privacy as performant as possible, the number of individuals $N$
providing data and the number of data per person $n$ should be linked in a way
ensuring that $n^{2}<<N\lesssim n^{3}$. We also see that the choice
$\alpha\sim\alpha_{f}/(2n)$ impacts the rate of estimation as
$\sqrt{N\bar{\alpha}^{2}}\sim c\sqrt{N/n^{2}}$. In the best situation,
corresponding to $N\sim n^{3}$, this rate is of magnitude at most
$\sqrt{n^{3}/n^{2}}\sim N^{1/6}$, showing a slow rate of estimation, as a
function of the number of individuals.

##  4 Concluding remarks

This section proposes a discussions arising from the insights gained through
this paper. As highlighted earlier, we introduced the concept of local
differential privacy (LDP) for diffusion processes and proposed an estimator
for parameter estimation of the drift coefficient of i.i.d. diffusion under
LDP constraints. Our main findings include the consistency and asymptotic
normality of the proposed estimator. The asymptotic normality is achieved at
two distinct convergence rates, dependent on whether the term due to the
privacy constraint is the primary contribution or not, leading to a dichotomy
(up to a logarithmic term). Specifically, when $r_{n,N}$ (defined as in (1))
tends to $\infty$, the contribution of privacy constraints is significant. On
the other hand, if $r_{n,N}\sqrt{\log(L_{n})}\rightarrow 0$, privacy is
negligible, and results align with those in the case without privacy. In the
threshold case, where $r_{n,N}$ converges towards a constant, the proposed
estimator converges to the sum of the two random variables obtained in the two
previous cases. This implies that the general limit of the estimator depends
on whether ’significant’ or ’negligible’ privacy dominates.
In the following table, we compare the conditions and results in the case of
negligible or significant contributions of privacy, recalling that
$r_{n,N}=\frac{L_{n}^{2}\log(n)}{\sqrt{\bar{\alpha}^{2}}}$.

| Negligible Privacy | Significant Privacy
---|---|---
Consistency |
Assumptions |  A1\- A4
Perfect privacy not allowed | $\frac{N^{-1/2}}{L_{n}}r_{n,N}\rightarrow 0$
Asymptotic normality |  |
Dichotomy privacy | $\sqrt{\log(L_{n})}r_{n,N}\rightarrow 0$ | $r_{n,N}\rightarrow\infty$
Spline approximation A6 |  $a>3$, $\frac{\sqrt{N}}{L_{n}^{a-2}}\rightarrow 0$ |  $a>3$, $\frac{\sqrt{N\bar{\alpha}^{2}}}{L_{n}^{a}\log(n)}\rightarrow 0$
Discretization A7 | $\sqrt{N\Delta_{n}}\rightarrow 0$ | $\frac{\sqrt{N\Delta_{n}\bar{\alpha}^{2}}}{L_{n}^{2}\log(n)}\rightarrow 0$
Privacy ratio A8 | Not needed | Needed
Result for $n,N\rightarrow\infty$ | $\sqrt{N}(\widehat{\theta}_{n}^{N}-\theta^{\star})\xrightarrow{\mathcal{L}}{\mathcal{N}}(0,2\Sigma_{0}^{-1})$ | $\frac{\sqrt{N\bar{\alpha}^{2}}}{4L_{n}^{2}{\color[rgb]{0,0,0}(a+1)}\log(n)\sqrt{T}}(\widehat{\theta}_{n}^{N}-\theta^{\star})\xrightarrow{\mathcal{L}}\mathcal{N}(0,\widehat{v}(S)\Sigma_{0}^{-2})$

Here, we address potential extensions and limitations of our main results.
Firstly, it is important to note that in the presence of privacy constraints,
the levels of privacy $\alpha_{j}$ are commonly assumed to be smaller than 1.
Consequently, the condition of negligible privacy (i.e.
$\sqrt{\log(L_{n})}r_{n,N}\rightarrow 0$) is quite stringent and rarely met in
practice. Therefore, the case of ’significant’ privacy emerges as potentially
the most interesting. In such case, we obtain a convergence rate slower than
$\sqrt{N\bar{\alpha}^{2}}$ mainly due to the necessity of introducing a grid
with size $L_{n}$ and the subsequent need for spline approximation. It may be
possible to enhance the convergence rate by adopting a different approach, at
least in certain models, and the rate optimality of the estimators is an
unsolved problem. For instance, in the case where $b$ is polynomial in
$\theta$, the spline approximation is exact, and it is possible to show that
the conditions as in A6 are no longer necessary. This would enable us to prove
our results even without the condition $L_{n}\rightarrow\infty$, allowing us
the flexibility to choose $L_{n}$ as a constant.
Another remark stems from the dependence among data related to the same
individual, impacting the effective privacy. This deviation from the privacy
level introduced in Section 2.2 results from the extra information carried by
side channels, as detailed in Section 3.3. This discrepancy leads to a
connection between $N$ and $n$, prompting us to ponder what would happen in
the case where the time horizon $T$ approaches infinity. While studying the
fixed time horizon seems natural from a practical standpoint as it aligns with
the model that better fits reality, from a mathematical perspective,
understanding the behavior as $T$ approaches infinity in the ergodic case
becomes intriguing. In this scenario, allowing for more distant (and less
dependent) observations, such as in the case where the discretization step is
fixed, would reduce the amount of retrievable information from the side
channel, ensuring better effective privacy.

In conclusion, from our current knowledge this project is a first instance of
local differential privacy for continuous time stochastic processes. We
believe that it can be an informative starting point, aiding statisticians in
comprehending some challenges associated with local differential privacy for
evolving time-based data. Additionally, it offers some insights into potential
directions for future research.

##  5 Preliminary results

Before proving our main results, let us introduce some notation and provide
some tools that will be useful in the sequel.
In particular, we will start by providing a detailed introduction about spline
functions.

###  5.1 Splines, some tools

Here are some basic refreshments on B-splines. For a more detailed exposé, the
reader can refer to the work of Lyche et al. [36] from which our presentation
is widely inspired.

#### Definition and properties of B-splines

B-splines are piecewise polynomial functions characterized by :

  * •

A knot vector $\bm{t}=(t_{1},\dots,t_{M})$ that is a nondecreasing sequence of
$M$ elements of some interval $I\subset{\mathbb{R}}$,

  * •

An integer $p$ called the degree of the spline.

In our frame, we exclusively consider B-splines whose associated knot sequence
$\bm{t}$ satisfies $M=(a+1)({\color[rgb]{0,0,0}\Lambda}+3)$ for some
${\color[rgb]{0,0,0}\Lambda},a\in{\mathbb{N}}^{*}$ and such that the knots are
repeated as follows:

$t_{1}=\dots=t_{2a+2},\quad{\color[rgb]{0,0,0}t_{(\ell+1)(a+1)+1}=\dots=t_{(\ell+2)(a+1)},\;\ell\in\\{1,\dots,{\color[rgb]{0,0,0}\Lambda}-1\\},\quad\text{and}\quad
t_{(a+1)({\color[rgb]{0,0,0}\Lambda}+1)+1}=\dots=t_{M}.}$

For the sake of convenience we introduce an auxiliary sequence, i.e. the
sequence of interpolation points
$\bm{\xi}=(\xi_{\ell},\ell\in\\{-1,\dots,{\color[rgb]{0,0,0}\Lambda+1}\\})$
built from $\bm{t}$ as

$\xi_{\ell}=:t_{(a+1)(\ell+1)+1}=\dots=t_{(a+1)(\ell+2)},\;\ell\in\\{-1,\dots,{\color[rgb]{0,0,0}\Lambda}+1\\}.$
(17)

Note that $\xi_{-1}=\xi_{0}$ and
$\xi_{{\color[rgb]{0,0,0}\Lambda}}=\xi_{{\color[rgb]{0,0,0}\Lambda}+1}$: the
points $\xi_{-1}$ and $\xi_{{\color[rgb]{0,0,0}\Lambda}+1}$ have only been
introduced for convenience. To sum up,

${}\bm{t}=(\underbrace{\xi_{-1},\dots,\xi_{-1}}_{a+1\,\text{times}},\underbrace{\xi_{0},\dots,\xi_{0}}_{a+1\,\text{times}},\dots,\underbrace{\xi_{{\color[rgb]{0,0,0}\Lambda}+1},\dots,\xi_{{\color[rgb]{0,0,0}\Lambda}+1}}_{a+1\,\text{times}}).$
(18)

From now on, we assume that the knots are equally spaced, i.e. for all
$\ell\in\\{{\color[rgb]{0,0,0}0},\dots,{\color[rgb]{0,0,0}\Lambda}-1\\}$

$\xi_{\ell+1}-\xi_{\ell}=1/{\color[rgb]{0,0,0}\Lambda}.$

We recall the definition of B-splines given in Lyche et al. [36]:

######  Definition 1.

Let
$h\in\\{1,\dots,({\color[rgb]{0,0,0}a+1})({\color[rgb]{0,0,0}\Lambda}+1)\\}$.
The $h$-th B-spline $B_{h,p,\bm{t}}:I\rightarrow{\mathbb{R}}$ with degree
$p\leq 2a+1$ is identically zero if $t_{h+p+1}=t_{h}$ and is otherwise defined
recursively by

$B_{h,p,\bm{t}}:=\frac{\cdot-
t_{\color[rgb]{0,0,0}h}}{t_{h+p}-t_{h}}B_{h,p-1,\bm{t}}+\frac{t_{h+p+1}-\cdot}{t_{h+p+1}-t_{h+1}}B_{h+1,p-1,\bm{t}},$
(19)

starting with

$B_{{\color[rgb]{0,0,0}h},0,\bm{t}}:=\left\\{\begin{array}[]{cl}1&\text{on}\;[t_{\color[rgb]{0,0,0}h},t_{{\color[rgb]{0,0,0}h}+1})\\\
0&\text{otherwise}.\end{array}\right.$

Note that the definition of the spline $B_{h,p,\bm{t}}$ relies only on the
sequence of $p+2$ knots $t_{h}\leq t_{h+1}\leq\dots\leq t_{h+p+1}$ and that
the support of $B_{h,p,\bm{t}}$ is $[t_{h},t_{h+p+1}]$. In the sequel, unless
stated otherwise, we consider B-spline of order $p=2a+1$. For the sake of
readability, we will simply denote $B_{h}$ for $B_{h,2a+1,\bm{t}}$; i.e. the
B-spline with order $p=2a+1$ and underlying knot sequence $\bm{t}$ defined in
Equation (18). It will be helpful to use the notation:

${\color[rgb]{0,0,0}B_{i}^{k}:=B_{h,2a+1,\bm{t}},\text{ with
$h=(i+1)(a+1)+k+1$ }}$ (20)

where $k\in\\{0,\dots,a\\}$ and
$i\in\\{-1,\dots,{\color[rgb]{0,0,0}\Lambda}-1\\}$. Note that the support of
$B_{i}^{k}$ is $[\xi_{i},\xi_{i+2}]$. We will refer to $B_{i}^{k}$ as the
$p$-order B-splines which starts at position $i$ and whose shape is $k$. For
$i\in\\{0,\dots,{\color[rgb]{0,0,0}\Lambda}-2\\}$ the functions $B_{i}^{k}$
are $a$-differentiable, recalling that the smoothness of a spline function
$B_{h,p,\bm{t}}$ is equal to the order $p$ minus the maximal multiplicity of a
knot appearing in the sequence $t_{h},\dots,t_{h+p+2}$. The spline functions
$B_{-1}^{k}$ are right continuous at $\xi_{0}$ and $a$-differentiable on
$(\xi_{0},\xi_{1}]$. Following the convention in [36], to avoid the asymmetry
of splines decomposition on the closed interval
$[\xi_{0},\xi_{\color[rgb]{0,0,0}\Lambda}]$, we set
$B_{{\color[rgb]{0,0,0}\Lambda}-1}^{k}(\xi_{{\color[rgb]{0,0,0}\Lambda}})=\lim_{x\to\xi_{\Lambda^{-}}}B_{{\color[rgb]{0,0,0}\Lambda}-1}^{k}(x)$.
With this modification, the functions $B_{{\color[rgb]{0,0,0}\Lambda}-1}^{k}$
are $a$-differentiable on
$[\xi_{{\color[rgb]{0,0,0}\Lambda}-1},\xi_{\color[rgb]{0,0,0}\Lambda})$ and
left continuous at $\xi_{\color[rgb]{0,0,0}\Lambda}$. The spline functions
$(B_{i}^{k})_{i,k}$ restricted to $[\xi_{0},\xi_{\color[rgb]{0,0,0}\Lambda}]$
generate a linear space of piecewise polynomial functions with degree $2a+1$,
regularity of class ${\mathcal{C}}^{a}$ on
$(\xi_{0},\xi_{\color[rgb]{0,0,0}\Lambda})$, and admitting one-sided
derivatives at the boundaries of this interval.

######  Remark .

As knots are equally spaced, i.e.
$\xi_{\ell+1}-\xi_{\ell}=1/{\color[rgb]{0,0,0}\Lambda}$, it is easy to see
from (19) that the spline functions are sharing a common scaling factor
${\color[rgb]{0,0,0}\Lambda}$. Applying Proposition 11 in [36], we also get

$\sup_{x\in[\xi_{0},\xi_{\color[rgb]{0,0,0}\Lambda}]}\Big{|}\frac{\partial^{m}}{\partial
x^{m}}B_{i}^{k}(x)\Big{|}\leq c{\color[rgb]{0,0,0}\Lambda}^{m}$ (21)

for $m\in\\{0,\dots,a\\},\leavevmode\nobreak\
i\in\\{-1,\dots,\Lambda-1\\},\leavevmode\nobreak\ k\in\\{0,\dots,a\\}$.

#### Hermite interpolation

We denote $I^{\neq,{\color[rgb]{0,0,0}\Lambda}+1}$ the set consisting of
$({\color[rgb]{0,0,0}\Lambda}+1)$-tuples
$\bm{\xi}=(\xi_{0},\dots,\xi_{{\color[rgb]{0,0,0}\Lambda}})$ of $I$ with
$\xi_{0}<\xi_{1}<\cdots<\xi_{\color[rgb]{0,0,0}{\color[rgb]{0,0,0}\Lambda}}$.

