# The LAN property for McKean-Vlasov models in a mean-field regime

Laetitia Della Maestra and Marc Hoffmann  Laetitia Della Maestra, Université
Paris-Dauphine & PSL, CNRS, CEREMADE, 75016 Paris, France
<EMAIL_ADDRESS>Marc Hoffmann, Université Paris-Dauphine &
PSL, CNRS, CEREMADE, 75016 Paris, France<EMAIL_ADDRESS>
###### Abstract.

We establish the local asymptotic normality (LAN) property for estimating a
multidimensional parameter in the drift of a system of $N$ interacting
particles observed over a fixed time horizon in a mean-field regime
$N\rightarrow\infty$. By implementing the classical theory of Ibragimov and
Hasminski, we obtain in particular sharp results for the maximum likelihood
estimator that go beyond its simple asymptotic normality thanks to Hájek’s
convolution theorem and strong controls of the likelihood process that yield
asymptotic minimax optimality (up to constants). Our structural results shed
some light to the accompanying nonlinear McKean-Vlasov experiment, and enable
us to derive simple and explicit criteria to obtain identifiability and non-
degeneracy of the Fisher information matrix. These conditions are also of
interest for other recent studies on the topic of parametric inference for
interacting diffusions.

Mathematics Subject Classification (2010): 62C20, 62F12, 62F99, 62M99 .

Keywords: Parametric estimation; LAN property; maximum likelihood estimation;
statistics and PDE; interacting particle systems; McKean-Vlasov models.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation
    2. 1.2 Setting
    3. 1.3 Results and organisation of the paper
  2. 2 Construction and properties of the statistical model
    1. 2.1 Notation
    2. 2.2 Model assumptions
    3. 2.3 The companion McKean-Vlasov product experiment
    4. 2.4 Identifiability and non-degeneracy of the Fisher information
  3. 3 Main results
    1. 3.1 The LAN property
    2. 3.2 Maximum likelihood estimation and properties
  4. 4 Examples
    1. 4.1 McKean-like models
    2. 4.2 Generalised linear like models
    3. 4.3 A double layer potential model
    4. 4.4 A genuinely non-linear example
  5. 5 Proof of the main results
    1. 5.1 Preliminaries: couplings
    2. 5.2 Proof of Theorem 17
    3. 5.3 Proof of Theorem 19
  6. 6 Remaining proofs
    1. 6.1 Proof of Proposition 9
    2. 6.2 Proof of Proposition 10
    3. 6.3 Proof of Proposition 11
    4. 6.4 Proof of Proposition 16
  7. 7 Appendix
    1. 7.1 Proof of Lemma 7
    2. 7.2 Proof of Lemma 8
    3. 7.3 Proof of Lemma 20
    4. 7.4 Proof of Lemma 21

##  1\. Introduction

###  1.1. Motivation

Collective dynamics models are becoming increasingly popular in modelling
complex stochastic systems, with a versatiliy of applications, ranging from
mathematical biology (neurosciences, Baladron et al. [2], structured models in
population dynamics, Mogilner et al. [40], Burger et al. [8]) to social
sciences (opinion dynamics, Chazelle et al. [13], cooperative behaviours,
Canuto et al. [9]) and finance (systemic risk, Fouque and Sun [17]), or more
recently, mean-field games (Cardaliguet et al. [10], Cardaliaguet and Lehalle
[11]). Whereas stochastic systems of interacting particles and associated
nonlinear Markov processes in the sense of McKean [38] date back to the 1960’s
and have been studied extensively over more than half a century, see e.g. [7,
44, 45, 39, 47] among a myriad of references, the development of statistical
inference in this setting is only emerging, (with some notable exceptions like
Löcherbach [35] in large time or e.g. Kasonga [27] or Bishwal [4]) in a mean-
field limit. Recently, Giesecke et al. [22] and Sharrock, Kantas, Parpas and
Grigorios [43] revisit the work of Kasonga and consider a parametric framework
where convergent and asymptotically normal contrast estimators are
constructed. Several other parametric frameworks (that consider various
observation schemes and asymptotic frameworks) have also been recently
considered, like [14, 34, 49] or Genon-Catalot and Laredo [20, 21]. There also
exist recent results in nonparametric inference: we mention our work [16] and
Belometsny et al. [3], together with studies in identification like [31, 32,
33] or learning [30, 36, 37].

The present paper, close in spirit to [4, 22, 27] and [43] (in their so-called
offline case) considers a parametric framework in a mean-field regime over a
fixed time horizon. We take a deeper look at the asymptotic structure of the
associated statistical experiment, in the sense of local asymptotic normality
or LAN, in order to derive strong results for the maximum likelihood, both in
asymptotic distribution and in an asymptotic minimax sense (up to constants)
for various loss functions. For simplicity, we keep-up with continuous
observations, but we briefly explain how to move to a discrete data setting.
Also, we look for simple and explicit criteria that enable us to verify
identifiability and non-degeneracy of the model. This is a non-trivial issue
in the context of nonlinear McKean-Vlasov models that is usually a bit
overlooked in the literature.

###  1.2. Setting

We have a parameter of interest $\vartheta$ lying in a compact set
$\Theta\subset\mathbb{R}^{p}$ (with non empty interior), for some fixed $p\geq
1$. For some fixed time horizon $T>0$, we continuously observe a stochastic
system of $N$ interacting particles

(1) $X^{(N)}=(X_{t}^{1},\ldots,X_{t}^{N})_{t\in[0,T]},$

evolving in an Euclidean ambient space $\mathbb{R}^{d}$, that solves

(2)
$\left\\{\begin{array}[]{l}dX_{t}^{i}=b(\vartheta;t,X_{t}^{i},\mu^{(N)}_{t})dt+\sigma(t,X_{t}^{i})dB_{t}^{i},\;\;1\leq
i\leq N,\;t\in[0,T],\\\
\mathcal{L}(X_{0}^{1},\ldots,X_{0}^{N})=\mu_{0}^{\otimes
N},\end{array}\right.$

where $\mu^{(N)}_{t}=N^{-1}\sum_{i=1}^{N}\delta_{X_{t}^{i}}$ is the empirical
measure of the system. The $(B_{t}^{i})_{t\in[0,T]}$ are independent
$\mathbb{R}^{d}$-valued Brownian motions. The initial condition $\mu_{0}$, the
drift $b$ and the diffusion coefficient $\sigma$ are at least sufficiently
regular so that

$\mu^{(N)}=(\mu^{(N)}_{t})_{t\in[0,T]}\rightarrow\mu=(\mu_{t})_{t\in[0,T]}$

weakly as $N\rightarrow\infty$, where $\mu$ is a family of probability
measures that solves (in a weak sense) the parabolic nonlinear equation

(3)
$\left\\{\begin{array}[]{ll}\partial_{t}\mu+\mathrm{div}\big{(}b(\vartheta;\cdot,\mu)\mu\big{)}=\tfrac{1}{2}\sum_{k,k^{\prime}=1}^{d}\partial_{kk^{\prime}}^{2}\big{(}c_{kk^{\prime}}\mu\big{)},\;t\in[0,T],\\\
\mu_{t=0}=\mu_{0},\end{array}\right.$

with $c=\sigma\sigma^{\top}$. We will write
$\mu^{\vartheta}=(\mu^{\vartheta}_{t})_{t\in[0,T]}$ to emphasise the
dependence in $\vartheta$. In this context, we are interested in estimating
from data (1) the parameter $\vartheta\in\Theta$ of the function
$(\vartheta;t,x,\nu)\mapsto b(\vartheta;t,x,\nu)\in\mathbb{R}^{d}$.
Asymptotics are taken as $N\rightarrow\infty$.

A particular case of interest that covers many examples is when the dependence
in the measure variable for $b$ is linear: we then have

(4)
$b(\vartheta;t,X_{t}^{i},\mu^{(N)}_{t})=\int_{\mathbb{R}^{d}}\widetilde{b}(\vartheta;X_{t}^{i},y)\mu_{t}^{(N)}(dy)=N^{-1}\sum_{j=1}^{N}\widetilde{b}(\vartheta;X_{t}^{i},X_{t}^{j}),$

for some function
$\widetilde{b}:\Theta\times\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$.
A typical form is
$\widetilde{b}(\vartheta;t,x,y)=G_{\vartheta}(x)+F_{\vartheta}(x-y)$ where
$G_{\vartheta},F_{\vartheta}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ play the
role of a common external force to the system and an interaction force
respectively.

###  1.3. Results and organisation of the paper

In Section 2, we rigorously construct the (sequence of) statistical
experiment(s) generated by the observation (1) under the dynamics (2) that we
denote $(\mathcal{E}^{N})_{N\geq 1}$. It is well defined and regular in the
classical sense of Ibragimov and Hasminski [24] under strong integrability of
the initial condition $\mu_{0}$ and standard smoothness assumptions on the
drift $\vartheta\mapsto b(\vartheta;\cdot)$ and the diffusion matrix
$c=\sigma\sigma^{\top}$, see Assumptions 1, 2, 3 and 4 and Proposition 6. The
deep study of the identifiability of $\mathcal{E}^{N}$ and the non-degeneracy
of its information matrix $\mathbb{I}_{\mathcal{E}^{N}}(\vartheta)$ is
simplified via the accompanying experiment $\mathcal{G}^{\otimes N}$, where
$\mathcal{G}$ is generated by the continuous observation of a solution to the
McKean-Vlasov equation

$\left\\{\begin{array}[]{l}dX_{t}=b(\vartheta;t,X_{t},\mu_{t}^{\vartheta})dt+\sigma(t,X_{t})dB_{t},\;t\in[0,T],\\\
\mathcal{L}(X_{0})=\mu_{0},\end{array}\right.$

for a standard Brownian motion $(B_{t})_{t\in[0,T]}$ on $\mathbb{R}^{d}$ and
where $\mu_{t}^{\vartheta}$ is the marginal distribution of the solution at
time $t$. In particular, in the case of representation 4 we have that
$\mathcal{E}^{N}$ and $\mathcal{G}^{\otimes N}$ do not separate asymptotically
by a simple entropy argument, see Proposition 10, and we always have the
convergence of the corresponding Fisher information matrices:

$N^{-1}\mathbb{I}_{\mathcal{E}^{N}}(\vartheta)\rightarrow\mathbb{I}_{\mathcal{G}}(\vartheta)$

in a mean-field limit $N\rightarrow\infty$, as established in Proposition 11.
This approximation is the gateway to obtain explicit identifiability and non-
degeneracy criteria, as detailed in Section 2.4. In particular, under
additional regularity assumptions, we obtain a quite simple criterion for
$\mathbb{I}_{\mathcal{G}}(\vartheta)$ to be non-degenerate in Proposition 15,
namely the property that one of the functions

(5)
$x\mapsto\nabla_{\vartheta}(c^{-1/2}b)^{j}(\vartheta;0,x,\mu_{0})^{\top}z,\;\;\;j=1,\ldots,d$

is not identically vanishing, for every $z\in\mathbb{R}^{p}$ with $|z|=1$,
with $c^{-1/2}$ a square root of $c=\sigma\sigma^{\top}$. We use the notation
$f=(f^{j})_{1\leq j\leq d}$ componentwise, the $f^{j}$ being real-valued
functions. In particular, (5) has the advantage to only relate to the initial
condition $\mu_{0}$ in the measure argument and not the whole
$(\mu_{t}^{\vartheta})_{t\in[0,T]}$ which is (almost) never explicit. Having a
simple criterion to achieve the non-degeneracy of the Fisher information seems
to have been a bit overlooked in the literature (where it is usually simply
assumed to hold true) and our result is thus of interest for other studies.

