# Nonparametric estimation of the diffusion coefficient from S.D.E. paths

Eddy Ella-Mintsa
LAMA, Université Gustave Eiffel.

###### Abstract

Consider a diffusion process $X=(X_{t})_{t\in[0,1]}$ observed at discrete
times and high frequency, solution of a stochastic differential equation whose
drift and diffusion coefficients are assumed to be unknown. In this article,
we focus on the nonparametric esstimation of the diffusion coefficient. We
propose ridge estimators of the square of the diffusion coefficient from
discrete observations of $X$ and that are obtained by minimization of the
least squares contrast. We prove that the estimators are consistent and derive
rates of convergence as the size of the sample paths tends to infinity, and
the discretization step of the time interval $[0,1]$ tend to zero. The
theoretical results are completed with a numerical study over synthetic data.

Keywords. Nonparametric estimation, diffusion process, diffusion coefficient,
least squares contrast, repeated observations.

MSC: 62G05; 62M05; 60J60

##  1 Introduction

Let $X=(X_{t})_{t\in[0,1]}$ be a one dimensional diffusion process with finite
horizon time, solution of the following stochastic differential equation:

$dX_{t}=b(X_{t})dt+\sigma(X_{t})dW_{t},\ \ X_{0}=0$ (1)

where $(W_{t})_{t\geq 0}$ is a standard Brownian motion. The drift function
$b$ and the diffusion coefficient $\sigma$ are assumed to be unknown Lipschitz
functions. We denote by $(\mathcal{F}_{t})_{t\in[0,1]}$ the natural filtration
of the diffusion process $X$. The goal of the article is to construct, from
$N$ discrete observations $\bar{X}^{j}=(X^{j}_{k\Delta_{n}})_{0\leq k\leq
n},\leavevmode\nobreak\ \leavevmode\nobreak\ 1\leq j\leq N$ with time step
$\Delta_{n}=1/n$, a nonparametric estimator of the square of the diffusion
coefficient $\sigma^{2}(.)$. We are in the framework of high frequency data
since the time step $\Delta_{n}$ tends to zero as $n$ tends to infinity.
Furthermore, we consider estimators of $\sigma^{2}(.)$ built from a single
diffusion path ($N=1$), and those built on $N$ paths when
$N\rightarrow\infty$. In this paper, we first propose a ridge estimator of
$\sigma^{2}(.)$ on a compact interval. Secondly, we focus on a nonparametric
estimation of $\sigma^{2}(.)$ on the real line $\mathbb{R}$. We measure the
risk of any estimator $\widehat{\sigma}^{2}$ of the square of the diffusion
coefficient $\sigma^{2}$ by
$\mathbb{E}\left[\|\widehat{\sigma}^{2}-\sigma^{2}\|^{2}_{n,N}\right]$, where
$\|\widehat{\sigma}^{2}-\sigma^{2}\|^{2}_{n,N}:=(Nn)^{-1}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{\left(\widehat{\sigma}^{2}(X^{j}_{k\Delta})-\sigma^{2}(X^{j}_{k\Delta})\right)^{2}}}$
is an empirical norm defined from the sample paths.

##### Related works.

There is a large literature on the estimation of coefficients of diffusion
processes, and we focus on the papers studying the estimation of $\sigma^{2}$.

Estimation of the diffusion coefficient has been considered in the parametric
case (see e.g. Genon-Catalot & Jacod (1993), Jacod (1993), Genon-Catalot &
Jacod (1994), Clement (1997), Genon-Catalot _et al._ (1999), Gloter (2000),
Sørensen (2002)). In the nonparametric case, estimators of the diffusion
coefficient from discrete observations are proposed under various frameworks.

First, the diffusion coefficient is constructed from one discrete observation
of the diffusion process ($N=1$) in long time ($T\rightarrow\infty$) (see e.g.
Hoffmann (1999a), Comte _et al._ (2007), Schmisser (2012), Schmisser (2019)),
or in short time ($T=1$) (see e.g. Genon-Catalot _et al._ (1992), Soulier
(1993), Hoffmann (1997), Florens (1998), Hoffmann (1999b)). Note that in short
time ($T<\infty$), only the diffusion coefficient can be estimated
consistently from a single discrete path contrary to the drift function whose
consistent estimation relies on repeated discrete observations of the
diffusion process (see e.g. Comte & Genon-Catalot (2020a), Denis _et al._
(2021)). For the case of short time diffusion processes (for instance $T=1$),
estimators of a time-dependent diffusion coefficients $t\mapsto\sigma^{2}(t)$
have been proposed. In this context, Genon-Catalot _et al._ (1992) built a
nonparametric estimator of $t\mapsto\sigma^{2}(t)$ and studied its $L_{2}$
risk using wavelets methods, Soulier (1993) studies the $L_{p}$ risk of a
kernel estimator of $\sigma^{2}(t)$, and Hoffmann (1997) derived a minimax
rate of convergence of order $n^{-ps/(1+2s)}$ where $s>1$ is the smoothness
parameter of the Besov space $\mathcal{B}^{s}_{p,\infty}([0,1])$ (see later in
the paper). For the space-dependent diffusion coefficient
$x\mapsto\sigma^{2}(x)$, a first estimator based on kernels and built from a
single discrete observation of the diffusion process with $T=1$ is proposed in
Florens-Zmirou (1993). The estimator has been proved to be consistent under a
condition on the bandwidth, but a rate of convergence of its risk of
estimation has not been established.

Secondly, the diffusion coefficient is built in short time ($T<\infty$) from
$N$ repeated discrete observations with $N\rightarrow\infty$. In Denis _et
al._ (2022), a nonparametric estimator of $\sigma^{2}$ is proposed from
repeated discrete observations on the real line $\mathbb{R}$ when the time
horizon $T=1$. The estimator has been proved to be consistent with a rate of
order $N^{-1/5}$ over the space of Lipschitz functions.

Two main methods are used to build consistent nonparametric estimators of
$x\mapsto\sigma^{2}(x)$. The first method is the one using kernels (see e.g.
Florens (1998), Bandi & Phillips (2003), Renò (2006), Gourieroux _et al._
(2017), Schmisser (2019), Park & Wang (2021)), the other method consists in
estimating $\sigma^{2}$ as solution of a nonparametric regression model using
the least squares approach. Since the diffusion coefficient is assumed to
belong to an infinite dimensional space, the method consists in projecting
$\sigma^{2}$ into a finite dimensional subspace, estimating the projection and
making a data-driven selection of the dimension by minimizing a penalized
least squares contrast (see e.g. Hoffmann (1999b), Hoffmann (1999a), Schmisser
(2012), Comte (2017), Schmisser (2019), Denis _et al._ (2022)).

##### Main contribution.

In this article, we assume to have at our disposal $N$ $\mathrm{i.i.d.}$
discrete observations of length $n$ of the diffusion process $X$. The main
objectives of this paper are the following.

  1. 1.

Construct a consistent and implementable ridge estimator of $\sigma^{2}$ from
a single diffusion path ($N=1$) using the least squares approach. We derive
rates of convergence of the risk of estimation of the ridge estimators built
on a compact interval and on the real line $\mathbb{R}$ over a Hölder space,
taking advantage of the properties of the local time of the diffusion process,
and its link with the transition density.

  2. 2.

We extend the result to the estimation of $\sigma^{2}$ on repeated
observations of the diffusion process ($N\rightarrow\infty$). We prove that
the estimators built on a compact interval and on $\mathbb{R}$ are more
efficient considering their respective rates compared to nonparametric
estimators built from a single diffusion path.

  3. 3.

Focusing on the support of the diffusion coefficient, we consider an
intermediate case between a compact interval and $\mathbb{R}$ by proposing a
ridge estimator of $\sigma^{2}$ restricted to the compact interval
$[-A_{N},A_{N}]$ where $A_{N}\rightarrow\infty$ as $N\rightarrow\infty$. The
benefit of this approach is that the resulting projection estimator can reach
a faster rate of convergence compared to the rate obtained on the real line
$\mathbb{R}$.

  4. 4.

Finally, we propose adaptive estimators of $\sigma^{2}$ based on a data-driven
selection of the dimension through the minimization of the penalized least
squares contrast in different settings.

We sum up below the rates of convergence (up to a log-factor) of the ridge
estimators of $\sigma^{2}_{|I}$ with $I\subseteq\mathbb{R}$ over a Hölder
space defined in the next section with a smoothness parameter $\beta\geq 1$.

Estimation interval |  $N=1$ and $n\rightarrow+\infty$ |  $N\rightarrow+\infty$ and $n\rightarrow+\infty$
---|---|---
$I=[-A,A],\leavevmode\nobreak\ \leavevmode\nobreak\ A>0$ | $n^{-\beta/(2\beta+1)}$ | $(Nn)^{-\beta/(2\beta+1)}$
$I=[-A_{N},A_{N}],\leavevmode\nobreak\ \leavevmode\nobreak\ A_{N}\underset{N\rightarrow+\infty}{\longrightarrow}{+\infty}$ | xxxx | $(Nn)^{-\beta/(2\beta+1)},\leavevmode\nobreak\ \leavevmode\nobreak\ N\propto n$
$I=\mathbb{R}$ | $n^{-\beta/(4\beta+1)}$ | $(Nn)^{-\beta/(4\beta+1)}+n^{-2}$
Table 1: Rates of convergence of the square root of the risk of estimation
$\mathbb{E}\left[\left\|\widehat{\sigma}^{2}-\sigma^{2}_{|I}\right\|^{2}_{n,N}\right]$
of the non-adaptive estimators $\widehat{\sigma}^{2}$ of the square of the
diffusion coefficient $\sigma^{2}_{|I}$ built from one diffusion path ($N=1$)
on the left column, and from repeated observations of the diffusion process
($N\rightarrow\infty$) on the right column. For the precise results, see
Sections 3 and 4.

##### Outline of the paper.

In Section 2, we define our framework with the key assumptions on the
coefficients of the diffusion process ensuring for instance that Equation (1)
admits a unique strong solution. Section 3 is devoted to the non-adaptive
estimation of the diffusion coefficient from one diffusion path both on a
compact interval and on the real line $\mathbb{R}$. In Section 4, we extend
the study to the non-adaptive estimation of the diffusion coefficient from
repeated observations of the diffusion process. We propose in Section 5,
adaptive estimators of the diffusion coefficient, and Section 6 complete the
study with numerical evaluation of the performance of estimators. We prove our
theoretical results in Section 8.

##  2 Framework and assumptions

Consider a diffusion process $X=(X_{t})_{t\in[0,1]}$, solution of Equation (1)
whose drift and diffusion coefficient satisfy the following assumption.

######  Assumption 2.1.

  1. 1.

There exists a constant $L_{0}>0$ such that $b$ and $\sigma$ are
$L_{0}-$Lipschitz functions on $\mathbb{R}$.

  2. 2.

There exist constants $\sigma_{0},\sigma_{1}>0$ such that :
$\sigma_{0}\leq\sigma(x)\leq\sigma_{1},\ \ \forall x\in\mathbb{R}$.

  3. 3.

$\sigma\in\mathcal{C}^{2}\left(\mathbb{R}\right)$ and there exist $C>0$ and
$\alpha\geq 0$ such that:

$\left|\sigma^{\prime}(x)\right|+\left|\sigma^{\prime\prime}(x)\right|\leq
C\left(1+|x|^{\alpha}\right),\ \ \forall x\in\mathbb{R}.$

Under Assumption 2.1, $X=(X_{t})_{t\in[0,1]}$ is the unique strong solution of
Equation (1), and this unique solution admits a transition density
$(t,x)\mapsto p_{X}(t,x)$. Besides, we draw from Assumption 2.1 that

$\forall q\geq 1,\ \
\mathbb{E}\left[\underset{t\in[0,1]}{\sup}{|X_{t}|^{q}}\right]<\infty.$ (2)

###  2.1 Definitions and notations

We suppose to have at our disposal, a sample $D_{N,n}=\left\\{\bar{X}^{j},\
j=1,\cdots,N\right\\}$ constituted of $N$ independent copies of the discrete
observation $\bar{X}=(X_{k\Delta_{n}})_{0\leq k\leq n}$ of the diffusion
process $X$ where $\Delta_{n}=1/n$ is the time-step. The objective is to
construct, from the sample $D_{N,n}$, a nonparametric estimator of the square
$\sigma^{2}$ of the diffusion coefficient on an interval
$I\subseteq\mathbb{R}$. In the sequel, we consider two main cases, the first
one being the estimation of $\sigma^{2}$ on the interval $I$ from a single
path ($N=1$ and $n\rightarrow\infty$). For the second case, we assume that
both $N$ and $n$ tend to infinity.

For each measurable function $h$, such that
$\mathbb{E}\left[h^{2}(X_{t})\right]<\infty$ for all $t\in[0,1]$, we define
the following empirical norms:

$\|h\|^{2}_{n}:=\mathbb{E}_{X}\left[\frac{1}{n}\sum_{k=0}^{n-1}{h^{2}\left(X_{k\Delta_{n}}\right)}\right],\
\ \
\|h\|^{2}_{n,N}:=\frac{1}{Nn}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{h^{2}\left(X^{j}_{k\Delta_{n}}\right)}}.$
(3)

For all $h\in\mathbb{L}^{2}(I)$, we have

$\displaystyle\|h\|^{2}_{n}=\int_{I}{h^{2}(x)\frac{1}{n}\sum_{k=0}^{n-1}{p_{X}(k\Delta_{n},x_{0},x)}dx}=\int_{I}{h^{2}(x)f_{n}(x)dx},$

where $f_{n}:x\mapsto\frac{1}{n}\sum_{k=0}^{n-1}{p_{X}(k\Delta_{n},x)}$ is a
density function. For the case of non-adaptive estimators of $\sigma^{2}$, we
also establish bounds of the risks of the estimators based on the empirical
norm $\|.\|_{n}$ or the $\mathbb{L}^{2}-$norm $\|.\|$ when the estimation
interval $I$ is compact.

For any integers $p,q\geq 2$ and any matrix $M\in\mathbb{R}^{p\times q}$, we
denote by ${}^{t}M$, the transpose of $M$.

###  2.2 Spaces of approximation

We propose projection estimators of $\sigma^{2}$ on a finite-dimensional
subspace. To this end, we consider for each $m\geq 1$, a $m$-dimensional
subspace $\mathcal{S}_{m}$ given as follows:

$\mathcal{S}_{m}:=\mathrm{Span}\left(\phi_{\ell},\ \
\ell=0,\cdots,m-1\right),\leavevmode\nobreak\ \leavevmode\nobreak\ m\geq 1$
(4)

where the functions $(\phi_{\ell},\leavevmode\nobreak\ \ell\in\mathbb{N})$ are
continuous, linearly independent and bounded on $I$. Furthermore, we need to
control the $\ell^{2}-$norm of the coordinate vectors of elements of
$\mathcal{S}_{m}$, which leads to the following constrained subspace,

$\mathcal{S}_{m,L}:=\left\\{h=\sum_{\ell=0}^{m-1}{a_{\ell}\phi_{\ell}},\ \
\sum_{\ell=0}^{m-1}{a^{2}_{\ell}}=\|\mathbf{a}\|^{2}_{2}\leq mL,\ \
\mathbf{a}=\left(a_{0},\cdots,a_{m-1}\right),\ \ L>0\right\\}.$ (5)

Note that $\mathcal{S}_{m,L}\subset\mathcal{S}_{m}$ and $\mathcal{S}_{m,L}$ is
no longer a vector space. The control of the coordinate vectors allows to
establish an upper bound of the estimation error that tends to zero as
$n\rightarrow\infty$ or $N,n\rightarrow\infty$. In fact, we prove in the next
sections that the construction of consistent estimators of $\sigma^{2}$
requires the functions $h=\sum_{\ell=0}^{m-1}{a_{\ell}\phi_{\ell}}$ to be
bounded, such that

$\|h\|_{\infty}\leq\underset{\ell=0,\ldots,m-1}{\max}{\|\phi_{\ell}\|_{\infty}}\leavevmode\nobreak\
\|\mathbf{a}\|_{2}.$

This condition is satisfied for the functions of the constrained subspaces
$\mathcal{S}_{m,L}$ with $m\geq 1$. In this article, we work with the
following bases.

##### [B] The B-spline basis

This is an exemple of a non-orthonormal basis defined on a compact interval.
Let $A>0$ be a real number, and suppose (without restriction) that $I=[-A,A]$.
Let $K,M\in\mathbb{N}^{*}$, and consider
$\mathbf{u}=\left(u_{-M},\cdots,u_{K+M}\right)$ a knots vector such that
$u_{-M}=\cdots=u_{-1}=u_{0}=-A$, $u_{K+1}=\cdots=u_{K+M}=A$, and for all
$i=0,\cdots,K$,

$\displaystyle u_{i}=-A+i\frac{2A}{K}.$

One calls B-spline functions, the piecewise polynomial functions
$\left(B_{\ell}\right)_{\ell=-M,\cdots,K-1}$ of degree $M$, associated with
the knots vector $\mathbf{u}$ (see Györfi _et al._ (2006), Chapter 14). The
B-spline functions are linearly independent smooths functions returning zero
for all $x\notin[-A,A]$, and satisfying some smoothness conditions established
in Györfi _et al._ (2006). Thus, we consider approximation subspaces
$\mathcal{S}_{K+M}$ defined by

$\displaystyle\mathcal{S}_{K+M}=\mathrm{Span}\left\\{B_{\ell},\
\ell=-M,\cdots,K-1\right\\}$

of dimension $\dim(\mathcal{S}_{K+M})=K+M$, and in which, each function
$h=\sum_{\ell=-M}^{K-1}{a_{\ell}B_{\ell}}$ is $M-1$ times continuously
differentiable thanks to the properties of the spline functions (see Györfi
_et al._ (2006)). Besides, the spline basis is included in the definition of
both the subspace $\mathcal{S}_{m}$ and the constrained subspace
$\mathcal{S}_{m,L}$ (see Equations (4) and (5)) with $m=K+M$ and for any
coordinates vector $(a_{-M},\ldots,a_{K-1})\in\mathbb{R}^{K+M}$,

$\sum_{\ell=-M}^{K-1}{a_{\ell}B_{\ell}}=\sum_{\ell=0}^{m-1}{a_{\ell-M}B_{\ell-M}}.$

The integer $M\in\mathbb{N}^{*}$ is fixed, while $K$ varies in the set of
integers $\mathbb{N}^{*}$. If we assume that $\sigma^{2}$ belongs to the
Hölder space $\Sigma_{I}(\beta,R)$ given as follows:

$\Sigma_{I}(\beta,R):=\left\\{h\in\mathcal{C}^{\lfloor\beta\rfloor+1}(I),\
\left|h^{(\ell)}(x)-h^{(\ell)}(y)\right|\leq R|x-y|^{\beta-l},\ \ x,y\in
I\right\\},$

where $\beta\geq 1$, $\ell=\lfloor\beta\rfloor$ and $R>0$, then the unknown
function $\sigma^{2}_{|I}$ restricted to the compact interval $I$ can be
approximated in the constrained subspace $\mathcal{S}_{K+M,L}$ spanned by the
spline basis. This approximation results to the following bias term:

$\underset{h\in\mathcal{S}_{K+M,L}}{\inf}{\|h-\sigma^{2}_{|I}\|^{2}_{n}}\leq
C|I|^{2\beta}K^{-2\beta}$ (6)

where the constant $C>0$ depends on $\beta,R$ and $M$, and $|I|=\sup I-\inf
I$. The above result is a modification of Lemma D.2 in Denis _et al._ (2021).

##### [F] The Fourier basis

The subspace $\mathcal{S}_{m}$ can be spanned by the Fourier basis

$\\{f_{\ell},\leavevmode\nobreak\ \leavevmode\nobreak\
\ell=0,\ldots,m-1\\}=\\{1,\sqrt{2}\cos(2\pi jx),\sqrt{2}\sin(2\pi
jx),\leavevmode\nobreak\ j=1,...,d\\}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{with}\leavevmode\nobreak\ \leavevmode\nobreak\ m=2d+1.$

The above Fourier basis is defined on the compact interval $[0,1]$. The
definition can be extended to any compact interval, replacing the bases
functions $x\mapsto f_{\ell}(x)$ by $x\mapsto 1/(\max I-\min
I)f_{\ell}(\frac{x-\min I}{\max I-\min I})$. We use this basis to build the
estimators of $\sigma^{2}$ on a compact interval $I\subset\mathbb{R}$.