######  Definition 2 (Hermite interpolation).

Let $a\in{\mathbb{N}}^{*}$,
$\bm{\xi}=(\xi_{\ell},\ell\in\\{0,\dots,{\color[rgb]{0,0,0}\Lambda}\\})\in
I^{\neq,{\color[rgb]{0,0,0}\Lambda}+1}$,
$\mathfrak{a}_{\ell}^{(k)}\in{\mathbb{R}}$ for
$\ell\in\\{0,\dots,\Lambda\\},\,k\in\\{0,\dots,a\\}$ and
$\mathfrak{a}=(\mathfrak{a}_{\ell}^{(k)},\,\ell\in\\{0,\dots,{\color[rgb]{0,0,0}\Lambda}\\},\,k\in\\{0,\dots,a\\})$.
If it exists, a function
$H_{\bm{\xi}}({\color[rgb]{0,0,0}\mathfrak{a}}):[\xi_{0},\xi_{\color[rgb]{0,0,0}\Lambda}]\rightarrow{\mathbb{R}}$
of class ${\mathcal{C}}^{a}$ such that

${\color[rgb]{0,0,0}H_{\bm{\xi}}({\mathfrak{a}})(\xi_{\ell})=\mathfrak{a}_{\ell}^{(k)}\,;\,\forall
k=0,\dots,a,\,\ell=0,\dots,{\color[rgb]{0,0,0}\Lambda},}$

is called a Hermite interpolation of
$\mathfrak{a}=(\mathfrak{a}_{\ell}^{(k)},\,\ell\in\\{0,\dots,{\color[rgb]{0,0,0}\Lambda}\\},\,k\in\\{0,\dots,a\\})$
at $\bm{\xi}=(\xi_{\ell},\ell\in\\{0,\dots,{\color[rgb]{0,0,0}\Lambda}\\})$
with order $a$.

By Mummy [34], it is possible to relate Hermite interpolation with the
B-splines functions defined in (20). We set $\xi_{-1}=\xi_{0}$,
$\xi_{{\color[rgb]{0,0,0}\Lambda+1}}=\xi_{{\color[rgb]{0,0,0}\Lambda}}$ and
define for $-1\leq i\leq{\Lambda}-1$ and $0\leq k\leq a$,

$g_{i}^{k}(x)=\frac{1}{{\color[rgb]{0,0,0}(2a+1)!}}(x-\xi_{i})^{\beta_{1}^{k}}(x-\xi_{i+1})^{\beta_{2}^{k}}(x-\xi_{i+2})^{\beta_{3}^{k}}$
(22)

with

$\beta_{1}^{k}=(a+1)-k-1=a-k,\;\beta_{2}^{k}=a+1,\;\beta_{3}^{k}=k.\\\ $

The following result can be found in [34].

######  Proposition 1.

Let
$\mathfrak{a}=(\mathfrak{a}_{\ell}^{(k)},\,\ell\in\\{0,\dots,\Lambda\\},\,k\in\\{0,\dots,a\\})$.
The function
$H_{\bm{\xi}}(\mathfrak{a}):{\color[rgb]{0,0,0}[\xi_{0},\xi_{\color[rgb]{0,0,0}\Lambda}]}\rightarrow{\mathbb{R}}$
such that

$H_{\bm{\xi}}(\mathfrak{a})=\sum_{i=-1}^{{\color[rgb]{0,0,0}\Lambda}-1}\sum_{k=0}^{a}c_{i}^{k}B_{i}^{k},$
(23)

where

$c_{i}^{k}(\mathfrak{a})=\sum_{v=0}^{{\color[rgb]{0,0,0}a}}\frac{(-1)^{v}}{v!}(g_{i}^{k})^{(p-v)}(\xi_{i+1})\mathfrak{a}_{i+1}^{v},$
(24)

$p=2a+1$ and the $g_{i}^{k}$ are given by (22), is a Hermite interpolation of
$\mathfrak{a}$ with order $a$.

Remark that although the formula (23) defines a function on the real line, the
function $H_{\xi}(\mathfrak{a})$ is equal to zero outside of
$[\xi_{0},\xi_{\color[rgb]{0,0,0}\Lambda}]$, and it is thus generally
${\mathcal{C}}^{a}$ only on the interval
$[\xi_{0},\xi_{\color[rgb]{0,0,0}\Lambda}]$.

For any function $f\in{\mathcal{C}}^{a}$, we define its projection $H_{\xi}f$
on the spline space by setting

$H_{\xi}f=H_{\xi}(\mathfrak{a})\text{\, with
$\mathfrak{a}_{i}^{k}=f^{(k)}(\xi_{i})$ for $0\leq
i\leq{\color[rgb]{0,0,0}\Lambda}$ and $0\leq k\leq a$.}$ (25)

When a function $f$ is already an element of the space generated by the
splines functions, then (23)–(24) is the usual representation of splines given
for instance in Section 1.2.4 of [36].

######  Remark .

Hermite interpolation preserves polynomial functions with order not larger
than $a$, i.e. for any polynomial function with degree $k\leq a$,

${\color[rgb]{0,0,0}H_{\bm{\xi}}P(x)=P(x)\quad\text{ for
$x\in[\xi_{0},\xi_{\color[rgb]{0,0,0}\Lambda}]$}.}$

The map $H_{\bm{\xi}}$ is linear as stated in the following lemma:

######  Lemma 3.

Let $\phi,\psi:I\rightarrow{\mathbb{R}}$ and their B-spline interpolating
functions at $\bm{\xi}$ respectively denoted by $H_{\bm{\xi}}\phi$ and
$H_{\bm{\xi}}\psi$. The B-spline interpolating function of $(f+g)$ denoted by
$H_{\bm{\xi}}(f+g)$ satisfies

$H_{\bm{\xi}}(\phi+\psi)=H_{\bm{\xi}}\phi+H_{\bm{\xi}}\psi.$

###### Proof.

By the characterization (23)–(25) of $H_{\bm{\xi}}$, we have

$H_{\bm{\xi}}(\phi+\psi)=\sum_{i=-1}^{{\color[rgb]{0,0,0}\Lambda}-1}\sum_{k=0}^{a}c_{i}^{k}(\phi+\psi)B_{i}^{k},$

where

$\displaystyle c_{i}^{k}(\phi+\psi)$
$\displaystyle=\sum_{m=0}^{a}\frac{(-1)^{m}}{m!}(g_{i}^{k})^{(p-m)}(\xi_{i+1})(\phi+\psi)^{(m)}(\xi_{i+1})$
$\displaystyle=\sum_{m=0}^{a}\frac{(-1)^{m}}{m!}(g_{i}^{k})^{(p-m)}(\xi_{i+1})\phi^{(m)}(\xi_{i+1})+\sum_{m=0}^{a}\frac{(-1)^{m}}{m!}(g_{i}^{k})^{(p-m)}(\xi_{i+1})\psi^{(m)}(\xi_{i+1})$
$\displaystyle=c_{i}^{k}(\phi)+c_{i}^{k}(\psi).$

Then,

$H_{\bm{\xi}}(\phi+\psi)=\sum_{i=-1}^{{\color[rgb]{0,0,0}\Lambda}-1}\sum_{k=0}^{a}c_{i}^{k}(\phi+\psi)B_{i}^{k}=\sum_{i=-1}^{{\color[rgb]{0,0,0}\Lambda}-1}\sum_{k=0}^{a}c_{i}^{k}(\phi)B_{i}^{k}+\sum_{i=-1}^{{\color[rgb]{0,0,0}\Lambda}-1}\sum_{k=0}^{a}c_{i}^{k}(\psi)B_{i}^{k}=H_{\bm{\xi}}(\phi)+H_{\bm{\xi}}(\psi).$

Hence the result. ∎

The following results provide insights on the interpolation $H_{\bm{\xi}}$.

######  Lemma 4.

For all collections $\bm{\xi}\in I^{\neq,{\color[rgb]{0,0,0}\Lambda}+1}$, with
$\xi_{\ell+1}-\xi_{\ell}=1/\Lambda$ for $\ell=0,\dots,\Lambda-1$, and
$\mathfrak{a}:=(\mathfrak{a}_{\ell}^{(k)},\,\ell\in\\{0,\dots,{\color[rgb]{0,0,0}\Lambda}\\},\,k\in\\{0,\dots,a\\})$,
the following holds true:

${\color[rgb]{0,0,0}\sup_{x\in[\xi_{0},\xi_{\color[rgb]{0,0,0}\Lambda}]}\Big{|}\frac{\partial^{u}}{\partial
x^{u}}H_{\bm{\xi}}(\mathfrak{a})(x)\Big{|}\leq
c{\color[rgb]{0,0,0}\Lambda}^{u}\sup_{\ell,k}|\mathfrak{a}_{\ell}^{(k)}|,}$

where $H_{\bm{\xi}}(\mathfrak{a})$ is Hermite interpolant to the data
$\mathfrak{a}$ at $\bm{\xi}$.

###### Proof.

From Proposition 1, the function $H_{\bm{\xi}}(\mathfrak{a})$ is defined by

$H_{\bm{\xi}}(\mathfrak{a})=\sum_{i=-1}^{{\color[rgb]{0,0,0}\Lambda}-1}\sum_{k=0}^{a}\sum_{v=0}^{{\color[rgb]{0,0,0}a}}\frac{(-1)^{v}}{v!}(g_{i}^{k})^{(p-v)}(\xi_{i+1})\mathfrak{a}_{i+1}^{(v)}B_{i}^{k}.$

We recall that $p=2a+1$. Using (21) and noting that
$|(g_{i}^{k})^{(p-v)}(\xi_{i+1})|$ is bounded by $1$, we get

$\displaystyle{\color[rgb]{0,0,0}\sup_{x\in[\xi_{0},\xi_{\color[rgb]{0,0,0}\Lambda}]}}\Big{|}\frac{\partial^{u}}{\partial
x^{u}}H_{\bm{\xi}}(\mathfrak{a})(x)\Big{|}$
$\displaystyle\leq\sum_{i=-1}^{{\color[rgb]{0,0,0}\Lambda}-1}\sum_{k=0}^{a}\sum_{v=0}^{{\color[rgb]{0,0,0}a}}{\color[rgb]{0,0,0}\frac{1}{v!}|\mathfrak{a}_{i+1}^{(v)}|}{\color[rgb]{0,0,0}\sup_{x\in[\xi_{0},\xi_{\color[rgb]{0,0,0}\Lambda}]}}\Big{|}\frac{\partial^{u}}{\partial
x^{u}}B_{i}^{k}(x)\Big{|}$ $\displaystyle\leq
c{\color[rgb]{0,0,0}\Lambda^{u}}{\color[rgb]{0,0,0}\sup_{\ell,k}|\mathfrak{a}_{\ell}^{(k)}|}$

for some constant $c>0$ that does not depend on ${\color[rgb]{0,0,0}\Lambda}$.
∎

######  Proposition 2.

Let $\bm{\xi}=(\xi_{\ell},\ell\in\\{0,\dots,{\color[rgb]{0,0,0}\Lambda}\\})\in
I^{\neq,{\color[rgb]{0,0,0}\Lambda}+1}$ such that
$\xi_{\ell+1}-\xi_{\ell}=1/{\color[rgb]{0,0,0}\Lambda}$ for
$\ell=0,\dots,{\color[rgb]{0,0,0}\Lambda}-1$, and let $f:I\to\mathbb{R}$ a
function of class ${\mathcal{C}}^{a+1}$. We denote its Hermite interpolation
$H_{\bm{\xi}}f$ at $\bm{\xi}$ defined as (25). For all $k\in\\{0,\dots,a\\}$,
there exists a constant $c>0$ independent of $f$ such that

$\sup_{x\in[\xi_{0},\xi_{\color[rgb]{0,0,0}\Lambda}]}\left|\frac{\partial^{k}}{\partial
x^{k}}f(x)-\frac{\partial^{k}}{\partial
x^{k}}H_{\xi}f(x)\right|\leq\frac{c}{{\color[rgb]{0,0,0}\Lambda}^{a+1-k}}\sup_{x\in[\xi_{0},\xi_{\color[rgb]{0,0,0}\Lambda}]}\left|\frac{\partial^{a+1}}{\partial
x^{a+1}}f(x)\right|.$ (26)

###### Proof.

The proof is postponed to Subsection 7.2 ∎

######  Remark .

If the function $f$ in the previous proposition is a polynomial function of
order at most $a$, then we see that the right hand side in (26) is zero. This
is not surprising as we know that the Hermite approximation preserves
polynomial functions.