In Section 3, we state the main results of the paper, Theorem 17, where we
establish the LAN property: if we reparametrise the experiments via
$\vartheta=\vartheta_{0}+N^{-1/2}u$ locally around a fixed point
$\vartheta_{0}$, with $u\in\mathbb{R}^{p}$ being now the unknown parameter,
then both $\mathcal{E}^{N}$ and $\mathcal{G}^{\otimes N}$ look like a Gaussian
shift: we observe

$Y^{N}=u+\mathbb{I}_{\mathcal{G}}(\vartheta_{0})^{-1/2}\xi,$

where $\xi$ is a standard Gaussian random vector in $\mathbb{R}^{p}$. This has
important consequences in terms of existence and properties of optimal
procedures: we have Hájek’s convolution theorem (Corollary 18), namely for any
estimator $\widehat{\vartheta}_{N}$,

(6)
$\liminf_{N\rightarrow\infty}\sup_{|\vartheta^{\prime}-\vartheta|\leq\delta}\mathbb{E}_{\mathbb{P}_{\vartheta^{\prime}}^{N}}\big{[}w\big{(}N^{1/2}\mathbb{I}_{\mathcal{G}}(\vartheta)^{1/2}(\widehat{\vartheta}_{N}-\vartheta^{\prime})\big{)}\big{]}\geq\mathbb{E}[w(\xi)],$

for small enough $\delta>0$, where $\mathbb{P}_{\vartheta^{\prime}}^{N}$ is
the distribution of the data when the parameter is $\vartheta^{\prime}$ and
$w$ is an arbitrary loss function satisfying some regularity properties. The
bound (6) is achieved by the maximum likelihood estimator
$\widehat{\vartheta}_{N}^{\,\tt{mle}}$ obtained by maximising the contrast

(7)
$\vartheta\mapsto\ell^{N}(\vartheta;X^{(N)})=\sum_{i=1}^{N}\int_{0}^{T}\Big{(}(c^{-1}b)(\vartheta;t,X^{i}_{t},\mu^{(N)}_{t})^{\top}dX^{i}_{t}-\tfrac{1}{2}|(c^{-1/2}b)(\vartheta;t,X^{i}_{t},\mu^{(N)}_{t})|^{2}dt\Big{)}.$

This implies in particular the convergence

(8)
$\sqrt{N}\big{(}\widehat{\vartheta}_{N}^{\,\tt{mle}}-\vartheta\big{)}\rightarrow\mathcal{N}\big{(}0,\mathbb{I}_{\mathcal{G}}(\vartheta)^{-1}\big{)}$

in distribution. Moreover, we have in Theorem 19 the minimax asymptotic
optimality of $\widehat{\vartheta}_{N}^{\,\tt{mle}}$, in the sense that

$\mathcal{R}_{w}^{N}(\widehat{\vartheta}_{N}^{\,\tt{mle}};\Theta)=\inf_{\widehat{\vartheta}_{N}}\mathcal{R}_{w}^{N}(\widehat{\vartheta}_{N};\Theta)(1+o(1))$

where
$\mathcal{R}_{w}^{N}(\widehat{\vartheta}_{N};\Theta)=\sup_{\vartheta\in\Theta}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}[w(N^{1/2}\mathbb{I}_{\mathcal{G}}(\vartheta)^{1/2}(\widehat{\vartheta}_{N}-\vartheta))]$
is the classical minimax risk. Thus the LAN property enables us to obtain
considerably stronger results than simply (8). In Section 4, we investigate
several non-trivial examples that generalise the results of [27], and where
our identifiability and non-degeneracy criteria easily apply. We treat in
particular the case of a kinetic mean-field double layer potential that may
serve as a representative model for swarming models, see in particular [6] and
the references therein. The proofs are delayed until Sections 5 and 6, with an
appendix (Section 7) that contains useful technical results.

In practice, maximising the function (7) is not feasible, since only discrete
data are available. It is then reasonable to replace the ideal observation (1)
by the more realistic

$X^{(N,m)}=\big{(}X_{t}^{1},\ldots,X_{t}^{N}\big{)}_{t\in\\{t_{0}^{m},\ldots,t_{m}^{m}\\}},$

where $(0=t_{0}^{m}<t_{1}^{m}<\ldots<t_{m}^{m}=T)$ is a subdivision of $[0,T]$
with mesh

$\max_{1\leq j\leq m}(t_{j}^{m}-t_{j-1}^{m})\leq Cm^{-1}.$

We thus have $(m+1)\times N$ data with values in $\mathbb{R}^{d}$. We may then
replace (7) by

$\displaystyle\vartheta$ $\displaystyle\mapsto
N^{-1}\sum_{i=1}^{N}\sum_{j=0}^{m}\Big{(}(c^{-1}b)(\vartheta;t_{j}^{m},X^{i}_{t_{j-1}^{m}},\mu^{(N)}_{t_{j-1}^{m}})^{\top}(X^{i}_{t_{j}^{m}}-X^{i}_{t_{j-1}^{m}})$
$\displaystyle-\tfrac{1}{2}|(c^{-1/2}b)(\vartheta;t_{j-1}^{m},X^{i}_{t_{j-1}^{m}},\mu^{(N)}_{t_{j-1}^{m}})|^{2}(t_{j}^{m}-t_{j-1}^{m})\Big{)}.$

Assuming the function $(t,x)\mapsto(c^{-1/2}b)(\vartheta;t,x,\mu_{t}^{(N)})$
to be smooth, we may safely expect the discrete approximation to be close to
its continuous counterpart up to an additional error of order $m^{-1/2}$, by
standard high-frequency discretisation techniques, see the textbooks of Jacod
and co-authors [1, 25, 26]. In particular, if $m\gg N$, the same results as
for continuous observations are likely to hold true.

##  2\. Construction and properties of the statistical model

###  2.1. Notation

The dimension $d\geq 1$ of the state space $\mathbb{R}^{d}$ and the dimension
$p\geq 1$ of the parameter space $\Theta$ as well as the time horizon $T>0$
are fixed once for all. We write $|\cdot|$ for the Euclidean distance on
$\mathbb{R}^{q}$ ($q=p,d$ or any other integer, depending on the context) or
for a matrix norm on $\mathbb{R}^{p}\otimes\mathbb{R}^{p}$ fixed throughout.

We consider functions that are mappings defined on products of metric spaces
(typically $\Theta\times[0,T]\times\mathbb{R}^{d}\times\mathcal{P}_{1}$ or
subsets of these) with values in $\mathbb{R}$ or $\mathbb{R}^{d}$. Here,
$\mathcal{P}_{1}$ denotes the set of probability measures on $\mathbb{R}^{d}$
with a first moment, endowed with the Wasserstein $1$-metric

$\mathcal{W}_{1}(\mu,\nu)=\inf_{m\in\Gamma(\mu,\nu)}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\big{|}x-y\big{|}m(dx,dy)=\sup_{|\phi|_{\mathrm{Lip}}\leq
1}\int_{\mathbb{R}^{d}}\phi\,d\big{(}\mu-\nu\big{)},$

where $\Gamma(\mu,\nu)$ denotes the set of probability measures on the product
space $\mathbb{R}^{d}\times\mathbb{R}^{d}$ with marginals $\mu$ and $\nu$. For
a probability measure $\mu$ on $\mathbb{R}^{d}$, we also set

$\mathfrak{m}_{r}(\mu)=\int_{\mathbb{R}^{d}}|y|^{r}\mu(dy)$

for its moment of order $r\geq 1$ and we say that $\mu\in\mathcal{P}_{r}$ if
$\mathfrak{m}_{r}(\mu)$ is finite. All the functions in the paper are
implicitly measurable with respect to the Borel-sigma field induced by the
product topology. A $\mathbb{R}^{d}$-valued function $f$ is written
componentwise as $f=(f^{k})_{1\leq k\leq d}$ where the $f^{k}$ are real-
valued. We denote by $\partial_{\vartheta_{k}}$, $\nabla_{\vartheta}$,
$\partial^{2}_{\vartheta_{k}\vartheta_{l}}$ respectively the partial
derivative of a function with respect to the $k$-th component $\vartheta_{k}$,
the gradient of a real-valued function with respect to $\vartheta$, the second
order partial derivative of a function with respect to the $k$-th and $l$-th
components $\vartheta_{k},\vartheta_{l}$.

Finally, we repeatedly use the notation $C$ for a positive number that does
not depend on $N$, nor $\vartheta$, that may vary from line to line and that
we call a constant, although it usually depends on some other (fixed)
quantities of the model. In most cases, it is explicitly computable.

###  2.2. Model assumptions

#### Well-posedness of the model and its associated statistical experiment

We work under the following strong integrability property for the initial
condition $\mu_{0}$.

######  Assumption 1.

For every $r\geq 1$, we have $\mu_{0}\in\mathcal{P}_{r}$.

As for the diffusion matrix
$\sigma:[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}\otimes\mathbb{R}^{d}$,
we make the following strong ellipticity and Lipschitz smoothness assumption.

######  Assumption 2.

The diffusion matrix $\sigma$ is measurable and for some $C\geq 0$, we have

$|\sigma(t,x^{\prime})-\sigma(t,x)|\leq C|x^{\prime}-x|.$

Moreover, $c=\sigma\sigma^{\top}$ is such that
$\sigma_{-}^{2}|y|^{2}\leq(c(t,x)y)^{\top}y\leq\sigma_{+}^{2}|y|^{2}$ for some
$\sigma_{\pm}>0$.

As for the drift part
$b:\Theta\times[0,T]\times\mathbb{R}^{d}\times\mathcal{P}_{1}\rightarrow\mathbb{R}^{d}$,
we work under usual Lipschitz smoothness assumptions.

######  Assumption 3.

The drift b is measurable and for some $C\geq 0$, we have

$\sup_{t\in[0,T],\vartheta\in\Theta}\big{|}b(\vartheta;t,x^{\prime},\nu^{\prime})-b(\vartheta;t,x,\nu)\big{|}\leq
C\big{(}|x^{\prime}-x|+\mathcal{W}_{1}(\nu^{\prime},\nu)\big{)},$

and there exists some $\vartheta_{0}\in\Theta$ such that

$b_{0}=\sup_{t\in[0,T]}|b(\vartheta_{0};t,0,\delta_{0})|<\infty\;.$

We let $|b|_{\mathrm{Lip}}$ denote the smallest $C\geq 0$ for which Assumption
3 holds.

Assumptions 1, 2, 3 together are sufficient to guarantee the well-posedness of
the statistical model: there exists a unique weak solution to (2) for every
$\vartheta\in\Theta$ hence the data $X^{(N)}$ of (1) is well-defined. More
precisely, we let $\mathcal{C}^{N}=\mathcal{C}([0,T],(\mathbb{R}^{d})^{N})$
denote the space of continuous functions on $(\mathbb{R}^{d})^{N}$, equipped
with the natural filtration $(\mathcal{F}_{t})_{t\in[0,T]}$ induced by the
canonical mappings

$X_{t}^{(N)}(\omega)=\big{(}X^{1}_{t}(\omega),\ldots,X^{N}_{t}(\omega)\big{)}=\omega_{t}.$

For $\mu_{0}\in\mathcal{P}_{1}$ and $\vartheta\in\Theta$, the probability
$\mathbb{P}^{N}_{\vartheta}$ on $(\mathcal{C}^{N},\mathcal{F}^{N})$ under
which the canonical process $X^{(N)}=(X_{t}^{(N)})_{t\in[0,T]}$ is a solution
of (2) for the initial condition $\mu_{0}^{\otimes N}$ is uniquely defined
under Assumptions 1, 2 and 3. Recommended reference (that covers our set of
assumptions) is the textbook by Carmona and Delarue [12] or the lectures notes
of Lacker [28]. Moreover, for every $\vartheta\in\Theta$, the parabolic
nonlinear equation (3) has a unique probability solution
$\mu=(\mu_{t}^{\vartheta})_{t\in[0,T]}$ and we have the weak convergence
$\mu_{t}^{(N)}\rightarrow\mu_{t}^{\vartheta}$ under
$\mathbb{P}_{\vartheta}^{N}$, for every $\vartheta\in\Theta$.

We thus study under Assumptions 1, 2, 3 the (sequence of) statistical
experiment(s) generated by the observation (1) under the dynamics (2) and that
we realise as

$(\mathcal{E}^{N})_{N\geq
1}=\Big{(}\mathcal{C}^{N},\mathcal{F}^{N},\big{(}\mathbb{P}_{\vartheta}^{N},\vartheta\in\Theta\big{)}\Big{)}_{N\geq
1}.$

Note that at that stage, we do not impose any identifiability assumption i.e.
we do not assume that the mapping $\vartheta\mapsto\mathbb{P}_{\vartheta}^{N}$
is one-to-one. We will discuss that matter together with the non-degeneracy of
the model later in Section 2.4.

#### Regularity of the experiment $\mathcal{E}^{N}$

In order to study the regularity of the model, we need specific smoothness
properties for the function $\vartheta\mapsto b(\vartheta,\cdot)$.

######  Assumption 4.

There exist $r_{1},r_{2}\geq 1$ and $C>0$ such that for every point
$\vartheta$ in the interior of $\Theta$, the function $\vartheta\mapsto
b(\vartheta;t,x,\nu)$ is twice differentiable and for every
$1\leq\ell,\ell^{\prime}\leq p$,

$\sup_{t\in[0,T]}(|\partial_{\vartheta_{\ell}}b(\vartheta;t,x,\nu)|+|\partial_{\vartheta_{\ell}\vartheta_{\ell^{\prime}}}^{2}b(\vartheta;t,x,\nu)|)\leq
C(1+|x|^{r_{1}}+\mathfrak{m}_{r_{2}}(\nu)),$
$\sup_{t\in[0,T]}|\partial_{\vartheta_{\ell}}b(\vartheta;t,x^{\prime},\nu^{\prime})-\partial_{\vartheta_{\ell}}b(\vartheta;t,x,\nu)|\leq
C(|x^{\prime}-x|+\mathcal{W}_{1}(\nu^{\prime},\nu)).$

The smoothness properties of the map $\vartheta\mapsto b(\vartheta;\cdot)$
granted by Assumption 4 enables us to explore further the regularity of the
experiment $\mathcal{E}^{N}$. First, note that we have a log-likelihood by
setting

(9)
$\ell^{N}(\vartheta;X^{(N)})=\sum_{i=1}^{N}\int_{0}^{T}(c^{-1}b)(\vartheta;t,X^{i}_{t},\mu^{(N)}_{t})^{\top}dX^{i}_{t}-\frac{1}{2}\sum_{i=1}^{N}\int_{0}^{T}|(c^{-1/2}b)(\vartheta;t,X^{i}_{t},\mu^{(N)}_{t})|^{2}dt,$

where $c^{-1/2}$ is fixed once for all. Indeed, by Girsanov’s theorem again,
the laws $\mathbb{P}_{\vartheta}^{N}$ are all absolutely continuous w.r.t.
$\mathbb{W}^{N}$, defined as the unique probability on
$(\mathcal{C}^{N},\mathcal{F}^{N})$ under which the processes

$\Big{(}\int_{0}^{t}c^{-1/2}(s,X_{s}^{i})dX_{s}^{i}\Big{)}_{t\in[0,T]},\;\;1\leq
i\leq N$

are independent standard Brownian motions on $\mathbb{R}^{d}$, together with
$\mathcal{L}(X_{0}^{1},\ldots,X_{0}^{N})=\mu_{0}^{\otimes N}$. In turn, for
every $\vartheta\in\Theta$,

$\frac{d\mathbb{P}_{\vartheta}^{N}}{d\mathbb{W}^{N}}(X^{(N)})=\exp\big{(}\ell^{N}(\vartheta;X^{(N)})\big{)}$

holds $\mathbb{W}^{N}$-almost-surely. We further write
$\mathcal{L}^{N}(\vartheta;X^{(N)})=\exp\big{(}\ell^{N}(\vartheta;X^{(N)})\big{)}$
for the likelihood process, indexed by the parameter $\vartheta\in\Theta$. We
recall one possible classical definition of a regular statistical experiment,
following [24].