Define for all $s\geq 1$ and for any compact interval $I\subset\mathbb{R}$,
the Besov space $\mathcal{B}^{s}_{2,\infty}(I)$ which is a space of functions
$f\in L^{2}(I)$ such that the $\lfloor s\rfloor^{th}$ derivative $f^{(\lfloor
s\rfloor)}$ belongs to the space $\mathcal{B}^{s-\lfloor
s\rfloor}_{2,\infty}(I)$ given by

$\mathcal{B}^{s-\lfloor s\rfloor}_{2,\infty}(I)=\left\\{f\in
L^{2}(I)\leavevmode\nobreak\ \mathrm{and}\leavevmode\nobreak\
\frac{w_{2,f}(t)}{t^{s-\lfloor s\rfloor}}\in
L^{\infty}(I\cap\mathbb{R}^{+})\right\\}$

where for $s-\lfloor s\rfloor\in(0,1)$, $w_{2,f}(t)=\underset{|h|\leq
t}{\sup}{\|\tau_{h}f-f\|_{2}}$ with $\tau_{h}f(x)=f(x-h)$, and for $s-\lfloor
s\rfloor=1$, $w_{2,f}(t)=\underset{|h|\leq
t}{\sup}{\|\tau_{h}f+\tau_{-h}f-2f\|_{2}}$. Thus, if we assume that the
function $\sigma^{2}_{|I}$ belongs to the Besov space
$\mathcal{B}^{s}_{2,\infty}$, then it can be approximated in a constrained
subspace $\mathcal{S}_{m,L}$ spanned by the Fourier basis. Moreover, under
Assumption 2.1 and from Lemma 12 in Barron _et al._ (1999), there exists a
constant $C>0$ depending on the constant $\tau_{1}$ of Equation (12), the
smoothness parameter $s$ of the Besov space such that

$\underset{h\in\mathcal{S}_{m,L}}{\inf}{\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n}}\leq\tau_{1}\underset{h\in\mathcal{S}_{m,L}}{\inf}{\left\|h-\sigma^{2}_{|I}\right\|^{2}}\leq
C\left|\sigma^{2}_{|I}\right|^{2}_{\beta}m^{-2\beta}$ (7)

where $|\sigma^{2}_{|I}|_{s}$ is the semi-norm of $\sigma^{2}_{|I}$ in the
Besov space $\mathcal{B}^{s}_{2,\infty}(I)$.

Note that for all $\beta\geq 1$, the Hölder space $\Sigma_{I}(\beta,R)$ and
the Besov space $\mathcal{B}^{\beta}_{2,\infty}$ satisfy:

$L^{\infty}(\mathbb{R})\cap\Sigma_{I}(\beta,R)\subset\mathcal{B}^{\beta}_{\infty,\infty}(I)\subset\mathcal{B}^{\beta}_{2,\infty}(I)$

(see DeVore & Lorentz (1993), Chap. 2 page 16). As a result, we rather
consider in the sequel the Hölder space $\Sigma_{I}(\beta,R)$ which can also
be approximated by the Fourier basis.

##### [H] The Hermite basis

The basis is defined from the Hermite functions $(h_{j},j\geq 0)$ defined on
$\mathbb{R}$ and given for all $j\geq 0$ and for all $x\in\mathbb{R}$ by:

$\displaystyle h_{j}(x)=c_{j}H_{j}(x),\ \ \mathrm{where}\ \
H_{j}(x)=(-1)^{j}\exp\left(\frac{x^{2}}{2}\right)\frac{d^{j}}{dx^{j}}\left(\mathrm{e}^{-x^{2}/2}\right)\
\ \mathrm{and}\ \ c_{j}=\left(2^{j}j!\sqrt{\pi}\right)^{-1/2}.$

The polynomials $H_{j}(x),\ j\geq 0$ are the Hermite polynomials, and
$(h_{j},j\geq 0)$ is an orthonormal basis of $L^{2}(\mathbb{R})$. Furthermore,
for all $j\geq 1$ and $x\in\mathbb{R}$, $|h_{j}(x)|\leq c|x|\exp(-c_{0}x^{2})$
for $x^{2}\geq(3/2)(4j+3)$ where $c,c_{0}>0$ are constants independent of $j$
(see Comte & Genon-Catalot (2020b), Proof of Proposition 3.5). We use the
Hermite basis in the sequel for the estimation of $\sigma^{2}$ on the real
line $\mathbb{R}$.

If one assumes that $\sigma^{2}$ belongs to the Sobolev space
$W^{s}_{f_{n}}(\mathbb{R},R)$ given for all $s\geq 1$ by

$W^{s}_{f_{n}}(\mathbb{R},R):=\left\\{g\in
L^{2}(\mathbb{R},f_{n}(x)dx),\leavevmode\nobreak\ \forall\leavevmode\nobreak\
\ell\geq 1,\leavevmode\nobreak\ \|g-g_{\ell}\|^{2}_{n}\leq
R\ell^{-s}\right\\}$

where for each $\ell\geq 1$, $g_{\ell}$ is the
$L^{2}(\mathbb{R},f_{n}(x)dx)-$orthogonal projection of $g$ on the
$\ell-$dimensional vector space $\mathcal{S}_{\ell}$ spanned by the Hermite
basis. Consider a compact interval $I\subset\mathbb{R}$ and the following
spaces:

$\displaystyle W^{s}(I,R):=$ $\displaystyle\leavevmode\nobreak\ \left\\{g\in
L^{2}(I),\leavevmode\nobreak\
\sum_{j=0}^{\infty}{j^{s}\left<g,\phi_{j}\right>^{2}\leq R}\right\\},$
$\displaystyle W^{s}_{f_{n}}(I,R):=$ $\displaystyle\leavevmode\nobreak\
\left\\{g\in L^{2}(I,f_{n}(x)dx),\leavevmode\nobreak\
\forall\leavevmode\nobreak\ \ell\geq 1,\leavevmode\nobreak\
\|g-g_{\ell}\|^{2}_{n}\leq R\ell^{-s}\right\\}$

where $(\phi_{j})_{j\geq 0}$ is an orthonormal basis defined on $I$ and for
all $\ell\geq 1$, $g_{\ell}$ is the orthogonal projection of $g$ onto
$\mathcal{S}_{\ell}=\mathrm{Span}(h_{j},\leavevmode\nobreak\ j\leq\ell)$ of
dimension $\ell\geq 1$ (see e.g. Comte & Genon-Catalot (2021)). Then, for all
$g\in W^{s}(I,R)$, we have

$g=\sum_{j=0}^{\infty}{\left<g,\phi_{j}\right>\phi_{j}}\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\|g-g_{\ell}\|^{2}=\sum_{j=\ell+1}^{\infty}{\left<g,\phi_{j}\right>^{2}}\leq\ell^{-s}\sum_{j=\ell+1}^{\infty}{j^{s}\left<g,\phi_{j}\right>^{2}}\leq
R\ell^{-s}.$

We have $W^{s}_{f_{n}}(I,R)=W^{s}(I,R)$ as the empirical norm $\|.\|_{n}$ and
the $L^{2}-$norm $\|.\|$ are equivalent. The space
$W^{s}_{f_{n}}(\mathbb{R},R)$ is an extension of the space
$W^{s}_{f_{n}}(I,R)$ wher $I=\mathbb{R}$ and $(\phi_{j})_{j\geq 0}$ is the
Hermite basis.

######  Remark 2.2.

The B-spline basis is used for the estimation of $\sigma^{2}$ on a compact
interval on one side ($N=1$ and $N>1$), and on the real line on the other side
restricting $\sigma^{2}$ on the compact interval $[-\log(n),\log(n)]$ for
$N=1$, or $[-\log(N),\log(N)]$ for $N>1$, and bounding the exit probability of
the process $X$ from the interval $[-\log(N),\log(N)]$ (or
$[-\log(n),\log(n)]$) by a negligible term with respect to the estimation
error. In a similar context, the Fourier basis is used as an othonormal basis
to built nonparametric estimators of $\sigma^{2}$ on a compact interval and on
$\mathbb{R}$, both for $N=1$ and for $N>1$. The main goal is to show that, in
addition to the spline basis which is not orthogonal, we can built projection
estimators of $\sigma^{2}$ on orthonormal bases that are consistent. The
advantage of the Hermite basis compared to the Fourier basis is its definition
on the real line $\mathbb{R}$. As a result, we use the Hermite basis to
propose for $N>1$, a projection estimator of $\sigma^{2}$ whose support is the
real line $\mathbb{R}$.

######  Remark 2.3.

Denote by $\mathcal{M}$, the set of possible values of the dimension $m\geq 1$
of the approximation subspace $\mathcal{S}_{m}$. If
$\left(\phi_{0},\cdots,\phi_{m-1}\right)$ is an orthonormal basis, then for
all $m,m^{\prime}\in\mathcal{M}$ such that $m<m^{\prime}$, we have
$\mathcal{S}_{m}\subset\mathcal{S}_{m^{\prime}}$. For the case of the B-spline
basis, one can find a subset $\mathcal{K}\subset\mathcal{M}$ of the form

$\mathcal{K}=\left\\{2^{q},\ q=0,\cdots,q_{\max}\right\\}$

such that for all $K,K^{\prime}\in\mathcal{K}$, $K<K^{\prime}$ implies
$\mathcal{S}_{K+M}\subset\mathcal{S}_{K^{\prime}+M}$ (see for example Denis
_et al._ (2021)). The nesting of subspaces $\mathcal{S}_{m},\ m\in\mathcal{M}$
is of great importance in the context of adaptive estimation of the diffusion
coefficient and the establishment of upper-bounds for the risk of adaptive
estimators.

In the sequel, we denote by $[\mathbf{F}],\leavevmode\nobreak\ [\mathbf{H}]$
and $[\mathbf{B}]$ the respective collection of subspaces spanned by the
Fourier basis, the Hermite basis and the B-spline basis.

###  2.3 Ridge estimators of the square of the diffusion coefficient

We establish from Equation (1) and the sample $D_{N,n}$ the regression model
for the estimation of $\sigma^{2}$. For all $j\in[\\![1,N]\\!]$ and
$k\in[\\![0,n-1]\\!]$, define

$U^{j}_{k\Delta_{n}}:=\frac{\left(X^{j}_{(k+1)\Delta_{n}}-X^{j}_{k\Delta_{n}}\right)^{2}}{\Delta_{n}}.$

The increments $U^{j}_{k\Delta_{n}}$ are approximations in discrete times of
$\frac{d\left<X,X\right>_{t}}{dt}$ since, from Equation (1), one has
$d\left<X,X\right>_{t}=\sigma^{2}(X_{t})dt$. From Equation (1), we obtain the
following regression model,

$U^{j}_{k\Delta_{n}}=\sigma^{2}(X^{j}_{k\Delta_{n}})+\zeta^{j}_{k\Delta_{n}}+R^{j}_{k\Delta_{n}},\leavevmode\nobreak\
\leavevmode\nobreak\ \forall(j,k)\in[\\![1,N]\\!]\times[\\![0,n-1]\\!]$

where $U^{j}_{k\Delta_{n}}$ is the response variable,
$\zeta^{j}_{k\Delta_{n}}$ and $R^{j}_{k\Delta_{n}}$ are respectively the error
term and a negligible residual whose explicit formulas are given in Section 8.

We consider the least squares contrast $\gamma_{n,N}$ defined for all
$m\in\mathcal{M}$ and for all function $h\in\mathcal{S}_{m,L}$ by

$\gamma_{n,N}(h):=\frac{1}{Nn}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{\left(U^{j}_{k\Delta}-h(X^{j}_{k\Delta_{n}})\right)^{2}}}.$
(8)

For each dimension $m\in\mathcal{M}$, the projection estimator
$\widehat{\sigma}^{2}_{m}$ of $\sigma^{2}$ over the subspace
$\mathcal{S}_{m,L}$ satisfies:

$\widehat{\sigma}^{2}_{m}\in\underset{h\in\mathcal{S}_{m,L}}{\arg\min}{\
\gamma_{n,N}(h)}.$ (9)

Indeed, for each dimension $m\in\mathcal{M}$, the estimator
$\widehat{\sigma}^{2}_{m}$ of $\sigma^{2}$ given in Equation (9) satisfies
$\widehat{\sigma}^{2}_{m}=\sum_{\ell=0}^{m-1}{\widehat{a}_{\ell}\phi_{\ell}}$,
where

$\widehat{\mathbf{a}}=\left(\widehat{a}_{0},\cdots,\widehat{a}_{m-1}\right):=\underset{\|\mathbf{a}\|^{2}_{2}\leq
mL}{\arg\min}{\left\|\mathbf{U}-\mathbf{F}_{m}\mathbf{a}\right\|^{2}_{2}}$
(10)

with
${\bf{}^{t}U}=\left(U^{1}_{0},\ldots,U^{1}_{(n-1)\Delta_{n}},\ldots,U^{N}_{0},\ldots,U^{N}_{(n-1)\Delta_{n}}\right)$
and the matrix $\mathbf{F}_{m}$ is defined as follows

${\bf
F}_{m}:=\left({}^{t}(\phi_{\ell}(X^{j}_{0}),\ldots,\phi_{\ell}(X^{j}_{(n-1)\Delta_{n}}))\right)_{\underset{1\leq
j\leq N}{0\leq\ell\leq m-1}}\in\mathbb{R}^{Nn\times m}.$

The vector of coefficients $\widehat{\mathbf{a}}$ is unique and called the
ridge estimator of $\mathbf{a}$ because of the $\ell^{2}$ constraint on the
coordinate vectors (see Hastie _et al._ (2001) Chap. 3 page 61).

##  3 Estimation of the diffusion coefficient from a single diffusion path

This section focuses on the nonparametric estimation of the square of the
diffusion coefficient $\sigma^{2}$ on an interval $I\subseteq\mathbb{R}$ when
only a single diffusion path is observed at discrete times ($N=1$). It is
proved in the literature that one can construct consistent estimators of the
diffusion coefficient from one path when the time horizon $T$ is finite (see
e.g. Hoffmann (1999b)). Two cases are considered. First, we propose a ridge
estimator of $\sigma^{2}$ on a compact interval $I\subset\mathbb{R}$, say for
example $I=[-1,1]$. Secondly, we extend the study to the estimation of
$\sigma^{2}$ on the real line $I=\mathbb{R}$.

###  3.1 Non-adaptive estimation of the diffusion coefficient on a compact
interval

In this section, we consider the estimator $\widehat{\sigma}^{2}_{m}$ of the
compactly supported square of the diffusion coefficient $\sigma^{2}_{|I}$ on
the constrained subspaces $\mathcal{S}_{m,L}$ from the observation of a single
diffusion path.

Since the interval $I\subset\mathbb{R}$ is compact, the immediate benefit is
that the density function $f_{n}$ defined from the transition density of the
diffusion process $\bar{X}=(X_{k\Delta})$ is bounded from below. In fact,
there exist constants $\tau_{0},\tau_{1}\in(0,1]$ such that

$\forall x\in I,\ \ \ \tau_{0}\leq f_{n}(x)\leq\tau_{1},$ (11)

(see Denis _et al._ (2021)). Thus, for each function $h\in\mathbb{L}^{2}(I)$,

$\tau_{0}\|h\|^{2}\leq\|h\|^{2}_{n}\leq\tau_{1}\|h\|^{2}$ (12)

where $\|.\|$ is the $\mathbb{L}^{2}-$norm. Equation (12) allows to establish
global rates of convergence of the risk of the ridge estimators
$\widehat{\sigma}^{2}_{m}$ of $\sigma^{2}_{|I}$ with $m\in\mathcal{M}$ using
the $L^{2}-$norm $\|.\|$ which is, in this case, equivalent with the empirical
norm $\|.\|_{n}$.

To establish an upper-bound of the risk of estimation that tends to zero as
$n$ tends to infinity, we need to establish equivalence relations between the
pseudo-norms $\|.\|_{n,1}\leavevmode\nobreak\ (N=1)$ and $\|.\|_{X}$ on one
side, and $\|.\|_{X}$ and the $L^{2}-$norm $\|.\|$ on the other side, where
the random pseudo-norm $\|.\|_{X}$ is defined for each function
$h\in\mathbb{L}^{2}(I)$ by

$\|h\|^{2}_{X}:=\int_{0}^{1}{h^{2}(X_{s})ds}.$ (13)

Define for $x\in\mathbb{R}$, the local time $\mathcal{L}^{x}$ of the diffusion
process $X=(X_{t})_{t\in[0,1]}$ by

$\mathcal{L}^{x}=\underset{\varepsilon\rightarrow
0}{\lim}{\frac{1}{2\varepsilon}\int_{0}^{1}{\mathds{1}_{(x-\varepsilon,x+\varepsilon)}(X_{s})ds}}.$
(14)

In general, the local time of a continuous semimartingale is a.s. càdlàg (see
e.g. Revuz & Yor (2013)). But, for diffusion processes and under Assumption
2.1, the local time $\mathcal{L}^{x}$ is bicontinuous at any point
$x\in\mathbb{R}$ (see Lemma 8.5 in Section 8). Furthermore, we obtain the
following result.

######  Lemma 3.1.

Under Assumption 2.1, and for any continuous and integrable function $h$, it
yields,

  1. 1.

$\int_{0}^{1}{h(X_{s})ds}=\int_{\mathbb{R}}{h(x)\mathcal{L}^{x}dx}$.

  2. 2.

For all $x\in\mathbb{R}$,
$\mathbb{E}(\mathcal{L}^{x})=\int_{0}^{1}{p_{X}(s,x)ds}$.

In Lemma 3.1, we remark that there is a link between the local time and the
transition density of the diffusion process. Thus, if we consider the pseudo-
norm $\|.\|_{X}$ depending on the process $X=(X_{t})_{t\in[0,1]}$ and given in
Equation (13), and using Lemma 3.1, we obtain that,

$\mathbb{E}\left[\|h\|_{X}^{2}\right]=\int_{\mathbb{R}}{h^{2}(x)\mathbb{E}\left[\mathcal{L}^{x}\right]dx}=\int_{\mathbb{R}}{h^{2}(x)\int_{0}^{1}{p_{X}(s,x)ds}dx}\geq\tau_{0}\|h\|^{2}.$
(15)

where $\int_{0}^{1}{p_{X}(s,x)ds}\geq\tau_{0}>0$ (see Denis _et al._ (2021),
Lemma 4.3), and $\|h\|^{2}$ is the $\mathbb{L}^{2}-$norm of $h$.

######  Theorem 3.2.

Set $L=\log(n)$. Suppose that $\sigma^{2}$ is approximated in one of the
collections $[\mathbf{B}]$ and $[\mathbf{F}]$. Under Assumption 2.1, it yields

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n,1}\right]\leq
3\underset{h\in\mathcal{S}_{m,L}}{\inf}{\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n}}+C\left(\frac{m}{n}+\frac{m^{2\gamma+1}\log(n)}{n^{\gamma/2}}+\Delta^{2}_{n}\right)$
$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n}\right]\leq\frac{34\tau_{1}}{\tau_{0}}\underset{h\in\mathcal{S}_{m,L}}{\inf}{\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n}}+C^{\prime}\left(\frac{m}{n}+\frac{m^{2\gamma+1}\log(n)}{n^{\gamma/2}}+\Delta^{2}_{n}\right)$

where the number $\gamma>1$ comes from the use of the Hölder inequality. The
constant $C>0$ depends on $\sigma_{1}$ and the constant $C^{\prime}>0$ depends
on $\sigma_{1},\tau_{0}$ and $\tau_{1}$.

We observe that the upper-bound of the risk of estimation of
$\widehat{\sigma}^{2}_{m}$ is composed of the bias term, which quantifies the
cost of approximation of $\sigma^{2}_{|I}$ in the constrained space
$\mathcal{S}_{m,L}$, the estimation error $\mathrm{O}(m/n)$ and the cost of
the time discretization $\mathrm{O}(\Delta^{2}_{n})$ are established on a
random event in which the pseudo-norms $\|.\|_{n,1}$ and $\|.\|_{X}$ are
equivalent, and whose probability of the complementary times
$\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{\infty}$ is
bounded by the term
$\mathrm{O}\left(\frac{m^{2\gamma+1}\log(n)}{n^{\gamma/2}}\right)$ (see Lemma
8.9 and proof of Theorem 3.2).

The next result proves that the risk of estimation can reach a rate of
convergence of the same order than the rate established in Hoffmann (1999b) if
the parameter $\gamma>1$ is chosen such that the term
$\mathrm{O}(m^{2\gamma+1}\log(n)/n^{\gamma/2})$ is of the same order than the
estimation error of order $m/n$. Note that the risk
$\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n}$ is random
since

$\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n}=\mathbb{E}_{X}\left[\frac{1}{n}\sum_{k=0}^{n-1}{(\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I})(X_{k\Delta})}\right]$

and the estimator $\widehat{\sigma}^{2}_{m}$ is built from an independent copy
$\bar{X}^{1}$ of the discrete times process $\bar{X}$. Thus, the expectation
$\mathbb{E}$ relates to the estimator $\widehat{\sigma}^{2}_{m}$.

######  Corollary 3.3.

Suppose that $\sigma^{2}\in\Sigma_{I}(\beta,R)$ with $\beta>3/2$, and
$\gamma=2(2\beta+1)/(2\beta-3)$. Assume that $K_{\mathrm{opt}}\propto
n^{1/(2\beta+1)}$ for $[\mathbf{B}]$ ($m_{\mathrm{opt}}=K_{\mathrm{opt}}+M$),
and $m_{\mathrm{opt}}\propto n^{1/(2\beta+1)}$ for $[\mathbf{F}]$. Under
Assumptions 2.1, it yields,

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m_{\mathrm{opt}}}-\sigma^{2}_{|I}\right\|^{2}_{n,1}\right]=\mathrm{O}\left(\log(n)n^{-2\beta/(2\beta+1)}\right)$
$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m_{\mathrm{opt}}}-\sigma^{2}_{|I}\right\|^{2}_{n}\right]=\mathrm{O}\left(\log(n)n^{-2\beta/(2\beta+1)}\right).$

Note that we obtain the exact same rates when considering the risk of
$\widehat{\sigma}^{2}_{m_{\mathrm{opt}}}$ defined with the
$\mathbb{L}^{2}-$norm equivalent to the empirical norm $\|.\|_{n}$. Moreover,
these rates of convergence are of the same order than the optimal rate
$n^{-s/(2s+1)}$ established in Hoffmann (1999b) over a Besov ball.