In our asymptotics, we will see that the number of point $\Lambda$ will be
asymptotically equivalent to $L_{n}$. Hence, it will tend to infinity and so
it will be convenient to use the scaling and translation properties (see
Section 1.1.1 in [36]) of the spline functions to represent $B_{\ell}^{k}$.
Assuming that $\xi_{\ell+1}-\xi_{\ell}=1/{\color[rgb]{0,0,0}\Lambda}$ for
$\ell\in\\{0,\dots,{\color[rgb]{0,0,0}\Lambda}-1\\}$, we can represent, for
$k\in\\{0,\dots,a\\}$,
$\ell\in\\{0,\dots,{\color[rgb]{0,0,0}\Lambda}-{\color[rgb]{0,0,0}1}\\}$,

$B_{\ell}^{k}(x)=\overline{B}_{k}({\color[rgb]{0,0,0}\Lambda}(x-\xi_{\ell})),$

where $\overline{B}_{k}$ is the spline function of order $p=2a+1$ constructed
on the knots $0,1,2$ where the knot $0$ is repeated $a+1-k$ times, while the
knot $1$ is repeated $a+1$ times and the knot 2 is repeated $k+1$ times. We
have the following result:

######  Lemma 5.

With previous notation, for all $x\in{\color[rgb]{0,0,0}\mathbb{R}}$,

$\displaystyle\sum_{k=0}^{a}\overline{B}_{k}^{\prime}(x)=(2a+1)\binom{2a}{a}\Big{[}x^{a}(1-x)^{a}{\color[rgb]{0,0,0}\mathbf{1}_{[0,1]}(x)}-(x-1)^{a}(2-x)^{a}{\color[rgb]{0,0,0}\mathbf{1}_{[1,2]}(x)}\Big{]}.$

###### Proof.

The proof is postponed to Subsection 7.3. ∎

#### Public data extension by B-splines

Starting from the notation introduced in the previous subsection on B-splines,
we consider the case where $I=\Theta={\color[rgb]{0,0,0}[0,1]}$,
$\Lambda=L_{n}-1$ and $\xi_{\ell}=\theta_{\ell}$ for all
$\ell\in\\{0,\dots,\Lambda\\}$, i.e. the sequence of interpolation points (17)
is here the sequence of grid points
$(\theta_{\ell},\,\ell\in\\{0,\dots,L_{n}-1\\})$. Now, the operator $H_{\Xi}$
given in (10) is the operator $H_{\xi}$ detailed in the previous subsection.
We recall that for any $(i,j)\in\\{1,\dots,N\\}\times\\{1,\dots,n\\}$, the
public data $Z_{j}^{i}=(Z_{j}^{i,(k)}(\theta))_{\theta\in\Xi,0\leq k\leq
a}\in\mathbb{R}^{L_{n}\times(a+1)}$ is available. It encapsulates some proxy
for the values of $f(\theta;X^{i}_{t_{j-1}},X^{i}_{t_{j}})$, where $\theta$ is
an element of the grid, as well as for its derivatives up to order $a$.
Applying for any $(i,j)\in\\{1,\dots,N\\}\times\\{1,\dots,n\\}$ the
Proposition 1 to the data $Z_{j}^{i}$ we define
$H_{\Xi}Z_{j}^{i}:=H_{\xi}Z_{j}^{i}$ the Hermite interpolation of $Z_{j}^{i}$
on the grid $\Xi$, i.e. such that for all $k\in\left\\{0,\dots,a\right\\}$,
$\theta\in\Xi=\\{\theta_{0},\dots,\theta_{L_{n}-1}\\}$,

$(H_{\color[rgb]{0,0,0}\Xi}Z_{j}^{i})^{(k)}(\theta)={\color[rgb]{0,0,0}Z_{j}^{i,{(k)}}(\theta).}$

To lighten the notation we will write $H$ for $H_{\Xi}$.

###  5.2 Technical results

In the following, it will be particularly convenient to keep track of the size
of the different terms we will be working with. The definition of the function
$R_{t,n}^{N}$ as below will help us in such sense.
Define ${\mathcal{F}}_{t}^{N}=\sigma\big{(}X_{s}^{i},\,s\leq
t,\,i\in\\{1,\dots,N\\}\big{)}\vee\sigma\big{(}\mathcal{E}_{u}^{i,\ell,(k)},\,t_{u}\leq
t,\,i\in\\{1,\dots,N\\},\,\ell\in\\{0,\dots,L_{n}-1\\},k\in\\{0,...,a\\}\big{)}$
and ${\mathbb{E}}_{t}[\cdot]:={\mathbb{E}}[\cdot|{\mathcal{F}}_{t}^{N}]$. For
a set of random variables $(R^{N}_{t,n})$ and $\tilde{k}\geq 0$, the notation
$R^{N}_{t,n}=R_{t}(\Delta_{n}^{\tilde{k}})$ means that $R^{N}_{t,n}$ is
${\mathcal{F}}_{t}^{N}$-measurable and the set
$(R^{N}_{t,n}/\Delta_{n}^{\tilde{k}})$ is bounded in $L^{q}$ for all $q\geq
1$, uniformly in $t,n,N$. Hence, it exists a constant $C_{q}>0$ such that

${}{\mathbb{E}}\bigg{[}\Big{|}\frac{R^{N}_{t,n}}{\Delta_{n}^{\tilde{k}}}\Big{|}^{q}\bigg{]}\leq
C_{q}$ (27)

for any $t,n,N,q\geq 1$. If this remainder term also depends on the individual
$i\in\\{0,\dots,N\\}$, we assume that the control is uniform in $i$. Thanks to
the definition just provided, the function $R$ has the following useful
property

$R_{t}(\Delta_{n}^{\tilde{k}})=\Delta_{n}^{\tilde{k}}R_{t}(1).$

We underline that the equation above does not entail the linearity of $R$, as
in the left and the right hand side above the two functions $R$ are not
necessarily the same but just two functions on which the control in (27) is
satisfied.

Let us state a technical lemma that gathers some moment inequalities we will
use several times in the sequel. For the interested reader, its proof can be
found for example in Lemma 5.1 of [4].
In the sequel the notation $c$ refers to a general constant and its value may
change from line to line.

######  Lemma 6.

Assume that Assumption 1 is satisfied. Then, for all $p\geq 1$, $0\leq s<t\leq
T$ such that $t-s\leq 1$ and $i\in\\{1,\dots,N\\}$, the following holds true.

  * •

$\sup_{t\in[0,T]}{\mathbb{E}}[|X_{t}^{i}|^{p}]<c$.

  * •

${\mathbb{E}}[|X_{t}^{i}-X_{s}^{i}|^{p}]\leq c(t-s)^{\frac{p}{2}}$.

  * •

${\mathbb{E}}_{s}[|X_{t}^{i}-X_{s}^{i}|^{p}]\leq
c(t-s)^{\frac{p}{2}}{\color[rgb]{0,0,0}R_{s}(1)}$.

The following lemma will be useful in studying the asymptotic behavior of the
elements that will come into play. Its proof is provided in Section 7.

######  Lemma 7.

Suppose that Assumption 1 is in hold. Let $g:\mathbb{R}\rightarrow\mathbb{R}$
satisfy, for some $c>0$, $k\in\mathbb{N}$, $x,y\in{\mathbb{R}}$

$|g(x)-g(y)|\leq c|x-y|(1+|x|+|y|)^{k}.$

Then, the following convergence in probability holds true

$\frac{\Delta_{n}}{N}\sum_{i=1}^{N}\sum_{j=1}^{n}g(X^{i}_{t_{j-1}})\rightarrow\int_{0}^{T}{\mathbb{E}}[g(X_{s})]ds.$

In the main body of our paper, we will frequently rely on the fact that
$\varphi_{\tau_{n}}(f^{i,\ell,(k)}_{j})$ can be approximated as $1$. To
establish this, the following lemma will be of utmost importance. Its proof is
deferred to Section 7.

######  Lemma 8.

Assume that Assumptions 1\- 3 are in order. Recall that
$f(\theta;X_{t_{j-1}}^{i},X_{t_{j}}^{i})$ has been given in (5) and $\tau_{n}$
has been chosen as $\sqrt{\Delta_{n}}\log(n)$. Then, for any $r\geq 2$ and any
$k\geq 0$, there exist constants $c_{1},c_{2}>0$ such that,

$\mathbb{P}_{t_{j-1}}(|f^{(k)}(\theta;X_{t_{j-1}}^{i},X_{t_{j}}^{i})|>\tau_{n})\leq\frac{\Delta_{n}^{r/2}}{(\log(n))^{r}}R_{t_{j-1}}(1)+c_{1}\exp\Big{(}-c_{2}(\log(n))^{2}\Big{)}.$

##  6 Proof of main results

###  6.1 Consistency

Let us start by providing the proof of the consistency as gathered in Theorem
1. This heavily relies on the fact that we can move from the contrast
$S^{N,\text{pub}}_{n}$ as defined in (11) to the contrast
$S_{n}^{N,0}(\theta)$ we would have had in absence of privacy constraints and
the latter is given by

${}S_{n}^{N,0}(\theta):=\sum_{i=1}^{N}\sum_{j=1}^{n}f(\theta;X_{t_{j-1}}^{i},X_{t_{j}}^{i}),$
(28)

where we recall

$f(\theta;X_{t_{j-1}}^{i},X_{t_{j}}^{i})=\frac{2b(\theta,X_{t_{j-1}}^{i})(X_{t_{j}}^{i}-X_{t_{j-1}}^{i})-\Delta_{n}b^{2}(\theta,X_{t_{j-1}}^{i})}{{\color[rgb]{0,0,0}\sigma}^{2}(X_{t_{j-1}}^{i})}.$
(29)

Then, the following proposition provides a bound, for any
${\color[rgb]{0,0,0}u\in\\{0,\dots,a\\}},$ on the quantity

$E_{n,N}^{(u)}:=\sup_{\theta\in{\color[rgb]{0,0,0}[\theta_{0},\theta_{L_{n}-1}]}}\Big{|}\frac{\partial^{u}}{\partial\theta^{u}}({\color[rgb]{0,0,0}S_{n}^{N,\text{pub}}(\theta)}-S_{n}^{N,0}(\theta))\Big{|}.$

######  Proposition 3.

Assume that Assumptions 1\- 3 are in order. Then, for any $p\geq 2$ and for
any ${\color[rgb]{0,0,0}u\in\\{0,\dots,a\\}},$

$\displaystyle\left\|E_{n,N}^{(u)}\right\|_{p}$
$\displaystyle=O\bigg{(}N\Big{(}\frac{1}{L_{n}}\Big{)}^{a-u-1}\bigg{)}+O\bigg{(}L_{n}^{u+1}\sqrt{N}\log(n)\sqrt{\frac{\log(L_{n})}{\bar{\alpha}^{2}}}\bigg{)}{\color[rgb]{0,0,0}+O\bigg{(}\frac{L_{n}^{u}N}{n^{r}}\bigg{)},}$

where the constant $r>0$ can be chosen arbitrarily large, and where we recall
that $\bar{\alpha}^{2}$ is such that
$1/\bar{\alpha}^{2}=n^{-1}\sum_{j=1}^{n}1/\alpha_{j}^{2}$.

###### Proof.

Recall that, according to Section 3.1, the contrast function
$S^{N,\text{pub}}$ is defined by

$S^{N,\text{pub}}_{n}(\theta)=\sum_{i=1}^{n}\sum_{j=1}^{N}H((Z_{j}^{i,(k)}(\theta_{\ell}))_{\ell,k})(\theta),$

where $H$ is the interpolation operator (10), fully described in Section 5.1.
The private contrast function

$S^{N,0}_{n}(\theta)=\sum_{i=1}^{n}\sum_{j=1}^{N}f(\theta;X_{t_{j-1}}^{i},X_{t_{j}}^{i})$

admits the spline projection

$HS^{N,0}_{n}(\theta)={\color[rgb]{0,0,0}H\Big{(}}\sum_{i=1}^{n}\sum_{j=1}^{N}f(\cdot;X_{t_{j-1}}^{i},X_{t_{j}}^{i})\Big{)}(\theta)=\sum_{i=1}^{n}\sum_{j=1}^{N}H((f_{j}^{i,\ell,(k)})_{\ell,k})(\theta),$

where we recall $f_{j}^{i,\ell,(k)}$ is as in (8).

The proof is now divided in two steps. In the first we evaluate the error
committed by approximating $S_{n}^{N,0}(\theta)$ with
$HS_{n}^{N,0}{\color[rgb]{0,0,0}(\theta)}$, while in the second we move from
$HS_{n}^{N,0}(\theta)$ to $S_{n}^{N,\text{pub}}{\color[rgb]{0,0,0}(\theta)}$,
the contrast we propose in presence of privacy constraints. More formally, the
proposition is a consequence of (30) and (38) below.

Step 1:
In this part of the proof, we show that, for any
${\color[rgb]{0,0,0}u\in\\{0,\dots,a\\}},$

${\color[rgb]{0,0,0}{\mathbb{E}}}\left[\sup_{\theta\in{\color[rgb]{0,0,0}[\theta_{0},\theta_{L_{n}-1}]}}\left\lvert\frac{\partial^{u}}{\partial\theta^{u}}(HS^{N,0}_{n}(\theta)-S^{N,0}_{n}(\theta))\right\rvert^{p}\right]\leq
c\left(\frac{1}{L_{n}}\right)^{(a-u-1)p}N^{p}.$ (30)

We apply Proposition 2 to the function $H{\color[rgb]{0,0,0}S}^{N,0}_{n}$ with
$\Lambda=L_{n}-1$. Note that here and in the subsequent discussions, for the
sake of simplicity in notation, we will replace $L_{n}-1$ with $L_{n}$, as
they are asymptotically equivalent. It follows

$\displaystyle\sup_{\theta}\left\lvert\frac{\partial^{u}}{\partial\theta^{u}}\Big{(}HS_{n}^{N,0}{\color[rgb]{0,0,0}(\theta)}-S_{n}^{N,0}{\color[rgb]{0,0,0}(\theta)}\Big{)}\right\rvert\leq
c\Big{(}\frac{1}{L_{n}}\Big{)}^{a-u-1}\times\sup_{\theta}\Big{|}\frac{\partial^{a+1}}{\partial\theta^{a+1}}S_{n}^{N,0}(\theta)\Big{|},$

where the supremum on $\theta$, here and below, is on
$\theta\in[\theta_{0},\theta_{L_{n}-1}]$. Then,

$\displaystyle{}{\mathbb{E}}\bigg{[}\sup_{\theta}\Big{|}\frac{\partial^{u}}{\partial\theta^{u}}\Big{(}HS_{n}^{N,0}(\theta)-S_{n}^{N,0}(\theta)\Big{)}\Big{|}^{p}\bigg{]}\leq
c\Big{(}\frac{1}{L_{n}}\Big{)}^{(a-u-1)p}{\mathbb{E}}\bigg{[}\sup_{\theta}\Big{|}\frac{\partial^{a+1}}{\partial\theta^{a+1}}S_{n}^{N,0}(\theta)\Big{|}^{p}\bigg{]}.$
(31)