######  Definition 5.

The dominated (sequence of) experiment(s) $(\mathcal{E}^{N})_{N\geq 1}$ is
regular if

  * (i)

$\vartheta\mapsto\mathcal{L}^{N}(\vartheta;X^{(N)})$ is differentiable for
every $\vartheta$ in (the interior of) $\Theta$, $\mathbb{W}^{N}$-almost
surely,

  * (ii)

$\vartheta\mapsto\nabla_{\vartheta}\mathcal{L}^{N}(\vartheta;X^{(N)})$ is
continuous in quadratic $\mathbb{W}^{N}$-mean, for every $\vartheta$ in (the
interior of) $\Theta$,

  * (iii)

we have finite Fisher information

$\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|\nabla_{\vartheta}\ell^{N}(\vartheta;X^{(N)})|^{2}\big{]}<\infty$

for every $\vartheta$ in (the interior of) $\Theta$.

######  Proposition 6.

Under Assumptions 1, 2, 3 and 4 the (sequence of) experiment(s)
$(\mathcal{E}^{N})_{N\geq 1}$ is regular.

###### (Sketch of) Proof.

By exchanging the order of the differentiation with respect to $\vartheta$ and
the stochastic integral we have

$\displaystyle\partial_{\vartheta_{k}}\ell^{N}(\vartheta;X^{(N)})$
$\displaystyle=\sum_{i=1}^{N}\int_{0}^{T}\partial_{\vartheta_{k}}(c^{-1}b)(\vartheta;t,X^{i}_{t},\mu^{(N)}_{t})^{\top}dX^{i}_{t}$
$\displaystyle-\sum_{i=1}^{N}\int_{0}^{T}\partial_{\vartheta_{k}}(c^{-1/2}b)(\vartheta;t,X^{i}_{t},\mu^{(N)}_{t})^{\top}(c^{-1/2}b)(\vartheta;t,X^{i}_{t},\mu^{(N)}_{t})dt.$

We obtain the representation

(10)
$\partial_{\vartheta_{k}}\ell^{N}(\vartheta;X^{(N)})=\sum_{i=1}^{N}\int_{0}^{T}\partial_{\vartheta_{k}}(c^{-1/2}b)(\vartheta;t,X^{i}_{t},\mu^{(N)}_{t})^{\top}dB^{i,N,\vartheta}_{t},$

where the

$(B^{i,N,\vartheta}_{t})_{t\in[0,T]}=\Big{(}\int_{0}^{t}c^{-1/2}(s,X_{s}^{i})(dX_{s}^{i}-b(\vartheta;s,X_{s}^{i},\mu_{s}^{(N)})ds)\Big{)}_{t\in[0,T]},\;\;1\leq
i\leq N$

are independent Brownian motions on $\mathbb{R}^{d}$ under
$\mathbb{P}_{\vartheta}^{N}$. The properties (i), (ii) and (iii) are then a
simple consequence of Assumption 4 together with the following moment bound,

######  Lemma 7.

Under Assumptions 1, 2, 3, for every $r\geq 1$, we have

$\sup_{\vartheta\in\Theta,t\in[0,T],N\geq
1}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}[|X_{t}^{i}|^{r}]<\infty.$

Note that $\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}[|X_{t}^{i}|^{r}]$ does not
depend on $i$. The proof of Lemma 7 is given in Appendix 7.1. ∎

Finally, we have a notion of Fisher information matrix by setting

$\mathbb{I}_{\mathcal{E}^{N}}(\vartheta)=\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\nabla_{\vartheta}\ell^{N}(\vartheta;X^{(N)})\nabla_{\vartheta}\ell^{N}(\vartheta;X^{(N)})^{\top}\big{]}.$

Thanks to (10), we also have

(11)
$\mathbb{I}_{\mathcal{E}^{N}}(\vartheta)=\Big{(}\sum_{i=1}^{N}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\Big{[}\int_{0}^{T}\partial_{\vartheta_{\ell}}(c^{-1/2}b)(\vartheta;t,X^{i}_{t},\mu^{(N)}_{t})\partial_{\vartheta_{\ell^{\prime}}}(c^{-1/2}b)(\vartheta;t,X^{i}_{t},\mu^{(N)}_{t})^{\top}dt\Big{]}\Big{)}_{1\leq\ell,\ell^{\prime}\leq
p}.$

###  2.3. The companion McKean-Vlasov product experiment

We let $\mathcal{C}=\mathcal{C}([0,T],\mathbb{R}^{d})$ denote the space of
continuous functions on $\mathbb{R}^{d}$, equipped with the natural filtration
$(\mathcal{F}_{t})_{0\leq t\leq T}$ induced by the canonical mapping
$X_{t}(\omega)=\omega_{t}$. For every $\vartheta\in\Theta$, we let
$\overline{\mathbb{P}}_{\vartheta}$ denote the unique law under which the
process

$(B_{t}^{\vartheta})_{t\in[0,T]}=\Big{(}\int_{0}^{t}c^{-1/2}(s,X_{s})(dX_{s}-b(\vartheta;s,X_{s},\mu_{s}^{\vartheta})ds)\Big{)}_{t\in[0,T]}$

is a standard Brownian motion on $\mathbb{R}^{d}$, appended with the condition
$\mathcal{L}(X_{0})=\mu_{0}$, and
$\mu^{\vartheta}=(\mu_{t}^{\vartheta})_{t\in[0,T]}$ is a probability solution
of (3). The family $(\overline{\mathbb{P}}_{\vartheta})_{\vartheta\in\Theta}$
is well-defined under Assumptions 1, 2, 3. In particular, the canonical
process $X$ on $(\mathcal{C},\mathcal{F}_{T})$ is a solution to the McKean-
Vlasov equation

(12)
$\left\\{\begin{array}[]{l}dX_{t}=b(\vartheta;t,X_{t},\mu_{t}^{\vartheta})dt+\sigma(t,X_{t})dB_{t}^{\vartheta},\;t\in[0,T],\\\
\mathcal{L}(X_{0})=\mu_{0}.\end{array}\right.$

The following result is the counterpart of Lemma 7. Note in particular that
the marginals of $\overline{\mathbb{P}}_{\vartheta}$ coincide with the
solution $\mu^{\vartheta}=(\mu_{t}^{\vartheta})_{t\in[0,T]}$ of the Fokker-
Planck equation (3).

######  Lemma 8.

Under Assumptions 1, 2, 3, for every $r\geq 1$, we have

$\sup_{\vartheta\in\Theta,t\in[0,T]}\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}}[|X_{t}|^{r}]=\sup_{\vartheta\in\Theta,t\in[0,T]}\int_{\mathbb{R}^{d}}|x|^{r}\mu_{t}^{\vartheta}(dx)<\infty.$

The proof is given in Section 7.2. We also have the following smoothness
property in the parameter $\vartheta$, proof of which is delayed until Section
6.1.

######  Proposition 9.

Under Assumption 1, 2, 3 and 4, the mapping
$\vartheta\mapsto\mu_{t}^{\vartheta}$ is Lipschitz continuous in the
Wasserstein-1 metric $\mathcal{W}_{1}$, uniformly in $t\in[0,T]$.

We next consider the limit experiment

$\mathcal{G}=\big{(}\mathcal{C},\mathcal{F}_{T},(\overline{\mathbb{P}}_{\vartheta})_{\vartheta\in\Theta}\big{)}$

and its $N$-fold counterpart

$\mathcal{G}^{\otimes
N}=\Big{(}\mathcal{C}^{N},\mathcal{F}_{T}^{N},(\overline{\mathbb{P}}_{\vartheta}^{\otimes
N})_{\vartheta\in\Theta}\Big{)}$

that serves as an approximation for the experiment $\mathcal{E}^{N}$. Inspired
by classical propagation of chaos techniques (see in particular [29]), we can
easily show that the measures $\mathbb{P}_{\vartheta}^{N}$ and
$\overline{\mathbb{P}}_{\vartheta}^{\otimes N}$ are indistinguishable when the
drift is of the form

(13)
$b(\vartheta;t,x,\nu)=\int_{\mathbb{R}^{d}}\widetilde{b}(\vartheta;t,x,y)\nu(dy),$

for some kernel
$\widetilde{b}(\vartheta;\cdot):[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$
such that

(14)
$\sup_{t\in[0,T],\vartheta\in\Theta}\big{|}\widetilde{b}(\vartheta;t;x;y)\big{|}\leq
C(1+|x|^{r_{1}}+|y|^{r_{2}})$

for some $r_{1},r_{2}\geq 1$, a situation that covers most of our examples,
see Section 4 below. More precisely, we have the following

######  Proposition 10.

Under Assumptions 1, 2, 3, if $b$ has moreover the form (13)-(14), we have

(15)
$\limsup_{N\rightarrow\infty}\sup_{\vartheta\in\Theta}\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}^{\otimes
N}}\Big{[}\log\frac{d\overline{\mathbb{P}}_{\vartheta}^{\otimes
N}}{d\mathbb{P}_{\vartheta}^{N}}\Big{]}<\infty.$

In particular, if

$\sup_{\vartheta\in\Theta}\int_{0}^{T}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|\widetilde{b}(\vartheta;t,x,y)|^{2}(\mu_{t}^{\vartheta}\otimes\mu_{t}^{\vartheta})(dx,dy)dt<4,$

then

(16)
$\limsup_{N\rightarrow\infty}\sup_{\vartheta\in\Theta}\|\mathbb{P}_{\vartheta}^{N}-\overline{\mathbb{P}}_{\vartheta}^{\otimes
N}\|_{TV}<1,$

where $\|\cdot\|_{TV}$ denotes total variation distance.

The proof is given in Section 6.2. Some remarks are in order: 1) The estimate
(15) tells us that it is impossible to statistically discriminate between
$\mathbb{P}_{\vartheta}^{N}$ and $\overline{\mathbb{P}}_{\vartheta}^{\otimes
N}$ asymptotically. More precisely, inequality (16) shows in particular that
provided $\widetilde{b}$ is not too big or $T$ not too large, then there
exists no test of the null
$H_{0}:\mathbb{P}_{\vartheta}^{N}=\overline{\mathbb{P}}_{\vartheta}^{\otimes
N}$ against the alternative
$H_{1}:\mathbb{P}_{\vartheta}^{N}\neq\overline{\mathbb{P}}_{\vartheta}^{\otimes
N}$ with asymptotically arbitrarily small first and second kind error in the
limit $N\rightarrow\infty$. 2) We will actually prove a stronger result in
Section 3 below, showing that both $(\mathcal{E}^{N})_{N\geq 1}$ and
$(\mathcal{G}^{\otimes N})_{N\geq 1}$ share the LAN property, with same
asymptotic Fisher information. 3) Finally, (15) may hold in wider generality
when the dependence in the measure variable in the drift is nonlinear, as soon
as we have some differentiability in the following sense: there exists
$\partial_{\nu}b(\vartheta;t,x,\cdot):\mathbb{R}^{d}\times\mathcal{P}_{1}\rightarrow\mathbb{R}^{d}$
such that

$b(\vartheta;t,x,\nu)-b(\vartheta;t,x,\nu^{\prime})=\int_{0}^{1}\partial_{\nu}b(\vartheta;t,x,y,\lambda\nu+(1-\lambda)\nu^{\prime})(\nu-\nu^{\prime})(dy)$

for every $\nu,\nu^{\prime}\in\mathcal{P}_{1}$ and
$\partial_{\nu}b(\vartheta;t,x,\cdot)$ satisfies additional smoothness
properties. Iterating the operator $\partial_{\nu}$, if
$\partial_{\nu}^{k}b(\vartheta;t,x,\cdot):(\mathbb{R}^{d})^{k}\times\mathcal{P}_{1}\rightarrow\mathbb{R}^{d}$
exists and satisfies some smoothness and integrability properties, we may
expect (15) to hold as soon as $k\geq d/2$. We refer to Assumption 4 and
Proposition 19 of [16] where this approach is developed.