###  3.2 Non-adaptive estimation of the diffusion coefficient on the real line

In this section, we propose a ridge estimator of $\sigma^{2}$ on the real line
$\mathbb{R}$, built from one diffusion path. In this context, the main
drawback is that the density function
$f_{n}:x\mapsto\frac{1}{n}\sum_{k=0}^{n-1}{p_{X}(k\Delta,x)}$ is no longer
lower bounded. Consequently, the empirical norm $\|.\|_{n}$ is no longer
equivalent to the $L_{2}-$norm $\|.\|$ and the consistency of the estimation
error is no longer ensured under the only assumptions made in the previous
sections. Consider the truncated estimator $\widehat{\sigma}^{2}_{m,L}$ of
$\sigma^{2}$ given by

$\widehat{\sigma}^{2}_{m,L}(x)=\widehat{\sigma}^{2}_{m}(x)\mathds{1}_{\widehat{\sigma}^{2}_{m}(x)\leq\sqrt{L}}+\sqrt{L}\mathds{1}_{\widehat{\sigma}^{2}_{m}(x)>\sqrt{L}}.$
(16)

Thus, the risk of the ridge estimator $\widehat{\sigma}^{2}_{m,L}$ is upper-
bounded as follows:

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n,1}\right]\leq$
$\displaystyle\leavevmode\nobreak\
\mathbb{E}\left[\left\|(\widehat{\sigma}^{2}_{m,L}-\sigma^{2})\mathds{1}_{[-\log(n),\log(n)]}\right\|^{2}_{n,1}\right]+\mathbb{E}\left[\left\|(\widehat{\sigma}^{2}_{m,L}-\sigma^{2})\mathds{1}_{[-\log(n),\log(n)]^{c}}\right\|^{2}_{n,1}\right]$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
\mathbb{E}\left[\left\|(\widehat{\sigma}^{2}_{m,L}-\sigma^{2})\mathds{1}_{[-\log(n),\log(n)]}\right\|^{2}_{n,1}\right]+4\log^{2}(n)\underset{t\in[0,1]}{\sup}{\mathbb{P}(|X_{t}|>\log(n))}.$

The first term on the r.h.s. is equivalent to the risk of a ridge estimator of
$\sigma^{2}$ on the compact interval $[-\log(n),\log(n)]$. The second term on
the r.h.s. is upper-bounded using Lemma 8.4. We derive below, an upper-bound
of the risk of estimation of $\widehat{\sigma}^{2}_{m}$.

######  Theorem 3.4.

Suppose that $L=\log^{2}(n)$. Under Assumption 2.1, it yields,

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n,1}\right]\leq\underset{h\in\mathcal{S}_{m,L}}{\inf}{\|h-\sigma^{2}\|^{2}_{n}}+C\sqrt{\frac{m^{q}\log^{2}(n)}{n}}$

where $C>0$ is a constant, $q=1$ for the collection $[\mathbf{B}]$, and $q=2$
for the collection $[\mathbf{F}]$.

We first remark that the upper-bound of the risk of the truncated estimator of
$\sigma^{2}$ differs with respect to each of the chosen bases. This contrast
comes from the fact that the Fourier basis $\\{f_{\ell},\leavevmode\nobreak\
\ell=0,\ldots,m-1\\}$ and the spline basis $\\{B_{\ell-M},\leavevmode\nobreak\
\ell=0,\ldots,m-1\\}$ satisfy

$\sum_{\ell=0}^{m-1}{f_{\ell}(x)}\leq C_{f}m,\leavevmode\nobreak\
\mathrm{and}\leavevmode\nobreak\ \sum_{\ell=0}^{m-1}{B_{\ell-M}(x)}=1.$

Secondly, the estimation error is not as fine as the one established in
Theorem 3.2 where $\sigma^{2}$ is estimated on a compact interval. In fact, on
the real line $\mathbb{R}$, the pseudo-norm $\|.\|_{X}$ can no longer be
equivalent to the $\mathbb{L}^{2}-$norm since the transition density is not
bounded from below on $\mathbb{R}$. Consequently, we cannot take advantage of
the exact method used to establish the risk bound obtained in Theorem 3.2
which uses the equivalence relation between the pseudo-norms $\|.\|_{n,1}$ and
$\|.\|_{X}$ on one side, and $\|.\|_{X}$ and the $\mathbb{L}^{2}-$norm $\|.\|$
on the other side. Moreover, we can also notice that the term of order
$1/n^{2}$ does not appear since it is dominated by the estimation error.

We obtain below rates of convergence of the ridge estimator of $\sigma^{2}$
for each of the collections $[\mathbf{B}]$ and $[\mathbf{F}]$.

######  Corollary 3.5.

Suppose that $\sigma^{2}\in\Sigma_{I}(\beta,R)$ with $\beta\geq 1$

##### For [B].

Assume that $K\propto n^{1/(4\beta+1)}$. Under Assumptions 2.1, there exists a
constant $C>0$ depending on $\beta$ and $\sigma_{1}$ such that

$\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n,1}\right]\leq
C\log^{2\beta}(n)n^{-2\beta/(4\beta+1)}.$

##### For [F].

Assume that $m\propto n^{1/2(2\beta+1)}$. Under Assumptions 2.1, it yields,

$\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n,1}\right]\leq
C\log(n)n^{-\beta/(2\beta+1)}$

where the constant $C>0$ depends on $\beta$ and $\sigma_{1}$.

As we can remark, the obtained rates are slower than the ones established in
Section 3.1 where $\sigma^{2}$ is estimated on a compact interval. This result
is the immediate consequence of the result of Theorem 3.4.

##  4 Estimation of the diffusion coefficient from repeated diffusion paths

We now focus on the estimation of the (square) of the diffusion coefficient
from i.i.d. discrete observations of the diffusion process
($N\rightarrow\infty$).

###  4.1 Non-adaptive estimation of the diffusion coefficient on a compact
interval

We study the rate of convergence of the ridge estimators
$\widehat{\sigma}^{2}_{m}$ of $\sigma^{2}_{|I}$ from $D_{N,n}$ when $I$ is a
compact interval. The next theorem gives an upper-bound of the risk of our
estimators $\widehat{\sigma}^{2}_{m},\leavevmode\nobreak\ m\in\mathcal{M}$.

######  Theorem 4.1.

Suppose that $L=\log(Nn)$ and
$\mathcal{M}=\left\\{1,\ldots,\sqrt{\min(n,N)}/\log(Nn)\right\\}$. Under
Assumption 2.1 and for all $m\in\mathcal{M}$, there exist constants $C>0$ and
$C^{\prime}>0$ depending on $\sigma_{1}$ such that,

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n,N}\right]\leq$
$\displaystyle\leavevmode\nobreak\
3\underset{h\in\mathcal{S}_{m,L}}{\inf}{\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n}}+C\left(\frac{m}{Nn}+m\log(Nn)\exp\left(-C\sqrt{\min(n,N)}\right)+\Delta^{2}_{n}\right)$
$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n}\right]\leq$
$\displaystyle\leavevmode\nobreak\
34\underset{h\in\mathcal{S}_{m,L}}{\inf}{\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n}}+C^{\prime}\left(\frac{m}{Nn}+m\log(Nn)\exp\left(-C\sqrt{\min(n,N)}\right)+\Delta^{2}_{n}\right).$

Note that the result of Theorem 4.1 is independent of the choice of the basis
that generate the approximation space $\mathcal{S}_{m}$. The first term on the
right-hand side represents the approximation error of the initial space, the
second term $\mathrm{O}\left(m/(Nn)\right)$ is the estimation error, and the
last term characterizes the cost of the time discretization. The next result
is derived from Theorem 4.1.

######  Corollary 4.2.

Suppose that $\sigma^{2}\in\Sigma_{I}(\beta,R)$ with $\beta>3/2$. Moreover,
assume that $K_{\mathrm{opt}}\propto(Nn)^{1/(2\beta+1)}$ for $[\mathbf{B}]$
($m_{\mathrm{opt}}=K_{\mathrm{opt}}+M$), and
$m_{\mathrm{opt}}\propto(Nn)^{1/(2\beta+1)}$ for $[\mathbf{F}]$. Under
Assumptions 2.1, it yields,

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m_{\mathrm{opt}}}-\sigma^{2}_{|I}\right\|^{2}_{n,N}\right]=$
$\displaystyle\leavevmode\nobreak\
\mathrm{O}\left((Nn)^{-2\beta/(2\beta+1)}\right)$
$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m_{\mathrm{opt}}}-\sigma^{2}_{|I}\right\|^{2}_{n}\right]=$
$\displaystyle\leavevmode\nobreak\
\mathrm{O}\left((Nn)^{-2\beta/(2\beta+1)}\right).$

The obtained result shows that the nonparametric estimators of
$\sigma^{2}_{|I}$ based on repeated observations of the diffusion process are
more efficient when $N,n\rightarrow\infty$. Note that the same rate is
obtained if the risk of $\widehat{\sigma}^{2}_{m_{\mathrm{opt}}}$ is defined
with the $\mathbb{L}^{2}-$norm $\|.\|$ equivalent to the empirical norm
$\|.\|_{n}$.

The rate obtained in Corollary 4.2 is established for $\beta>3/2$. If we
consider for example the collection ${\bf[B]}$ and assume that
$\beta\in[1,3/2]$, then $K_{\mathrm{opt}}\propto(Nn)^{1/(2\beta+1)}$ belongs
to $\mathcal{M}$ for $n\propto\sqrt{N}/\log^{4}(N)$ and we have

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m_{\mathrm{opt}}}-\sigma^{2}_{|I}\right\|^{2}_{n,N}\right]\leq
C(Nn)^{-2\beta/(2\beta+1)}.$

Under the condition $n\propto\sqrt{N}/\log^{4}(N)$ imposed on the length of
diffusion paths, the obtained rate is of order $n^{-3\beta/(2\beta+1)}$ (up to
a log-factor) which is equivalent to $N^{-3\beta/2(2\beta+1)}$ (up to a log-
factor).

###  4.2 Non-adaptive estimation of the diffusion coefficient on the real line

Consider a ridge estimator of $\sigma^{2}$ on $\mathbb{R}$ built from $N$
independent copies of the diffusion process $X$ observed in discrete times,
where both $N$ and $n$ tend to infinity. For each $m\in\mathcal{M}$, we still
denote by $\widehat{\sigma}^{2}_{m}$ the ridge estimators of $\sigma^{2}$ and
$\widehat{\sigma}^{2}_{m,L}$ the truncated estimators of $\sigma^{2}$ given in
Equation (16). We establish, through the following theorem, the first risk
bound that highlights the main error terms.

######  Theorem 4.3.

Suppose that $L=\log^{2}(N)$. Under Assumptions 2.1 and for any dimension
$m\in\mathcal{M}$, the following holds:

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n,N}\right]\leq$
$\displaystyle\leavevmode\nobreak\
2\underset{h\in\mathcal{S}_{m,L}}{\inf}{\|h-\sigma^{2}\|^{2}_{n}}+C\left(\sqrt{\frac{m^{q}\log^{2}(N)}{Nn}}+\Delta^{2}_{n}\right)$

where $C>0$ is a constant depending on the upper bound $\sigma_{1}$ of the
diffusion coefficient. Moreover, $q=1$ for the collection $[\mathbf{B}]$ and
$q=2$ for the collection $[\mathbf{H}]$.

If we consider the risk of $\widehat{\sigma}^{2}_{m,L}$ using the empirical
norm $\|.\|_{n}$, then we obtain

$\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n}\right]\leq
2\underset{h\in\mathcal{S}_{m,L}}{\inf}{\|h-\sigma^{2}\|^{2}_{n}}+C\left(\sqrt{\frac{m^{q}\log^{2}(N)}{Nn}}+\frac{m^{2}\log^{3}(N)}{N}+\Delta^{2}_{n}\right)$
(17)

The risk bound given in Equation (17) is a sum of four error terms. The first
term is the approximation error linked to the choice of the basis, the second
term is the estimation error given in Theorem 4.3, the third term
$m^{2}\log^{3}(N)/N$ comes from the relation linking the empirical norm
$\|.\|_{n}$ to the pseudo-norm $\|.\|_{n,N}$ (see Lemma 8.7), and the last
term is the cost of the time-discretization.

We derive, in the next result, rates of convergence of the risk bound of the
truncated ridge estimators $\widehat{\sigma}^{2}_{m,L}$ based on the
collections $[\mathbf{B}]$ and $[\mathbf{H}]$ respectively.

######  Corollary 4.4.

Suppose that $\sigma^{2}\in\Sigma_{I}(\beta,R)$ with $\beta\geq
1,\leavevmode\nobreak\ I=[-\log(N),\log(N)]$, and
$K\propto(Nn)^{1/(4\beta+1)}$ for $[\mathbf{B}]$, and $\sigma^{2}\in
W^{s}_{f_{n}}(\mathbb{R},R)$ with $s\geq 1$ and $m\propto(Nn)^{1/2(2s+1)}$ for
$[\mathbf{H}]$. Under Assumption 2.1, the following holds:

$\mathrm{For}\leavevmode\nobreak\ [\mathbf{B}]\leavevmode\nobreak\
\leavevmode\nobreak\
\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n,N}\right]\leq
C\left(\log^{2\beta}(N)(Nn)^{-2\beta/(4\beta+1)}+\frac{1}{n^{2}}\right),$
$\mathrm{For}\leavevmode\nobreak\ [\mathbf{H}]\leavevmode\nobreak\
\leavevmode\nobreak\
\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n,N}\right]\leq
C\left(\log^{3}(N)(Nn)^{-s/(2s+1)}+\frac{1}{n^{2}}\right).$

where $C>0$ is a constant depending on $\beta$ and $\sigma_{1}$ for
$[\mathbf{B}]$, or $s$ and $\sigma_{1}$ for $[\mathbf{H}]$.

The obtained rates are slower compared to the rates established in Section 4.1
for the estimation of $\sigma^{2}_{|I}$ where the interval
$I\subset\mathbb{R}$ is compact. In fact, the method used to establish the
rates of Theorem 4.3 from which the rates of Corollary 4.4 are obtained, does
not allow us to derive rates of order $(Nn)^{-\alpha/(2\alpha+1)}$ (up to a
log-factor) with $\alpha\geq 1$ (e.g. $\alpha=\beta,s$). Finally, if we
consider the risk defined with the empirical norm $\|.\|_{n}$, then from
Equation (17) with $n\propto N$ and assuming that $m\propto N^{1/4(s+1)}$ for
$[\mathbf{H}]$ or $K\propto N^{1/4(\beta+1)}$ for $[\mathbf{B}]$, we obtain

$\displaystyle[\mathbf{B}]:\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n}\right]\leq$
$\displaystyle\leavevmode\nobreak\ C\log^{2\beta}(N)(Nn)^{-\beta/2(\beta+1)},$
$\displaystyle[\mathbf{H}]:\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n}\right]\leq$
$\displaystyle\leavevmode\nobreak\ C\log^{3}(N)(Nn)^{-s/2(s+1)},$

where $C>0$ is a constant depending on $\sigma_{1}$ and on the smoothness
parameter. We can see that the obtained rates are slower compared to the
results of Corollary 4.4 for $n\propto N$. The deterioration of the rates
comes from the additional term of order $m^{2}\log^{3}(N)/N$ which is now
regarded as the new estimation error since it dominates the other term in each
case as $N\rightarrow\infty$.

###  4.3 Non-adaptive estimation of the diffusion coefficient on a compact
interval depending on the sample size

This section combines the two first sections 4.1 and 4.2 focusing on the
estimation of $\sigma^{2}$ on the compact interval $[-A_{N},A_{N}]$ where
$(A_{N})$ is a strictly positive sequence such that $A_{N}\rightarrow\infty$
as $N\rightarrow\infty$. Consequently, we obtain that the estimation interval
tends to $\mathbb{R}$ as the sample size $N$ tends to infinity.

Define from the observations and for each dimension $m\in\mathcal{M}$, the
following matrices:

$\widehat{\Psi}_{m}:=\left(\frac{1}{Nn}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{\phi_{\ell}\left(X^{j}_{k\Delta}\right)\phi_{\ell^{\prime}}\left(X^{j}_{k\Delta}\right)}}\right)_{0\leq\ell,\ell^{\prime}\leq
m-1},$ (18)

$\Psi_{m}:=\mathbb{E}\left(\widehat{\Psi}_{m}\right)=\left(\mathbb{E}\left[\frac{1}{n}\sum_{k=0}^{n-1}{\phi_{\ell}\left(X_{k\Delta}\right)\phi_{\ell^{\prime}}\left(X_{k\Delta}\right)}\right]\right)_{0\leq\ell,\ell^{\prime}\leq
m-1}.$ (19)

These two matrices play an essential role in the construction of a consistent
projection estimator of $\sigma^{2}$ over any approximation subspace
$\mathcal{S}_{m}$ spanned by the basis
$\left(\phi_{0},\cdots,\phi_{m-1}\right)$. Furthermore, for all
$h=\sum_{\ell=0}^{m-1}{a_{\ell}\phi_{\ell}}\in\mathcal{S}_{m}$, we have:

$\displaystyle\|h\|^{2}_{n,N}={{}^{t}}\mathbf{a}\widehat{\Psi}_{m}\mathbf{a},\
\ \
\|h\|^{2}_{n}=\mathbb{E}\left(\|h\|^{2}_{n,N}\right)={{}^{t}}\mathbf{a}\Psi_{m}\mathbf{a},$

where $\mathbf{a}=\left(a_{0},\cdots,a_{m-1}\right)$. The Gram matrix
$\Psi_{m}$ is invertible under the spline basis (see Denis _et al._ (2022))
and the Hermite basis (see Comte & Genon-Catalot (2021)). We define for any
invertible matrix $M$, the operator norm $\left\|M^{-1}\right\|_{\mathrm{op}}$
of $M^{-1}$ given by
$\left\|M^{-1}\right\|_{\mathrm{op}}=1/\inf\\{\lambda_{j}\\}$ where the
$\lambda_{j}$ are eigenvalues of $M$.

For all dimension $m\in\mathcal{M}$, the matrices $\widehat{\Psi}_{m}$ and
$\mathbf{F}_{m}$ satisfy:

$\displaystyle\widehat{\Psi}_{m}={{}^{t}}\mathbf{F}_{m}\mathbf{F}_{m}.$

Consider the ridge estimator $\widehat{\sigma}^{2}_{m}$ of
$\sigma^{2}_{A_{N}}=\sigma^{2}\mathds{1}_{[-A_{N},A_{N}]}$, with
$m\in\mathcal{M}$ and $A_{N}\rightarrow\infty$ as $N\rightarrow\infty$. The
estimator $\widehat{\sigma}^{2}_{m}$ can reach a faster rate of convergence if
the Gram matrix $\Psi_{m}$ given in Equation (19) satisfies the following
condition,

$\mathcal{L}(m)\left(\left\|\Psi^{-1}_{m}\right\|_{\mathrm{op}}\vee
1\right)\leq C\frac{N}{\log^{2}(N)},\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{where}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{L}(m):=\underset{x\in\mathbb{R}}{\sup}{\sum_{\ell=0}^{m-1}{\phi^{2}_{\ell}(x)}}<\infty$
(20)

where $C>0$ is a constant. In fact, the optimal rate of convergence is
achieved on a random event $\Omega_{n,N,m}$ in which the two empirical norms
$\|.\|_{n,N}$ and $\|.\|_{n}$ are equivalent (see Comte & Genon-Catalot
(2020a), Denis _et al._ (2021)). Then, Condition (20) is used to upper-bound
$\mathbb{P}\left(\Omega^{c}_{n,N,m}\right)$ by a negligible term with respect
to the considered rate (see Comte & Genon-Catalot (2020a)). Note that in
Equation (20), the square on $\log(N)$ is justified by the fact that the value
of constant $C>0$ is unknown, and that the spline basis is not othonormal (see
Denis _et al._ (2022), proof of Lemma 7.8). The assumption of Equation (20) is
also made in Comte & Genon-Catalot (2020a) on the operator norm of
$\Psi^{-1}_{m}$ based on an orthonormal basis with the bound
$\mathbf{c}N/\log(N)$ where the value of $\mathbf{c}$ is known, and chosen and
such that the upper-bound of $\mathbb{P}(\Omega^{c}_{n,N,m})$ is negligible
with respect to the estimation error. In our framework, since the transition
density is approximated by Gaussian densities, we derive the following result.