We now aim to demonstrate that, for some constant $c>0$,

${}{\mathbb{E}}\bigg{[}\sup_{\theta}\Big{|}\frac{\partial^{a+1}}{\partial\theta^{a+1}}S_{n}^{N,0}(\theta)\Big{|}^{p}\bigg{]}\leq
cN^{p}.$ (32)

Once established, it will yield the desired result for the approximation of
$S_{n}^{N,0}$ with its spline function, thereby concluding the proof of Step
1. To achieve this, we begin by analyzing
$\partial^{a+1}S_{n}^{N,0}(\theta)/\partial\theta^{a+1}$. Using the
definitions of $S_{n}^{N,0}$ as given in (28) and the function $f$ in (29),
one obtains

$\displaystyle\Big{|}\frac{\partial^{a+1}}{\partial\theta^{a+1}}S_{n}^{N,0}(\theta)\Big{|}$
$\displaystyle{\color[rgb]{0,0,0}=}\left\lvert\sum_{i=1}^{N}\sum_{j=1}^{n}\frac{\partial^{a+1}}{\partial\theta^{a+1}}f(\theta;X^{i}_{t_{j}},X^{i}_{t_{j}-1})\right\rvert$
$\displaystyle\leq\bigg{\lvert}\sum_{i=1}^{N}\sum_{j=1}^{n}\frac{\partial^{a+1}}{\partial\theta^{a+1}}\frac{2b(\theta,X_{t_{j-1}}^{i})\big{(}X_{t_{j}}^{i}-X_{t_{j-1}}^{i}\big{)}}{{\color[rgb]{0,0,0}\sigma}^{2}(X_{t_{j-1}}^{i})}\bigg{\rvert}+\bigg{\lvert}\sum_{i=1}^{N}\sum_{j=1}^{n}\frac{\partial^{a+1}}{\partial\theta^{a+1}}\frac{\Delta_{n}b^{2}(\theta,X_{t_{j-1}}^{i})}{{\color[rgb]{0,0,0}\sigma}^{2}(X_{t_{j-1}}^{i})}\bigg{\rvert}$
$\displaystyle=:\left\lvert S_{n}^{N,(a+1,1)}(\theta)\right\rvert+\left\lvert
S_{n}^{N,(a+1,2)}(\theta)\right\rvert.$

From Assumption 2, ${\color[rgb]{0,0,0}\sigma}^{2}$ is lower bounded by
${\color[rgb]{0,0,0}\sigma}^{2}_{\min}$. Moreover, Assumption 3 implies that
$\sup_{\theta,i,j}|\frac{\partial^{a+1}}{\partial\theta^{a+1}}b^{2}(\theta,X_{t_{j-1}}^{i})|^{p}$
is bounded by some constant $c$. Then,

$\displaystyle{}\sup_{{\color[rgb]{0,0,0}\theta}}\Big{|}S_{n}^{N,(a+1,2)}(\theta)\Big{|}^{p}$
$\displaystyle\leq
c(\Delta_{n}\,n\,N)^{p}\sup_{\theta,i,j}\Big{|}\frac{\partial^{a+1}}{\partial\theta^{a+1}}b^{2}(\theta,X_{t_{j-1}}^{i})\Big{|}^{p}$
$\displaystyle\leq cN^{p}.$ (33)

Let us now move to the analysis of $S_{n}^{N,(a+1,1)}(\theta)$. We want to
control
${\mathbb{E}}[\sup_{{\color[rgb]{0,0,0}\theta}}|S_{n}^{N,(a+1,1)}(\theta)|^{p}]$
and so, in order to deal with the $\sup$ inside the expectation, we use a
Kolmogorov-type argument. Observe we can write $S_{n}^{N,(a+1,1)}$ in an
integral way by introducing the function
$\psi_{s}(n):=\sup\\{j{\color[rgb]{0,0,0}\in\\{0,\dots,n\\}}:t_{j}\leq s\\}$.
Then,

${}S_{n}^{N,(a+1,1)}(\theta)=\sum_{i=1}^{N}\int_{0}^{T}\frac{\partial^{a+1}}{\partial\theta^{a+1}}\frac{b^{2}(\theta,X_{\psi_{n}(s)}^{i})}{{\color[rgb]{0,0,0}\sigma}^{2}(X_{\psi_{n}(s)}^{i})}dX_{s}^{i}.$
(34)

Let us set
$g(\theta,\theta^{\prime},x)=\frac{\partial^{a+1}}{\partial\theta^{a+1}}\Big{(}\frac{2b}{{\color[rgb]{0,0,0}\sigma}^{2}}\Big{)}(\theta,x)-\frac{\partial^{a+1}}{\partial\theta^{a+1}}\Big{(}\frac{2b}{{\color[rgb]{0,0,0}\sigma}^{2}}\Big{)}(\theta^{\prime},x)$.
Thanks to Rosenthal inequality for centered i.i.d. variables (see Theorem 3 in
[45]), it follows that, for all $\theta,\theta^{\prime}\in\Theta$,

$\displaystyle{\mathbb{E}}\bigg{[}\Big{|}S_{n}^{N,(u,1)}(\theta)$
$\displaystyle-
S_{n}^{N,(u,1)}(\theta^{\prime})-\Big{(}{\mathbb{E}}\Big{[}S_{n}^{N,(u,1)}(\theta)-S_{n}^{N,(u,1)}(\theta^{\prime})\Big{]}\Big{)}\Big{|}^{p}\bigg{]}$
$\displaystyle={\mathbb{E}}\bigg{[}\Big{\lvert}\sum_{i=1}^{N}\int_{0}^{T}g(\theta,\theta^{\prime},X^{i}_{\psi_{n}(s)})dX^{i}_{s}-\sum_{i=1}^{N}{\mathbb{E}}\Big{[}\int_{0}^{T}g(\theta,\theta^{\prime},X^{i}_{\psi_{n}(s)})dX^{i}_{s}\Big{]}\Big{\rvert}^{p}\bigg{]}$
$\displaystyle\leq\left(\sum_{i=1}^{N}{\mathbb{E}}\bigg{[}\Big{\lvert}\int_{0}^{T}g(\theta,\theta^{\prime},X^{i}_{\psi_{n}(s)})dX^{i}_{s}-{\mathbb{E}}\Big{[}\int_{0}^{T}g(\theta,\theta^{\prime},X^{i}_{\psi_{n}(s)})dX^{i}_{s}\Big{]}\Big{\rvert}^{2}\bigg{]}\right)^{p/2}$
$\displaystyle\quad+\sum_{i=1}^{N}{\mathbb{E}}\bigg{[}\Big{\lvert}\int_{0}^{T}g(\theta,\theta^{\prime},X^{i}_{\psi_{n}(s)})dX^{i}_{s}-{\mathbb{E}}\Big{[}\int_{0}^{T}g(\theta,\theta^{\prime},X^{i}_{\psi_{n}(s)})dX^{i}_{s}\Big{]}\Big{\rvert}^{p}\bigg{]}$
$\displaystyle\leq
cN^{{\color[rgb]{0,0,0}p}/2-1}\sum_{i=1}^{N}{\mathbb{E}}\bigg{[}\Big{\lvert}\int_{0}^{T}g(\theta,\theta^{\prime},X^{i}_{\psi_{n}(s)})dX^{i}_{s}-{\mathbb{E}}\Big{[}\int_{0}^{T}g(\theta,\theta^{\prime},X^{i}_{\psi_{n}(s)})dX^{i}_{s}\Big{]}\Big{\rvert}^{2}\bigg{]}^{p/2}$
$\displaystyle\quad+\sum_{i=1}^{N}{\mathbb{E}}\bigg{[}\Big{\lvert}\int_{0}^{T}g(\theta,\theta^{\prime},X^{i}_{\psi_{n}(s)})dX^{i}_{s}-{\mathbb{E}}\Big{[}\int_{0}^{T}g(\theta,\theta^{\prime},X^{i}_{\psi_{n}(s)})dX^{i}_{s}\Big{]}\Big{\rvert}^{p}\bigg{]},$
having used Jensen’s inequality on the first sum, as $p\geq 2$. Then, it is
upper bounded by $\displaystyle
c(N^{p/2-1}+1)\sum_{i=1}^{N}{\mathbb{E}}\bigg{[}\Big{\lvert}\int_{0}^{T}g(\theta,\theta^{\prime},X^{i}_{\psi_{n}(s)})dX^{i}_{s}-{\mathbb{E}}\big{[}\int_{0}^{T}g(\theta,\theta^{\prime},X^{i}_{\psi_{n}(s)})dX^{i}_{s}\big{]}\Big{\rvert}^{p}\bigg{]}$
$\displaystyle\leq
c(N^{p/2-1}+1)\sum_{i=1}^{N}{\mathbb{E}}\bigg{[}\Big{\lvert}\int_{0}^{T}g(\theta,\theta^{\prime},X^{i}_{\psi_{n}(s)})dX^{i}_{s}\Big{\rvert}^{p}\bigg{]},$
(35)

where we used again Jensen inequality in the two last lines, with $p>1$.
Observe that, by the dynamics of $X_{s}^{i}$, we have

$\displaystyle{}{\mathbb{E}}\bigg{[}{\color[rgb]{0,0,0}\Big{|}}\int_{0}^{T}\Big{(}g(\theta,\theta^{\prime},X_{\psi_{n}(s)}^{i})\Big{)}dX_{s}^{i}{\color[rgb]{0,0,0}\Big{|}^{p}}\bigg{]}$
$\displaystyle\leq
c{\mathbb{E}}\bigg{[}{\color[rgb]{0,0,0}\Big{|}}\int_{0}^{T}\Big{(}g(\theta,\theta^{\prime},X_{\psi_{n}(s)}^{i})\Big{)}b(X_{s}^{i},\theta^{\star})ds{\color[rgb]{0,0,0}\Big{|}^{p}}\bigg{]}$
$\displaystyle\quad+c{\mathbb{E}}\bigg{[}{\color[rgb]{0,0,0}\Big{|}}\int_{0}^{T}\Big{(}g(\theta,\theta^{\prime},X_{\psi_{n}(s)}^{i})\Big{)}{\color[rgb]{0,0,0}\sigma}(X_{s}^{i})dW_{s}^{i}{\color[rgb]{0,0,0}\Big{|}^{p}}\bigg{]}$
$\displaystyle\leq
cT^{p-1}\int_{0}^{T}{\mathbb{E}}\big{[}{\color[rgb]{0,0,0}|}g(\theta,\theta^{\prime},X_{\psi_{n}(s)}^{i}){\color[rgb]{0,0,0}|}^{p}\big{]}ds$
(36)
$\displaystyle\quad+cT^{\frac{p}{2}-1}\int_{0}^{T}{\mathbb{E}}\big{[}{\color[rgb]{0,0,0}|}g(\theta,\theta^{\prime},X_{\psi_{n}(s)}^{i}){\color[rgb]{0,0,0}|}^{p}|{\color[rgb]{0,0,0}\sigma}(X_{s}^{i})|^{p}\big{]}ds,$

where we have used Burkholder-Davis-Gundy and Jensen inequalities as well as
the boundedness of the drift. Remark now that

$\big{|}g(\theta,\theta^{\prime},X_{\psi_{n}(s)}^{i})\big{|}^{p}=\left\lvert\frac{\partial^{a+1}}{\partial\theta^{a+1}}\Big{(}\frac{2b}{{\color[rgb]{0,0,0}\sigma}^{2}}\Big{)}(\theta,X_{\psi_{n}(s)}^{i})-\frac{\partial^{a+1}}{\partial\theta^{a+1}}\Big{(}\frac{2b}{{\color[rgb]{0,0,0}\sigma}^{2}}\Big{)}(\theta^{\prime},X_{\psi_{n}(s)}^{i})\right\rvert^{p}\\\
=\big{|}\theta-\theta^{\prime}\big{|}^{p}\Big{|}\frac{\partial^{a+2}}{\partial\theta^{a+2}}\Big{(}\frac{2b}{{\color[rgb]{0,0,0}\sigma}^{2}}\Big{)}\big{(}\tau\theta+(1-\tau)\theta^{\prime},X_{\psi_{n}(s)}^{i}\big{)}\Big{|}^{p}$

for some $\tau\in[0,1]$. Replacing this in (6.1) and recalling that $T$ is a
fixed constant, it is easy to check that (6.1) is upper bounded by

$\displaystyle
c|\theta-\theta^{\prime}|^{p}\int_{0}^{T}\big{(}{\mathbb{E}}[R_{\psi_{n}(s)}(1)]+{\mathbb{E}}[R_{\psi_{n}(s)}(1)R_{s}(1)]\big{)}{\color[rgb]{0,0,0}ds}$
$\displaystyle\leq$ $\displaystyle c|\theta-\theta^{\prime}|^{p},$

where the last follows from the definition of function $R_{t}$, as in (27).
Then, going back to (35), we obtain

${\mathbb{E}}\bigg{[}\Big{|}S_{n}^{N,(a+1,1)}(\theta)-S_{n}^{N,(a+1,1)}(\theta^{\prime})-\Big{(}{\mathbb{E}}\Big{[}S_{n}^{N,(a+1,1)}(\theta)-S_{n}^{N,(a+1,1)}(\theta^{\prime})\Big{]}\Big{)}\Big{|}^{p}\bigg{]}\\\
\leq c|\theta-\theta^{\prime}|^{p}(N^{\frac{p}{2}}+N)\leq
c|\theta-\theta^{\prime}|^{p}N^{\frac{p}{2}},$

as $p\geq 2$ and $N>1$. Moreover, thanks to (34) it is

$\Big{|}{\mathbb{E}}\big{[}S_{n}^{N,(u,1)}(\theta)-S_{n}^{N,(u,1)}(\theta^{\prime})\big{]}\Big{|}\leq
c\sum_{i=1}^{N}\Big{|}{\mathbb{E}}\Big{[}\int_{0}^{T}(g(\theta,\theta^{\prime},X_{\psi_{n}(s)}^{i}))dX_{s}^{i}\Big{]}\Big{|}\leq
cN|\theta-\theta^{\prime}|,$

the validity of the last inequality can be easily confirmed by closely
following the reasoning presented above. It follows

${\mathbb{E}}\bigg{[}\Big{|}S_{n}^{N,(u,1)}(\theta)-S_{n}^{N,(u,1)}(\theta^{\prime})\Big{|}^{p}\bigg{]}\leq
c|\theta-\theta^{\prime}|^{p}N^{p}.$

Let us introduce the continuity modulus $\omega_{h}$ i.e.