We also have a log-likelihood in the experiment $\mathcal{G}^{\otimes N}$ by
setting

(17)
$\overline{\ell}^{N}(\vartheta;X^{(N)})=\sum_{i=1}^{N}\int_{0}^{T}(c^{-1}b)(\vartheta;t,X^{i}_{t},\mu^{\vartheta}_{t})^{\top}dX^{i}_{t}-\frac{1}{2}\sum_{i=1}^{N}\int_{0}^{T}|(c^{-1/2}b)(\vartheta;t,X^{i}_{t},\mu^{\vartheta}_{t})|^{2}dt.$

This is the same argument as before: the laws
$\overline{\mathbb{P}}_{\vartheta}^{\otimes N}$ are all absolutely continuous
w.r.t. $\mathbb{W}^{N}$, and for every $\vartheta\in\Theta$,

$\frac{d\overline{\mathbb{P}}_{\vartheta}^{\otimes
N}}{d\mathbb{W}^{N}}(X^{(N)})=\exp\big{(}\overline{\ell}^{N}(\vartheta;X^{(N)})\big{)}$

holds $\mathbb{W}^{N}$-almost-surely.

Finally under Assumptions 1, 2, 3 and 4, the (sequence of) experiment(s)
$\mathcal{G}^{\otimes N}$ is also a regular model and its (normalised) Fisher
information
$\mathbb{I}_{\mathcal{G}}(\vartheta)=N^{-1}\mathbb{I}_{\mathcal{G}^{\otimes
N}}(\vartheta)$ is given by

$\displaystyle
N^{-1}\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}}\big{[}\nabla_{\vartheta}\overline{\ell}^{N}(\vartheta;X^{(N)})\nabla_{\vartheta}\overline{\ell}^{N}(\vartheta;X^{(N)})^{\top}\big{]}$
$\displaystyle=\sum_{j=1}^{d}\int_{0}^{T}\int_{\mathbb{R}^{d}}\nabla_{\vartheta}(c^{-1/2}b)^{j}(\vartheta;t,x,\mu^{\vartheta}_{t})\nabla_{\vartheta}(c^{-1/2}b)^{j}(\vartheta;t,x,\mu^{\vartheta}_{t})^{\top}\mu^{\vartheta}_{t}(dx)dt$
$\displaystyle=\Big{(}\int_{0}^{T}\int_{\mathbb{R}^{d}}\partial_{\vartheta_{\ell}}(c^{-1/2}b)(\vartheta;t,x,\mu^{\vartheta}_{t})\partial_{\vartheta_{\ell^{\prime}}}(c^{-1/2}b)(\vartheta;t,x,\mu^{\vartheta}_{t})^{\top}\mu_{t}^{\vartheta}(dx)dt\Big{)}_{1\leq\ell,\ell^{\prime}\leq
p}.$

Moreover, the mapping $\vartheta\mapsto\mathbb{I}_{\mathcal{G}}(\vartheta)$ is
smooth and appears as the (normalised) asymptotic information of
$\mathcal{E}^{N}$:

######  Proposition 11.

Under Assumptions 1, 2, 3 and 4, the mapping
$\vartheta\mapsto\mathbb{I}_{\mathcal{G}}(\vartheta)$ is Lipschitz continuous.
Moreover, for every $\vartheta$ in (the interior of) $\Theta$, we have

$N^{-1}\mathbb{I}_{\mathcal{E}^{N}}(\vartheta)\rightarrow\mathbb{I}_{\mathcal{G}}(\vartheta)$

as $N\rightarrow\infty$, where $\mathbb{I}_{\mathcal{E}^{N}}(\vartheta)$ is
the Fisher information matrix of the experiment $\mathcal{E}^{N}$ defined in
(11) above.

The proof is given in Section 6.3.

###  2.4. Identifiability and non-degeneracy of the Fisher information

#### Motivation

In the preceding section, we have built $\mathcal{E}^{N}$ and
$\mathcal{G}^{\otimes N}$ (equivalently $\mathcal{G}$) as possibly redundant,
in the sense that the mappings $\vartheta\mapsto\mathbb{P}_{\vartheta}^{N}$
and $\vartheta\mapsto\overline{\mathbb{P}}_{\vartheta}$ are not necessarily
one-to-one on $\Theta$. Having a well-posed parametrisation is required since
we wish to have at least consistent estimators. Arguing asymptotically, we
only need to work in the limit model $\mathcal{G}$.

Also, asymptotic identifiability is somehow linked to the non-degeneracy of
the (normalised) Fisher information matrix $\mathbb{I}_{\mathcal{G}}$.
Following [42], see also [48], we say that a point $\vartheta$ in (the
interior of) $\Theta$ is regular if
$\vartheta^{\prime}\mapsto\mathbb{I}_{\mathcal{G}}(\vartheta^{\prime})$ has
constant rank in a neighbourhood of $\vartheta$ and the experiment
$\mathcal{G}$ is called locally identifiable at $\vartheta$ if the mapping
$\vartheta^{\prime}\mapsto\overline{\mathbb{P}}_{\vartheta^{\prime}}$ is
injective in a neighbourhood of $\vartheta$. We have the following classical
result (that goes back at least to Cramer [15]):

######  Proposition 12 (Theorem 1 in [42]).

If $\vartheta$ is regular, then $\mathcal{G}$ is locally identifiable at
$\vartheta$ if and only if $\mathbb{I}_{\mathcal{G}}(\vartheta)$ has full
rank.

Unfortunately, there is no hope to obtain a global result that links the two
notions unless in very specific cases, see Proposition 16 below. We next give
ad-hoc assumptions that give sufficient (and independent) condition for both
identifiability and non-degeneracy of the Fisher information.

#### An identifiability assumption

We first have a relatively weak assumption that guarantees global
identifiability in $\mathcal{G}$.

######  Assumption 13.

For all $\vartheta\in\Theta$, for $\overline{\mathbb{P}}_{\vartheta}$-almost
all $\omega$, for all $\vartheta^{\prime}\neq\vartheta$, the functions
$t\mapsto b(\vartheta;t,X_{t}(\omega),\mu^{\vartheta}_{t})$ and $t\mapsto
b(\vartheta^{\prime};t,X_{t}(\omega),\mu^{\vartheta^{\prime}}_{t})$ are not
$dt$-a.e. equal.

Assumption 13 is relatively standard in the literature of statistics of random
processes and minimal (see e.g. [19] in a somewhat analogous context). Indeed,
by Girsanov’s theorem, for two different parameters
$\vartheta,\vartheta^{\prime}\in\Theta$, the laws
$\overline{\mathbb{P}}_{\vartheta}$ and
$\overline{\mathbb{P}}_{\vartheta^{\prime}}$ are absolutely continuous and

$\displaystyle\log\frac{d\overline{\mathbb{P}}_{\vartheta}}{d\overline{\mathbb{P}}_{\vartheta^{\prime}}}(X)$
$\displaystyle=\int_{0}^{T}((c^{-1}b)(\vartheta;s,X_{s},\mu^{\vartheta}_{s})-(c^{-1}b)(\vartheta^{\prime};s,X_{s},\mu^{\vartheta}_{s}))^{\top}dX^{i}_{s}$
$\displaystyle-\frac{1}{2}\int_{0}^{T}(|(c^{-1/2}b)(\vartheta;s,X^{i}_{s},\mu^{\vartheta}_{s})|^{2}-|(c^{-1/2}b)(\vartheta^{\prime};s,X^{i}_{s},\mu^{\vartheta}_{s})|^{2})ds.$

Having Assumption 13 fail for some $\vartheta^{\prime}$ implies
$\overline{\mathbb{P}}_{\vartheta}\big{(}\frac{d\overline{\mathbb{P}}_{\vartheta}}{d\overline{\mathbb{P}}_{\vartheta^{\prime}}}(X)=1\big{)}$,
i.e.
$\overline{\mathbb{P}}_{\vartheta}=\overline{\mathbb{P}}_{\vartheta^{\prime}}$.
Assumption 13 may be difficult to check in practice. Yet, it is satisfied as
soon as the mapping $\vartheta\mapsto\big{(}(t,x)\mapsto
b(\vartheta;t,x,\mu_{t}^{\vartheta})\big{)}$ is one-to-one. Also, for certain
form of the likelihood, we have other criteria, see Proposition 16 below.

#### Non-degeneracy of the information

We need some notation. For any $\vartheta,\vartheta^{\prime}\in\Theta$ such
that the segment
$[\vartheta,\vartheta^{\prime}]=\\{\vartheta+\lambda(\vartheta^{\prime}-\vartheta),\lambda\in[0,1]\\}\subset\Theta$
and a function $\phi$ defined on $\Theta$, we set

$\phi([\vartheta,\vartheta^{\prime}])=\int_{0}^{1}\phi(\vartheta+\lambda(\vartheta^{\prime}-\vartheta))d\lambda.$

######  Definition 14.

The statistical experiment $\mathcal{G}$ is non-degenerate if

(18)
$\inf_{[\vartheta,\vartheta^{\prime}]\subset\Theta}\mathsf{det}\,\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}}\big{[}\nabla_{\vartheta}\overline{\ell}^{1}([\vartheta,\vartheta^{\prime}])\nabla_{\vartheta}\overline{\ell}^{1}([\vartheta,\vartheta^{\prime}])^{\top}\big{]}>0,$

where $\mathsf{det}$ denotes the determinant.

Equivalently, we can rewrite (18) as

$\inf\mathsf{det}\,\Big{(}\sum_{j=1}^{d}\;\int_{0}^{T}\int_{\mathbb{R}^{d}}\nabla_{\vartheta}(c^{-1/2}b)^{j}([\vartheta,\vartheta^{\prime}];t,x,\mu^{\vartheta}_{t})\nabla_{\vartheta}(c^{-1/2}b)^{j}([\vartheta,\vartheta^{\prime}];t,x,\mu^{\vartheta}_{t})^{\top}\mu_{t}^{\vartheta}(dx)dt\Big{)}>0,$

where the infimum is taken over all segments
$[\vartheta,\vartheta^{\prime}]\subset\Theta$. Obviously, if $\mathcal{G}$ is
non-degenerate, taking $\vartheta=\vartheta^{\prime}$, Definition 14 boils
down to

(19)
$\inf_{\vartheta\in\Theta}\mathsf{det}\,\mathbb{I}_{\mathcal{G}}(\vartheta)>0$

i.e. $\vartheta\mapsto\mathbb{I}_{\mathcal{G}}(\vartheta)$ has full rank
uniformly in $\vartheta$ and we find back the usual non-degeneracy of the
Fisher information. The somewhat stronger non-degeneracy criterion that we
pick in Definition 14 enables us to check the assumptions of the theory of
Ibragimov and Hasminski for obtaining sharp properties for the maximum
likelihood estimator (see in particular Step 2 of the proof of Theorem 19 in
Section 5.3 below). In explicit examples, proving (18) is no more difficult
than proving (19), see Section 4 below.

#### Checking (18) or (19) in practice

A special difficulty for the statistical analysis of $\mathcal{E}^{N}$ or
rather $\mathcal{G}$ lies in the asymptotic form (12) with the presence of
$(\mu_{t}^{\vartheta})_{0\leq t\leq T}$ in the drift, which is never explicit,
except in very special cases with a specific moment structure in the measure
dependence, see Section 4 below.

It is noteworthy that (18) can usually be tested in a simple way given an
explicit parametrisation. Indeed, Definition 14 is equivalent to show that for
every segment $[\vartheta,\vartheta^{\prime}]\subset\Theta$,

$\inf_{[\vartheta,\vartheta^{\prime}]\subset\Theta}\min_{|z|=1}\sum_{j=1}^{d}\int_{0}^{T}\;\int_{\mathbb{R}^{d}}(\nabla_{\vartheta}\big{(}c^{-1/2}b)^{j}([\vartheta,\vartheta^{\prime}];t,x,\mu^{\vartheta}_{t})^{\top}z\big{)}^{2}\mu^{\vartheta}_{t}(dx)dt>0.$

Under Assumptions 1, 2, 3, we have that
$\mu_{t}^{\vartheta}(dx)=\mu_{t}^{\vartheta}(x)dx$ is absolutely continuous on
$\mathbb{R}^{d}$ for $t>0$, and we may pick a version $\mu_{t}^{\vartheta}$ of
the density that is continuous and positive on $\mathbb{R}^{d}$. This follows
from classical Gaussian tail estimates for the solution of parabolic
equations. We refer for example to Corollary 8.2.2 of [5]. By a simple
continuity argument, it is then sufficient to show that there cannot exist a
segment $[\vartheta,\vartheta^{\prime}]\subset\Theta$ and some $|z|=1$, such
that the function

$x\mapsto\int_{0}^{T}\sum_{j=1}^{d}\big{(}\nabla_{\vartheta}(c^{-1/2}b)^{j}([\vartheta,\vartheta^{\prime}];t,x,\mu^{\vartheta}_{t})^{\top}z\big{)}^{2}dt$

vanishes asymptotically, or, as soon as we have continuity in $t$ as
$t\rightarrow 0$, if one of the functions

$x\mapsto\nabla_{\vartheta}(c^{-1/2}b)^{j}([\vartheta,\vartheta^{\prime}];0,x,\mu_{0})^{\top}z,\;\;j=1,\ldots,d$

does not identically vanishes. This last criterion has the advantage to avoid
the term $\mu_{t}^{\vartheta}$ for $t>0$. We gather these observations in the
following:

######  Proposition 15.

Work under Assumptions 1, 2, 3 and 4. Assume moreover that the functions

$t\mapsto\nabla_{\vartheta}(c^{-1/2}b)^{j}([\vartheta,\vartheta^{\prime}];t,x,\mu^{\vartheta}_{t}),\;\;j=1,\ldots,d$

are all continuous at $t=0$ for every
$[\vartheta,\vartheta^{\prime}]\subset\Theta$ and a.e.-almost
$x\in\mathbb{R}^{d}$.