######  Lemma 4.5.

Suppose that $n\propto N$ and that the spline basis is constructed on the
interval $[-A_{N},A_{N}]$ with $A_{N}>0$. Under Assumption 2.1 , for all
$m\in\mathcal{M}$ and for all $w\in\mathbb{R}^{m}$ such that $\|w\|_{2,m}=1$,
there exists a constant $C>0$ such that

$\displaystyle\mathrm{For}\leavevmode\nobreak\ \leavevmode\nobreak\
[\mathbf{H}]:\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle\leavevmode\nobreak\
w^{\prime}\Psi_{m}w\geq\frac{C}{\log(N)}\exp\left(-\frac{3c_{\sigma}(4m+3)}{2\left(1-\log^{-1}(N)\right)}\right),$
$\displaystyle\mathrm{For}\leavevmode\nobreak\ \leavevmode\nobreak\
[\mathbf{B}]:\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle\leavevmode\nobreak\
w^{\prime}\Psi_{m}w\geq\frac{CA_{N}}{m\log(N)}\exp\left(-c_{\sigma}A^{2}_{N}\right),$

where the constant $c_{\sigma}>1$ that comes from the approximation of the
transition density, depends on the diffusion coefficient $\sigma$.

The result of Lemma 4.5 implies for the Hermite basis that

$\left(\left\|\Psi^{-1}_{m}\right\|_{\mathrm{op}}\vee
1\right)\leq\frac{\log(N)}{C}\exp\left(\frac{3c_{\sigma}(4m+3)}{2\left(1-\log^{-1}(N)\right)}\right)$

where the upper-bound is an exponentially increasing sequence of $N$ since the
dimension $m\in\mathcal{M}$ has a polynomial growth with respect to $N$. Thus,
Condition (20) cannot be satisfied for the Hermite basis in our framework.
Considering the spline basis, one has $\mathcal{L}(m)=\mathcal{L}(K+M)\leq 1$
and there exists a constant $C>0$ such that

$\left\|\Psi^{-1}_{m}\right\|_{\mathrm{op}}\leq
C\frac{m\log(N)}{A_{N}}\exp\left(c_{\sigma}A^{2}_{N}\right).$ (21)

For $K\propto\left(N^{2/(2\beta+1)}A_{N}\right)$, Condition (20) is satisfied
if the estimation interval $[-A_{N},A_{N}]$ is chosen such that
$A_{N}=o\left(\sqrt{\log(N)}\right)$. In the next theorem, we prove that the
spline-based ridge estimator of $\sigma^{2}_{A_{N}}$ reaches a faster rate of
convergence compared to the result of Corollary 4.4 for the collection
$[\mathbf{B}]$.

######  Theorem 4.6.

Suppose that $N\propto n$ and consider the ridge estimator
$\widehat{\sigma}^{2}_{A_{N},m}$ of $\sigma^{2}_{A_{N}}$ based on the spline
basis. Furthermore, suppose that $L=\log(N)$,
$A_{N}=o\left(\sqrt{\log(N)}\right)$ and $K\propto(Nn)^{1/(2\beta+1)}A_{N}$
($m=K+M$). Under Assumptions 2.1 and for $\sigma^{2}\in\Sigma_{I}(\beta,R)$
with $I=[-A_{N},A_{N}]$, the following holds:

$\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{A_{N},m}-\sigma^{2}_{A_{N}}\right\|^{2}_{n,N}\right]\leq
C\log^{\beta}(N)(Nn)^{-2\beta/(2\beta+1)}$

where $C>0$ is a constant depending on $\beta$.

The above result shows that the risk of the ridge estimator of
$\sigma^{2}_{A_{N}}$ on $[-A_{N},A_{N}]$ reaches a rate of order
$(Nn)^{-\beta/(2\beta+1)}$ (up tp a log-factor) thanks to Condition (20) which
allows us to take advantage of the equivalence relation between the empirical
norms $\|.\|_{n}$ and $\|.\|_{n,N}$ given in Equation (3) to derive a finer
estimation error (see proof of Theorem 4.6). Note that the obtained result
depends on an appropriate choice of the estimation interval $[-A_{N},A_{N}]$
which tends to $\mathbb{R}$ as $N$ tends to infinity. Therefore, any choice of
$A_{N}$ such that $A_{N}/\sqrt{\log(N)}\longrightarrow+\infty$ cannot lead to
a consistent estimation error since Equation (20) is no longer satisfied for
the upper-bounding of $\mathbb{P}\left(\Omega^{c}_{n,N,m}\right)$ by a term
that tends to zero as $N\rightarrow\infty$. Thus, the assumption
$A_{N}=o\left(\sqrt{\log(N)}\right)$ is a necessary and sufficient condition
for the validation of Condition (20) which leads, together with Assumption
2.1, to the result of Theorem 4.6. Finally, under the assumptions of Theorem
4.6 and considering the risk of $\widehat{\sigma}^{2}_{A_{N},m}$ based on the
empirical norm $\|.\|_{n}$, we also obtain

$\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{A_{N},m}-\sigma^{2}_{A_{N}}\right\|^{2}_{n}\right]=\mathrm{O}\left(\log^{\beta}(N)(Nn)^{-2\beta/(2\beta+1)}\right).$

In fact, under Condition (20), the estimator $\widehat{\sigma}^{2}_{A_{N},m}$
satisfies the results of Theorem 4.1 with $I=[-A_{N},A_{N}]$ and
$A_{N}=o\left(\sqrt{\log(N)}\right)$, which implies rates of the same order
for the two empirical norms.

##  5 Adaptive estimation of the diffusion coefficient from repeated
observations

In this section, we suppose that $n\propto N$ and we propose a adaptive ridge
estimator of $\sigma^{2}$ by selecting an optimal dimension from the sample
$D_{N}$. In fact, consider the estimator
$\widehat{\sigma}^{2}_{\widehat{K},L}$ where $\widehat{K}$ satisfies:

$\widehat{K}:=\underset{K\in\mathcal{K}}{\arg\min}{\left\\{\gamma_{n,N}\left(\widehat{\sigma}^{2}_{K}\right)+\mathrm{pen}(K)\right\\}}$
(22)

and the penalty function $\mathrm{pen}:K\mapsto\mathrm{pen}(K)$ is established
using the chaining technique of Baraud _et al._ (2001). We derive below the
risk of the adaptive estimator of $\sigma^{2}_{|I}$ when the interval
$I\subset\mathbb{R}$ is compact and the sample size $N\rightarrow\infty$.

######  Theorem 5.1.

Suppose that $N\propto n,\leavevmode\nobreak\ L=\log(N)$ and consider the
collection ${\bf[B]}$ with

$K\in\mathcal{K}=\left\\{2^{q},\leavevmode\nobreak\
q=0,1,\ldots,q_{\max}\right\\}\subset\mathcal{M}=\left\\{1,\ldots,\sqrt{N}/\log(N)\right\\}.$

Under Assumption 2.1 , there exists a constant $C>0$ such that,

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{\widehat{K},L}-\sigma^{2}_{|I}\right\|^{2}_{n,N}\right]\leq
34\underset{K\in\mathcal{K}}{\inf}{\left\\{\underset{h\in\mathcal{S}_{K+M,L}}{\inf}{\|h-\sigma^{2}_{|I}\|^{2}_{n}}+\mathrm{pen}(K)\right\\}}+\frac{C}{Nn}$

where $\mathrm{pen}(K)=\kappa(K+M)\log(N)/Nn$ with $\kappa>0$ a numerical
constant.

We deduce from Corollary 4.2 and its assumptions that the adaptive estimator
$\widehat{\sigma}^{2}_{\widehat{K},L}$ satisfies:

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{\widehat{K},L}-\sigma^{2}_{|I}\right\|^{2}_{n}\right]=\mathrm{O}\left((Nn)^{-2\beta/(2\beta+1)}\right).$

This result is justified since the penalty term is of the same order (up to a
log-factor) than the estimation error established in Theorem 4.1.

Considering the adaptive estimator of $\sigma^{2}$ on the real line
$I=\mathbb{R}$ when the sample size $N\rightarrow\infty$, we obtain the
following result.

######  Theorem 5.2.

Suppose that $N\propto n$ and $L=\log(N)$, and consider the collection
$[\mathbf{B}]$ with

$K\in\mathcal{K}=\left\\{2^{q},\leavevmode\nobreak\
q=0,1,\ldots,q_{\max}\right\\}\subset\mathcal{M}=\left\\{1,\ldots,\sqrt{N}/\log(N)\right\\}.$

Under Assumption 2.1 and for $N$ large enough, the exists a constant $C>0$
such that,

$\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{\widehat{K},L}-\sigma^{2}\right\|^{2}_{n,N}\right]\leq
3\underset{K\in\mathcal{K}}{\inf}\left\\{\underset{h\in\mathcal{S}_{K+M,L}}{\inf}{\left\|h-\sigma^{2}\right\|^{2}_{n}}+\mathrm{pen}(K)\right\\}+\frac{C}{Nn}.$

where $\mathrm{pen}(K)=\kappa^{\prime}\frac{(K+M)\log(N)}{Nn}$ with
$\kappa^{\prime}>0$ a numerical constant.

We have a penalty term of the same order than the one obtained in Theorem 5.1
where $\sigma^{2}$ is estimated on a compact interval. One can deduce that the
adaptive estimator reaches a rate of the same order than the rate of the non-
adaptive estimator given in Corollary 4.4 for the collection $[\mathbf{B}]$.

If we consider the adaptive estimator of the compactly supported diffusion
coefficient built from a single diffusion path, we obtain below an upper-bound
of its risk of estimation.

######  Theorem 5.3.

Suppose that $N=1,\leavevmode\nobreak\ L=\sqrt{\log(n)}$ and consider the
collection $[\mathbf{B}]$ with

$K\in\mathcal{K}=\left\\{2^{q},\leavevmode\nobreak\
q=0,\ldots,q_{\max}\right\\}\subset\mathcal{M}=\left\\{1,\ldots,\sqrt{n}/\log(n)\right\\}.$

Under Assumption 2.1, it yields

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{\widehat{K},L}-\sigma^{2}_{|I}\right\|^{2}_{n,1}\right]\leq$
$\displaystyle\leavevmode\nobreak\
3\underset{K\in\mathcal{K}}{\inf}{\left\\{\underset{h\in\mathcal{S}_{K+M,L}}{\inf}{\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n}}+\mathrm{pen}(K)\right\\}}+\frac{C}{n}.$

where $C>0$ is a constant depending on $\tau_{0}$, and
$\mathrm{pen}(K)=\kappa\frac{(K+M)\log(n)}{n}$ with $\kappa>0$ a numerical
constant.

We deduce from Theorem 5.3 that if we assume that
$\sigma^{2}\in\Sigma_{I}(\beta,R)$, then the adaptive estimator
$\widehat{\sigma}^{2}_{\widehat{K},L}$ reaches a rate of order
$n^{-\beta/(2\beta+1)}$ (up to a log-factor). The result of this theorem is
almost a deduction of the result of Theorem 5.1, the slight difference being
the use, in the proofs, of the local time of the process and the equivalence
relation between the pseudo-norm $\|.\|_{n,1}$ with the pseudo-norm
$\|.\|_{X}$ instead of the empirical norm $\|.\|_{n}$ considered in the proof
of Theorem 5.1.

##  6 Numerical study

This section is devoted to the numerical study on a simulation scheme. Section
6.1 focuses on the presentation of the chosen diffusion models. In Section
6.2, we describe the scheme for the implementation of the ridge estimators. We
mainly focus on the B-spline basis for the numerical study, and in Section
6.3, we add a numerical study on the performance of the Hermite-based ridge
estimator of $\sigma^{2}$ on $\mathbb{R}$. Finally, we compare the efficiency
of our estimator built on the real line $\mathbb{R}$ from a single path with
that of the Nadaraya-Watson estimator proposed in Florens-Zmirou (1993).

###  6.1 Models and simulations

Recall that the time horizon is $T=1$ and $X_{0}=0$. Consider the following
diffusion models:

  * Model $1$ Ornstein-Uhlenbeck: $b(x)=1-x,\leavevmode\nobreak\ \leavevmode\nobreak\ \sigma(x)=1$

  * Model $2$: $b(x)=1-x,\leavevmode\nobreak\ \leavevmode\nobreak\ \sigma(x)=1-x^{2}$

  * Model $3$: $b(x)=1-x,\leavevmode\nobreak\ \leavevmode\nobreak\ \sigma(x)=\frac{1}{3+\sin(2\pi x)}+\cos^{2}\left(\frac{\pi}{2}x\right)$

Model $1$ is the commonly used Ornstein-Uhlenbeck model, known to be a simple
diffusion model satisfying Assumption 2.1. Model $2$ does not satisfy
Assumption 2.1. Model $3$ satisfies Assumption 2.1 with a multimodal diffusion
coefficient.

The size $N$ of the sample $D_{N}$ takes values in the set
$\\{1,10,100,1000\\}$ where the length $n$ of paths varies in the set
$\\{100,250,500,1000\\}$. As we work with the spline basis, the dimension
$m=K+M$ of the approximation space is chosen such that $M=3$ and $K$ takes
values in $\mathcal{K}=\\{2^{p},\ p=0,\cdots,5\\}$ so that the subspaces are
nested inside each other. We are using R for the simulation of diffusion paths
via the function sde.sim of sde package, (see Iacus (2009) for more details on
the simulation of SDEs).

###  6.2 Implementation of the ridge estimators

In this section, we assess the quality of estimation of the adaptive estimator
$\widehat{\sigma}^{2}_{\widehat{m}}$ in each of the $3$ models through the
computation of its risk of estimation. We compare the performance of the
adaptive estimator with that of the oracle estimator
$\widehat{\sigma}^{2}_{m^{*}}$ where $m^{*}$ is given by:

$m^{*}:=\underset{m\in\mathcal{M}}{\arg\min}{\
\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}\right\|^{2}_{n,N}}.$ (23)

For the spline basis, we have $m^{*}=K^{*}+M$ with $M=3$. Finally, we complete
the numerical study with a representation of a set of $10$ estimators of
$\sigma^{2}$ for each of the $3$ models.

We evaluate the MISE of the spline-based adaptive estimators
$\widehat{\sigma}^{2}_{\widehat{K}}$ by repeating $100$ times the following
steps:

  1. 1.

Simulate samples $D_{N,n}$ and $D_{N^{\prime},n}$ with
$N\in\\{1,10,100,1000\\}$, $N^{\prime}=100$ and $n\in\\{100,250,1000\\}$.

  2. 2.

For each $K\in\mathcal{K}$, and from $D_{N,n}$, compute estimators
$\widehat{\sigma}^{2}_{K}$ given in Equations (9) and (10).

  3. 3.

Select the optimal dimension $\widehat{K}\in\mathcal{K}$ using Equation (22)
and compute $K^{*}$ from Equation (23)

  4. 4.

Using $D_{N^{\prime},n}$, evaluate
$\left\|\widehat{\sigma}^{2}_{\widehat{K}}-\sigma^{2}\right\|^{2}_{n,N^{\prime}}$
and
$\left\|\widehat{\sigma}^{2}_{K^{*}}-\sigma^{2}\right\|^{2}_{n,N^{\prime}}$.

We deduce the risks of estimation considering the average values of
$\left\|\widehat{\sigma}^{2}_{\widehat{m}}-\sigma^{2}\right\|^{2}_{n,N^{\prime}}$
and
$\left\|\widehat{\sigma}^{2}_{m^{*}}-\sigma^{2}\right\|^{2}_{n,N^{\prime}}$
over the $100$ repetitions. Note that we consider in this section, the
estimation of $\sigma^{2}$ on the compact interval $I=[-1,1]$ and on the real
line $\mathbb{R}$. The unknown parameters $\kappa$ and $\kappa^{\prime}$ in
the penalty functions given in Theorem 5.1 and Theorem 5.2 respectively, are
numerically calibrated (details are given in Appendix References), and we
choose $\kappa=4$ and $\kappa^{\prime}=5$ as their respective values.

###  6.3 Numerical results

We present in this section the numerical results of the performance of the
spline-based adaptive estimators of $\sigma^{2}_{|I}$ with
$I\subseteq\mathbb{R}$ together with the performance of the oracle estimators.
We consider the case $I=[-1,1]$ for the compactly supported diffusion
coefficient, and the case $I=\mathbb{R}$.

Tables 2 and 3 present the numerical results of estimation of
$\sigma^{2}_{|I}$ from simulated data following the steps given in Section
6.2.

Models | Intervals | Estimators | $N=10$ | $N=100$ | $N=1000$
---|---|---|---|---|---
Model 1 | $[-1,1]$ | $\widehat{\sigma}^{2}_{\widehat{K},L}$ | $0.0102\ \ (0.0083)$ | $0.0009\ \ (0.0009)$ | $0.0002\ \ (0.0001)$
$\widehat{\sigma}^{2}_{K^{*},L}$ | $0.0094\ \ (0.0065)$ | $0.0009\ \ (0.0009)$ | $0.0002\ \ (0.0001)$
$\mathbb{R}$ | $\widehat{\sigma}^{2}_{\widehat{K},L}$ | $0.0096\ (0.0062)$ | $0.0009\ \ (0.0008)$ | $0.0003\ \ (0.0002)$
$\widehat{\sigma}^{2}_{K^{*},L}$ | $0.0093\ \ (0.0057)$ | $0.0009\ \ (0.0008)$ | $0.0003\ \ (0.0002)$
Model 2 | $[-1,1]$ | $\widehat{\sigma}^{2}_{\widehat{K},L}$ | $0.0048\ \ (0.0052)$ | $0.0019\ \ (0.0008)$ | $0.0005\ \ (0.0002)$
$\widehat{\sigma}^{2}_{K^{*},L}$ | $0.0039\ \ (0.0043)$ | $0.0009\ \ (0.0005)$ | $0.0005\ \ (0.0002)$
$\mathbb{R}$ | $\widehat{\sigma}^{2}_{\widehat{K},L}$ | $0.0195\ (0.0140)$ | $0.0057\ \ (0.0006)$ | $0.0012\ \ (0.0002)$
$\widehat{\sigma}^{2}_{K^{*},L}$ | $0.0048\ \ (0.0064)$ | $0.0025\ \ (0.0021)$ | $0.0010\ \ (0.0003)$
Model 3 | $[-1,1]$ | $\widehat{\sigma}^{2}_{\widehat{K},L}$ | $0.0521\ \ (0.0191)$ | $0.0176\ \ (0.0070)$ | $0.0073\ \ (0.0021)$
$\widehat{\sigma}^{2}_{K^{*},L}$ | $0.0260\ \ (0.0081)$ | $0.0073\ \ (0.0030)$ | $0.0048\ \ (0.0009)$
$\mathbb{R}$ | $\widehat{\sigma}^{2}_{\widehat{K},L}$ | $0.1132\ (0.0595)$ | $0.0319\ \ (0.0031)$ | $0.0179\ \ (0.0054)$
$\widehat{\sigma}^{2}_{K^{*},L}$ | $0.0351\ \ (0.0169)$ | $0.0319\ \ (0.0031)$ | $0.0116\ \ (0.0051)$
Table 2: Assessment of $\mathrm{MISEs}$ (mean and standard deviation between brackets) of both the adaptive estimator $\widehat{\sigma}^{2}_{\widehat{K},L}$ and the oracle estimator $\widehat{\sigma}^{2}_{K^{*},L}$ from diffusion paths of size $n=100$. Models | Intervals | Estimators | $N=10$ | $N=100$ | $N=1000$
---|---|---|---|---|---
Model 1 | $[-1,1]$ | $\widehat{\sigma}^{2}_{\widehat{K},L}$ | $0.0047\ \ (0.0037)$ | $0.0003\ \ (0.0002)$ | $0.0001\ \ (0.00003)$
$\widehat{\sigma}^{2}_{K^{*},L}$ | $0.0042\ \ (0.0030)$ | $0.0003\ \ (0.0002)$ | $0.0001\ \ (0.00003)$
$\mathbb{R}$ | $\widehat{\sigma}^{2}_{\widehat{K},L}$ | $0.0053\ (0.0037)$ | $0.0003\ \ (0.0002)$ | $0.0001\ \ (0.00004)$
$\widehat{\sigma}^{2}_{K^{*},L}$ | $0.0050\ \ (0.0031)$ | $0.0003\ \ (0.0002)$ | $0.0001\ \ (0.00004)$
Model 2 | $[-1,1]$ | $\widehat{\sigma}^{2}_{\widehat{K},L}$ | $0.0027\ \ (0.0019)$ | $0.0003\ \ (0.0002)$ | $0.0002\ \ (0.00004)$
$\widehat{\sigma}^{2}_{K^{*},L}$ | $0.0018\ \ (0.0019)$ | $0.0002\ \ (0.0001)$ | $0.0001\ \ (0.00004)$
$\mathbb{R}$ | $\widehat{\sigma}^{2}_{\widehat{K},L}$ | $0.0091\ (0.0077)$ | $0.0028\ \ (0.0025)$ | $0.0008\ \ (0.0002)$
$\widehat{\sigma}^{2}_{K^{*},L}$ | $0.0020\ \ (0.0023)$ | $0.0021\ \ (0.0023)$ | $0.0002\ \ (0.00004)$
Model 3 | $[-1,1]$ | $\widehat{\sigma}^{2}_{\widehat{K},L}$ | $0.0306\ \ (0.0150)$ | $0.0058\ \ (0.0012)$ | $0.0010\ \ (0.0003)$
$\widehat{\sigma}^{2}_{K^{*},L}$ | $0.0216\ \ (0.0067)$ | $0.0023\ \ (0.0020)$ | $0.0010\ \ (0.0003)$
$\mathbb{R}$ | $\widehat{\sigma}^{2}_{\widehat{K},L}$ | $0.0560\ (0.0313)$ | $0.0275\ \ (0.0049)$ | $0.0069\ \ (0.0049)$
$\widehat{\sigma}^{2}_{K^{*},L}$ | $0.0261\ \ (0.0127)$ | $0.0096\ \ (0.0051)$ | $0.0065\ \ (0.0041)$
Table 3: Assessment of $\mathrm{MISEs}$ of both the adaptive estimator
$\widehat{\sigma}^{2}_{\widehat{K},L}$ and the oracle estimator
$\widehat{\sigma}^{2}_{K^{*},L}$ from diffusion paths of size $n=250$.