$\omega_{h}(f)=\underset{|\theta-\theta^{\prime}|\leq
h}{\sup}|f(\theta)-f(\theta^{\prime})|.$

We apply Kolmogorov’s criterion as given by Theorem 2.1 in Chapter 1 in [43],
to get for all $|h|\leq 1$,

$E[\omega_{h}(S_{n}^{N,(a+1,1)})]\leq ch^{1-\varepsilon}N,$

where $\varepsilon>0$ is any fixed constant. Recall now that the interval
$[\theta_{0},\theta_{L_{n}}{\color[rgb]{0,0,0}-1}]\subset\Theta=[0,1]$ has a
radius smaller that $1$,

$\displaystyle{}{\mathbb{E}}\Big{[}\sup_{\theta}\big{|}S_{n}^{N,(a+1,1)}(\theta)\big{|}^{p}\Big{]}$
$\displaystyle\leq
c{\mathbb{E}}\Big{[}\sup_{\theta}\big{|}S_{n}^{N,(a+1,1)}(\theta)-S_{n}^{N,(a+1,1)}(\theta^{\star})\big{|}^{p}\Big{]}+c{\mathbb{E}}\Big{[}\big{|}S_{n}^{N,(a+1,1)}(\theta^{\star})\big{|}^{p}\Big{]}$
$\displaystyle\leq c\omega_{1}(S_{n}^{N,(a+1,1)})^{p}+cN^{p}\leq cN^{p},$ (37)

as we wanted. The bounds gathered in (6.1) and (6.1) yield (32) and therefore
conclude the proof of Step 1, as (30) is then a consequence of (31).

Step 2:
From now on, let us compare $HS_{n}^{N,0}$ with $S_{n}^{N,\text{pub}}$, and we
aim to show that for all $u\in\\{0,\dots,a\\}$,

$\left\|{\sup_{\theta\in[\theta_{0},\theta_{L_{n}{\color[rgb]{0,0,0}-1}}]}\left\lvert\frac{\partial^{u}}{\partial\theta^{u}}(HS_{n}^{N,0}(\theta)-S_{n}^{N,\text{pub}}(\theta))\right\rvert}\right\|_{p}=O\bigg{(}L_{n}^{u+1}\sqrt{N}\log(n)\sqrt{\frac{\log(L_{n})}{\bar{\alpha}^{2}}}\bigg{)}+O\bigg{(}\frac{L_{n}^{u}N}{n^{r}}\bigg{)}.$
(38)

From the definitions of $HS^{N,0}_{n}$ and $S_{n}^{N,\text{pub}}$ with the
linearity of the operator $H$,

$\displaystyle HS_{n}^{N,0}-S_{n}^{N,\text{pub}}$
$\displaystyle=H\bigg{(}\Big{(}\sum_{i=1}^{N}\sum_{j=1}^{n}f^{i,\ell,(k)}_{j})_{k,\ell}\Big{)}\bigg{)}-H\bigg{(}\Big{(}\sum_{i=1}^{N}\sum_{j=1}^{n}Z^{i,(k)}_{j}(\theta_{\ell})\Big{)}_{k,\ell}\bigg{)}$
$\displaystyle=H\bigg{(}\Big{(}\sum_{i=1}^{N}\sum_{j=1}^{n}\beta_{j}^{i,(k)}(\theta_{\ell})\Big{)}_{k,\ell}\bigg{)}-H\bigg{(}\Big{(}\sum_{i=1}^{N}\sum_{j=1}^{n}\mathcal{E}_{j}^{i,\ell,(k)}\big{)}_{k,\ell}\bigg{)}=:H(\bm{\beta})-H(\bm{\mathcal{E}})$
(39)

where

$\beta_{j}^{i,(k)}(\theta_{\ell})=f^{i,\ell,(k)}_{j}\big{[}1-\varphi_{\tau_{n}}(f_{j}^{i,\ell,(k)})\big{]},$

and we used (7)–(8) and recall that $\mathcal{E}_{j}^{i,\ell,(k)}$ are the
Laplace random variables introduced in Section 3.1. Let

${}\Omega^{n}:=\Big{\\{}\omega\in\Omega\,:\,\varphi_{\tau_{n}}\big{(}f_{j}^{i,\ell,(k)}\big{)}(\omega)=1,\,\forall
k\leq a,\forall i\in\\{1,\dots,N\\},\forall
j\in\\{1,\dots,n\\},\forall\ell\in\\{0,\dots,L_{n}-1\\}\Big{\\}}.$ (40)

Note that on $\Omega^{n}$ we have
$HS_{n}^{N,0}-S_{n}^{N,\text{pub}}=-H(\bm{\mathcal{E}})$, whereas on
$(\Omega^{n})^{c}$,
$HS_{n}^{N,0}-S_{n}^{N,\text{pub}}=H(\bm{\beta})-H(\bm{\mathcal{E}})$. Then,
our goal consists in finding a bound on

$\displaystyle{}\left\|{\color[rgb]{0,0,0}\sup_{\theta}}\Big{|}\frac{\partial^{u}}{\partial\theta^{u}}(HS_{n}^{N,0}(\theta)-S_{n}^{N,\text{pub}})(\theta)\Big{|}\right\|_{p}$
$\displaystyle=\left\|{\color[rgb]{0,0,0}\sup_{\theta}}\Big{|}\frac{\partial^{u}}{\partial\theta^{u}}(HS_{n}^{N,0}(\theta){\color[rgb]{0,0,0}-}S_{n}^{N,\text{pub}}(\theta))\Big{|}1_{\Omega^{n}}\right\|_{p}$
$\displaystyle+\left\|{\color[rgb]{0,0,0}\sup_{\theta}}\Big{|}\frac{\partial^{u}}{\partial\theta^{u}}(HS_{n}^{N,0}(\theta)-S_{n}^{N,\text{pub}}(\theta))\Big{|}1_{(\Omega^{n})^{c}}\right\|_{p}$
$\displaystyle\leq\left\|{\color[rgb]{0,0,0}\sup_{\theta}}\Big{|}\frac{\partial^{u}}{\partial\theta^{u}}H(\bm{\mathcal{E}})(\theta)\Big{|}\right\|_{p}+\left\|{\color[rgb]{0,0,0}\sup_{\theta}}\Big{|}\frac{\partial^{u}}{\partial\theta^{u}}(H(\bm{\beta})(\theta)-H(\bm{\mathcal{E}})(\theta))\Big{|}1_{(\Omega^{n})^{c}}\right\|_{p}.$
(41)

Let us start considering an upper bound on $H(\bm{\beta})$. Lemma 4 provides

${}\Big{\|}\frac{\partial^{u}}{\partial\theta^{u}}H(\bm{\beta})\Big{\|}_{\infty}=\Big{\|}\frac{\partial^{u}}{\partial\theta^{u}}H\Big{(}\big{(}\sum_{i=1}^{N}\sum_{j=1}^{n}\beta_{j}^{i,(k)}(\theta_{\ell})\big{)}_{\ell,k}\Big{)}\Big{\|}_{\infty}\leq
cL_{n}^{u}\sup_{\ell,k}\Big{|}\sum_{i=1}^{N}\sum_{j=1}^{n}\beta_{j}^{i,(k)}(\theta_{\ell})\Big{|}.$
(42)

Observe that, according to the definition of $\beta_{j}^{i,(k)}$ and
$f^{i,\ell,(k)}_{j}$ it is

$\displaystyle\sup_{\ell,k}\Big{|}\sum_{i=1}^{N}\sum_{j=1}^{n}\beta_{j}^{i,(k)}(\theta_{\ell})\Big{|}$
$\displaystyle\leq\sum_{i=1}^{N}\sum_{j=1}^{n}2\sup_{\ell,k}|f^{(k)}(\theta_{\ell},X_{t_{j-1}}^{i},X_{t_{j}}^{i})|$
$\displaystyle\leq
c\sum_{i=1}^{N}\sum_{j=1}^{n}\Big{[}|X_{t_{j}}^{i}-X_{t_{j-1}}^{i}|+\Delta_{n}\Big{]},$

where we used the definition of $f$ recalled in (29) and the fact that
${\color[rgb]{0,0,0}\sigma}^{2}$ is lower bounded thanks to Assumption 2 and
that the derivatives of $b$ with respect of $\theta$ are bounded because of
Assumption 3. Then, second point of Lemma 6 ensures that

$\displaystyle\Big{\|}\sup_{\ell,k}\Big{|}\sum_{i=1}^{N}\sum_{j=1}^{n}\beta_{j}^{i,(k)}(\theta_{\ell})\Big{|}\Big{\|}_{p}$
$\displaystyle\leq
c\sum_{i=1}^{N}\sum_{j=1}^{n}\big{\|}|X_{t_{j}}^{i}-X_{t_{j-1}}^{i}|+\Delta_{n}\big{\|}_{p}$
$\displaystyle\leq
c\sum_{i=1}^{N}\sum_{j=1}^{n}\big{(}\Delta_{n}^{1/2}+\Delta_{n})\leq
cN\sqrt{n},$ (43)

having also used that $n=T/\Delta_{n}$ and $T$ is a fixed constant. From (42)
and (43) follows

${}{\mathbb{E}}\bigg{[}\sup_{{\color[rgb]{0,0,0}\theta}}\Big{|}\frac{\partial^{u}}{\partial\theta^{u}}|H(\bm{\beta})(\theta)|\Big{|}^{p}\bigg{]}\leq
cL_{n}^{up}{\mathbb{E}}\Big{[}\sup_{\ell,k}\Big{|}\sum_{i=1}^{N}\sum_{j=1}^{n}\beta_{j}^{i,(k)}(\theta_{\ell})\Big{|}^{p}\Big{]}\leq
cL_{n}^{up}N^{p}n^{p/2},$ (44)

that concludes the analysis of $H(\bm{\beta})$.
Let us now study $H(\bm{\mathcal{E}})$. Lemma 4 with the definition of
$H(\bm{\mathcal{E}})$ through (39) provides

${}\Big{\|}\frac{\partial^{u}}{\partial\theta^{u}}H(\bm{\mathcal{E}})\Big{\|}_{\infty}\leq
cL_{n}^{u}\sup_{\ell,k}\Big{|}\sum_{i=1}^{N}\sum_{j=1}^{n}\mathcal{E}_{j}^{i,\ell,(k)}\Big{|},$
(45)

where we recall that
$\mathcal{E}_{j}^{i,\ell,(k)}\sim[2\tau_{n}L_{n}(a+1)/\alpha_{j}]{\mathcal{L}}(1)$
and $\tau_{n}=\sqrt{\Delta_{n}}\log(n)$. Let us now introduce
$\overset{\circ}{\mathcal{E}}_{j}^{i,\ell,(k)}$, i.i.d. random variables whose
law is ${\mathcal{L}}(1/\alpha_{j})$. It is

${}\mathcal{E}_{j}^{i,\ell,(k)}={2\tau_{n}L_{n}{\color[rgb]{0,0,0}(a+1)}}\overset{\circ}{\mathcal{E}}_{j}^{i,\ell,(k)}=2\sqrt{\Delta_{n}}\log(n)L_{n}{\color[rgb]{0,0,0}(a+1)}\overset{\circ}{\mathcal{E}}_{j}^{i,\ell,(k)},$
(46)

where we have also replaced the definition of
$\tau_{n}=\sqrt{\Delta_{n}}\log(n)=\sqrt{T/n}\log(n)$.
Then, for any fixed $k,\ell$ we have

$\displaystyle{}\sum_{i=1}^{N}\sum_{j=1}^{n}\mathcal{E}_{j}^{i,\ell,(k)}$
$\displaystyle={{\color[rgb]{0,0,0}2\sqrt{T}(a+1)}\log(n)L_{n}\sqrt{N}}\frac{1}{\sqrt{N}}\frac{1}{\sqrt{n}}{\color[rgb]{0,0,0}\sum_{i=1}^{N}}\sum_{j=1}^{n}\overset{\circ}{\mathcal{E}}_{j}^{i,\ell,(k)}.$
(47)

Let us now state a control on the sum of Laplace random variables. Its proof
can be found in Section 7.

######  Lemma 9.

Let $U\in{\mathbb{N}}^{*}$ and let us introduce a collection of independent
random variables ${\color[rgb]{0,0,0}\bm{\mathcal{U}}=(\mathcal{U}_{h})_{1\leq
h\leq U}}$ such that for $h\in\\{1,\dots,U\\}$,
$\mathcal{U}_{h}\sim\mathcal{L}(1/{\color[rgb]{0,0,0}\gamma_{h}})$. The
harmonic mean of $\gamma_{1},...,\gamma_{U}$ is denoted by $\bar{\gamma}^{2}$
and satisfies
$1/{\color[rgb]{0,0,0}\bar{\gamma}^{2}}:=U^{-1}\sum_{h=1}^{U}1/{\color[rgb]{0,0,0}\gamma_{h}}^{2}$,
and we set $S_{U}:=\sum_{h=1}^{U}{\color[rgb]{0,0,0}\mathcal{U}_{h}}$. Then,

${\mathbb{P}}\bigg{(}\frac{|S_{U}|}{\sqrt{U}}\geq\lambda\bigg{)}\leq\left\\{\begin{array}[]{rcl}2e^{-\lambda^{2}{\color[rgb]{0,0,0}\bar{\gamma}^{2}}/8}&\text{if}&0\leq\lambda\leq
2\gamma_{\max}\sqrt{U}/{\color[rgb]{0,0,0}\bar{\gamma}}^{2}\\\
{\color[rgb]{0,0,0}2}e^{-{\color[rgb]{0,0,0}\gamma_{\max}}\lambda\sqrt{U}/4}&\text{if}&\lambda\geq
2\gamma_{\max}\sqrt{U}/\bar{\gamma}^{2},\end{array}\right.$

where we have also introduced
${\color[rgb]{0,0,0}\gamma_{\max}}:=\max_{h=1,\dots,U}\gamma_{h}$.