If, for every $[\vartheta,\vartheta^{\prime}]\subset\Theta$ and any
$z\in\mathbb{R}^{p}$ with $|z|=1$, one of the functions

(20)
$x\mapsto\nabla_{\vartheta}(c^{-1/2}b)^{j}([\vartheta,\vartheta^{\prime}];0,x,\mu_{0})^{\top}z,\;\;\;j=1,\ldots,d$

does not identically vanishes, then $\mathcal{G}$ is non-degenerate in the
sense of Definition 14.

We specifically apply this criterion in the examples Section 4 and check that
the criterion (20) is particularly simple to establish when the dependence in
the measure argument of the function $b$ is of the form (13).

#### A case of equivalence between global identifiability and non-degeneracy
of the information

We revisit Theorem 3 in [42] to obtain the following criterion:

######  Proposition 16.

Work under Assumptions 1, 2, 3 and 4. Assume that the log-likelihood
$\ell^{N}(\vartheta;X^{(N)})$ in $\mathcal{E}^{N}$ defined by (9) has the form

(21)
$\ell^{N}(\vartheta,;X^{N})=\vartheta^{\top}G^{N}(X^{(N)})+\vartheta^{\top}H^{N}(X^{(N)})\vartheta,$

where $G^{N}$ and $H^{N}$ are functions of the trajectory $X^{(N)}$ with
values in $\mathbb{R}^{p}$ and $\mathbb{R}^{p}\otimes\mathbb{R}^{p}$
respectively, and $(H^{N})^{\top}=H^{N}$ is symmetric. If
$\Theta_{0}\subset\Theta$ is a convex set such that
$\mathbb{I}_{\mathcal{G}}(\vartheta)$ is non-singular for every
$\vartheta\in\Theta_{0}$, then, both $(\mathcal{E}^{N})_{N\geq 1}$ and
$\mathcal{G}$ are identifiable on $\Theta_{0}$.

By identifiability of the sequence of experiment $(\mathcal{E}^{N})_{N\geq
1}$, we mean injectivity of the mapping
$\vartheta\mapsto(\mathbb{P}_{\vartheta}^{N})_{N\geq 1}$ (i.e. simultaneously
for every $N\geq 1$). The proof is given in Section 6.4. In the specific case
of McKean type models that date back to [38, 44, 46] and widely used in
practice (see e.g. [12, 17] or [27] in statistics), we have in some instances
a representation like (21) and explicit formulas for
$\mathbb{I}_{\mathcal{G}}(\vartheta)$, which gives global identifiability for
free as soon as $\mathbb{I}_{\mathcal{G}}(\vartheta)$ is non-degenerate. See
the examples in Section 4.

##  3\. Main results

###  3.1. The LAN property

The local asymptotic normality property of a statistical model characterises
its regularity: it expresses the fact that the experiment locally resembles a
Gaussian shift in an optimal scale driven by the Fisher information. It has
powerful consequences in terms of properties of optimal procedures via the
celebrated Hájek convolution theorem [23]. More precisely the sequence of
experiments $(\mathcal{E}_{N})_{N\geq 1}$ satisfies the LAN property at
$\vartheta\in\Theta$ with information rate
$N\mathbb{I}_{\mathcal{G}}(\vartheta)$ if

(22)
$\log\frac{d\mathbb{P}_{\vartheta+(N\mathbb{I}_{\mathcal{G}}(\vartheta))^{-1/2}u}^{N}}{d\mathbb{P}_{\vartheta}^{N}}=u^{\top}\xi_{\vartheta}^{N}-\tfrac{1}{2}|u|^{2}+r_{N}(\vartheta,u),$

where $\xi_{\vartheta}^{N}$ converges in distribution under
$\mathbb{P}_{\vartheta}^{N}$ to standard Gaussian variable in $\mathbb{R}^{p}$
and $r_{N}(\vartheta,u)\rightarrow 0$ in
$\mathbb{P}_{\vartheta}^{N}$-probability. Of course, the convergence (22) is
meaningful only if
$\vartheta+(N\mathbb{I}_{\mathcal{G}}(\vartheta))^{-1/2}u\in\Theta$ and is
well-defined, i.e. if $\mathsf{det}\,\mathbb{I}_{\mathcal{G}}(\vartheta)>0$.
This is granted for instance for $\vartheta$ in the interior of $\Theta$ for
large enough $N$ and under (19).

######  Theorem 17.

Work under Assumptions 1, 2, 3, 4 and 13. Assume moreover that $\mathcal{G}$
is non-degenerate according to Definition 14. For every $\vartheta$ in (the
interior of) $\Theta$, the sequence of experiments $(\mathcal{E}^{N})_{N\geq
1}$ is locally asymptotically normal at $\vartheta$ with information rate
$N\mathbb{I}_{\mathcal{G}}(\vartheta)$.

The same result holds for $(\mathcal{G}^{\otimes N})_{N\geq 1}$.

Several remarks are in order: 1) Theorem 17 is the most powerful result one
can obtain about the structure of $(\mathcal{E}^{N})_{N\geq 1}$ and
$(\mathcal{G}^{\otimes N})_{N\geq 1}$: it tells us that around a given point
$\vartheta_{0}$, if we parametrise locally the experiment via
$\vartheta=\vartheta_{0}+N^{-1/2}u$ with $u\in\mathbb{R}^{p}$ being the
unknown parameter, then the experiments look like the simplest possible
experiment, namely a Gaussian shift

$Y^{N}=u+\mathbb{I}_{\mathcal{G}}(\vartheta_{0})^{-1/2}\xi+o(1)$

where $\xi$ is a standard normal $\mathcal{N}(0,\mathrm{Id}_{\mathbb{R}^{p}})$
and $o(1)$ is a small term that vanishes in $\mathbb{P}_{\vartheta}^{N}$ or
$\overline{\mathbb{P}}_{\vartheta}^{\otimes N}$ probability, locally uniformly
in $u$. 2) The fact that both $(\mathcal{E}^{N})_{N\geq 1}$ and
$(\mathcal{G}^{\otimes N})_{N\geq 1}$ share the LAN property with same
asymptotic Fisher variance quantifies their asymptotic similarity, see in
particular Proposition 10. 3) The LAN property has several consequences in
terms of strong properties of the maximum likelihood estimator, see Theorem 19
below. In particular, the first simple consequence is given in terms of exact
asymptotic minimax lower bounds: call a centrally symmetric function
$w:\mathbb{R}^{p}\rightarrow[0,\infty)$ such that the sets $\\{w<c\\},c>0$ are
all convex a polynomial loss function if it admits a polynomial majorant.

######  Corollary 18.

In the setting of Theorem 17, let $w$ be a polynomial loss function. Then, for
any estimator $\widehat{\vartheta}_{N}$ in $\mathcal{E}^{N}$ and any
sufficiently small $\delta>0$, for every $\vartheta$ in (the interior of)
$\Theta$ for which $\mathsf{det}\,\mathbb{I}_{\mathcal{G}}(\vartheta)>0$, we
have

$\liminf_{N\rightarrow\infty}\sup_{|\vartheta^{\prime}-\vartheta|\leq\delta}\mathbb{E}_{\mathbb{P}_{\vartheta^{\prime}}^{N}}\big{[}w\big{(}N^{1/2}\mathbb{I}_{\mathcal{G}}(\vartheta)^{1/2}(\widehat{\vartheta}_{N}-\vartheta^{\prime})\big{)}\big{]}\geq(2\pi)^{-p/2}\int_{\mathbb{R}^{p}}w(x)\exp(-\tfrac{1}{2}|x|^{2})dx.$

The same result holds true for $\widehat{\vartheta}_{N}$ in
$\mathcal{G}^{\otimes N}$ replacing $\mathbb{P}_{\vartheta}^{N}$ by
$\overline{\mathbb{P}}_{\vartheta}^{\otimes N}$.

Corollary 18 is a simple application of Hájek convolution theorem, given the
LAN property of Theorem 17, see e.g. Theorem II.12.1 (an in particular Remark
III.12.1) in [24]. It provide with a sharp local asymptotically minimax bound,
up to constants. We shall see below that the maximum likelihood estimator
achieves this bound.

###  3.2. Maximum likelihood estimation and properties

We elaborate on the properties of the maximum likelihood estimator by relying
on (a uniform version of) the LAN property of Theorem 17. It implies several
fine results that go beyond the usual asymptotic weak expansions given by an
ad-hoc study of the form of the estimator, as is usually the case in the
literature.

######  Theorem 19.

Work under Assumptions 1, 2, 3, 4 and 13. Then, for large enough $N$, the
solution $\widehat{\vartheta}_{N}^{\,\tt{mle}}$ to

(23)
$\mathcal{L}^{N}(\widehat{\vartheta}_{N}^{\,\tt{mle}};X^{(N)})=\sup_{\vartheta\in\Theta}\mathcal{L}^{N}(\vartheta;X^{(N)})$

is well-defined. Moreover, the following asymptotic upper bounds are valid:

  1. (i)

if $\mathcal{G}$ is non-degenerate in the sense of Definition 14,

$\sqrt{N}\big{(}\widehat{\vartheta}_{N}^{\,\tt{mle}}-\vartheta\big{)}\rightarrow\mathcal{N}\big{(}0,\mathbb{I}_{\mathcal{G}}(\vartheta)^{-1}\big{)}$

in $\mathbb{P}_{\vartheta}^{N}$-distribution as $N\rightarrow\infty$.

  2. (ii)

For every polynomial loss function $w$ and any $\vartheta$ in the interior of
$\Theta$, we have exact local asymptotic minimax optimality:

$\limsup_{N\rightarrow\infty}\sup_{|\vartheta^{\prime}-\vartheta|\leq\delta}\mathbb{E}_{\mathbb{\mathbb{P}}_{\vartheta^{\prime}}^{N}}\big{[}w\big{(}N^{1/2}\mathbb{I}_{\mathcal{G}}(\vartheta)^{1/2}\big{(}\widehat{\vartheta}_{N}^{\,\tt{mle}}-\vartheta^{\prime}\big{)}\big{)}\big{]}\rightarrow(2\pi)^{-p/2}\int_{\mathbb{R}^{p}}w(x)\exp(-\tfrac{1}{2}|x|^{2})dx$

as $\delta\rightarrow 0$.

  3. (iii)

For every polynomial loss function $w$ and any (non empty) open set
$\Theta_{0}\subset\Theta$, we have global asymptotic minimax optimality:

$\mathcal{R}_{w}^{N}(\widehat{\vartheta}_{N}^{\,\tt{mle}};\Theta_{0})=\inf_{\widehat{\vartheta}_{N}}\mathcal{R}_{w}^{N}(\widehat{\vartheta}_{N};\Theta_{0})(1+o(1))$

as $N\rightarrow\infty$, where

$\mathcal{R}_{w}^{N}(\widehat{\vartheta}_{N};\Theta_{0})=\sup_{\vartheta\in\Theta_{0}}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}w\big{(}N^{1/2}\mathbb{I}_{\mathcal{G}}(\vartheta)^{1/2}(\widehat{\vartheta}_{N}-\vartheta)\big{)}\big{]}.$

Some further remarks: 1) We find back the classical asymptotic properties (i)
of the maximum likelihood estimator that are given in the literature, but the
result is appended by a much stronger convergence in (ii), that matches in
particular the lower bound of Corollary 18. 2) We finally obtain global
asymptotic minimax optimality by (iii), which is the parametric analog (in a
much more precise way) of our minimax results of Section 4 in [16] in the
nonparametric case.

##  4\. Examples

In this section, we elaborate on specific examples that appear in the
literature and in applications. We first revisit the linear McKean model
studied at length in [27]. We slightly extend in Section 4.1 his example (1.3)
from $p=2$ to $p=3$. In Section 4.2, we develop an example of a generalised
linear form and show in particular how our identifiability and non-degeneracy
criteria of Section 2.4 are easily implementable and avoid to use the
machinery of [27]. In Section 4.3, we develop a non-trivial example of kinetik
mean-field model with a double layer potential that may serve in many
applications, like swarming models or more general individual based-models,
see [6] and the references therein. We finally develop a genuinely non-linear
example, i.e. when the measure argument is not linear like in (4), as for
instance in the examples of [41]. Assumption 1 is in force throughout.