The results of Table 2 and Table 3 show that the adapted estimator
$\widehat{\sigma}^{2}_{\widehat{K}}$ is consistent, since its MISE tends to
zero as both the size $N$ of the sample $D_{N,n}$ and the length $n$ of paths
are larger. Moreover, note that in most cases, the ridge estimators of the
compactly supported diffusion coefficients perform better than those of the
non-compactly supported diffusion functions. As expected, we observe that the
oracle estimator has generally a better performance compared to the adaptive
estimator. Nonetheless, we can remark that the performances are very close in
several cases, highlighting the efficiency of the data-driven selection of the
dimension.

An additional important remark is the significant influence of the length $n$
of paths on the performance of $\widehat{\sigma}^{2}_{\widehat{K}}$ and
$\widehat{\sigma}^{2}_{K^{*},L}$ (by comparison of Table 2 with Table 3),
which means that estimators built from higher frequency data are more
efficient. A similar remark is made for theoretical results obtained in
Sections 4.2 and 4.1.

##### Performance of the Hermite-based estimator of the diffusion coefficient

We focus on the estimation of $\sigma^{2}$ on $\mathbb{R}$ and assess the
performance of its Hermite-based estimator (see Section 4.2). We present in
Table 4, the performance of the oracle estimator
$\widehat{\sigma}^{2}_{m^{*},L}$.

Models | Intervals | Estimators | $N=10,\ n=100$ | $N=100,\ n=100$ | $N=100,\ n=250$
---|---|---|---|---|---
Model 1 | $\mathbb{R}$ | $\widehat{\sigma}^{2}_{K^{*},L}$ | $0.0082\ (0.0059)$ | $0.0015\ \ (0.0008)$ | $0.0006\ \ (0.0004)$
Model 2 | $\mathbb{R}$ | $\widehat{\sigma}^{2}_{K^{*},L}$ | $0.0058\ (0.0111)$ | $0.0007\ \ (0.0004)$ | $0.0003\ \ (0.0002)$
Model 3 | $\mathbb{R}$ | $\widehat{\sigma}^{2}_{K^{*},L}$ | $0.0188\ (0.0151)$ | $0.0077\ \ (0.0037)$ | $0.0040\ \ (0.0036)$
Table 4: Assessment of $\mathrm{MISEs}$ of the Hermite-based oracle estimator
$\widehat{\sigma}^{2}_{K^{*},L}$ of the square of the diffusion coefficient.

From the numerical results of Table 4, we observe that the Hermite-based
estimator of $\sigma^{2}$ is consistent as the sample size $N$ and the length
$n$ paths take larger values.

##### Estimation of the diffusion coefficient from one path

Consider ridge estimators of $\sigma^{2}_{|I}$ with $I=[-1,1]$. For the case
of the adaptive estimators of $\sigma^{2}_{|I}$, the dimension $\widehat{K}$
is selected such that

$\widehat{K}=\underset{K\in\mathcal{K}}{\arg\min}{\gamma_{n}(\widehat{\sigma}^{2}_{\widehat{K}})+\mathrm{pen}(K)}$
(24)

where $\mathrm{pen}(K)=\kappa(K+M)\log(n)/n$ with $\kappa>0$. We choose the
numerical constant $\kappa=4$ and we derive the numerical performance of the
adaptive estimator of $\sigma^{2}_{|I}$.

Models | Intervals | Estimators | $n=100$ | $n=1000$
---|---|---|---|---
Model 1 | $[-1,1]$ | $\widehat{\sigma}^{2}_{\widehat{K},L}$ | $0.1751\ \ (0.1921)$ | $0.0915\ \ (0.1925)$
$\widehat{\sigma}^{2}_{K^{*},L}$ | $0.1563\ \ (0.1776)$ | $0.0783\ \ (0.1699)$
Model 2 | $[-1,1]$ | $\widehat{\sigma}^{2}_{\widehat{K},L}$ | $0.1721\ \ (0.3483)$ | $0.1365\ \ (0.5905)$
$\widehat{\sigma}^{2}_{K^{*},L}$ | $0.0987\ \ (0.1644)$ | $0.0552\ \ (0.2409)$
Model 3 | $[-1,1]$ | $\widehat{\sigma}^{2}_{\widehat{K},L}$ | $0.2184\ \ (0.2780)$ | $0.2106\ \ (0.5790)$
$\widehat{\sigma}^{2}_{K^{*},L}$ | $0.1263\ \ (0.1486)$ | $0.0751\ \ (0.1469)$
Table 5: Evaluation of MISEs of adaptive estimators
$\widehat{\sigma}^{2}_{\widehat{K},L}$ built from a single diffusion path
($N=1$) for each of the three models.

Table 5 gives the numerical performances of both the adaptive estimator and
the oracle estimator of $\sigma^{2}_{|I}$ on the compact interval $I=[-1,1]$
and from a single diffusion path. From the obtained results, we see that the
estimators are numerically consistent. However, we note that the convergence
is slow (increasing $n$ from $100$ to $1000$), which highlights the
significant impact of the number $N$ of paths on the efficiency of the ridge
estimator.

##### Comparison of the efficiency of the ridge estimator of the diffusion
coefficient with its Nadaraya-Watson estimator.

Consider the adaptive estimator $\widehat{\sigma}^{2}_{\widehat{K}}$ of the
square of the diffusion coefficient buit on the real line $\mathbb{R}$ from a
single diffusion path ($N=1$), where the dimension $\widehat{K}$ is selected
using Equation (24). For the numerical assessment, we use the interval
$I=[-10^{6},10^{6}]$ to approximate the real line $\mathbb{R}$, and then, use
Equation (24) for the data-driven selection of the dimension.

We want to compare the efficiency of $\widehat{\sigma}^{2}_{\widehat{K}}$ with
that of the Nadaraya-Watson estimator of $\sigma^{2}$ given from a diffusion
path $\bar{X}=(X_{k/n})_{1\leq k\leq n}$ and for all $x\in\mathbb{R}$ by

$S_{n}(x)=\frac{\sum_{k=1}^{n-1}{K\left(\frac{X_{k/n}-x}{h_{n}}\right)[X_{(k+1)/n}-X_{k/n}]^{2}/n}}{\sum_{k=1}^{n}{K\left(\frac{X_{k/n}-x}{h_{n}}\right)}}$

where $K$ is a positive kernel function, and $h_{n}$ is the bandwidth. Thus,
the estimator $S_{n}(x)$ is consistent under the condition
$nh^{4}_{n}\rightarrow 0$ as $n$ tends to infinity (see Florens-Zmirou
(1993)). We use the function ksdiff() of the R-package sde to compute the
Nadaraya-Watson estimator $S_{n}$.

Models | Ridge estimator | Nadaraya-Watson estimator
---|---|---
Model 1 | $0.0020\ \ (0.0023)$ | $0.9377\ \ (0.0017)$
Model 2 | $0.1323\ \ (0.0794)$ | $0.5086\ \ (0.0885)$
Model 3 | $0.4077\ \ (0.1178)$ | $1.3175\ \ (0.3039)$
Table 6: This table shows the loss errors of the ridge estimator
$\widehat{\sigma}^{2}_{\widehat{K},L}$ on $\mathbb{R}$ and the Nadaraya-Watson
estimator $S_{n}$ of $\sigma^{2}$ built from a diffusion path ($N=1)$ of
length $n=1000$.

We remark from the results of Table 6 that our ridge estimator is more
efficient. Note that for the kernel estimator $S_{n}$, the bandwidth is
computed using the rule of thumb of Scott (see Odell-Scott (1992)). The
bandwidth is proportional to $n^{-1/(d+4)}$ where $n$ is the number of points,
and $d$ is the number of spatial dimensions.

###  6.4 Concluding remarks

The results of our numerical study show that our ridge estimators built both
on a compact interval and on the real line are consistent as $N$ and $n$ take
larger values, or as only $n$ takes larger values when the estimators are
built from a single path. These results are in accordance with the theoretical
results established in the previous sections. Moreover, as expected, we
obtained the consistency of the Hermite-based estimators of $\sigma^{2}$ on
the real line $\mathbb{R}$. Nonetheless, we only focus on the Hermite-based
oracle estimator since we did not establish a risk bound of the corresponding
adaptive estimator. Finally, we remark that the ridge estimator of
$\sigma^{2}$ built from a single path performs better than its Nadaraya-Watson
kernel estimator proposed in Florens-Zmirou (1993) and implemented in the
R-package sde.

Figure 1: Bundles of $10$ estimators $\widehat{\sigma}^{2}_{\widehat{K},L}$
(in green) of the true diffusion coefficient $\sigma^{2}_{|I}$ restricted on
the compact interval $I=[-1,1]$ (in red) of each of Models $1,2,3$ (from top
to bottom) using samples of size $N=1000$ with diffusion paths of length
$n=500$.

##  7 Conclusion

In this article, we have proposed ridge-type estimators of the diffusion
coefficient on a compact interval from a single diffusion path. We took
advantage of the local time of the diffusion process to prove the consistency
of non-adaptive estimators of $\sigma^{2}$ and derive a rate of convergence of
the same order than the optimal rate established in Hoffmann (1999b). We also
propose an estimator of $\sigma^{2}$ on the real line from a single path. We
proved its consistency using the method described in Section 3.2, and derive a
rate of convergence order $n^{-\beta/(4\beta+1)}$ over a Hölder space for the
collection $[\mathbf{B}]$. Then, we extended the study to the estimation of
$\sigma^{2}$ from repeated discrete observations of the diffusion process. We
establish rates of convergence of the ridge estimators both on a compact
interval and on $\mathbb{R}$. We complete the study proposing adaptive
estimators of $\sigma^{2}$ on a compact interval for $N=1$ and
$N\rightarrow\infty$, and on the real line $\mathbb{R}$ for
$N\rightarrow\infty$.

A perspective on the estimation of the diffusion coefficient could be the
establishment of a minimax rate of convergence of the compactly supported
(square of the) diffusion coefficient from repeated discrete observations of
the diffusion process. The case of the non-compactly supported diffusion
coefficient may be a lot more challenging, since the transition density of the
diffusion process is no longer lower-bounded. This new fact can lead to
different rates of convergence depending on the considered method (see Section
4).

## Acknowledgements

I would like to thank my supervisors, Christophe Denis, Charlotte Dion-Blanc,
and Viet-Chi Tran, for their sound advice, guidance and support throughout
this research project.

##  8 Proofs

In this section, we prove our main results of Sections 3, 4 and 5. To simplify
our notations, we set $\Delta_{n}=\Delta(=1/n)$ and constants are generally
denoted by $C>0$ or $c>0$ whose values can change from a line to another.
Moreover, we use the notation $C_{\alpha}$ in case we need to specify the
dependency of the constant $C$ on a parameter $\alpha$.

###  8.1 Technical results

Recall first some useful results on the local time and estimates of the
transition density of diffusion processes.

######  Lemma 8.1.

For all integer $q\geq 1$, there exists $C^{*}>0$ depending on $q$ such that
for all $0\leq s<t\leq 1$,

$\mathbb{E}\left[\left|X_{t}-X_{s}\right|^{2q}\right]\leq C^{*}(t-s)^{q}.$

The proof of Lemma 8.1 is provided in Denis _et al._ (2022).

######  Proposition 8.2.

Under Assumptions 2.1, there exist constants $c_{\sigma}>1$, $C>1$ such that
for all $t\in(0,1]$, $x\in\mathbb{R}$,

$\frac{1}{C\sqrt{t}}\exp\left(-c_{\sigma}\frac{x^{2}}{t}\right)\leq
p_{X}(t,x)\leq\dfrac{C}{\sqrt{t}}\exp\left(-\frac{x^{2}}{c_{\sigma}t}\right).$

The proof of Proposition 8.2 is provided in Gobet (2002), Proposition 1.2.

######  Proposition 8.3.

Let $h$ be a $L_{0}$-lipschitz function. Then there exists
$\tilde{h}\in\mathcal{S}_{K_{N},M}$, such that

$|\tilde{h}(x)-h(x)|\leq C\frac{\log(N)}{K_{N}},\;\;\forall
x\in(-\log(N),\log(N)),$

where $C>0$ depends on $L_{0}$, and $M$.

The proof of Proposition 8.3 is provided in Denis _et al._ (2022). The finite-
dimensional vector space $\mathcal{S}_{K_{N},M}=\mathcal{S}_{K_{N}+M}$ is
introduced in Section 2.

######  Lemma 8.4.

Under Assumption 2.1, there exist $C_{1},C_{2}>0$ such that for all $A>0$,

$\sup_{t\in[0,1]}\mathbb{P}\left(\left|X_{t}\right|\geq
A\right)\leq\frac{C_{1}}{A}\exp(-C_{2}A^{2}).$

The proof of Lemma 8.4 is provided in Denis _et al._ (2022), Lemma 7.3.

######  Lemma 8.5.

Under Assumption 2.1, the following holds:

$\forall\leavevmode\nobreak\ x\in\mathbb{R},\leavevmode\nobreak\
\leavevmode\nobreak\ \mathcal{L}^{x}=\mathcal{L}^{x_{-}}\leavevmode\nobreak\
\leavevmode\nobreak\ a.s.$

where $\mathcal{L}^{x_{-}}=\underset{\varepsilon\rightarrow
0}{\lim}{\mathcal{L}^{x-\varepsilon}}$.

The result of Lemma 8.5 justifies the definition of the local time
$\mathcal{L}^{x}$, for $x\in\mathbb{R}$, given in Equation (14).

###### Proof.

From Revuz & Yor (2013), Theorem 1.7, we have

$\forall\leavevmode\nobreak\ x\in\mathbb{R},\leavevmode\nobreak\
\leavevmode\nobreak\
\mathcal{L}^{x}-\mathcal{L}^{x_{-}}=2\int_{0}^{1}{\mathds{1}_{X_{s}=x}dX_{s}}=2\int_{0}^{1}{\mathds{1}_{X_{s}=x}b(X_{s})ds}+2\int_{0}^{1}{\mathds{1}_{X_{s}=x}\sigma(X_{s})dW_{s}}.$

For all $x\in\mathbb{R}$ and for all $s\in[0,1]$, we have for all
$\varepsilon>0$,

$\displaystyle\mathbb{P}(X_{s}=x)=$ $\displaystyle\leavevmode\nobreak\
\underset{\varepsilon\rightarrow 0}{\lim}{\leavevmode\nobreak\
\mathbb{P}(X_{s}\leq x+\varepsilon)}-\underset{\varepsilon\rightarrow
0}{\lim}{\leavevmode\nobreak\ \mathbb{P}(X_{s}\leq
x-\varepsilon)}=\underset{\varepsilon\rightarrow 0}{\lim}{\leavevmode\nobreak\
F_{s}(x+\varepsilon)}-\underset{\varepsilon\rightarrow
0}{\lim}{\leavevmode\nobreak\ F_{s}(x-\varepsilon)}$ $\displaystyle=$
$\displaystyle\leavevmode\nobreak\ F_{s}(x)-F_{s}(x^{-})$ $\displaystyle=$
$\displaystyle\leavevmode\nobreak\ 0$

Thus, for all $x\in\mathbb{R}$,

$\displaystyle\mathbb{E}\left[\left|\mathcal{L}^{x}-\mathcal{L}^{x_{-}}\right|\right]\leq$
$\displaystyle\leavevmode\nobreak\
2\int_{0}^{1}{|b(x)|\mathbb{P}(X_{s}=x)ds}+2\mathbb{E}\left[\left|\int_{0}^{1}{\mathds{1}_{X_{s}=x}\sigma(X_{s})dW_{s}}\right|\right]$
$\displaystyle=$ $\displaystyle\leavevmode\nobreak\
2\mathbb{E}\left[\left|\int_{0}^{1}{\mathds{1}_{X_{s}=x}\sigma(X_{s})dW_{s}}\right|\right].$

Using the Cauchy Schwartz inequality, we conclude that

$\displaystyle\mathbb{E}\left[\left|\mathcal{L}^{x}-\mathcal{L}^{x_{-}}\right|\right]\leq$
$\displaystyle\leavevmode\nobreak\
2\sqrt{\mathbb{E}\left(\int_{0}^{1}{\mathds{1}_{X_{s}=x}\sigma^{2}(X_{s})ds}\right)}=2\sigma(x)\int_{0}^{1}{\mathbb{P}(X_{s}=x)ds}=0.$

Using the Markov inequality, we have

$\forall\leavevmode\nobreak\ \varepsilon>0,\leavevmode\nobreak\
\leavevmode\nobreak\
\mathbb{P}(|\mathcal{L}^{x}-\mathcal{L}^{x_{-}}|>\varepsilon)\leq\frac{1}{\varepsilon}\mathbb{E}\left[\left|\mathcal{L}^{x}-\mathcal{L}^{x_{-}}\right|\right]=0.$

We finally conclude that for all $x\in\mathbb{R}$,

$\mathbb{P}(\mathcal{L}^{x}\neq\mathcal{L}^{x_{-}})=\mathbb{P}(|\mathcal{L}^{x}-\mathcal{L}^{x_{-}}|>0)=0.$

∎

###  8.2 Proofs of Section 3

####  8.2.1 Proof of Lemma 3.1

###### Proof.

The proof is divided into two parts for each of the two results to be proven.

##### First result.

Since the function $h$ is continuous on $\mathbb{R}$, let $H$ be a primitive
of $h$ on $\mathbb{R}$. We deduce that for all $s\in[0,1]$,

$h(X_{s})=\underset{\varepsilon\rightarrow
0}{\lim}{\frac{H(X_{s}+\varepsilon)-H(X_{s}-\varepsilon)}{2\varepsilon}}=\underset{\varepsilon\rightarrow
0}{\lim}{\frac{1}{2\varepsilon}\int_{X_{s}-\varepsilon}^{X_{s}+\varepsilon}{h(x)dx}}=\underset{\varepsilon\rightarrow
0}{\lim}{\frac{1}{2\varepsilon}\int_{-\infty}^{+\infty}{h(x)\mathds{1}_{(x-\varepsilon,x+\varepsilon)}(X_{s})dx}}.$

Finally, since $h$ is integrable on $\mathbb{R}$ and using the theorem of
dominated convergence, we obtain

$\int_{0}^{1}{h(X_{s})ds}=\int_{-\infty}^{+\infty}{h(x)\underset{\varepsilon\rightarrow
0}{\lim}{\frac{1}{2\varepsilon}\int_{0}^{1}{\mathds{1}_{(x-\varepsilon,x+\varepsilon)}(X_{s})ds}}dx}=\int_{-\infty}^{+\infty}{h(x)\mathcal{L}^{x}dx}.$

##### Second result.

Fix $t\in(0,1]$ and consider
$P_{X}:(t,x)\mapsto\int_{-\infty}^{x}{p_{X}(t,y)dy}$ the cumulative density
function of the random variable $X_{t}$ of the density function $x\mapsto
p_{X}(t,x)$. We have:

$\displaystyle\forall\leavevmode\nobreak\ x\in\mathbb{R},\leavevmode\nobreak\
\leavevmode\nobreak\ \mathbb{E}(\mathcal{L}^{x})=$
$\displaystyle\underset{\varepsilon\rightarrow
0}{\lim}{\frac{1}{2\varepsilon}\int_{0}^{1}{\mathbb{E}\left[\mathds{1}_{(x-\varepsilon,x+\varepsilon)}(X_{s})\right]ds}}=\underset{\varepsilon\rightarrow
0}{\lim}{\frac{1}{2\varepsilon}\int_{0}^{1}{\mathbb{P}\left(x-\varepsilon\leq
X_{s}\leq x+\varepsilon\right)ds}}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}{\underset{\varepsilon\rightarrow
0}{\lim}{\frac{P_{X}(s,x+\varepsilon)-P_{X}(s,x-\varepsilon)}{2\varepsilon}}ds}$
$\displaystyle=$ $\displaystyle\int_{0}^{1}{p_{X}(s,x)ds}.$

∎

####  8.2.2 Proof of Theorem 3.2

Let $\Omega_{n,m}$ be the random event in which the two pseudo-norms
$\|.\|_{n,1}$ and $\|.\|_{X}$ are equivalent and given by

$\Omega_{n,m}:=\underset{g\in\mathcal{S}_{m}\setminus\\{0\\}}{\bigcap}\left\\{\left|\frac{\|g\|^{2}_{n,1}}{\|g\|^{2}_{X}}-1\right|\leq\frac{1}{2}\right\\}.$

The proof of Theorem 3.2 relies on the following lemma.