We apply this lemma to (47). To this end, we define the sequence
$(\gamma_{h})_{h=1,\dots,nN}$ such that for all $h\in\\{1,\dots,nN\\}$ that
writes $h=j+kN$ with $j\in\\{1,\dots,n\\}$ and $k\in\\{0,\dots,N-1\\}$, we
have $\gamma_{h}=\alpha_{j}$ and also set
$\mathcal{U}_{h}=\overset{\circ}{\mathcal{E}}^{i,\ell,(k)}_{j}$. Then, for any
fixed $k\in\\{0,\dots,a\\},\ell\in\\{0,\dots,L_{n}{\color[rgb]{0,0,0}-1}\\}$,
we have

$\sum_{i=1}^{N}\sum_{j=1}^{n}\mathcal{E}_{j}^{i,\ell,(k)}=2\sqrt{T}(a+1)\log(n)L_{n}\sqrt{N}\frac{1}{\sqrt{U}}\sum_{h=1}^{U}\mathcal{U}_{h}^{\ell,(k)},$

where we have introduced $U:=Nn$. Using the definition of the $\gamma_{h}$’s,
we get

$\frac{1}{\bar{\alpha}^{2}}=\frac{1}{n}\sum_{j=1}^{n}\frac{1}{\alpha_{j}^{2}}=\frac{1}{Nn}\sum_{i=1}^{N}\sum_{j=1}^{n}\frac{1}{\alpha_{j}^{2}}=\frac{1}{U}\sum_{h=1}^{U}\frac{1}{\gamma_{h}^{2}}=\frac{1}{\bar{\gamma}^{2}}.$

Noting that we also have $\alpha_{\max}=\gamma_{\max}$, the application of
Lemma 9 directly provides

$\displaystyle{\mathbb{P}}\bigg{(}\frac{1}{{\color[rgb]{0,0,0}\sqrt{Nn}}}\sum_{i=1}^{N}\sum_{j=1}^{n}\overset{\circ}{\mathcal{E}}_{j}^{i,\ell,(k)}\geq\lambda\bigg{)}$
$\displaystyle={\mathbb{P}}\bigg{(}\frac{1}{{\color[rgb]{0,0,0}\sqrt{U}}}\sum_{h=1}^{U}\mathcal{U}_{h}^{\ell,(k)}\geq\lambda\bigg{)}$
$\displaystyle\leq\left\\{\begin{array}[]{rcl}2e^{-\lambda^{2}\bar{\alpha}^{2}/8}&\text{if}&0\leq\lambda\leq\frac{2\alpha_{\max}\sqrt{Nn}}{\bar{\alpha}^{2}}\\\
{\color[rgb]{0,0,0}2}e^{-\alpha_{\max}\lambda\sqrt{Nn}/4}&\text{if}&\lambda\geq\frac{{\color[rgb]{0,0,0}2}\alpha_{\max}\sqrt{Nn}}{\bar{\alpha}^{2}}.\end{array}\right.$
(50)

Let us now introduce a threshold $M_{n}$ that will be better specified later
and satisfies $\overline{\alpha}M_{n}\to\infty$. Then,

$\displaystyle{\mathbb{E}}\bigg{[}\sup_{\ell,k}\Big{|}\frac{1}{\sqrt{Nn}}\sum_{i=1}^{N}\sum_{j=1}^{n}\overset{\circ}{\mathcal{E}}_{j}^{i,\ell,(k)}\Big{|}^{p}\bigg{]}$
$\displaystyle={\mathbb{E}}\Bigg{[}\sup_{\ell,k}\Big{|}\frac{1}{\sqrt{Nn}}\sum_{i=1}^{N}\sum_{j=1}^{n}\overset{\circ}{\mathcal{E}}_{j}^{i,\ell,(k)}\Big{|}^{p}\mathbf{1}_{\big{\\{}\sup_{\ell,k}|\frac{1}{\sqrt{Nn}}\sum_{i=1}^{N}\sum_{j=1}^{n}\overset{\circ}{\mathcal{E}}_{j}^{i,\ell,(k)}|<M_{n}\big{\\}}}\Bigg{]}$
$\displaystyle\qquad+{\mathbb{E}}\Bigg{[}\sup_{\ell,k}\Big{|}\frac{1}{\sqrt{Nn}}\sum_{i=1}^{N}\sum_{j=1}^{n}\overset{\circ}{\mathcal{E}}_{j}^{i,\ell,(k)}\Big{|}^{p}\mathbf{1}_{\big{\\{}\sup_{\ell,k}|\frac{1}{\sqrt{Nn}}\sum_{i=1}^{N}\sum_{j=1}^{n}\overset{\circ}{\mathcal{E}}_{j}^{i,\ell,(k)}|\geq
M_{n}\big{\\}}}\Bigg{]}$ $\displaystyle\leq
M_{n}^{p}+{\mathbb{E}}\Bigg{[}\sup_{\ell,k}\Big{|}\frac{1}{\sup_{\ell,k}\sqrt{Nn}}\sum_{i=1}^{N}\sum_{j=1}^{n}\overset{\circ}{\mathcal{E}}_{j}^{i,\ell,(k)}\Big{|}^{p}\mathbf{1}_{\big{\\{}|\frac{1}{\sqrt{Nn}}\sum_{i=1}^{N}\sum_{j=1}^{n}\overset{\circ}{\mathcal{E}}_{j}^{i,\ell,(k)}|\geq
M_{n}\big{\\}}}\Bigg{]}$ $\displaystyle\leq
M_{n}^{p}+{\color[rgb]{0,0,0}p}\int_{M_{n}}^{\infty}\lambda^{p-1}{\mathbb{P}}\bigg{(}\sup_{\ell,k}\Big{|}\frac{1}{\sqrt{Nn}}\sum_{i=1}^{N}\sum_{j=1}^{n}\overset{\circ}{\mathcal{E}}_{j}^{i,\ell,(k)}\Big{|}\geq\lambda\bigg{)}d\lambda.$

We now use that

$\mathbb{P}\bigg{(}\sup_{\ell\in\\{0,\dots,L_{n}-1\\},\,k\in\\{0,\dots,a\\}}\Big{|}\frac{1}{\sqrt{Nn}}\sum_{i=1}^{N}\sum_{j=1}^{n}\overset{\circ}{\mathcal{E}}_{j}^{i,\ell,(k)}\Big{|}\geq\lambda\Big{)}\leq\sum_{\ell=0}^{L_{n}-1}\sum_{k=0}^{a}\mathbb{P}\Big{(}\Big{|}\frac{1}{\sqrt{Nn}}\sum_{i=1}^{N}\sum_{j=1}^{n}\overset{\circ}{\mathcal{E}}_{j}^{i,\ell,(k)}\Big{|}\geq\lambda\bigg{)}.$

From (50) follows

$\displaystyle{\mathbb{E}}\Bigg{[}\sup_{\ell,k}\Big{|}\frac{1}{\sqrt{Nn}}\sum_{i=1}^{N}\sum_{j=1}^{n}\overset{\circ}{\mathcal{E}}_{j}^{i,\ell,(k)}\Big{|}^{p}\Bigg{]}$
$\displaystyle\leq
M_{n}^{p}+\sum_{\ell=0}^{L_{n}-1}\sum_{k=0}^{a}\int_{M_{n}}^{\infty}{\color[rgb]{0,0,0}p}\lambda^{p-1}{\mathbb{P}}\bigg{(}\frac{1}{\sqrt{Nn}}\sum_{i=1}^{N}\sum_{j=1}^{n}\overset{\circ}{\mathcal{E}}_{j}^{i,\ell,(k)}\geq\lambda\bigg{)}d\lambda$
$\displaystyle\leq
M_{n}^{p}+{\color[rgb]{0,0,0}p}L_{n}(a+1)\int_{M_{n}}^{\frac{2\alpha_{\max}\sqrt{Nn}}{\bar{\alpha}^{2}}}2\lambda^{p-1}e^{-\lambda^{2}\bar{\alpha}^{2}/8}d\lambda$
$\displaystyle\qquad+{\color[rgb]{0,0,0}p}L_{n}(a+1)\int_{\frac{2\alpha_{\max}\sqrt{Nn}}{\bar{\alpha}^{2}}}^{\infty}2\lambda^{p-1}e^{-(\lambda\alpha_{\max}\sqrt{Nn})/4}d\lambda,$

where the first integral is zero if
$M_{n}>\frac{2\alpha_{\max}\sqrt{Nn}}{\bar{\alpha}^{2}}$. Now remark that this
first integral is bounded by

$\int_{M_{n}}^{\infty}2\lambda^{p-1}e^{-\lambda^{2}\bar{\alpha}^{2}/8}d\lambda=O\big{(}M_{n}^{{\color[rgb]{0,0,0}p}}e^{-M_{n}^{2}\bar{\alpha}^{2}/8}\big{)}.$

Regarding the second integral, we apply the change of variable
$\lambda^{\prime}:=\lambda\alpha_{\max}\sqrt{Nn}$, that provides it is equal
to

$(\alpha_{\max}\sqrt{Nn})^{-p}\int_{\frac{2\alpha_{\max}^{2}{Nn}}{\bar{\alpha}^{2}}}^{\infty}2(\lambda^{\prime})^{p-1}e^{-\lambda^{\prime}/4}d\lambda^{\prime}=O\Big{(}(\alpha_{\max}\sqrt{Nn})^{-p}\Big{(}\frac{\alpha_{\max}^{2}{Nn}}{\bar{\alpha}^{2}}\Big{)}^{p-1}e^{-\frac{\alpha_{\max}^{2}{Nn}}{4\bar{\alpha}^{2}}}\Big{)}.$

It yields

$\displaystyle{\mathbb{E}}\Bigg{[}\sup_{\ell,k}\Big{|}\frac{1}{\sqrt{Nn}}\sum_{i=1}^{N}\sum_{j=1}^{n}\overset{\circ}{\mathcal{E}}_{j}^{i,\ell,(k)}\Big{|}^{p}\Bigg{]}$
$\displaystyle\leq
M_{n}^{p}+cL_{n}M_{n}^{{\color[rgb]{0,0,0}p}}e^{-\frac{M_{n}^{2}\bar{\alpha}^{2}}{8}}+c\frac{L_{n}}{(\alpha_{\max}^{2}Nn)^{p/2}}\big{(}\frac{\alpha_{\max}Nn}{\bar{\alpha}^{2}}\big{)}^{p-1}e^{-\frac{\alpha_{\max}^{2}{Nn}}{4\bar{\alpha}^{2}}}.$

Comparing the first two terms in the right hand side of the equation above
leads us to the choice $M_{n}^{2}=\kappa\log(L_{n})/\bar{\alpha}^{2}$, for a
constant $\kappa$ arbitrarily large. It implies

$\displaystyle{\mathbb{E}}\Bigg{[}\sup_{\ell,k}\Big{|}\frac{1}{\sqrt{Nn}}\sum_{i=1}^{N}\sum_{j=1}^{n}\overset{\circ}{\mathcal{E}}_{j}^{i,\ell,(k)}\Big{|}^{p}\Bigg{]}$
$\displaystyle\leq
c\Big{(}\frac{\log(L_{n})}{\bar{\alpha}^{2}}\Big{)}^{\frac{p}{2}},$ (51)

as the the second and third terms are negligible compared to the first using
$L_{n}=o(n^{r})$ for some $r>0$. Combining (45) and (51) and recalling the
definition (46) of the variables
$\overset{\circ}{\mathcal{E}}_{j}^{i,\ell,(k)}$, we deduce

${}{\mathbb{E}}\Bigg{[}\sup_{{\color[rgb]{0,0,0}\theta}}\Big{|}\frac{\partial^{u}}{\partial\theta^{u}}|H({\bm{\mathcal{E}}})(\theta)|\Big{|}^{p}\Bigg{]}\leq
cL_{n}^{up}(\sqrt{N}\log(n)L_{n})^{p}\Big{(}\frac{\log(L_{n})}{\bar{\alpha}^{2}}\Big{)}^{\frac{p}{2}}.$
(52)

Replacing (44) and (52) in (6.1) we obtain

$\displaystyle\left\|\sup_{{\color[rgb]{0,0,0}\theta}}|\frac{\partial^{u}}{\partial\theta^{u}}(HS_{n}^{N,0}(\theta)-{\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\theta))|\right\|_{p}$
$\displaystyle\leq
cL_{n}^{u+1}\sqrt{N}\log(n)\sqrt{\frac{\log(L_{n})}{\bar{\alpha}^{2}}}+cn^{-{\color[rgb]{0,0,0}r_{0}/2}}\bigg{[}L_{n}^{u+1}\sqrt{N}\log(n)\sqrt{\frac{\log(L_{n})}{\bar{\alpha}^{2}}}+L_{n}^{u}n^{\frac{1}{2}}N\bigg{]},$
(53)

having used Cauchy-Schwarz inequality and the fact that

$\mathbb{P}(\Omega_{n}^{c})\leq cn^{-r_{0}}$ (54)

for any $r_{0}\geq 2$. Indeed, thanks to Lemma 8,

$\displaystyle\mathbb{P}(\Omega_{n}^{c})$
$\displaystyle=\mathbb{P}\big{(}\exists\bar{k},\bar{i},\bar{\ell},\bar{j}:\,|f^{\bar{i},\bar{\ell},(\bar{k})}_{\bar{j}}|>\tau_{n}\big{)}$
$\displaystyle={\mathbb{E}}\Big{[}{\mathbb{E}}_{t_{\bar{j}{\color[rgb]{0,0,0}-1}}}\Big{[}\mathbf{1}_{|f^{\bar{i},\bar{\ell},(\bar{k})}_{\bar{j}}|>\tau_{n}}\Big{]}\Big{]}$
$\displaystyle\leq
c{\mathbb{E}}\Big{[}\frac{\Delta_{n}^{\frac{r_{0}}{2}}}{(\log(n))}R_{t_{\bar{j}{\color[rgb]{0,0,0}-1}}}(1)+\exp(-c(\log(n))^{2})\Big{]},$

that goes to zero at a rate given by any arbitrarily large exponent of $1/n$,
recalling that $\Delta_{n}=T/{\color[rgb]{0,0,0}n}$, with fixed $T$.
It concludes the proof of Step 2, as (38) is an immediate consequence of (53).
The proposition is therefore proven. ∎

We are now ready to prove the consistency of the proposed estimator based on
its analogous result in the privacy-free context and on the approximation
argument presented in Proposition 3.