###  4.1. McKean-like models

In many applications, (2) takes the explicit form

(24)
$dX_{t}^{i}=(\vartheta_{1}X_{t}^{i}+\vartheta_{2})dt-\vartheta_{3}N^{-1}\sum_{j=1}^{N}(X_{t}^{i}-X_{t}^{j})dt+dB_{t}^{i},\;\;i=1,\ldots,N$

with $X_{t}^{i}\in\mathbb{R}$. The parameter is
$\vartheta=(\vartheta_{1}\;\vartheta_{2}\;\vartheta_{3})^{\top}$. In [27] the
case $\vartheta_{2}=0$ is studied at length in particular. In our setting, we
can encompass a more general situation with $X_{t}^{i}\in\mathbb{R}^{d}$ for
some arbitrary $d\geq 1$ and replace $\vartheta_{3}$ by a parameter in
$\mathbb{R}^{d}\otimes\mathbb{R}^{d}$ as well as $\vartheta_{2}$ by a
parameter in $\mathbb{R}^{d}$. In this case, Assumptions 2, 3 and 4 are
readily checked. Likewise, the identifiability and non-degeneracy assumptions
can be obtained with some extra care on the initial condition. We elaborate on
a specific case below.

#### Likelihood equations

To keep-up with notational simplicity, we detail the case $p=3$ with
$\vartheta=(\vartheta_{1}\;\vartheta_{2}\;\vartheta_{3})^{\top}\in\Theta$ as a
compact subset of $\mathbb{R}^{3}$ for an ambient dimension $d=1$, with
$\vartheta_{1}\neq\vartheta_{3}$ and $\vartheta_{1}\neq 0$. Introduce

$\mathsf{A}_{t}^{N}(x)=\left(\begin{array}[]{ccc}x^{2}&x&-\langle\cdot-x,\mu_{t}^{(N)}\rangle^{2}\\\
x&1&0\\\
-\langle\cdot-x,\mu_{t}^{(N)}\rangle^{2}&0&\langle\cdot-x,\mu_{t}^{(N)}\rangle^{2}\\\
\end{array}\right),\;\;\mathsf{B}_{t}^{N}(x)=\left(\begin{array}[]{c}x\\\ 1\\\
\langle\cdot-x,\mu_{t}^{(N)}\rangle\end{array}\right),$

where we use the bracket notation $\langle\cdot,\nu\rangle$ to denote
integration w.r.t. the measure $\nu$. Define

(25)
$\mathsf{A}_{T}^{N}=\int_{0}^{T}\langle\mathsf{A}_{t}^{N}(x),\mu_{t}^{(N)}\rangle
dt$

and

(26)
$\mathsf{B}_{T}^{N}=N^{-1}\sum_{i=1}^{N}\int_{0}^{T}\mathsf{B}_{t}^{N}(X_{t}^{i})dX_{t}^{i}.$

Thanks to the linearity in $\vartheta$ of the drift
$b(\vartheta;t,x,\nu)=\vartheta_{1}x+\vartheta_{2}-\vartheta_{3}\int_{\mathbb{R}}(x-y)\nu(dy)$,
the likelihood equations are explicit and the maximum likelihood estimator
$\widehat{\vartheta}_{N}^{\,\tt{mle}}$ solves

(27)
$\mathsf{A}_{T}^{N}\widehat{\vartheta}_{N}^{\,\tt{mle}}\,=\mathsf{B}_{T}^{N}.$

Moreover, the Fisher information matrix is given by

$\mathbb{I}_{\mathcal{G}}(\vartheta)=\int_{0}^{T}\langle\mathsf{A}_{t}(\vartheta;x),\mu_{t}^{\vartheta}\rangle
dt,$

with

$\mathsf{A}_{t}(\vartheta;x)=\left(\begin{array}[]{ccc}x^{2}&x&-\langle\cdot-x,\mu_{t}^{\vartheta}\rangle^{2}\\\
x&1&0\\\
-\langle\cdot-x,\mu_{t}^{\vartheta}\rangle^{2}&0&\langle\cdot-x,\mu_{t}^{\vartheta}\rangle^{2}\end{array}\right).$

#### Non-degeneracy and identifiability

The property $\mathsf{det}\,\mathbb{I}_{\mathcal{G}}(\vartheta)>0$ can be
verified on the explicit form of its matrix:

$\displaystyle\mathsf{det}\,\mathbb{I}_{\mathcal{G}}(\vartheta)$
$\displaystyle=\int_{0}^{T}\mathsf{Var}(\mu_{t}^{\vartheta})dt\,\Big{(}T\int_{0}^{T}\overline{\mathfrak{m}}_{1}(\mu_{t}^{\vartheta})^{2}dt-\big{(}\int_{0}^{T}\overline{\mathfrak{m}}_{1}(\mu_{t}^{\vartheta})dt\big{)}^{2}\Big{)}$

with
$\mathsf{Var}(\nu)=\int_{\mathbb{R}}(x-\overline{\mathfrak{m}}_{1}(\nu))^{2}\nu(dx)$
and $\overline{\mathfrak{m}}_{1}(\nu)=\int_{\mathbb{R}^{d}}x\nu(dx)$.
Therefore $\mathsf{det}\,\mathbb{I}_{\mathcal{G}}(\vartheta)>0$ unless
$\mu_{t}^{\vartheta}$ is degenerate for all $t$ or stationary. In the case of
a linear equation of the type (24), the accompanying limiting measure
$\mu_{t}^{\vartheta}$ associated to the McKean-Vlasov equation

$dX_{t}=(\vartheta_{1}X_{t}+\vartheta_{2})dt-\vartheta_{3}(X_{t}-\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}}[X_{t}])dt+dB_{t}$

is a Gaussian process that can be thought of as an inhomogeneous Ornstein-
Uhlenbeck model for which we have closed-form moment formulas:

(28)
$\overline{\mathfrak{m}}_{1}(\mu_{t}^{\vartheta})=-\vartheta_{1}^{-1}\vartheta_{2}+(\overline{\mathfrak{m}}_{1}(\mu_{0})+\vartheta_{1}^{-1}\vartheta_{2})\exp(\vartheta_{1}t)$

and

(29)
$\mathfrak{m}_{2}(\mu_{t}^{\vartheta})=\exp\big{(}-2(\vartheta_{1}-\vartheta_{3})t\big{)}\mathrm{Var}(\mu_{0})+\frac{1-\exp(-2(\vartheta_{1}-\vartheta_{3})t)}{2(\vartheta_{1}-\vartheta_{3})}+\overline{\mathfrak{m}}_{1}(\mu_{t}^{\vartheta})^{2}.$

In particular, having

(30) $\overline{\mathfrak{m}}_{1}(\mu_{0})+\vartheta_{1}^{-1}\vartheta_{2}\neq
0,\;\;\vartheta_{1}\neq 0,\;\;\vartheta_{1}\neq\vartheta_{3}$

yields the non-degeneracy of $\mathbb{I}_{\mathcal{G}}(\vartheta)$ as well as
the non-degeneracy in the sense of Definition 14, since
$\nabla_{\vartheta}b(\vartheta;t,x,\nu)$ does not depend on $\vartheta$. Also,
the convergence
$\mathsf{A}_{T}^{N}\rightarrow\mathbb{I}_{\mathcal{G}}(\vartheta)$ in
$\mathbb{P}_{\vartheta}^{N}$-probability as $N\rightarrow\infty$ tells us that
(33) has a well defined and unique solution with $\mathbb{P}_{\vartheta}^{N}$
probability tending to one as $N\rightarrow\infty$. This last statement can be
quantified via the convergence Lemma 21 below. Writing
$\phi(x,\nu)=(x\;\;1\;\;-\int_{\mathbb{R}}(x-y)\nu(dy))$, the log-likelihood

(31)
$\ell^{N}(\vartheta;X^{(N)})=\sum_{i=1}^{N}\int_{0}^{T}\vartheta^{\top}\phi(X_{t}^{i},\mu_{t}^{(N)})dX_{t}^{i}-\frac{1}{2}\sum_{i=1}^{N}\int_{0}^{T}\big{(}\vartheta^{\top}\phi(X_{t}^{i},\mu_{t}^{(N)})\big{)}^{2}dt$

has representation (21) with

$G^{N}(X^{(N)})=\sum_{i=1}^{N}\int_{0}^{T}\phi(X_{t}^{i},\mu_{t}^{(N)})dX_{t}^{i}\;\;\text{and}\;\;H^{N}(X^{(N)})=-\frac{1}{2}\sum_{i=1}^{N}\int_{0}^{T}\phi(X_{t}^{i},\mu_{t}^{(N)})\phi(X_{t}^{i},\mu_{t}^{(N)})^{\top}dt$

and we obtain the identifiability of $\mathcal{E}^{N}$ and $\mathcal{G}$ over
compact parameter sets $\Theta\subset\mathbb{R}^{3}$ that are moreover convex
and satisfy the constraint (30) by Proposition 16. Finally, explicit formulas
for $\mathbb{I}_{\mathcal{G}}(\vartheta)$ and its inverse can be derived
thanks to (28) and (29).

###  4.2. Generalised linear like models

We push further the preceding linear structure by considering the following
model

$dX_{t}^{i}=\vartheta_{1}f(X_{t}^{i})dt+\vartheta_{2}N^{-1}\sum_{j=1}^{N}g(X_{t}^{i}-X_{t}^{j})dt+dB_{t}^{i},\;\;1\leq
i\leq N$

where $f$ and $g$ are known and Lipschitz continuous real-valued functions and
$X_{t}^{i}\in\mathbb{R}$ for simplicity. The parameter is
$\vartheta=(\vartheta_{1},\vartheta_{2})\in\mathbb{R}^{2}$. Thus $p=2$ which
again yields simple and explicit formulas. Assumptions 2, 3 and 4 are readily
checked. Writing $\phi(x,\nu)=(f(x)\;g\star\nu(x))^{\top}$, with
$g\star\nu(x)=\int_{\mathbb{R}}g(x-y)\nu(dy)$ the log-likelihood function
$\ell^{N}(\vartheta;X^{(N)})$ has again representation (31) hence (21) holds
as well and we obtain identifiability of $\mathcal{E}^{N}$ and $\mathcal{G}$
over compact parameter sets $\Theta\subset\mathbb{R}^{2}$ that are moreover
convex as soon as $\mathsf{det}\,\mathbb{I}_{\mathcal{G}}(\vartheta)>0$ by
Proposition 16.

#### Non-degeneracy

For proving non-degeneracy (that implies in particular
$\mathsf{det}\,\mathbb{I}_{\mathcal{G}}(\vartheta)>0$), we plan to apply
Proposition 15. We first notice that
$\nabla_{\vartheta}b(\vartheta;t,x,\nu)=\phi(x,\nu)$ does not depend on
$\vartheta$, hence
$\nabla_{\vartheta}b([\vartheta,\vartheta^{\prime}];t,x,\nu)=(f(x)\;g\star\nu(x))^{\top}$
does not depend on the segment $[\vartheta,\vartheta^{\prime}]$ either. The
continuity of $t\mapsto(f(x)\;g\star\mu_{t}^{\vartheta}(x))^{\top}$ follows
from the representation

$(f(x)\;g\star\mu_{t}^{\vartheta}(x))^{\top}=\big{(}f(x)\;\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}}[g(X_{t})]\big{)}^{\top},$

the fact that $t\mapsto X_{t}$ is continuous in probability at $t=0$ under
$\overline{\mathbb{P}}_{\vartheta}$ and Lebesgue dominated convergence. Now,
let $z=(z_{1},z_{2})$ with $z_{1}^{2}+z_{2}^{2}=1$. In order to obtain non-
degeneracy, it is sufficient by Proposition 15 to show that the function

$x\mapsto f(x)z_{1}+g\star\mu_{0}(x)z_{2}$

is non identically zero. If $f$ does not identically vanishes, we may assume
that $z_{2}\neq 0$. Then it is sufficient to have that

(32) $x\mapsto\lambda f(x)+g\star\mu_{0}(x)\;\;\text{does not vanish
identically for every}\;\;\lambda\neq 0.$

It is then very easy to build families of functions and initial condition
$(f,g,\mu_{0})$ such that (32) is satisfied. For instance, if
$\mu_{0}=\delta_{x_{0}}$ for some arbitrary $x_{0}$, then having $f$ non-
identically equal to a constant is sufficient.