######  Lemma 8.6.

Let $\gamma>1$ be a real number. Under Assumption 2.1, the following holds

$\mathbb{P}\left(\Omega^{c}_{n,m}\right)\leq
C\frac{m^{2\gamma}}{n^{\gamma/2}},$

where $C>0$ is a constant depending on $\gamma$.

The parameter $\gamma>1$ has to be chosen appropriately (i.e. such that
$m^{2\gamma}/n^{\gamma/2}=o(1/n)$) so that we obtain a variance term of the
risk of the estimator $\widehat{\sigma}^{2}_{m}$ of order $m\log(n)/n$ (see
Theorem 3.2 and Corollary 3.3).

###### Proof of Theorem 3.2 .

Recall that since $N=1$,
$\zeta^{1}_{k\Delta}=\zeta^{1,1}_{k\Delta}+\zeta^{1,2}_{k\Delta}+\zeta^{1,3}_{k\Delta}$
is the error term of the regression model, with:

$\zeta^{1,1}_{k\Delta}=\frac{1}{\Delta}\left[\left(\int_{k\Delta}^{(k+1)\Delta}{\sigma(X^{1}_{s})dW^{1}_{s}}\right)^{2}-\int_{k\Delta}^{(k+1)\Delta}{\sigma^{2}(X^{1}_{s})ds}\right],$
(25)
$\zeta^{1,2}_{k\Delta}=\frac{2}{\Delta}\int_{k\Delta}^{(k+1)\Delta}{((k+1)\Delta-s)\sigma^{\prime}(X^{1}_{s})\sigma^{2}(X^{1}_{s})dW^{1}_{s}},$
(26)
$\zeta^{1,3}_{k\Delta}=2b(X^{1}_{k\Delta})\int_{k\Delta}^{(k+1)\Delta}{\sigma\left(X^{1}_{s}\right)dW^{1}_{s}}.$
(27)

Besides, $R^{1}_{k\Delta}=R^{1,1}_{k\Delta}+R^{1,2}_{k\Delta}$, with:

$R^{1,1}_{k\Delta}=\frac{1}{\Delta}\left(\int_{k\Delta}^{(k+1)\Delta}{b(X^{1}_{s})ds}\right)^{2}+\frac{1}{\Delta}\int_{k\Delta}^{(k+1)\Delta}{((k+1)\Delta-s)\Phi(X^{1}_{s})ds}$
(28)
$R^{1,2}_{k\Delta}=\frac{2}{\Delta}\left(\int_{k\Delta}^{(k+1)\Delta}{\left(b(X^{1}_{s})-b(X^{1}_{k\Delta})\right)ds}\right)\left(\int_{k\Delta}^{(k+1)\Delta}{\sigma(X^{1}_{s})dW^{1}_{s}}\right)$
(29)

where

$\Phi:=2b\sigma^{\prime}\sigma+\left[\sigma^{\prime\prime}\sigma+\left(\sigma^{\prime}\right)^{2}\right]\sigma^{2}.$
(30)

By definition of the projection estimator $\widehat{\sigma}^{2}_{m}$ for each
$m\in\mathcal{M}$ (see Equation (9)), for all $h\in\mathcal{S}_{m,L}$, we
have:

$\gamma_{n,1}\left(\widehat{\sigma}^{2}_{m}\right)-\gamma_{n,1}(\sigma^{2}_{|I})\leq\gamma_{n,1}(h)-\gamma_{n,1}(\sigma^{2}_{|I}).$
(31)

Furthermore, for all $h\in\mathcal{S}_{m,L}$,

$\gamma_{n,1}(h)-\gamma_{n,1}(\sigma^{2}_{|I})=\left\|\sigma^{2}_{|I}-h\right\|^{2}_{n,1}+2\nu_{1}(\sigma^{2}_{|I}-h)+2\nu_{2}(\sigma^{2}_{|I}-h)+2\nu_{3}(\sigma^{2}_{|I}-h)+2\mu(\sigma^{2}_{|I}-h),$

where,

$\nu_{i}\left(h\right)=\frac{1}{n}\sum_{k=0}^{n-1}{h(X^{1}_{k\Delta})\zeta^{1,i}_{k\Delta}},\
\ i\in\\{1,2,3\\},\ \ \ \
\mu(h)=\frac{1}{n}\sum_{k=0}^{n-1}{h(X^{1}_{k\Delta})R^{1}_{k\Delta}},$ (32)

and $\zeta^{1,1}_{k\Delta},\ \zeta^{1,2}_{k\Delta},\ \zeta^{1,3}_{k\Delta}$
are given in Equations (25), (26), (27), and finally,
$R^{1}_{k\Delta}=R^{1,1}_{k\Delta}+R^{1,2}_{k\Delta}$ given in Equations (28)
and (29). Then, for all $m\in\mathcal{M}$, and for all
$h\in\mathcal{S}_{m,L}$, we obtain from Equation (31) that

$\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n,1}\leq\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n,1}+2\nu\left(\widehat{\sigma}^{2}_{m}-h\right)+2\mu\left(\widehat{\sigma}^{2}_{m}-h\right),\
\ \mathrm{with}\ \ \nu=\nu_{1}+\nu_{2}+\nu_{3}.$ (33)

Then, it comes,

$\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n,1}\right]\leq\underset{h\in\mathcal{S}_{m,L}}{\inf}{\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n}}+2\mathbb{E}\left[\nu\left(\widehat{\sigma}^{2}_{m}-h\right)\right]+2\mathbb{E}\left[\mu\left(\widehat{\sigma}^{2}_{m}-h\right)\right].$
(34)

Besides, for any $a,d>0$, using the inequality $xy\leq\eta x^{2}+y^{2}/\eta$
with $\eta=a,d$, we have,

$\begin{cases}2\nu\left(\widehat{\sigma}^{2}_{m}-h\right)\leq\frac{2}{a}\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{X}+\frac{2}{a}\left\|h-\sigma^{2}_{|I}\right\|^{2}_{X}+a\underset{h\in\mathcal{S}_{m},\
\|h\|_{X}=1}{\sup}{\nu^{2}(h)},\\\ \\\
2\mu\left(\widehat{\sigma}^{2}_{m}-h\right)\leq\frac{2}{d}\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n,1}+\frac{2}{d}\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n,1}+\frac{d}{n}\sum_{k=1}^{n}{(R^{1}_{k\Delta})^{2}}.\end{cases}$
(35)

#### Upper bound of $\frac{1}{n}\sum_{k=1}^{n}{(R^{1}_{k\Delta})^{2}}$

We have:

$\displaystyle\leavevmode\nobreak\ \forall k\in[\\![1,n]\\!],\
R^{1}_{k\Delta}=R^{1,1}_{k\Delta}+R^{1,2}_{k\Delta}+R^{1,3}_{k\Delta}\
\mathrm{with},$ $\displaystyle\leavevmode\nobreak\
R^{1,1}_{k\Delta}=\frac{1}{\Delta}\left(\int_{k\Delta}^{(k+1)\Delta}{b(X^{1}_{s})ds}\right)^{2},\
\
R^{1,2}_{k\Delta}=\frac{1}{\Delta}\int_{k\Delta}^{(k+1)\Delta}{((k+1)\Delta-s)\Phi(X^{1}_{s})ds}$
$\displaystyle\leavevmode\nobreak\
R^{1,3}_{k\Delta}=\frac{2}{\Delta}\left(\int_{k\Delta}^{(k+1)\Delta}{\left(b(X^{1}_{s})-b(X^{1}_{k\Delta})\right)ds}\right)\left(\int_{k\Delta}^{(k+1)\Delta}{\sigma(X^{1}_{s})dW^{1}_{s}}\right).$

For all $k\in[\\![1,n]\\!]$, using the Cauchy-Schwarz inequality and Equation
(2),

$\displaystyle\mathbb{E}\left[\left|R^{1,1}_{k\Delta}\right|^{2}\right]$
$\displaystyle\leq\mathbb{E}\left[\left(\int_{k\Delta}^{(k+1)\Delta}{b^{2}(X^{1}_{k\Delta})ds}\right)^{2}\right]\leq\Delta\mathbb{E}\left[\int_{k\Delta}^{(k+1)\Delta}{b^{4}(X^{1}_{k\Delta})ds}\right]\leq
C\Delta^{2}.$

Consider now the term $R^{1,2}_{k\Delta}$. From Equation (30), we have
$\Phi=2b\sigma^{\prime}\sigma+\left[\sigma^{\prime\prime}\sigma+\left(\sigma^{\prime}\right)^{2}\right]\sigma^{2}$
and according to Assumption 2.1, there exists a constant $C>0$ depending on
$\sigma_{1}$ and $\alpha$ such that

$\left|\Phi(X^{1}_{s})\right|\leq
C\left[(2+|X^{1}_{s}|)(1+|X^{1}_{s}|^{\alpha})+(1+|X^{1}_{s}|^{\alpha})^{2}\right].$

Then, from Equation (2) and for all $s\in(0,1]$,

$\mathbb{E}\left[\Phi^{2}(X^{1}_{s})\right]\leq
C\underset{s\in(0,1]}{\sup}{\mathbb{E}\left[(2+|X^{1}_{s}|)^{2}(1+|X^{1}_{s}|^{\alpha})^{2}+(1+|X^{1}_{s}|^{\alpha})^{4}\right]}<\infty$

and

$\displaystyle\mathbb{E}\left[\left|R^{1,2}_{k\Delta}\right|^{2}\right]$
$\displaystyle\leq\frac{1}{\Delta^{2}}\int_{k\Delta}^{(k+1)\Delta}{((k+1)\Delta-s)^{2}ds}\int_{k\Delta}^{(k+1)\Delta}{\mathbb{E}\left[\Phi^{2}\left(X^{1}_{s}\right)\right]ds}\leq
C\Delta^{2}$

Finally, under Assumption 2.1, from Equation (2) and using the Cauchy-Schwarz
inequality, we have

$\displaystyle\mathbb{E}\left[\left|R^{1,3}_{k\Delta}\right|^{2}\right]$
$\displaystyle\leq\frac{4}{\Delta^{2}}\mathbb{E}\left[\Delta\int_{k\Delta}^{(k+1)\Delta}{L^{2}_{0}\left|X^{1}_{s}-X^{1}_{k\Delta}\right|^{2}ds}\left(\int_{k\Delta}^{(k+1)\Delta}{\sigma(X^{1}_{s})dW_{s}}\right)^{2}\right]$
$\displaystyle\leq\frac{4}{\Delta}\sqrt{\mathbb{E}\left[L^{4}_{0}\Delta\int_{k\Delta}^{(k+1)\Delta}{\left|X^{1}_{s}-X^{1}_{k\Delta}\right|^{4}ds}\right]\mathbb{E}\left[\left(\int_{k\Delta}^{(k+1)\Delta}{\sigma(X^{1}_{s})dW_{s}}\right)^{4}\right]}$
$\displaystyle\leq C\Delta^{2}.$

As a result, there exists a constant $C>0$ such that,

$\mathbb{E}\left[\frac{1}{n}\sum_{k=1}^{n}{(R^{1}_{k\Delta})^{2}}\right]\leq
C\Delta^{2}.$ (36)

We set $a=d=8$ and considering the event $\Omega_{n,m}$ on which the empirical
norms $\|.\|_{X}$ and $\|.\|_{n,1}$ are equivalent, we deduce from Equations
(34), (35) and (36) that,

$\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n,1}\mathds{1}_{\Omega_{n,m}}\right]\leq
3\underset{h\in\mathcal{S}_{m}}{\inf}{\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n}}+C\mathbb{E}\left(\underset{h\in\mathcal{S}_{m},\|h\|_{X}=1}{\sup}{\nu^{2}(h)}\right)+C\Delta^{2}$
(37)

where $C>0$ is a constant depending on $\sigma_{1}$.

### Upper bound of $\mathbb{E}\left(\underset{h\in\mathcal{S}_{m},\
\|h\|_{X}=1}{\sup}{\nu^{2}(h)}\right)$

For all $h=\sum_{\ell=0}^{m-1}{a_{\ell}\phi_{\ell}}\in\mathcal{S}_{m}$ such
that $\|h\|^{2}_{X}=1$, we have $\|h\|^{2}\leq\frac{1}{\tau_{0}}$ (see
Equation (15)) and the coordinate vector
$\mathbf{a}=\left(a_{-M},\cdots,a_{K-1}\right)$ satisfies:

  * •

$\|\mathbf{a}\|^{2}_{2}\leq Cm\leavevmode\nobreak\ \leavevmode\nobreak\
(m=K+M)$ for the spline basis (see Denis _et al._ (2021), Lemma 2.6)

  * •

$\|\mathbf{a}\|^{2}_{2}\leq 1/\tau_{0}$ for an orthonormal basis since
$\|h\|^{2}=\|\mathbf{a}\|^{2}_{2}$.

Furthermore, using the Cauchy-Schwarz inequality, we have:

$\nu^{2}(h)=\left(\sum_{\ell=0}^{m-1}{a_{\ell}\nu\left(\phi_{\ell}\right)}\right)^{2}\leq\|\mathbf{a}\|^{2}_{2}\sum_{\ell=0}^{m-1}{\nu^{2}\left(\phi_{\ell}\right)}.$
(38)

Thus, since $\nu=\nu_{1}+\nu_{2}+\nu_{3}$, for all $\ell\in[\\![-M,K-1]\\!]$
and for all $i\in\\{1,2,3\\}$,

$\displaystyle\mathbb{E}\left[\nu^{2}_{i}\left(\phi_{\ell}\right)\right]=$
$\displaystyle\leavevmode\nobreak\
\frac{1}{n^{2}}\mathbb{E}\left[\left(\sum_{k=0}^{n-1}{\phi_{\ell}\left(X^{1}_{k\Delta}\right)\zeta^{1,i}_{k\Delta}}\right)^{2}\right].$

  1. 1.

Case $i=1$

Recall that
$\zeta^{1,1}_{k\Delta}=\frac{1}{\Delta}\left[\left(\int_{k\Delta}^{(k+1)\Delta}{\sigma(X^{1}_{s})}dW_{s}\right)^{2}-\int_{k\Delta}^{(k+1)\Delta}{\sigma^{2}(X^{1}_{s})ds}\right]$
where $W=W^{1}$. We fix a initial time $s\in[0,1)$ and set
$M^{s}_{t}=\int_{s}^{t}{\sigma(X^{1}_{u})dW_{u}},\ \forall t\geq s$.
$(M^{s}_{t})_{t\geq s}$ is a martingale and for all $t\in[s,1]$, we have:

$\displaystyle\left<M^{s},M^{s}\right>_{t}=\int_{s}^{t}{\sigma^{2}\left(X^{1}_{u}\right)du}.$

Then,
$\zeta^{1,1}_{k\Delta}=\frac{1}{\Delta}\left(M^{k\Delta}_{(k+1)\Delta}\right)^{2}-\left<M^{k\Delta},M^{k\Delta}\right>_{(k+1)\Delta}$
is also a $\mathcal{F}_{k\Delta}$-martingale, and, using the Burkholder-Davis-
Gundy inequality, we obtain for all $k\in[\\![0,n-1]\\!]$,

$\mathbb{E}\left[\zeta^{1,1}_{k\Delta}|\mathcal{F}_{k\Delta}\right]=0,\ \ \
\mathbb{E}\left[\left(\zeta^{1,1}_{k\Delta}\right)^{2}|\mathcal{F}_{k\Delta}\right]\leq\frac{C}{\Delta^{2}}\mathbb{E}\left[\left(\int_{k\Delta}^{(k+1)\Delta}{\sigma^{2}(X^{1}_{u})du}\right)^{2}\right]\leq
C\sigma^{4}_{1}.$ (39)

Then, using Equation (39) we have:

$\displaystyle\mathbb{E}\left[\nu^{2}_{1}\left(\phi_{\ell}\right)\right]=$
$\displaystyle\leavevmode\nobreak\
\frac{1}{n^{2}}\mathbb{E}\left[\sum_{k=0}^{n-1}{\phi^{2}_{\ell}\left(X^{1}_{k\Delta}\right)\left(\zeta^{1,1}_{k\Delta}\right)^{2}}\right]=\frac{1}{n^{2}}\mathbb{E}\left[\sum_{k=0}^{n-1}{\phi^{2}_{\ell}\left(X^{1}_{k\Delta}\right)\mathbb{E}\left[\left(\zeta^{1,1}_{k\Delta}\right)^{2}|\mathcal{F}_{k\Delta}\right]}\right]$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
\frac{C\sigma^{4}_{1}}{n^{2}}\mathbb{E}\left[\sum_{k=0}^{n-1}{\phi^{2}_{\ell}\left(X^{1}_{k\Delta}\right)}\right]$

and,

$\displaystyle\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\nu^{2}_{1}\left(\phi_{\ell}\right)\right]}\leq\frac{C\sigma^{4}_{1}}{n^{2}}\mathbb{E}\left[\sum_{k=0}^{n-1}{\sum_{\ell=0}^{m-1}{\phi^{2}_{\ell}\left(X^{1}_{k\Delta}\right)}}\right].$

One has:

$\begin{cases}\sum_{\ell=-M}^{K-1}{B^{2}_{\ell}\left(X^{1}_{\eta(s)}\right)}\leq
1\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{for}\leavevmode\nobreak\
\mathrm{the}\leavevmode\nobreak\ \mathrm{Spline}\leavevmode\nobreak\
\mathrm{basis}\leavevmode\nobreak\ \leavevmode\nobreak\ (m=K+M),\\\ \\\
\sum_{\ell=0}^{m-1}{\phi^{2}_{\ell}(X^{1}_{\eta(s)})}\leq
Cm\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{for}\leavevmode\nobreak\
\mathrm{an}\leavevmode\nobreak\ \mathrm{orthonormal}\leavevmode\nobreak\
\mathrm{basis}\leavevmode\nobreak\ \mathrm{with}\leavevmode\nobreak\
\leavevmode\nobreak\ C=\underset{0\leq\ell\leq
m-1}{\max}{\|\phi_{\ell}\|^{2}_{\infty}}.\end{cases}$ (40)

Thus, it comes that

     * •

$\sum_{\ell=-M}^{K-1}{\mathbb{E}\left[\nu^{2}_{1}\left(B_{\ell}\right)\right]}\leq
C/n\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{for}\leavevmode\nobreak\
\mathrm{the}\leavevmode\nobreak\ \mathrm{Spline}\leavevmode\nobreak\
\mathrm{basis,}$

     * •

$\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\nu^{2}_{1}\left(\phi_{\ell}\right)\right]}\leq
Cm/n\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{for}\leavevmode\nobreak\
\mathrm{an}\leavevmode\nobreak\ \mathrm{orthonormal}\leavevmode\nobreak\
\mathrm{basis},$

and,

$\mathbb{E}\left(\underset{h\in\mathcal{S}_{m},\
\|h\|^{2}_{X}=1}{\sup}{\nu^{2}_{1}(h)}\right)\leq C\frac{m}{n}$ (41)

where $C>0$ is a constant depending on $\sigma_{1}$ and the basis.

  2. 2.

Case $i=2$

Wa have
$\zeta^{1,2}_{k\Delta}=\frac{2}{\Delta}\int_{k\Delta}^{(k+1)\Delta}{\left((k+1)\Delta-s\right)\sigma^{\prime}\left(X^{1}_{s}\right)\sigma^{2}\left(X^{1}_{s}\right)dW_{s}}$
and,

$\displaystyle\mathbb{E}\left[\nu^{2}_{2}\left(\phi_{\ell}\right)\right]=$
$\displaystyle\leavevmode\nobreak\
4\mathbb{E}\left[\left(\sum_{k=0}^{n-1}{\phi_{\ell}\left(X^{1}_{k\Delta}\right)\int_{k\Delta}^{(k+1)\Delta}{(k+1)\Delta-s)}\sigma^{\prime}\left(X^{1}_{s}\right)\sigma^{2}\left(X^{1}_{s}\right)dW_{s}}\right)^{2}\right]$
$\displaystyle=$ $\displaystyle\leavevmode\nobreak\
4\mathbb{E}\left[\left(\int_{0}^{1}{\phi_{\ell}\left(X^{1}_{\eta(s)}\right)(\eta(s)+\Delta-s)\sigma^{\prime}\left(X^{1}_{s}\right)\sigma^{2}\left(X^{1}_{s}\right)dW_{s}}\right)^{2}\right]$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
C\sigma^{4}_{1}\Delta^{2}\mathbb{E}\left[\int_{0}^{1}{\phi^{2}_{\ell}\left(X^{1}_{\eta(s)}\right)ds}\right]$

where $C>0$ is a constant. We deduce for both the spline basis and any
orthonormal basis that there exists a constant $C>0$ depending on $\sigma_{1}$
such that:

$\mathbb{E}\left(\underset{h\in\mathcal{S}_{m},\
\|h\|^{2}_{X}=1}{\sup}{\nu^{2}_{2}(h)}\right)\leq C\frac{m}{n^{2}}.$ (42)

  3. 3.