###### Proof of Theorem 1.

In order to show the consistency we will prove that

${}\frac{1}{N}({\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\theta)-{\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\theta^{\star}))\xrightarrow{\mathbb{P}}{\color[rgb]{0,0,0}-}\int_{0}^{T}{\mathbb{E}}\bigg{[}\frac{\big{(}b(\theta,X_{s})-b(\theta^{\star},X_{s})\big{)}^{2}}{a^{2}(X_{s})}\bigg{]}ds=:C_{\infty}(\theta)$
(55)

uniformly in
$\theta\in{\color[rgb]{0,0,0}[\theta_{0},\theta_{L_{n}-1}]=[0,1-1/L_{n}]}$.
Let us stress that, as $\theta^{\star}\in(0,1)$, we have
$\theta^{\star}\in[0,1-1/L_{n}]$ for $n$ large enough.
Observe that one can write

$\frac{1}{N}({\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\theta)-{\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\theta^{\star}))=\frac{1}{N}({\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\theta)-S_{n}^{N,0}(\theta))-\frac{1}{N}({\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\theta^{\star})-S_{n}^{N,0}(\theta^{\star}))\\\
+\frac{1}{N}(S_{n}^{N,0}(\theta)-S_{n}^{N,0}(\theta^{\star})).$

Proposition 3 with $u=0$ ensures that the $L^{p}$ norm of the first two terms
here above is upper bounded, uniformly in $\theta$, by
$c(\frac{1}{L_{n}})^{a-1}+c\frac{L_{n}\log(n)}{\sqrt{N}}\sqrt{\frac{\log(L_{n})}{\bar{\alpha}^{2}}}{\color[rgb]{0,0,0}+c\frac{N}{n^{r_{0}}}}$,
for any fixed $r_{0}>0$. This term goes to $0$ under our hypothesis. It
follows they converge to $0$ in probability as well. Then, it is enough to
prove

${}\frac{1}{N}(S_{n}^{N,0}(\theta)-S_{n}^{N,0}(\theta^{\star}))\xrightarrow{\mathbb{P}}C_{\infty}(\theta)$
(56)

uniformly in $\theta$ to obtain (55). The derivation of (56) closely mirrors
the argumentation found in Steps 3 and 4 of Lemma 6.1 in [4]. It is worth
noting that our contrast function $S_{n}^{N,0}$ differs slightly from
$S_{n}^{N}$ in [4]. In that paper, indeed, the authors aim to jointly estimate
both the drift and the diffusion coefficient in a parametric manner, whereas
our focus here is solely on estimating the drift. However, similarly as in
Step 3 of Lemma 6.1 in [4], and using (28)–(29), the following decomposition
holds:

$\frac{1}{N}(S_{n}^{N,0}(\theta)-S_{n}^{N,0}(\theta^{\star}))=I_{n}^{N}(\theta)+2\rho_{n}^{N}(\theta),$

where

$\displaystyle I_{n}^{N}(\theta)$
$\displaystyle={\color[rgb]{0,0,0}-}\frac{\Delta_{n}}{N}\sum_{i=1}^{N}\sum_{j=1}^{n}\frac{(b(\theta,X_{t_{j}-1}^{i})-b(\theta^{\star},X_{t_{j}-1}^{i}))^{2}}{{\color[rgb]{0,0,0}\sigma}^{2}(X_{t_{j}-1}^{i})},$
$\displaystyle\rho^{N}_{n}(\theta)$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{n}(X_{t_{j}}^{i}-X_{t_{j}-1}^{i}-\Delta_{n}b(\theta^{\star},X_{t_{j}-1}^{i}))\frac{b(\theta,X_{t_{j}-1}^{i})-b(\theta^{\star},X_{t_{j}-1}^{i})}{{\color[rgb]{0,0,0}\sigma}^{2}(X_{t_{j}-1}^{i})}.$

Then, $I_{n}^{N}(\theta)\xrightarrow{\mathbb{P}}C_{\infty}(\theta)$ because of
Lemma 7 while $\rho^{N}_{n}(\theta)\xrightarrow{\mathbb{P}}0$ uniformly in
$\theta$ as in the proof of Lemma 6.1 in [4]. Moreover, Step 4 of Lemma 6.1 in
[4] ensures the tightness of the two sequences $\theta\mapsto
I_{n}^{N}(\theta)$ and $\theta\mapsto\rho_{n}^{N}(\theta)$, which concludes
the proof of (55) and implies in particular that

${}\sup_{\theta\in{\color[rgb]{0,0,0}[\theta_{0},\theta_{L_{n}-1}]}}\Big{|}\frac{1}{N}({\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\theta)-{\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\theta^{\star}))-C_{\infty}(\theta)\Big{|}\xrightarrow{\mathbb{P}}0.$
(57)

Such equation implies the consistency of $\hat{\theta}_{n}^{N}$. Indeed, the
identifiability condition stated in Assumption 4 implies that, for every
$\epsilon>0$, there exists $\eta>0$ such that
$C_{\infty}(\theta){\color[rgb]{0,0,0}<-\eta}$ for every $\theta$ such that
$|\theta-\theta^{\star}|\geq\epsilon$. Then,

$\left\\{|\hat{\theta}^{N}_{n}-\theta^{\star}|\geq\epsilon\right\\}\subset\left\\{C_{\infty}(\hat{\theta}^{N}_{n}){\color[rgb]{0,0,0}<-\eta}\right\\}.$

Observe that we can write

$C_{\infty}(\hat{\theta}^{N}_{n})=\Big{(}C_{\infty}(\hat{\theta}^{N}_{n})-\frac{1}{N}({\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\hat{\theta}^{N}_{n})-{\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\theta^{\star}))\Big{)}+\frac{1}{N}({\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\hat{\theta}^{N}_{n})-{\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\theta^{\star})).$

Now, the first converges to $0$ in probability because of (57), while the
second is non-negative because of the definition of $\hat{\theta}^{N}_{n}$. We
derive that the probability of the event
$\left\\{C_{\infty}(\hat{\theta}^{N}_{n}){\color[rgb]{0,0,0}<-\eta}\right\\}$
converges to $0$, which concludes the proof of the consistency. ∎

###  6.2 Asymptotic normality

In this section, we establish the asymptotic normality of the proposed
estimator. The proof follows a classical path, relying on the asymptotic
behavior of
$\partial_{\theta}{\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\theta)$ and
$\partial^{2}_{\theta}{\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\theta)$
(refer, for instance, to [26], Section 5a).
Particularly noteworthy is the observation that the behavior of the first
derivative of the contrast function varies depending on whether the
contribution of privacy is negligible in our estimation procedure (see
Propositions 4 and 5 below). Conversely, the behavior of the second derivative
remains consistent, irrespective of the privacy’s contribution, as elucidated
in Proposition 6 below. The proofs of Propositions 4, 5, and 6 can be found in
the subsequent subsections.

######  Proposition 4.

[Negligible contribution of privacy]
Assume that A1\- A3, A6.1 and A7.1 hold. Assume moreover that
${\color[rgb]{0,0,0}\sqrt{\log(L_{n})}r_{n,N}}\rightarrow 0$ for
$N,n,L_{n}\rightarrow\infty$. Then,

$\frac{1}{\sqrt{N}}\partial_{\theta}{\color[rgb]{0,0,0}S^{N,\mathrm{pub}}_{n}}(\theta^{\star})\xrightarrow{\mathcal{L}}{\color[rgb]{0,0,0}\mathcal{N}}\bigg{(}0,2\int_{0}^{T}{\mathbb{E}}\bigg{[}\Big{(}\frac{\partial_{\theta}b(\theta^{\star},X_{s})}{{\color[rgb]{0,0,0}\sigma}(X_{s})}\Big{)}^{2}\bigg{]}ds\bigg{)}=:{\color[rgb]{0,0,0}\mathcal{N}}(0,2\Sigma_{0}).$

When the contribution of the privacy is significant, instead, the following
proposition is in hold.

######  Proposition 5.

[Significant contribution of privacy]
Assume that A1\- A3, A6.2, A7.2 and A8 hold. Assume moreover that
${\color[rgb]{0,0,0}r_{n,N}}\rightarrow\infty$ for
$N,n,L_{n}\rightarrow\infty$. Define

$N_{n,N}:=\frac{\sqrt{\bar{\alpha}^{2}}}{\sqrt{N}L_{n}^{2}\log(n)}\frac{1}{4{\color[rgb]{0,0,0}(a+1)}\sqrt{T}\sqrt{v_{n}(\theta^{\star})}}\frac{\partial}{\partial\theta}{\color[rgb]{0,0,0}S^{N,\mathrm{pub}}_{n}}(\theta^{\star}).$

Then, we have

$(N_{n,N},v_{n}(\theta^{\star}))\xrightarrow{\mathcal{L}}({\color[rgb]{0,0,0}\mathcal{N}},\widehat{v}(S)),$

where $\mathcal{N}$ is a Gaussian $\mathcal{N}(0,1)$ variable independent of
$S$. We recall that $\bar{\alpha}^{2}$ is such that
$1/\bar{\alpha}^{2}=n^{-1}\sum_{j=1}^{n}1/\alpha_{j}^{2}$,
$v_{n}(\theta^{\star})$ and $\widehat{v}$ have respectively been introduced in
(15) and (14).

One can remark that the definition of the estimator guarantees that
$\partial_{\theta}{\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\hat{\theta}^{N}_{n})=0$.
Thanks to Taylor’s formula, it implies

${}(\hat{\theta}^{N}_{n}-\theta^{\star})\int_{0}^{1}\partial^{2}_{\theta}{\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\theta^{\star}+s(\hat{\theta}^{N}_{n}-\theta^{\star}))ds=-\partial_{\theta}{\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\theta^{\star}).$
(58)

The asymptotic behaviour of the second derivative of
${\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}$ is gathered in the following
proposition.

######  Proposition 6.

Suppose that Assumptions A1\- A4 are in hold. Assume moreover that $a>3$ and
${\color[rgb]{0,0,0}L_{n}^{3}}\log(n)\sqrt{\frac{\log(L_{n})}{N\bar{\alpha}^{2}}}\rightarrow
0$. Then, for any $N,n\rightarrow\infty$ we have

  1. 1.

$\frac{\partial^{2}_{\theta}{\color[rgb]{0,0,0}S^{N,\mathrm{pub}}_{n}}(\theta^{\star})}{N}\xrightarrow{\mathbb{P}}\Sigma_{0}$,

  2. 2.

$\frac{1}{N}\sup_{s\leq
1}|\partial^{2}_{\theta}{\color[rgb]{0,0,0}S^{N,\mathrm{pub}}_{n}}(\theta^{\star}+s(\hat{\theta}^{N}_{n}-\theta^{\star}))-\partial^{2}_{\theta}{\color[rgb]{0,0,0}S^{N,\mathrm{pub}}_{n}}(\theta^{\star})|\xrightarrow{\mathbb{P}}0$.

From the propositions stated above we easily deduce the asymptotic normality
as in Theorems 2 and 3.

###### Proof of Theorem 2.

In the case of negligible privacy contribution it looks convenient to write
(58) as

$(\hat{\theta}^{N}_{n}-\theta^{\star})=-\frac{\sqrt{\frac{1}{N}}\partial_{\theta}{\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\theta^{\star})}{\frac{1}{N}\int_{0}^{1}\partial^{2}_{\theta}{\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\theta^{\star}+s(\hat{\theta}^{N}_{n}-\theta^{\star}))ds}\quad\frac{1}{\sqrt{N}}.$

Then, Propositions 4 and 6 yield

$\sqrt{N}(\hat{\theta}^{N}_{n}-\theta^{\star})\xrightarrow[]{\mathcal{L}}\mathcal{N}\big{(}0,2(\Sigma_{0})^{-1}\big{)},$

as in the statement of Theorem 2. ∎

###### Proof of Theorem 3.

In the case of significant privacy contribution we can write (58) as

$(\hat{\theta}^{N}_{n}-\theta^{\star})=-\frac{\sqrt{\frac{\bar{\alpha}^{2}}{N}}\frac{1}{L_{n}^{2}\log(n)4{\color[rgb]{0,0,0}(a+1)}\sqrt{T}\sqrt{v_{n}(\theta^{\star})}}\partial_{\theta}{\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\theta^{\star})}{\frac{1}{N}\int_{0}^{1}\partial^{2}_{\theta}{\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\theta^{\star}+s(\hat{\theta}^{N}_{n}-\theta^{\star}))ds}\quad\frac{1}{\sqrt{N\bar{\alpha}^{2}}}L_{n}^{2}\log(n)4{\color[rgb]{0,0,0}(a+1)}\sqrt{T}\sqrt{v_{n}(\theta^{\star})}.$

From here we can deduce, thanks to Propositions 5 and 6 and the continuous
mapping theorem

$\frac{\sqrt{N\bar{\alpha}^{2}}}{L_{n}^{2}\log(n)4{\color[rgb]{0,0,0}(a+1)}\sqrt{T}\sqrt{v_{n}(\theta^{\star})}}(\hat{\theta}^{N}_{n}-\theta^{\star})\xrightarrow{\mathcal{L}}\mathcal{N}\big{(}0,(\Sigma_{0})^{-2}\big{)}$

jointly with the convergence
$v_{n}(\theta^{\star})\xrightarrow{\mathcal{L}}\widehat{v}(S)$. It concludes
the proof of Theorem 3. ∎

The proof of asymptotic normality is concluded by establishing the validity of
the propositions stated above. We will begin with the case where privacy is
negligible, as this will serve as a foundation for the more intricate proof of
the significant privacy case.