#### Likelihood equations

Finally, we explicitly solve the likelihood equations. Again, they are of a
simple form, and the maximum likelihood estimator
$\widehat{\vartheta}_{N}^{\,\tt{mle}}$ solves

(33)
$\mathsf{A}_{T}^{N}\widehat{\vartheta}_{N}^{\,\tt{mle}}\,=\mathsf{B}_{T}^{N},$

where $\mathsf{A}_{T}^{N}$ and $\mathsf{B}_{T}^{N}$ are defined via (25) and
(26), with

$\mathsf{A}_{t}^{N}(x)=\left(\begin{array}[]{cc}f(x)^{2}&f(x)\langle
g(x-\cdot),\mu_{t}^{(N)}\rangle\\\ f(x)\langle
g(x-\cdot),\mu_{t}^{(N)}\rangle&\langle
g(x-\cdot),\mu_{t}^{(N)}\rangle^{2}\end{array}\right),\;\;\mathsf{B}_{t}^{N}(x)=\left(\begin{array}[]{c}f(x)\\\
\langle g(x-\cdot),\mu_{t}^{(N)}\rangle\end{array}\right).$

Again, we have

$\mathbb{I}_{\mathcal{G}}(\vartheta)=\left(\begin{array}[]{cc}\int_{0}^{T}\langle
f^{2},\mu_{t}^{\vartheta}\rangle dt&\int_{0}^{T}\langle
f(g\star\mu_{t}^{\vartheta}),\mu_{t}^{\vartheta}\rangle dt\\\
\int_{0}^{T}\langle f(g\star\mu_{t}^{\vartheta}),\mu_{t}^{\vartheta}\rangle
dt&\int_{0}^{T}\langle(g\star\mu_{t}^{\vartheta})^{2},\mu_{t}^{\vartheta}\rangle
dt\end{array}\right),$

by taking the limit in $\mathbb{P}_{\vartheta}^{N}$-probability of
$\mathsf{A}_{T}^{N}=\int_{0}^{T}\langle\mathsf{A}_{t}^{N}(x),\mu_{t}^{(N)}\rangle
dt$. We also know that $\mathsf{det}\,\mathbb{I}_{\mathcal{G}}(\vartheta)>0$
by the non-degeneracy property established above thanks to Proposition 15.
Hence $\mathsf{A}_{T}^{N}$ is invertible with
$\mathbb{P}_{\vartheta}^{N}$-probability that goes to one as
$N\rightarrow\infty$ and $\widehat{\vartheta}_{N}^{\,\tt{mle}}$ is
asymptotically well defined.

###  4.3. A double layer potential model

We depart from the structure (21) of the likelihood as in the two preceding
linear-like models and study the model

$dX_{t}^{i}=N^{-1}\sum_{j=1}^{N}\nabla
U_{\vartheta}(X_{t}^{i}-X_{t}^{j})dt+dB_{t}^{i},\;\;1\leq i\leq N,$

with ambient state space $\mathbb{R}^{d}$ for $d\geq 1$ and where
$U_{\vartheta}:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is a family of pairwise
potentials of the form

$U_{\vartheta}(x)=-\vartheta_{1}\exp(-\vartheta_{2}|x|^{2})+\vartheta_{3}\exp(-\vartheta_{4}|x|^{2})$

modelling short range repulsion and long range attraction, where
$\vartheta_{1},\vartheta_{3}$ and $\vartheta_{2},\vartheta_{4}$ are
respectively the strengths and the lengths of attraction and repulsion. The
parameter is
$\vartheta=(\vartheta_{1},\ldots,\vartheta_{4})\in(0,\infty)^{4}$. As minimal
identifiability condition, we impose $\vartheta_{2}\neq\vartheta_{4}$. Such
models are commonly used (in their kinetic version) in swarming modelling, see
e.g. [6].

We have $b(\vartheta;t,x,\nu)=\nabla U_{\vartheta}\star\nu(x)$ and it is
readily verified that Assumptions 2, 3 and 4 are met. Here, there is no hope
to explicitly solve the likelihood equations, and a numerical scheme has to be
implemented. We further investigate the identifiability and non-degeneracy of
the model. Note that the assumptions on the drift and the initial condition
ensure that for every $\vartheta\in\Theta$, $\mu_{t}^{\vartheta}$ is
absolutely continuous with a nowhere vanishing density for $t>0$ (we refer to
[5] and in particular Corollary 8.2.2).

#### Identifiability

We study the injectivity of the mapping
$\vartheta\mapsto\overline{\mathbb{P}}_{\vartheta}$ via the injectivity of
$\vartheta\mapsto\big{(}(t,x)\mapsto\nabla
U_{\vartheta}\star\mu_{t}^{\vartheta})\big{)}$, see Assumption 13. If
$\vartheta,\vartheta^{\prime}$ are such that
$\overline{\mathbb{P}}_{\vartheta}=\overline{\mathbb{P}}_{\vartheta^{\prime}}$,
this also implies $\mu_{t}^{\vartheta}=\mu_{t}^{\vartheta^{\prime}}$ for every
$t\in[0,T]$. Hence the condition-

$\nabla U_{\vartheta}\star\mu_{t}^{\vartheta}=\nabla
U_{\vartheta^{\prime}}\star\mu_{t}^{\vartheta^{\prime}}$

for almost every $t\in[0,T]$ becomes

(34) $\nabla U_{\vartheta}\star\mu_{t}^{\vartheta}=\nabla
U_{\vartheta^{\prime}}\star\mu_{t}^{\vartheta}$

for almost every $t\in[0,T]$. Let
$\mathcal{F}(\nu)(\xi)=\int_{\mathbb{R}^{d}}e^{ix^{\top}\xi}\,\nu(dx)$ denote
a Fourier transform of $\nu$ (well defined if $\nu$ is a probability measure
or an integrable function). Since $\xi\mapsto\mathcal{F}(\mu_{t})(\xi)$ is
continuous and $\mathcal{F}(\mu_{t})(\xi)=1$ (the $\mu_{t}$ are all
probability measures on $\mathbb{R}^{d}$), there are infinitely many points
$\xi\in\mathbb{R}^{d}$ such that $\mathcal{F}(\mu_{t})(\xi)\neq 0$ . Applying
$\mathcal{F}$ on both side of (34), we obtain

$\mathcal{F}(\nabla U_{\vartheta})(\xi)=\mathcal{F}(\nabla
U_{\vartheta^{\prime}})(\xi)$

for such points $\xi$. Moreover

(35) $\nabla
U_{\vartheta}(x)=2\vartheta_{1}\vartheta_{2}x\exp(-\vartheta_{2}|x|^{2})-2\vartheta_{3}\vartheta_{4}x\exp(-\vartheta_{4}|x|^{2}),$

and

$\mathcal{F}(\nabla
U_{\vartheta})(\xi)=i\xi\pi^{d/2}\big{(}\vartheta_{1}\vartheta_{2}^{-d/2}\exp(-\tfrac{1}{4\vartheta_{2}}|\xi|^{2})-\vartheta_{3}\vartheta_{4}^{-d/2}\exp(-\tfrac{1}{4\vartheta_{4}}|\xi|^{2})\big{)}.$

It follows that the condition $\vartheta_{2}\neq\vartheta_{4}$ is sufficient
to achieve identifiability, i.e. $\vartheta=\vartheta^{\prime}$. Henceforth,
we may parametrise our model via any compact $\Theta\subset(0,\infty)^{4}$
such that $\vartheta_{2}\neq\vartheta_{4}$.

#### Non-degeneracy

We plan to apply Proposition 15. From (35), we have

$\nabla_{\vartheta}b(\vartheta;t,x,\nu)^{j}=\nabla_{\vartheta}(\nabla
U_{\vartheta}\star\nu(x))^{j}=G(\vartheta;x)^{j}\star\nu(x),$

with

$G(\vartheta;x)^{j}=\left(\begin{array}[]{c}2\vartheta_{2}x_{j}\exp(-\vartheta_{2}|x|^{2})\\\
2\vartheta_{1}x_{j}(1-\vartheta_{2}|x|^{2})\exp(-\vartheta_{2}|x|^{2})\\\
-2\vartheta_{4}x_{j}\exp(-\vartheta_{4}|x|^{2})\\\
-2\vartheta_{3}x_{j}(1-\vartheta_{4}|x|^{2})\exp(-\vartheta_{4}|x|^{2})\end{array}\right),\;\;j=1,\ldots,d$

The mapping

$t\mapsto\nabla_{\vartheta}b([\vartheta,\vartheta^{\prime}];t,x,\mu_{t}^{\vartheta})^{j}=\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}}\big{[}G([\vartheta,\vartheta^{\prime}];x-X_{t})^{j}\big{]}$

is continuous at $t=0$, as a simple consequence of the fact that $t\mapsto
X_{t}$ is continuous in $\overline{\mathbb{P}}_{\vartheta}$-probability at
$t=0$. It is then sufficient to prove that for any $z\in\mathbb{R}^{4}$ with
$z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2}=1$, one of the functions

$x\mapsto(G([\vartheta,\vartheta^{\prime}];\cdot)^{j}\star\mu_{0}(x))^{\top}z,\;\;j=1,\ldots,d.$

does not vanish identically. Assume on the contrary that
$(G([\vartheta,\vartheta^{\prime}];\cdot)^{j}\star\mu_{0})^{\top}z$ is
identically $0$ for every $1\leq j\leq d$. Then this is also the case for
$\mathcal{F}\big{(}(G([\vartheta,\vartheta^{\prime}];\cdot)^{j}\star\mu_{0})^{\top}z\big{)}$.
Assume now that $\mathcal{F}(\mu_{0})(\xi)\neq 0$ $d\xi$-a.e. Then from

$\mathcal{F}\big{(}(G([\vartheta,\vartheta^{\prime}];\cdot)^{j}\star\mu_{0})^{\top}z\big{)}(\xi)=\mathcal{F}\big{(}G([\vartheta,\vartheta^{\prime}];\cdot)^{j}(\xi)\big{)}^{\top}z\cdot\mathcal{F}(\mu_{0})(\xi)$

we must have

$\xi\mapsto\mathcal{F}(G([\vartheta,\vartheta^{\prime}];\cdot)^{j})(\xi)^{\top}z=0\;\;d\xi-a.e.$

for every $1\leq j\leq d$, or equivalently

$x\mapsto\big{(}G([\vartheta,\vartheta^{\prime}];x)^{j}\big{)}^{\top}z=0\;\;dx-a.e.$

This is not possible, as proved by an inspection of the equation

$\displaystyle\int_{0}^{1}\big{(}[\vartheta_{2},\vartheta_{2}^{\prime}]_{\lambda}x_{j}\exp(-[\vartheta_{2},\vartheta_{2}^{\prime}]_{\lambda}|x|^{2})z_{1}+[\vartheta_{1},\vartheta_{1}^{\prime}]_{\lambda}x_{j}(1-[\vartheta_{2},\vartheta_{2}^{\prime}]_{\lambda}|x|^{2})\exp(-[\vartheta_{2},\vartheta_{2}^{\prime}]_{\lambda}|x|^{2})z_{2}\big{)}d\lambda$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\big{(}[\vartheta_{4},\vartheta_{4}^{\prime}]_{\lambda}x_{j}\exp(-[\vartheta_{4},\vartheta_{4}^{\prime}]_{\lambda}|x|^{2})z_{3}+[\vartheta_{3},\vartheta_{3}^{\prime}]_{\lambda}x_{j}(1-[\vartheta_{4},\vartheta_{4}^{\prime}]_{\lambda}|x|^{2})\exp(-[\vartheta_{4},\vartheta_{4}^{\prime}]_{\lambda}|x|^{2})z_{4}\big{)}d\lambda,$

for almost every $x\in\mathbb{R}^{d}$, which further reduces to

$\displaystyle\big{(}[\vartheta_{2},\vartheta_{2}^{\prime}]_{\lambda_{x}}x_{j}z_{1}+[\vartheta_{1},\vartheta_{1}^{\prime}]_{\lambda_{x}}x_{j}(1-[\vartheta_{2},\vartheta_{2}^{\prime}]_{\lambda_{x}}|x|^{2})z_{2}\big{)}\exp(-[\vartheta_{2},\vartheta_{2}^{\prime}]_{\lambda_{x}}|x|^{2})$
(36) $\displaystyle=$
$\displaystyle\big{(}[\vartheta_{4},\vartheta_{4}^{\prime}]_{\lambda^{\prime}_{x}}x_{j}z_{3}+[\vartheta_{3},\vartheta_{3}^{\prime}]_{\lambda^{\prime}_{x}}x_{j}(1-[\vartheta_{4},\vartheta_{4}^{\prime}]_{\lambda^{\prime}_{x}}|x|^{2})z_{4}\big{)}\exp(-[\vartheta_{4},\vartheta_{4}^{\prime}]_{\lambda_{x}^{\prime}}|x|^{2})$

for almost every $x\in\mathbb{R}^{d}$, by the mean-value theorem for some
$\lambda_{x},\lambda_{x}^{\prime}\in[0,1]$, that also respectively depend on
$(\vartheta_{1},\vartheta_{2},\vartheta^{\prime}_{1},\vartheta_{2}^{\prime},z_{1},z_{2})$
and
$(\vartheta_{3},\vartheta_{4},\vartheta_{3}^{\prime},\vartheta_{4}^{\prime},z_{3},z_{4})$.
A simple sufficient condition is $\vartheta_{2}\neq\vartheta_{4}$: indeed, if
$\vartheta_{2}$ and $\vartheta_{4}$ take values in disjoint intervals for
instance, then for every $\vartheta,\vartheta^{\prime}\in\Theta$, with
$[\vartheta,\vartheta^{\prime}]\subset\Theta$, we have
$[\vartheta_{2},\vartheta_{2}^{\prime}]\cap[\vartheta_{4},\vartheta_{4}^{\prime}]=\emptyset$.
Then one easily checks that (36) cannot hold for sufficiently large $|x|$.