Case $i=3$

We have
$\zeta^{1,3}_{k\Delta}=2b\left(X^{1}_{k\Delta}\right)\int_{k\Delta}^{(k+1)\Delta}{\sigma\left(X^{1}_{s}\right)dW_{s}}$
and,

$\displaystyle\mathbb{E}\left[\nu^{2}_{3}\left(\phi_{\ell}\right)\right]=$
$\displaystyle\leavevmode\nobreak\
\frac{4}{n^{2}}\mathbb{E}\left[\left(\int_{0}^{1}{\phi_{\ell}\left(X^{1}_{\eta(s)}\right)b\left(X^{1}_{\eta(s)}\right)\sigma\left(X^{1}_{s}\right)dW_{s}}\right)^{2}\right]$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
\frac{4\sigma^{2}_{1}}{n^{2}}\mathbb{E}\left[\int_{0}^{1}{\phi^{2}_{\ell}\left(X^{1}_{\eta(s)}\right)b^{2}\left(X^{1}_{\eta(s)}\right)ds}\right]$

Since for all $x\in\mathbb{R},\ b^{2}(x)\leq C_{0}(1+x^{2})$ and
$\underset{t\in[0,1]}{\sup}{\mathbb{E}\left(|X_{t}|^{2}\right)}<\infty$, there
exists a constant $C>0$ depending on $\sigma_{1}$ such that:

$\mathbb{E}\left(\underset{h\in\mathcal{S}_{m},\
\|h\|^{2}_{X}=1}{\sup}{\nu^{2}_{3}(h)}\right)\leq C\frac{m}{n^{2}}.$ (43)

We finally obtain from Equations (41), (42) and (43) that there exists a
constant $C>0$ depending on $\sigma_{1}$ such that:

$\mathbb{E}\left(\underset{h\in\mathcal{S}_{m},\
\|h\|^{2}_{X}=1}{\sup}{\nu^{2}(h)}\right)\leq C\frac{m}{n}.$ (44)

We deduce from Equations (37) and (44) that there exists a constant $C>0$
depending on $\sigma_{1}$ such that,

$\mathbb{E}\left[\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\|^{2}_{n,1}\mathds{1}_{\Omega_{n,m}}\right]\leq
3\underset{h\in\mathcal{S}_{m,L}}{\inf}{\|\sigma^{2}_{|I}-h\|^{2}_{n}}+C\left(\frac{m}{n}+\Delta^{2}\right).$

For $n$ large enough, we have
$\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\|^{2}_{\infty}\leq 2mL$ since
$\|\widehat{\sigma}^{2}_{m}\|_{\infty}\leq\sqrt{mL}$. Then, from Lemma 8.6 and
for all $m\in\mathcal{M}$, there exists a constant $C>0$ depending on
$\sigma_{1}$ such that

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n,1}\right]$
$\displaystyle=\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n,1}\mathds{1}_{\Omega_{n,m}}\right]+\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n,1}\mathds{1}_{\Omega^{c}_{n,m}}\right]$
$\displaystyle\leq\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n,1}\mathds{1}_{\Omega_{n,m}}\right]+2mL\mathbb{P}\left(\Omega^{c}_{n,m}\right)$
$\displaystyle\leq
3\underset{h\in\mathcal{S}_{m,L}}{\inf}{\|\sigma^{2}_{|I}-h\|^{2}_{n}}+C\left(\frac{m}{n}+\frac{m^{2\gamma+1}L}{n^{\gamma/2}}+\Delta^{2}\right).$

Since the pseudo-norms $\|.\|_{n,1}$ and $\|.\|_{X}$ are equivalent on the
event $\Omega_{n,m}$, then, using Lemma 8.6, there exists a constant $C>0$
depending on $\sigma_{1}$ such that

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{X}\right]=$
$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{X}\mathds{1}_{\Omega_{n,m}}\right]+\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{X}\mathds{1}_{\Omega^{c}_{n,m}}\right]$
$\displaystyle\leq$ $\displaystyle
8\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n,1}\right]+10\underset{h\in\mathcal{S}_{m}}{\inf}{\left\|\sigma^{2}_{|I}-h\right\|^{2}_{n}}+2mL\mathbb{P}\left(\Omega^{c}_{n,m}\right)$
$\displaystyle\leq$ $\displaystyle
34\underset{h\in\mathcal{S}_{m,L}}{\inf}{\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n}}+C\left(\frac{m}{n}+\frac{m^{2\gamma+1}L}{n^{\gamma/2}}+\Delta^{2}\right).$

Finally, since the estimator $\widehat{\sigma}^{2}_{m}$ is built from a
diffusion path $\bar{X}^{1}$ independent of the diffusion process $X$, and
from Equations (15) and (12), the pseudo-norm $\|.\|_{X}$ depending on the
process $X$ and the empirical norm $\|.\|_{n}$ are equivalent
($\forall\leavevmode\nobreak\ h\in\mathbb{L}^{2}(I),\leavevmode\nobreak\
\|h\|^{2}_{n}\leq(\tau_{1}/\tau_{0})\mathbb{E}\left[\|h\|^{2}_{X}\right]$),
there exists a constant $C>0$ depending on $\sigma_{1},\tau_{0}$ and
$\tau_{1}$ such that

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n}\right]\leq$
$\displaystyle\frac{34\tau_{1}}{\tau_{0}}\underset{h\in\mathcal{S}_{m,L}}{\inf}{\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n}}+C\left(\frac{m}{n}+\frac{m^{2\gamma+1}L}{n^{\gamma/2}}+\Delta^{2}\right).$

∎

###### Proof of Lemma 8.6 .

The proof of this Lemma mainly focus on the spline basis and the Fourier basis
based on functions $\cos$ and $\sin$ which are Lipschitz functions. Thus, for
all $g=\sum_{\ell=0}^{m-1}{a_{\ell}\phi_{\ell}}\in\mathcal{S}_{m}$,

$\left|\|g\|^{2}_{n,1}-\|g\|^{2}_{X}\right|\leq\int_{0}^{1}{\left|g^{2}(X_{\eta(s)})-g^{2}(X_{s})\right|ds}\leq
2\|g\|_{\infty}\int_{0}^{1}{\left|g(X_{\eta(s)})-g(X_{s})\right|ds}.$ (45)

From Equation (15), one has
$\mathbb{E}\left[\|g\|^{2}_{X}\right]\geq\tau_{0}\|g\|^{2}$. Thus, if
$\|g\|^{2}_{X}=1$, then $\|g\|^{2}\leq 1/\tau_{0}$, and we deduce for all
$g=\sum_{\ell=0}^{m-1}{a_{\ell}\phi_{\ell}}$ that there exists a constant
$C>0$ such that

  * •

Spline basis: $\|g\|_{\infty}\leq\|a\|_{2}\leq C\sqrt{m}$ (see Denis _et al._
(2021))

  * •

Fourier basis: $\|g\|_{\infty}\leq C\sqrt{m}$ since $\|g\|=\|a\|_{2}$ and
$\sum_{\ell=0}^{m-1}{\phi^{2}_{\ell}}=\mathrm{O}(m)$.

Moreover, each $g\in\mathcal{S}_{m}$ such that $\|g\|^{2}_{X}=1$ is the
Lipschitz function with a Lipschitz coefficient
$L_{g}=\mathrm{O}\left(m^{3/2}\right)$. For the spline basis, this result is
obtained in Denis _et al._ (2021), proof of Lemma C.1 combined with Lemma 2.6.
For the Fourier basis, for all $x,y\in I$ and using the Cauchy Schwarz
inequality, we obtain

$\displaystyle|g(x)-g(y)|\leq$
$\displaystyle\sum_{\ell=0}^{m-1}{|a_{\ell}|.|\phi_{\ell}(x)-\phi_{\ell}(y)|}$
$\displaystyle\leq$ $\displaystyle 2\pi m\sqrt{m}\|\mathbf{a}\|_{2}|x-y|$
$\displaystyle\leq$ $\displaystyle\frac{2\pi}{\tau_{0}}m\sqrt{m}|x-y|.$

Back to Equation (45), there exists a constant $C>0$ such that

$\left|\|g\|^{2}_{n,1}-\|g\|^{2}_{X}\right|\leq
Cm^{2}\int_{0}^{1}{|X_{\eta(s)}-X_{s}|ds}$ (46)

We have:

$\displaystyle\Omega^{c}_{n,m}=\left\\{\omega\in\Omega,\leavevmode\nobreak\
\exists g\in\mathcal{S}_{m}\setminus\\{0\\},\leavevmode\nobreak\
\left|\frac{\|g\|^{2}_{n,1}}{\|g\|^{2}_{X}}-1\right|>\frac{1}{2}\right\\},$

and, using Equation (46), we obtain

$\underset{g\in\mathcal{S}_{m}\setminus\\{0\\}}{\sup}{\left|\frac{\|g\|^{2}_{n,1}}{\|g\|^{2}_{X}}-1\right|}=\underset{g\in\mathcal{S}_{m},\leavevmode\nobreak\
\|g\|^{2}_{X}=1}{\sup}{\left|\|g\|^{2}_{n,1}-\|g\|_{X}\right|}\leq
Cm^{2}\int_{0}^{1}{|X_{\eta(s)}-X_{s}|ds}.$

Finally, using the Markov inequality, the Hölder inequality, Equation (2), and
Lemma 8.1, we conclude that

$\displaystyle\mathbb{P}\left(\Omega^{c}_{n,m}\right)\leq$
$\displaystyle\leavevmode\nobreak\
\mathbb{P}\left(Cm^{2}\int_{0}^{1}{|X_{\eta(s)}-X_{s}|ds}\geq\frac{1}{2}\right)$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
Cm^{2\gamma}\int_{0}^{1}{\mathbb{E}\left[|X_{\eta(s)}-X_{s}|^{\gamma}\right]ds}$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
C\frac{m^{2\gamma}}{n^{\gamma/2}}$

with $\gamma\in(1,+\infty)$. ∎

####  8.2.3 Proof of Theorem 3.4

###### Proof.

Since $L=\log^{2}(n)$, we have

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n,1}\right]=$
$\displaystyle\leavevmode\nobreak\
\mathbb{E}\left[\left\|(\widehat{\sigma}^{2}_{m,L}-\sigma^{2})\mathds{1}_{[-\log(n),\log(n)]}\right\|^{2}_{n,1}\right]+\mathbb{E}\left[\left\|(\widehat{\sigma}^{2}_{m,L}-\sigma^{2})\mathds{1}_{[-\log(n),\log(n)]^{c}}\right\|^{2}_{n,1}\right]$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
\mathbb{E}\left[\left\|(\widehat{\sigma}^{2}_{m,L}-\sigma^{2})\mathds{1}_{[-\log(n),\log(n)]}\right\|^{2}_{n,1}\right]+2\log^{2}(n)\underset{t\in(0,1]}{\sup}{\mathbb{P}(|X_{t}|>\log(n))}.$

From Equation (34) (Proof of Theorem 3.2), for all $h\in\mathcal{S}_{m,L}$,

$\mathbb{E}\left[\left\|(\widehat{\sigma}^{2}_{m,L}-\sigma^{2})\mathds{1}_{[-\log(n),\log(n)]}\right\|^{2}_{n,1}\right]\leq\underset{h\in\mathcal{S}_{m,L}}{\inf}{\left\|h-\sigma^{2}\right\|^{2}_{n}}+2\sum_{i=1}^{3}{\mathbb{E}\left[\nu_{i}\left(\widehat{\sigma}^{2}_{m}-h\right)\right]}+2\mathbb{E}\left[\mu\left(\widehat{\sigma}^{2}_{m}-h\right)\right]$
(47)

where $\nu_{i},\leavevmode\nobreak\ i=1,2,3$ and $\mu$ are given in Equation
(32). For all $i\in\\{1,2,3\\}$ and for all $h\in\mathcal{S}_{m,L}$, one has

$\mathbb{E}\left[\nu_{i}\left(\widehat{\sigma}^{2}_{m,L}-h\right)\right]\leq\sqrt{2m\log^{2}(n)}\sqrt{\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\nu^{2}_{i}(\phi_{\ell})\right]}}.$
(48)

  1. 1.

Upper bound of
$\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\nu^{2}_{1}(\phi_{\ell})\right]}$

According to Equation (32), we have

$\forall\ell\in[\\![0,m-1]\\!],\
\nu_{1}(\phi_{\ell})=\frac{1}{n}\sum_{k=0}^{n-1}{\phi_{\ell}(X^{1}_{k\Delta})\zeta^{1,1}_{k\Delta}}$

where
$\zeta^{1,1}_{k\Delta}=\frac{1}{\Delta}\left[\left(\int_{k\Delta}^{(k+1)\Delta}{\sigma(X^{1}_{s})dW_{s}}\right)^{2}-\int_{k\Delta}^{(k+1)\Delta}{\sigma^{2}(X^{1}_{s})ds}\right]$
is a martingale satisfying

$\mathbb{E}\left[\zeta^{1,1}_{k\Delta}|\mathcal{F}_{k\Delta}\right]=0\ \
\mathrm{and}\ \
\mathbb{E}\left[\left(\zeta^{1,1}_{k\Delta}\right)^{2}|\mathcal{F}_{k\Delta}\right]\leq\frac{1}{\Delta^{2}}\mathbb{E}\left[\left(\int_{k\Delta}^{(k+1)\Delta}{\sigma^{2}(X^{1}_{s})ds}\right)^{2}\right]\leq
C\sigma^{4}_{1}$

with $C>0$ a constant, $W=W^{1}$ and $(\mathcal{F}_{t})_{t\geq 0}$ the natural
filtration of the martingale $(M_{t})_{t\in[0,1]}$ given for all $t\in[0,1]$
by $M_{t}=\int_{0}^{t}{\sigma(X^{1}_{s})dW_{s}}$. We derive that

$\displaystyle\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\nu^{2}_{1}(\phi_{\ell})\right]}=$
$\displaystyle\frac{1}{n^{2}}\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\left(\sum_{k=0}^{n-1}{\phi_{\ell}(X^{1}_{k\Delta})\zeta^{1,1}_{k\Delta}}\right)^{2}\right]}=\frac{1}{n^{2}}\mathbb{E}\left[\sum_{k=0}^{n-1}{\sum_{\ell=0}^{m-1}{\phi^{2}_{\ell}(X^{1}_{k\Delta})\left(\zeta^{1,1}_{k\Delta}\right)^{2}}}\right]$

since for all integers $k,k^{\prime}$ such that $k>k^{\prime}\geq 0$, we have

$\displaystyle\mathbb{E}\left[\phi_{\ell}(X^{1}_{k\Delta})\zeta^{1,1}_{k\Delta}\phi_{\ell}(X^{1}_{k^{\prime}\Delta})\zeta^{1,1}_{k^{\prime}\Delta}|\mathcal{F}_{k\Delta}\right]=\phi_{\ell}(X^{1}_{k\Delta})\zeta^{1,1}_{k^{\prime}\Delta}\phi_{\ell}(X^{1}_{k^{\prime}\Delta})\mathbb{E}\left[\zeta^{1,1}_{k\Delta}|\mathcal{F}_{k\Delta}\right]=0.$

For each $k\in[\\![0,n-1]\\!]$, we have

$\begin{cases}\sum_{\ell=0}^{m-1}{\phi_{\ell}(X^{1}_{k\Delta})}=\sum_{\ell=-M}^{K-1}{B_{\ell}(X^{1}_{k\Delta})}=1\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{for\leavevmode\nobreak\ the\leavevmode\nobreak\
spline\leavevmode\nobreak\ basis}\\\ \\\
\sum_{\ell=0}^{m-1}{\phi_{\ell}(X^{1}_{k\Delta})}\leq Cm\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{For\leavevmode\nobreak\ an\leavevmode\nobreak\
orthonormal\leavevmode\nobreak\ basis\leavevmode\nobreak\
with}\leavevmode\nobreak\ C=\underset{0\leq\ell\leq
m-1}{\max}{\|\phi_{\ell}\|_{\infty}}.\end{cases}$

Finally, there exists a constant $C>0$ such that

$\displaystyle\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\nu^{2}_{1}(\phi_{\ell})\right]}\leq\begin{cases}\frac{C}{n}\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{for\leavevmode\nobreak\ the\leavevmode\nobreak\
spline\leavevmode\nobreak\ basis}\\\ \\\ C\frac{m}{n}\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{for\leavevmode\nobreak\ an\leavevmode\nobreak\
orthonormal\leavevmode\nobreak\ basis}.\end{cases}$

  2. 2.

Upper bound of
$\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\nu^{2}_{2}(\phi_{\ell})\right]}$

For all $k\in[\\![0,n-1]\\!]$ and for all $s\in[0,1]$, set $\eta(s)=k\Delta$
if $s\in[k\Delta,(k+1)\Delta)$. We have:

$\displaystyle\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\nu^{2}_{2}(\phi_{\ell})\right]}$
$\displaystyle=4\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\left(\sum_{k=0}^{n-1}{\int_{k\Delta}^{(k+1)\Delta}{\phi_{\ell}(X^{1}_{k\Delta})((k+1)\Delta-s)\sigma^{\prime}(X^{1}_{s})\sigma^{2}(X^{1}_{s})dW_{s}}}\right)^{2}\right]}$
$\displaystyle=4\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\left(\int_{0}^{1}{\phi_{\ell}(X^{1}_{\eta(s)})(\eta(s)+\Delta-s)\sigma^{\prime}(X^{1}_{s})\sigma^{2}(X^{1}_{s})dW_{s}}\right)^{2}\right]}.$

We conclude that

$\displaystyle\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\nu^{2}_{2}(\phi_{\ell})\right]}\leq\begin{cases}\frac{C}{n^{2}}\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{for\leavevmode\nobreak\ the\leavevmode\nobreak\
spline\leavevmode\nobreak\ basis}\\\ \\\ C\frac{m}{n^{2}}\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{for\leavevmode\nobreak\ an\leavevmode\nobreak\
orthonormal\leavevmode\nobreak\ basis}.\end{cases}$

where the constant $C>0$ depends on the diffusion coefficient and the upper
bound of the basis functions.

  3. 3.

Upper bound of
$\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\nu^{3}_{2}(\phi_{\ell})\right]}$

We have:

$\displaystyle\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\nu^{2}_{3}(\phi_{\ell})\right]}$
$\displaystyle=\frac{4}{n^{2}}\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\left(\sum_{k=0}^{n-1}{\int_{k\Delta}^{(k+1)\Delta}{\phi_{\ell}(X^{1}_{k\Delta})b(X^{1}_{k\Delta})\sigma(X^{1}_{s})dW_{s}}}\right)^{2}\right]}$
$\displaystyle=\frac{4}{n^{2}}\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\left(\int_{0}^{1}{\phi_{\ell}(X^{1}_{\eta(s)})b(X^{1}_{\eta(s)})\sigma(X^{1}_{s})dW_{s}}\right)^{2}\right]}$
$\displaystyle\leq\frac{4}{n^{2}}\mathbb{E}\left[\int_{0}^{1}{\sum_{\ell=0}^{m-1}{\phi^{2}_{\ell}(X^{1}_{\eta(s)})b^{2}(X^{1}_{\eta(s)})\sigma^{2}(X^{1}_{s})ds}}\right].$

Since for all $x\in\mathbb{R},\ b(x)\leq C_{0}(1+x^{2})$ and
$\underset{t\in[0,1]}{\sup}{\mathbb{E}\left(|X_{t}|^{4}\right)}<\infty$, there
exists a constant $C>0$ depending on the diffusion coefficient such that

$\displaystyle\sum_{\ell=0}^{m-1}{\mathbb{E}\left[\nu^{2}_{3}(\phi_{\ell})\right]}\leq\begin{cases}\frac{C}{n^{2}}\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{for\leavevmode\nobreak\ the\leavevmode\nobreak\
spline\leavevmode\nobreak\ basis}\\\ \\\ C\frac{m}{n^{2}}\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{for\leavevmode\nobreak\ an\leavevmode\nobreak\
orthonormal\leavevmode\nobreak\ basis}.\end{cases}$

We finally deduce that from Equations (47) and (48) that for all
$h\in\mathcal{S}_{m,L}$,

$\begin{cases}\mathbb{E}\left[\|(\widehat{\sigma}^{2}_{m,L}-\sigma^{2})\mathds{1}_{[-\log(n),\log(n)]}\|^{2}_{n,1}\right]\leq\underset{h\in\mathcal{S}_{m}}{\inf}{\|h-\sigma^{2}\|^{2}_{n}}+C\sqrt{\frac{m\log^{2}(n)}{n}}+2\mathbb{E}\left[\mu(\widehat{\sigma}^{2}_{m,L}-h)\right]\leavevmode\nobreak\
\leavevmode\nobreak\ {\bf[B]}\\\
\mathbb{E}\left[\|(\widehat{\sigma}^{2}_{m,L}-\sigma^{2})\mathds{1}_{[-\log(n),\log(n)]}\|^{2}_{n,1}\right]\leq\underset{h\in\mathcal{S}_{m}}{\inf}{\|h-\sigma^{2}\|^{2}_{n}}+C\sqrt{\frac{m^{2}\log^{2}(n)}{n}}+2\mathbb{E}\left[\mu(\widehat{\sigma}^{2}_{m,L}-h)\right]\leavevmode\nobreak\
\leavevmode\nobreak\ {\bf[F]}\end{cases}$ (49)

where $C>0$ is a constant. It remains to obtain an upper bound of the term
$\mu(\widehat{\sigma}^{2}_{m,L}-h)$. For all $a>0$ and for all
$h\in\mathcal{S}_{m,L}$,

$\displaystyle 2\mu\left(\widehat{\sigma}^{2}_{m,L}-h\right)\leq$
$\displaystyle\frac{2}{a}\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n,1}+\frac{2}{a}\left\|h-\sigma^{2}\right\|^{2}_{n,1}+\frac{a}{n}\sum_{k=0}^{n-1}{\left(R^{1}_{k\Delta}\right)^{2}}$
$\displaystyle
2\mathbb{E}\left[\mu\left(\widehat{\sigma}^{2}_{m,L}-h\right)\right]\leq$
$\displaystyle\frac{2}{a}\mathbb{E}\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n,1}+\frac{2}{a}\underset{h\in\mathcal{S}_{m}}{\inf}{\|h-\sigma^{2}\|^{2}_{n}}+\frac{a}{n}\sum_{k=0}^{n-1}{\mathbb{E}\left[\left(R^{1}_{k\Delta}\right)^{2}\right]}.$

Using Equations (36), (49) and setting $a=4$, we deduce that there exists
constant $C>0$ depending on $\sigma_{1}$ such that,

$\begin{cases}\mathbb{E}\left[\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\|^{2}_{n,1}\right]\leq\underset{h\in\mathcal{S}_{m}}{\inf}{\|h-\sigma^{2}\|^{2}_{n}}+C\sqrt{\frac{m\log^{2}(n)}{n}}+2\log^{2}(n)\underset{t\in(0,1]}{\sup}{\mathbb{P}(|X_{t}|>A_{n})}\leavevmode\nobreak\
\leavevmode\nobreak\ {\bf[B]}\\\
\mathbb{E}\left[\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\|^{2}_{n,1}\right]\leq\underset{h\in\mathcal{S}_{m}}{\inf}{\|h-\sigma^{2}\|^{2}_{n}}+C\sqrt{\frac{m^{2}\log^{2}(n)}{n}}+2\log^{2}(n)\underset{t\in(0,1]}{\sup}{\mathbb{P}(|X_{t}|>A_{n})}\leavevmode\nobreak\
\leavevmode\nobreak\ {\bf[F]}.\end{cases}$ (50)

From Proposition 8.3,
$\underset{t\in(0,1]}{\sup}{\mathbb{P}(|X_{t}|>\log(n))}\leq\log^{-1}(n)\exp(-c\log^{2}(n))$
with $c>0$ a constant. Then, we obtain from Equation (50) that

$\begin{cases}\mathbb{E}\left[\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\|^{2}_{n,1}\right]\leq\underset{h\in\mathcal{S}_{m}}{\inf}{\|h-\sigma^{2}\|^{2}_{n}}+C\sqrt{\frac{m\log^{2}(n)}{n}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ {\bf[B]}\\\ \\\
\mathbb{E}\left[\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\|^{2}_{n,1}\right]\leq\underset{h\in\mathcal{S}_{m}}{\inf}{\|h-\sigma^{2}\|^{2}_{n}}+C\sqrt{\frac{m^{2}\log^{2}(n)}{n}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{\bf[F]}.\end{cases}$ (51)

∎

###  8.3 Proof of Section 4

The following lemma allows us to obtain a risk bound of
$\widehat{\sigma}^{2}_{m,L}$ defined with the empirical norm $\|.\|_{n}$ from
the risk bound defined from the pseudo norm $\|.\|_{n,N}$.