####  6.2.1 Proof of Proposition 4

###### Proof.

The proof of Proposition 4 heavily relies on the approximation gathered in
Proposition 3. We have indeed

$\frac{1}{\sqrt{N}}\partial_{\theta}{\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\theta^{\star})=\frac{1}{\sqrt{N}}(\partial_{\theta}{\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\theta^{\star})-\partial_{\theta}S_{n}^{N,0}(\theta^{\star}))+\frac{1}{\sqrt{N}}\partial_{\theta}S_{n}^{N,0}(\theta^{\star}).$

Proposition 3 ensures that

$\left\|\frac{1}{\sqrt{N}}(\partial_{\theta}{\color[rgb]{0,0,0}S^{N,\text{pub}}_{n}}(\theta^{\star})-\partial_{\theta}S_{n}^{N,0}(\theta^{\star}))\right\|_{p}\leq
c\frac{\sqrt{N}}{L_{n}^{a-2}}+cL_{n}^{2}\log(n)\sqrt{\frac{\log(L_{n})}{\bar{\alpha}^{2}}}{\color[rgb]{0,0,0}+c\frac{L_{n}\sqrt{N}}{n^{r}}},$

where $r>0$ can be chosen arbitrarily large. It goes to zero under the
hypothesis A6.1, $r_{n,N}\sqrt{\log(L_{n})}\to 0$ and the fact that both
$L_{n}$ and $N$ go to $\infty$ at a polynomial rate in $n$.

Then, Proposition 4 is proven once we show that
$\frac{1}{\sqrt{N}}\partial_{\theta}S_{n}^{N,0}(\theta^{\star})\xrightarrow{\mathcal{L}}\mathcal{N}(0,2\Sigma_{0})$.
One can easily check this is implied by Proposition 6.2 in [4]. Indeed,
similarly as in [4], we can write
${\color[rgb]{0,0,0}\frac{1}{\sqrt{N}}}\partial_{\theta}S_{n}^{N,0}(\theta)=\sum_{j=1}^{n}\xi_{j}^{(1)}(\theta)$,
with

$\xi_{j}^{(1)}(\theta):=\frac{1}{\sqrt{N}}\sum_{i=1}^{N}\frac{2\partial_{\theta}b(\theta,X^{i}_{t_{j-1}})}{{\color[rgb]{0,0,0}\sigma}^{2}(X^{i}_{t_{j-1}})}(X^{i}_{t_{j}}-X^{i}_{t_{j-1}}-\Delta_{n}b(\theta,X^{i}_{t_{j-1}})).$
(59)

Remark that the main differences compared to $\xi_{j,h}^{(1)}(\theta)$ in
Proposition 6.2 in [4] is that in our case the diffusion coefficient
${\color[rgb]{0,0,0}\sigma}$ does not depend on a second parameter
$\theta_{2}$ and the drift parameter is no longer in ${\mathbb{R}}^{p_{1}}$,
but simply in ${\mathbb{R}}$. Moreover, the convergence gathered in (37) in
Proposition 6.2 of [4] is now replaced by

$\sum_{j=1}^{n}{\mathbb{E}}_{t_{j-1}}[(\xi_{j}^{(1)}(\theta^{\star}))^{2}]\xrightarrow{\mathbb{P}}4\int_{0}^{T}{\mathbb{E}}\bigg{[}\bigg{(}\frac{\partial_{\theta}b(\theta^{\star},X_{s})}{{\color[rgb]{0,0,0}\sigma}(X_{s})}\bigg{)}^{2}\bigg{]}ds,$

that can be obtained using Lemma 7 and $N\Delta_{n}\to 0$ by Assumption A7.1.
Then, (36) and (40) in Proposition 6.2 of [4] provide, for some $\tilde{r}>0$,

$\sum_{j=1}^{n}{\mathbb{E}}_{t_{j-1}}[\xi_{j}^{(1)}(\theta^{\star})]\xrightarrow{\mathbb{P}}0,\qquad\sum_{j=1}^{n}{\mathbb{E}}_{t_{j-1}}[(\xi_{j}^{(1)}(\theta^{\star}))^{2+\tilde{r}}]\xrightarrow{\mathbb{P}}0.$

The proof of Proposition 4 is therefore concluded by application of Theorem
3.2 in [27]. ∎

####  6.2.2 Proof of Proposition 5

###### Proof.

We start by observing that, as a consequence of Lemma 3, for any
$i\in\\{1,\dots,N\\}$ and $j\in\\{1,\dots,n\\}$ it is

$H(Z_{j}^{i})=H\left((f_{j}^{i,\ell,(k)}\varphi_{\tau_{n}}(f_{j}^{i,\ell,(k)}))_{\ell,k}\right)+H\left((\mathcal{E}_{j}^{i,\ell,(k)})_{\ell,k}\right)$

where, as a reminder, $\mathcal{E}_{j}^{i,\ell,(k)}$ and $f_{j}^{i,\ell,(k)}$
are defined as in (6) and (8). Then, from (11) one has

$\displaystyle S_{n}^{N,\text{pub}}(\theta^{\star})$
$\displaystyle=\sum_{i=1}^{N}\sum_{j=1}^{n}H\left((f_{j}^{i,\ell,(k)}\varphi_{\tau_{n}}(f_{j}^{i,\ell,(k)}))_{\ell,k}\right)(\theta^{\star})+\sum_{i=1}^{N}\sum_{j=1}^{n}H\left((\mathcal{E}_{j}^{i,\ell,(k)})_{\ell,k}\right)(\theta^{\star})$
$\displaystyle=:H(\bm{\gamma}){\color[rgb]{0,0,0}(\theta^{\star})}+H(\bm{\mathcal{E}})(\theta^{\star}),$

where $\bm{\gamma}=(\bm{\gamma}^{\ell,k})_{\ell,k}$ and
$\bm{\mathcal{E}}=(\bm{\mathcal{E}}^{\ell,(k)})_{\ell,k}$ are given by

$\displaystyle\bm{{\gamma}}^{\ell,(k)}:=\sum_{i=1}^{N}\sum_{j=1}^{n}f_{j}^{i,\ell,(k)}\varphi_{\tau_{n}}(f_{j}^{i,\ell,(k)}),$
(60)
$\displaystyle\bm{\mathcal{E}}^{\ell,(k)}:=\sum_{i=1}^{N}\sum_{j=1}^{n}\mathcal{E}^{j,\ell,(k)}_{i}.$
(61)

We will see that the main contribution in
$\partial_{\theta}S_{n}^{N,\text{pub}}(\theta^{\star})$ comes from
$\partial_{\theta}H(\bm{\mathcal{E}})(\theta^{\star})$. By Proposition 1, the
Hermite interpolation of the Laplace noise term is given by

$H(\bm{\mathcal{E}})(\theta)=\sum_{\ell=-1}^{{\color[rgb]{0,0,0}L_{n}-2}}\sum_{m=0}^{a}c_{\ell}^{m}(\bm{\mathcal{E}})B_{j}^{m}(\theta),$

where the coefficients in the spline decomposition are given by

$c_{\ell}^{m}(\bm{\mathcal{E}})=\sum_{\nu=0}^{a}(-1)^{\nu}(g_{\ell}^{m})^{(2a+1-\nu)}(\theta_{\ell+1})\bm{\mathcal{E}}^{\ell+1,(\nu)},$
(62)

with the polynomial function
$g_{\ell}^{m}(\theta)=\frac{1}{(2a+1)!}(\theta-\theta_{\ell})^{a-m}(\theta-\theta_{\ell+1})^{a+1}(\theta-\theta_{\ell+2})^{m}$.
The spline function $B_{j}^{m}$ is supported on $[\theta_{j},\theta_{j+2}]$,
where we recall that we have set $\theta_{-1}=\theta_{0}$ and
$\theta_{L_{n}}=\theta_{L_{n}-1}$, by repeating the two endpoints.

As the spline expansion is constructed upon a high frequency grid, the
dominating term in the expression of the coefficients (62) is the one
corresponding to $\nu=0$. Hence, we isolate such contribution. Remarking that
the term corresponding to $\nu=0$ in the sum (62) is equal to
$\bm{\mathcal{E}}^{\ell+1,(0)}$, as $(g^{m}_{\ell})^{(2a+1)}\equiv 1$, this
leads us to split $H(\bm{\mathcal{E}})$ into the sum
$H(\bm{\mathcal{E}})=H^{0}(\bm{\mathcal{E}})+{\color[rgb]{0,0,0}\overline{H}}(\bm{\mathcal{E}})$
where the two functions $H^{0}(\bm{\mathcal{E}})$ and
${\color[rgb]{0,0,0}\overline{H}}(\bm{\mathcal{E}})$ are defined by

$\displaystyle H^{0}({\bm{\mathcal{E}}})$
$\displaystyle=\sum_{\ell=-1}^{{\color[rgb]{0,0,0}L_{n}-2}}\sum_{m=0}^{a}\bm{\mathcal{E}}^{\ell+1,(0)}B_{\ell}^{m}$
$\displaystyle{\color[rgb]{0,0,0}\overline{H}}({\bm{\mathcal{E}}})$
$\displaystyle=\sum_{\ell=-1}^{{\color[rgb]{0,0,0}L_{n}-2}}\sum_{m=0}^{a}\overline{c}_{\ell}^{m}({\bm{\mathcal{E}}})B_{\ell}^{m},$

where

$\overline{c}_{\ell}^{m}({\bm{\mathcal{E}}})=\sum_{\nu=1}^{a}(-1)^{\nu}(g_{\ell}^{m})^{(2a+1-\nu)}(\theta_{\ell+1})\bm{\mathcal{E}}^{\ell+1,(\nu)}.$
(63)

The proposition will be proved if we show

$\displaystyle\left(\frac{\sqrt{\bar{\alpha}^{2}}}{\sqrt{N}L_{n}^{2}\log(n)}\frac{1}{4{\color[rgb]{0,0,0}(a+1)}\sqrt{T}\sqrt{v_{n}(\theta^{\star})}}\frac{\partial}{\partial\theta}H^{0}(\bm{\mathcal{E}})(\theta^{\star}),v_{n}(\theta^{\star})\right)\xrightarrow{\mathcal{L}}({\color[rgb]{0,0,0}\mathcal{N}},\hat{v}(S)),$
(64)
$\displaystyle\frac{\sqrt{\bar{\alpha}^{2}}}{\sqrt{N}L_{n}^{2}\log(n)}\frac{1}{\sqrt{v_{n}(\theta^{\star})}}\frac{\partial}{\partial\theta}{\color[rgb]{0,0,0}\overline{H}}(\bm{\mathcal{E}})(\theta^{\star})\xrightarrow{\mathbb{P}}0,$
(65)
$\displaystyle\frac{\sqrt{\bar{\alpha}^{2}}}{\sqrt{N}L_{n}^{2}\log(n)}\frac{1}{\sqrt{v_{n}(\theta^{\star})}}\frac{\partial}{\partial\theta}H({\color[rgb]{0,0,0}\bm{\gamma}})(\theta^{\star})\xrightarrow{\mathbb{P}}0.$
(66)

We now study the asymptotic behaviour of
$\frac{\partial}{\partial\theta}H^{0}(\bm{\mathcal{E}})(\theta^{\star})$. We
recall that $\ell^{\star}_{n}\in\\{0,\dots,{\color[rgb]{0,0,0}L_{n}-2}\\}$
denotes the value of the index such that
$\theta^{\star}\in[\theta_{\ell^{\star}_{n}},\theta_{\ell^{\star}_{n}+1})$.
Since $\theta^{\star}\in(0,1)$, it is possible to exclude the case
${\color[rgb]{0,0,0}\ell^{\star}_{n}}=0$ or
${\color[rgb]{0,0,0}\ell^{\star}_{n}=L_{n}-2}$ for $n$ large enough. Using
that the support of $B^{m}_{j}$ is $[\theta_{j},\theta_{j+2}]$ we deduce, for
$\theta\in[\theta_{\ell^{\star}_{n}},\theta_{\ell^{\star}_{n}+1})$,

$H^{0}({\bm{\mathcal{E}}})(\theta)=\sum_{\ell\in\\{\ell_{n}^{\star}-1,\ell_{n}^{\star}\\}}\sum_{m=0}^{a}\bm{\mathcal{E}}^{\ell+1,(0)}B_{\ell}^{m}(\theta).$
(67)

We use the scaling properties of the spline functions, to represent
$B_{\ell}^{m}(\theta)$ for $\ell\in\\{1,\dots,{\color[rgb]{0,0,0}L_{n}-3}\\}$,
in the following way

$B_{\ell}^{m}(\theta)=\overline{B}^{m}(L_{n}(\theta-\theta_{\ell})),$ (68)

where, for $m\in\\{0,\dots,{\color[rgb]{0,0,0}a}\\}$, $\overline{B}^{m}$ is
the spline function of order $2a+1$ constructed on the knots $0$, $1$, $2$
where the knot $0$ is repeated $a+1-m$ times, while the knot $1$ is repeated
$a+1$ times and knot $2$ is repeated $m+1$ times. The spline function
$\overline{B}^{m}$ is supported on $[0,2)$. By differentiating (67) and using
the representation (68), we deduce

$\displaystyle\frac{\partial}{\partial\theta}H^{0}({\bm{\mathcal{E}}})(\theta^{\star})$
$\displaystyle=\sum_{\ell\in\\{\ell_{n}^{\star}-1,\ell_{n}^{\star}\\}}\sum_{m=0}^{a}\bm{\mathcal{E}}^{\ell+1,(0)}L_{n}\frac{\partial}{\partial\theta}{\color[rgb]{0,0,0}\left(\overline{B}^{m}\right)}(L_{n}(\theta^{\star}-\theta_{\ell}))$