######  Lemma 8.7.

Let $\widehat{\sigma}^{2}_{m,L}$ be the truncated projection estimator on
$\mathbb{R}$ of $\sigma^{2}$ over the subspace $\mathcal{S}_{m,L}$. Suppose
that $L=\log^{2}(N),\leavevmode\nobreak\ N>1$. Under Assumption 2.1, there
exists a constant $C>0$ independent of $m$ and $N$ such that

$\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n,N}\right]-2\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n}\right]\leq
C\frac{m^{2}\log^{3}(N)}{N}.$

The proof of Lemma 8.7 is provided in Denis _et al._ (2021), Theorem 3.3. The
proof uses the independence of the copies $\bar{X}^{1},\ldots,\bar{X}^{N}$ of
the process $X$ at discrete times, and the Bernstein inequality.

####  8.3.1 Proof of Theorem 4.1

For fixed $n$ and $N$ in $\mathbb{N}^{*}$, we set for all $m\in\mathcal{M}$,

$\Omega_{n,N,m}:=\underset{h\in\mathcal{S}_{m}\setminus\\{0\\}}{\bigcap}{\left\\{\left|\frac{\|h\|^{2}_{n,N}}{\|h\|^{2}_{n}}-1\right|\leq\frac{1}{2}\right\\}}.$
(52)

As we can see, the empirical norms $\|h\|_{n,N}$ and $\|h\|_{n}$ of any
function $h\in\mathcal{S}_{m}\setminus\\{0\\}$ are equivalent on
$\Omega_{n,N,m}$. More precisely, on the set $\Omega_{n,N,m}$, for all
$h\in\mathcal{S}_{m}\setminus\\{0\\}$, we have :
$\frac{1}{2}\|h\|^{2}_{n}\leq\|h\|^{2}_{n,N}\leq\frac{3}{2}\|h\|^{2}_{n}$. We
have the following result:

######  Lemma 8.8.

Under Assumption 2.1, the following holds:

  * •

If $n\geq N$ or $n\propto N$, then
$m\in\mathcal{M}=\left\\{1,\ldots,\sqrt{N}/\log(Nn)\right\\}$ and,

$\displaystyle\mathbb{P}\left(\Omega^{c}_{n,N,m}\right)\leq
2\exp(-C\sqrt{N}).$

  * •

If $n\leq N$, then
$m\in\mathcal{M}=\left\\{1,\ldots,\sqrt{n}/\log(Nn)\right\\}$ and

$\displaystyle\mathbb{P}\left(\Omega^{c}_{n,N,m}\right)\leq 2\exp(-C\sqrt{n})$

where $C>0$ is a constant.

###### Proof of Lemma 8.8 .

We have:

$\displaystyle\Omega^{c}_{n,N,m}=\left\\{\omega\in\Omega,\ \exists
h_{0}\in\mathcal{S}_{m},\
\left|\frac{\|h\|^{2}_{n,N}}{\|h\|^{2}_{n}}-1\right|>\frac{1}{2}\right\\},$

Denote by $\mathcal{H}_{m}=\left\\{h\in\mathcal{S}_{m},\leavevmode\nobreak\
\|h\|_{n}=1\right\\}$ and $\mathcal{H}^{\varepsilon}_{m}$ the
$\varepsilon-$net of $\mathcal{H}_{m}$ for any $\varepsilon>0$. We have

$\displaystyle\underset{h\in\mathcal{H}_{m}}{\sup}{\left|\frac{\|h\|^{2}_{n,N}}{\|h\|^{2}_{n}}-1\right|}=\underset{h\in\mathcal{H}_{m}}{\sup}{\left|\|h\|^{2}_{n,N}-1\right|}.$

Let $\varepsilon>0$ and let $\mathcal{H}^{\varepsilon}_{m}$ be the
$\varepsilon-$net of $\mathcal{H}_{m}$ w.r.t. the supremum norm
$\|.\|_{\infty}$. Then, for each $h\in\mathcal{H}_{m}$, there exists
$h_{\varepsilon}\in\mathcal{H}^{\varepsilon}_{m}$ such that
$\|h-h_{\varepsilon}\|_{\infty}\leq\varepsilon$. Then

$\displaystyle\left|\|h\|^{2}_{n,N}-1\right|\leq\left|\|h\|^{2}_{n,N}-\|h_{\varepsilon}\|^{2}_{n,N}\right|+\left|\|h_{\varepsilon}\|^{2}_{n,N}-1\right|$

and,

$\displaystyle\left|\|h\|^{2}_{n,N}-\|h_{\varepsilon}\|^{2}_{n,N}\right|\leq$
$\displaystyle\leavevmode\nobreak\
\frac{1}{Nn}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{\left|h(X^{j}_{k\Delta})-h_{\varepsilon}(X^{j}_{k\Delta})\right|\left(\|h\|_{\infty}+\|h_{\varepsilon}\|_{\infty}\right)}}\leq\left(\|h\|_{\infty}+\|h_{\varepsilon}\|_{\infty}\right)\varepsilon.$

Moreover, we have $\|h\|^{2},\|h_{\varepsilon}\|^{2}\leq 1/\tau_{0}$. Then,
there exists a constant $\mathbf{c}>0$ such that

$\displaystyle\begin{cases}\left|\|h\|^{2}_{n,N}-\|h_{\varepsilon}\|^{2}_{n,N}\right|\leq
2\sqrt{\frac{{\bf c}m}{\tau_{0}}}\varepsilon\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{for}\leavevmode\nobreak\
\mathrm{the}\leavevmode\nobreak\ \mathrm{spline}\leavevmode\nobreak\
\mathrm{basis}\leavevmode\nobreak\ (\mathrm{see\leavevmode\nobreak\
\textit{Lemma 2.6}\leavevmode\nobreak\ in\leavevmode\nobreak\
Denis\leavevmode\nobreak\ {\it et\leavevmode\nobreak\ al.}(2021)})\\\ \\\
\left|\|h\|^{2}_{n,N}-\|h_{\varepsilon}\|^{2}_{n,N}\right|\leq
2\sqrt{\frac{{\bf c}m}{\tau_{0}}}\varepsilon\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{for\leavevmode\nobreak\ an\leavevmode\nobreak\
orthonormal\leavevmode\nobreak\ basis}\leavevmode\nobreak\
\leavevmode\nobreak\ (\|h\|^{2}_{\infty}\leq(\underset{0\leq\ell\leq
m-1}{\max}{\|\phi_{\ell}\|^{2}_{\infty}})m\|h\|^{2}).\end{cases}$

Therefore, for all $\delta>0$ and for both the spline basis and any
orthonormal basis,

$\mathbb{P}\left(\underset{h\in\mathcal{H}_{m}}{\sup}{\left|\|h\|^{2}_{n,N}-1\right|}\geq\delta\right)\leq\mathbb{P}\left(\underset{h\in\mathcal{H}^{\varepsilon}_{m}}{\sup}{\left|\|h\|^{2}_{n,N}-1\right|}\geq\delta/2\right)+\mathds{1}_{4\varepsilon\sqrt{\frac{{\bf
c}m}{\tau_{0}}}\geq\delta}.$

We set $\delta=1/2$ and we choose $\varepsilon>0$ such that
$4\varepsilon\sqrt{\frac{{\bf c}m}{\tau_{0}}}<1/2$. Then, using the Hoeffding
inequality, there exists a constant $c>0$ depending on ${\bf c}$ and
$\tau_{0}$ such that

$\displaystyle\mathbb{P}\left(\Omega^{c}_{n,N,m}\right)\leq
2\mathcal{N}_{\infty}(\varepsilon,\mathcal{H}_{m})\exp\left(-c\frac{N}{m}\right)$
(53)

where $\mathcal{N}_{\infty}(\varepsilon,\mathcal{H}_{m})$ is the covering
number of $\mathcal{H}_{m}$ satisfying:

$\mathcal{N}_{\infty}(\varepsilon,\mathcal{H}_{m})\leq\left(\kappa\frac{\sqrt{m}}{\varepsilon}\right)^{m}$
(54)

where the constant $\kappa>0$ depends on $c>0$ (see Denis _et al._ (2021),
Proof of Lemma D.1). We set $\varepsilon=\frac{\kappa\sqrt{m^{*}}}{N}$ with
$m^{*}=\max{\mathcal{M}}$ and we derive from Equations (53) and (54) that

$\mathbb{P}\left(\Omega^{c}_{n,N,m}\right)\leq
2N^{m^{*}}\exp\left(-c\frac{N}{m^{*}}\right)=2\exp\left(-c\frac{N}{m^{*}}\left(1-\frac{m^{*2}\log(N)}{cN}\right)\right).$

  * •

If $n\geq N$, then
$m\in\mathcal{M}=\left\\{1,\ldots,\sqrt{N}/\log(Nn)\right\\}$. Since
$m^{*2}\log(N)/N\rightarrow 0$ as $N\rightarrow+\infty$, there exists a
constant $C>0$ such that

$\displaystyle\mathbb{P}\left(\Omega^{c}_{n,N,m}\right)\leq
2\exp\left(-C\sqrt{N}\right).$

  * •

If $n\leq N$, then
$m\in\mathcal{M}=\left\\{1,\ldots,\sqrt{n}/\log(Nn)\right\\},\leavevmode\nobreak\
m^{*2}\log(N)/N\leq\log(N)/\log^{2}(Nn)\rightarrow 0$ as
$N,n\rightarrow\infty$, and

$\mathbb{P}\left(\Omega^{c}_{n,N,m}\right)\leq 2\exp\left(-C\sqrt{n}\right).$

  * •

If $n\propto N$, then
$m\in\mathcal{M}=\left\\{1,\ldots,\sqrt{N}/\log(Nn)\right\\}$. Since
$m^{*2}\log(N)/N\rightarrow 0$ as $N\rightarrow+\infty$, there exists a
constant $C>0$ such that

$\displaystyle\mathbb{P}\left(\Omega^{c}_{n,N,m}\right)\leq
2\exp\left(-C\sqrt{N}\right).$

∎

###### Proof of Theorem 4.1 .

The proof of Theorem 4.1 extends the proof of Theorem 3.2 when $N$ tends to
infinity. Then, we deduce from Equation (37) that

$\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n,N}\mathds{1}_{\Omega_{n,N,m}}\right]\leq
3\underset{h\in\mathcal{S}_{m,L}}{\inf}{\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n}}+C\mathbb{E}\left(\underset{h\in\mathcal{S}_{m},\|h\|_{n}=1}{\sup}{\nu^{2}(h)}\right)+C\Delta^{2}$
(55)

where $C>0$ is a constant depending on $\sigma_{1}$, and
$\nu=\nu_{1}+\nu_{2}+\nu_{3}$ with

$\nu_{i}(h)=\frac{1}{Nn}\sum_{j=1}^{N}{\sum_{k=0}^{n-1}{h(X^{j}_{k\Delta})\zeta^{j,i}_{k\Delta}}},\leavevmode\nobreak\
\leavevmode\nobreak\ i=1,2,3$

and the $\zeta^{j,i}_{k\Delta}$’s are the error terms depending on each path
$X^{j},\leavevmode\nobreak\ j=1,\ldots,N$.

### Upper bound of $\mathbb{E}\left(\underset{h\in\mathcal{S}_{m},\
\|h\|_{n}=1}{\sup}{\nu^{2}(h)}\right)$

For all $h=\sum_{\ell=0}^{m-1}{a_{\ell}\phi_{\ell}}\in\mathcal{S}_{m}$ such
that $\|h\|_{n}=1$, we have $\|h\|^{2}\leq\frac{1}{\tau_{0}}$ and the
coordinate vector $a=\left(a_{0},\cdots,a_{m-1}\right)$ satisfies:

  * •

$\|a\|^{2}_{2}\leq CK\leq Cm$ for the spline basis (see Denis _et al._ (2021),
Lemma 2.6)

  * •

$\|a\|^{2}_{2}\leq 1/\tau_{0}$ for an orthonormal basis since
$\|h\|^{2}=\|a\|^{2}_{2}$.

Furthermore, using the Cauchy Schwartz inequality, we have:

$\displaystyle\nu^{2}(h)=\left(\sum_{\ell=0}^{m-1}{a_{\ell}\nu\left(\phi_{\ell}\right)}\right)^{2}\leq\|a\|^{2}_{2}\sum_{\ell=0}^{m-1}{\nu^{2}\left(\phi_{\ell}\right)}.$

Thus, for all $\ell\in[\\![0,m-1]\\!],\ \ \nu=\nu_{1}+\nu_{2}+\nu_{3}$ and for
all $i\in\\{1,2,3\\}$

$\displaystyle\mathbb{E}\left[\nu^{2}_{i}\left(\phi_{\ell}\right)\right]=$
$\displaystyle\leavevmode\nobreak\
\frac{1}{Nn^{2}}\mathbb{E}\left[\left(\sum_{k=0}^{n-1}{\phi_{\ell}\left(X^{1}_{k\Delta}\right)\zeta^{1,i}_{k\Delta}}\right)^{2}\right].$

We finally deduce from (41), (42) and (43) that there exists a constant $C>0$
depending on $\sigma_{1}$ such that:

$\mathbb{E}\left(\underset{h\in\mathcal{S}_{m},\
\|h\|_{n}=1}{\sup}{\nu^{2}(h)}\right)\leq C\frac{m}{Nn}.$ (56)

We deduce from (55) and (56) that there exists a constant $C>0$ such that,

$\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n,N}\mathds{1}_{\Omega_{n,N,m}}\right]\leq
3\underset{h\in\mathcal{S}_{m,L}}{\inf}{\left\|\sigma^{2}_{|I}-h\right\|^{2}_{n}}+C\left(\frac{m}{Nn}+\Delta^{2}\right).$
(57)

Since we have $\left\|\widehat{\sigma}^{2}_{m}\right\|_{\infty}\leq\sqrt{mL}$,
then for $m$ and $L$ large enough,
$\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{\infty}\leq
2mL$. There exists a constant $C>0$ such that for all $m\in\mathcal{M}$ and
for $m$ and $L$ large enough,

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n,N}\right]$
$\displaystyle=\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n,N}\mathds{1}_{\Omega_{n,N,m}}\right]+\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n,N}\mathds{1}_{\Omega^{c}_{n,N,m}}\right]$
$\displaystyle\leq\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n,N}\mathds{1}_{\Omega_{n,N,m}}\right]+2mL\mathbb{P}\left(\Omega^{c}_{n,N,m}\right).$

Then, from Equation (57), Lemma 8.8 and for
$m\in\mathcal{M}=\left\\{1,\ldots,\sqrt{\min(n,N)}/\sqrt{\log(Nn)}\right\\}$,
we have:

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n,N}\right]\leq$
$\displaystyle\leavevmode\nobreak\
3\underset{h\in\mathcal{S}_{m,L}}{\inf}{\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n}}+C\left(\frac{m}{Nn}+mL\exp\left(-C\sqrt{\min(n,N)}\right)+\Delta^{2}\right)$

where $C>0$ is a constant. Recall that the empirical norms $\|.\|_{n,N}$ and
$\|.\|_{n}$ are equivalent on $\Omega_{n,N,m}$, that is for all
$h\in\mathcal{S}_{m},\ \ \|h\|^{2}_{n}\leq 2\|h\|^{2}_{n,N}$. Thus, we have

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n}\right]=$
$\displaystyle\leavevmode\nobreak\
\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n}\mathds{1}_{\Omega_{n,N,m}}\right]+\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n}\mathds{1}_{\Omega^{c}_{n,N,m}}\right]$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n}\mathds{1}_{\Omega_{n,N,m}}\right]+2mL\mathbb{P}\left(\Omega^{c}_{n,N,m}\right).$

For all $h\in\mathcal{S}_{m,L}\subset\mathcal{S}_{m}$, we have:

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n}\mathds{1}_{\Omega_{n,N,m}}\right]\leq$
$\displaystyle\leavevmode\nobreak\
2\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-h\right\|^{2}_{n}\mathds{1}_{\Omega_{n,N,m}}\right]+2\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n}$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
4\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-h\right\|^{2}_{n,N}\mathds{1}_{\Omega_{n,N,m}}\right]+2\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n}$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
8\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n,N}\right]+10\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n}.$

We finally conclude that

$\displaystyle\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m}-\sigma^{2}_{|I}\right\|^{2}_{n}\right]\leq$
$\displaystyle
34\underset{h\in\mathcal{S}_{m,L}}{\inf}{\left\|h-\sigma^{2}_{|I}\right\|^{2}_{n}}+C\left(\frac{m}{Nn}+mL\exp\left(-C\sqrt{\min(n,N)}\right)+\Delta^{2}\right).$

∎

####  8.3.2 Proof of Theorem 4.3

###### Proof.

We consider the restriction $\sigma^{2}\mathds{1}_{[-\log(N),\log(N)]}$ of
$\sigma^{2}$ on the compact interval $[-\log(N),\log(N)]$ on which the spline
basis is built. Then we have:

$\mathbb{E}\left[\left\|\widehat{\sigma}^{2}_{m,L}-\sigma^{2}\right\|^{2}_{n}\right]=\mathbb{E}\left[\left\|(\widehat{\sigma}^{2}_{m,L}-\sigma^{2})\mathds{1}_{[-\log(N),\log(N)]}\right\|^{2}_{n}\right]+\mathbb{E}\left[\left\|(\widehat{\sigma}^{2}_{m,L}-\sigma^{2})\mathds{1}_{[-\log(N),\log(N)]^{c}}\right\|^{2}_{n}\right]$

and from Proposition 8.2, Lemma 8.4 and for $N$ large enough, there exists
constants $c,C>0$ such that

$\displaystyle\mathbb{E}\left[\left\|(\widehat{\sigma}^{2}_{m,L}-\sigma^{2})\mathds{1}_{[-\log(N),\log(N)]^{c}}\right\|^{2}_{n}\right]\leq$
$\displaystyle\frac{2L}{n}\sum_{k=0}^{n-1}{\mathbb{P}\left(|X_{k\Delta}|>\log(N)\right)}\leq
2L\underset{t\in[0,1]}{\sup}{\mathbb{P}\left(|X_{t}|\geq\log(N)\right)}$
$\displaystyle\leq$