If we only need to verify that $\mathbb{I}_{\mathcal{G}}(\vartheta)$ is non-
degenerate, it is sufficient to take $\vartheta=\vartheta^{\prime}$ in (36)
that simply becomes

$\displaystyle\big{(}\vartheta_{2}x_{j}z_{1}+\vartheta_{1}x_{j}(1-\vartheta_{2}|x|^{2})z_{2}\big{)}\exp(-\vartheta_{2}|x|^{2})=\big{(}\vartheta_{4}x_{j}z_{3}+\vartheta_{3}x_{j}(1-\vartheta_{4}|x|^{2})z_{4}\big{)}\exp(-\vartheta_{4}|x|^{2})$

for almost every $x\in\mathbb{R}^{d}$, in which case having
$\vartheta_{2}\neq\vartheta_{4}$ is sufficient (and somewhat easier to obtain
then the non-degeneracy of $\mathcal{G}$ in the sense of Definition 14). In
conclusion, as soon as $\mathcal{F}(\mu_{0})$ is non-vanishing almost
everywhere and $\Theta\subset(0,\infty)^{4}$ is a compact such that
$\vartheta_{2}\neq\vartheta_{4}$, we obtain non-degeneracy of $\mathcal{G}$.

###  4.4. A genuinely non-linear example

We end-up this section by inspecting an example where the parametrisation
$\nu\mapsto b(\vartheta;t,x,\nu)$ is genuinely non-linear in the measure
argument. Consider the model

$dX_{t}^{i}=F\Big{(}\vartheta
N^{-1}\sum_{j=1}^{N}g(X_{t}^{i}-X_{t}^{j})\Big{)}dt+dB_{t}^{i},\;\;1\leq i\leq
N,$

with $X_{t}^{i}\in\mathbb{R}$ and $\vartheta>0$ for simplicity. The functions
$F,g:\mathbb{R}\rightarrow\mathbb{R}$ are known and smooth, $g$ is
nonnegative, integrable, with positive mass and $F$ is one-to-one on the
positive axis. We have $b(\vartheta;t,x,\nu)=F(\vartheta g\star\nu(x))$.

The smoothness of $F$ and $g$ yields Assumptions 2, 3 and 4. As for the
identifiability of the model, assume that
$\overline{\mathbb{P}}_{\vartheta}=\overline{\mathbb{P}}_{\vartheta^{\prime}}$,
and so $\mu_{t}^{\vartheta}=\mu_{t}^{\vartheta^{\prime}}$ as well. If, for
almost every $x\in\mathbb{R}$, we have

$F(\vartheta
g\star\mu_{t}^{\vartheta}(x))=F(\vartheta^{\prime}g\star\mu_{t}^{\vartheta}(x)),$

then, since $g(x)\geq 0$ and $\mu_{t}^{\vartheta}(x)>0$ for every
$x\in\mathbb{R}$ and $t>0$, the function $g\star\mu_{t}^{\vartheta}$ is
positive on the whole real line $\mathbb{R}$. Since $F$ is one-to-one on the
positive axis, it follows that

$\vartheta
g\star\mu_{t}^{\vartheta}(x)=\vartheta^{\prime}g\star\mu_{t}^{\vartheta}(x)$

and that can only be true if $\vartheta=\vartheta^{\prime}$ since
$g\star\mu_{t}^{\vartheta}(x)$ is nowhere vanishing. As for the non-
degeneracy, we have

$\displaystyle\partial_{\vartheta}b(\vartheta;t,x,\mu_{t})$
$\displaystyle=g\star\mu_{t}^{\vartheta}(x)F^{\prime}\big{(}\vartheta
g\star\mu_{t}^{\vartheta}(x)\big{)}=\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}}[g(x-X_{t})]F^{\prime}\big{(}\vartheta\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}}[g(x-X_{t})]\big{)}$

which is continuous at $t=0$ by the continuity in
$\overline{\mathbb{P}}_{\vartheta}$\- probability of $t\mapsto X_{t}$ at
$t=0$. Finally, assume that $\mu_{0}$ has a positive density. Then $x\mapsto
g\star\mu_{0}(x)F^{\prime}\big{(}\vartheta g\star\mu_{0}(x)\big{)}$ is non-
identically vanishing since $g\star\mu_{0}$ is positive and $F^{\prime}$ is
non-vanishing on $(0,\infty)$. We conclude by Proposition 15.

##  5\. Proof of the main results

###  5.1. Preliminaries: couplings

For technical reasons, we will need certain couplings on the canonical space
$(\mathcal{C}^{N},\mathcal{F}^{N})$. We now fix $\vartheta\in\Theta$, and
introduce, for every $\vartheta^{\prime}\in\Theta$, the following two
processes

$X^{(N),\vartheta^{\prime}}=(X_{t}^{1,\vartheta^{\prime}},\ldots,X_{t}^{N,\vartheta^{\prime}})_{t\in[0,T]}$

and

$\overline{X}^{(N),\vartheta^{\prime}}=(\overline{X}_{t}^{1,\vartheta^{\prime}},\ldots,\overline{X}_{t}^{N,\vartheta^{\prime}})_{t\in[0,T]}$

defined on $(\mathcal{C}^{N},\mathcal{F}^{N})$ by

(37)
$X_{t}^{i,\vartheta^{\prime}}=X_{0}^{i}+\int_{0}^{t}b(\vartheta^{\prime};s,X_{s}^{i,\vartheta^{\prime}},\mu_{t}^{(N),\vartheta^{\prime}})ds+\int_{0}^{t}\sigma(s,X_{s}^{i,\vartheta^{\prime}})dB^{i,N,\vartheta}_{s},$

with
$\mu_{t}^{(N),\vartheta^{\prime}}=N^{-1}\sum_{i=1}^{N}\delta_{X_{t}^{i,\vartheta^{\prime}}}$
and

(38)
$\overline{X}_{t}^{i,\vartheta^{\prime}}=X_{0}^{i}+\int_{0}^{t}b(\vartheta^{\prime};s,\overline{X}_{s}^{i,\vartheta^{\prime}},\mu_{t}^{\vartheta^{\prime}})ds+\int_{0}^{t}\sigma(s,\overline{X}_{s}^{i,\vartheta^{\prime}})dB^{i,N,\vartheta}_{s},$

and

$B^{i,N,\vartheta}_{t}=\int_{0}^{t}c^{-1/2}(s,X_{s}^{i})(dX_{s}^{i}-b(\vartheta,s,X_{s}^{i},\mu_{s}^{(N)})ds).$

Note that $X^{(N),\vartheta^{\prime}}$ and
$\overline{X}^{(N),\vartheta^{\prime}}$ actually depend on $\vartheta$
pathwise via $(B_{t}^{i,N,\vartheta})_{t\in[0,1]}$ under
$\mathbb{P}_{\vartheta}^{N}$ but not their laws! Indeed, the
$(B_{t}^{i,N,\vartheta})_{t\in[0,1]}$ are standard Brownian motions under
$\mathbb{P}_{\vartheta}^{N}$. For notational simplicity, we omit the
dependence upon $\vartheta$ here. Otherwise, we write
$X_{t}^{i,\vartheta^{\prime},\vartheta}$ or
$\overline{X}_{t}^{i,\vartheta^{\prime},\vartheta}$. Thus
$X^{(N),\vartheta^{\prime}}$ and $\overline{X}^{(N),\vartheta^{\prime}}$ are
defined as functions of $X^{(N)}$ (as strong solutions of (37) and (38)) and
have law $\mathbb{P}_{\vartheta^{\prime}}^{N}$ and
$\overline{\mathbb{P}}_{\vartheta^{\prime}}^{\otimes N}$ under
$\mathbb{P}_{\vartheta}^{N}$. This is a convenient way to couple
$\mathbb{P}_{\vartheta^{\prime}}^{N}$ and
$\overline{\mathbb{P}}_{\vartheta^{\prime}}^{\otimes N}$ while still working
with the canonical process under $\mathbb{P}_{\vartheta}^{N}$. We write
$\mu_{t}^{(N)}=N^{-1}\sum_{i=1}^{N}\delta_{X_{t}^{i}}$, for the empirical
measure of the canonical process. We also introduce

$\overline{\mu}_{t}^{(N),\vartheta^{\prime}}=N^{-1}\sum_{i=1}^{N}\delta_{\overline{X}_{t}^{i,\vartheta^{\prime}}}$

and write $\mu_{t}^{(N),\vartheta^{\prime},\vartheta}$ and
$\overline{\mu}_{t}^{(N),\vartheta^{\prime},\vartheta}$ whenever we want to
emphasise that the coupling is constructed with the processes
$(B^{i,N,\vartheta}_{t})_{t\in[0,T]}$ for $1\leq i\leq N$. We have the
following approximation results:

######  Lemma 20.

For every $\vartheta,\vartheta^{\prime}\in\Theta$ and every $r\geq 1$, we have

(39)
$\sup_{t\in[0,T]}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\mathcal{W}_{1}\big{(}\mu^{(N),\vartheta^{\prime},\vartheta}_{t},\mu^{(N)}_{t}\big{)}^{r}\big{]}\leq\sup_{t\in[0,T]}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}N^{-1}\sum_{i=1}^{N}|X^{i,\vartheta^{\prime},\vartheta}_{t}-X^{i}_{t}|^{r}\big{]}\leq
C|\vartheta^{\prime}-\vartheta|^{r},$ (40)
$\sup_{t\in[0,T]}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\mathcal{W}_{1}\big{(}\bar{\mu}^{(N),\vartheta^{\prime},\vartheta}_{t},\overline{\mu}^{(N),\vartheta,\vartheta}_{t}\big{)}^{r}\big{]}\leq\sup_{t\in[0,T]}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}N^{-1}\sum_{i=1}^{N}|\overline{X}^{i,\vartheta^{\prime},\vartheta}_{t}-\overline{X}^{i,\vartheta,\vartheta}_{t}|^{r}\big{]}\leq
C|\vartheta^{\prime}-\vartheta|^{r}.$

There exists $\delta>0$ such that for every $r\geq 1$:

(41)
$\displaystyle\sup_{t\in[0,T],\vartheta\in\Theta}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\mathcal{W}_{1}\big{(}\mu^{(N)}_{t},\mu^{\vartheta}_{t}\big{)}^{r}\big{]}\leq
CN^{-\delta r},$ (42)
$\displaystyle\sup_{t\in[0,T],\vartheta\in\Theta}N^{-1}\sum_{i=1}^{N}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}[|X^{i}_{t}-\overline{X}^{i,\vartheta,\vartheta}_{t}|^{r}]\leq
CN^{-\delta r}$

as $N\rightarrow\infty$.

The proof is given in Appendix 7.3.

###  5.2. Proof of Theorem 17

We prove a slightly stronger result, namely a uniform type LAN condition,
following Chapter III of [24]. Let $(u_{N})_{N\geq 1}$ be a sequence of
$\mathbb{R}^{p}$ such that $u_{N}\rightarrow u$ and $(\vartheta_{N})_{N\geq
1}$ a sequence of $\Theta$ such that
$\vartheta_{N}+(N\mathbb{I}_{\mathcal{G}}(\vartheta_{N}))^{-1/2}u_{N}\in\Theta$
for large enough $N$ and such that $\vartheta_{N}\rightarrow\vartheta$, for
some $\vartheta$ such that $\mathbb{I}_{\mathcal{G}}(\vartheta)>0$. We claim
that

(43)
$\zeta_{N}(\vartheta_{N};u_{N})=\log\frac{d\mathbb{P}_{\vartheta_{N}+(N\mathbb{I}_{\mathcal{G}}(\vartheta_{N}))^{-1/2}u_{N}}^{N}}{d\mathbb{P}_{\vartheta_{N}}^{N}}=u^{\top}\Gamma_{N}-\tfrac{1}{2}|u|^{2}+r_{N}(\vartheta_{N},u_{N}),$

where $\Gamma_{N}\rightarrow\mathcal{N}(0,\mathrm{Id}_{\mathbb{R}^{p}})$ in
distribution under $\mathbb{P}_{\vartheta}^{N}$ and
$r_{N}(\vartheta_{N},u_{N})\rightarrow 0$ in
$\mathbb{P}_{\vartheta}^{N}$-probability. Clearly, (43) implies (22) and thus
Theorem 17. Note that since $\mathbb{I}_{\mathcal{G}}(\vartheta)>0$ we have
that $\mathbb{I}_{\mathcal{G}}(\vartheta_{N})$ is invertible for large enough
$N$, thanks to the continuity of the mapping
$\vartheta\mapsto\mathbb{I}_{\mathcal{G}}(\vartheta)$, recall Proposition 11.
The asymptotic expansion (43) is therefore meaningful for large enough $N$.

We further write $\mathbb{I}(\vartheta)$ for
$\mathbb{I}_{\mathcal{G}}(\vartheta)$.

Step 1. (Preliminary expansion.) We have

$\displaystyle\zeta_{N}(\vartheta_{N};u_{N})$
$\displaystyle=\sum_{i=1}^{N}\int_{0}^{T}\big{(}(c^{-1/2}b)(\vartheta_{N}+(N{\mathbb{I}(\vartheta_{N})})^{-1/2}u_{N};t,X^{i}_{t},\mu^{N}_{t})-(c^{-1/2}b)(\vartheta_{N};t,X^{i}_{t},\mu^{N}_{t})\big{)}^{\top}dB^{i,N,\vartheta_{N}}_{t}$
$\displaystyle-\frac{1}{2}\sum_{i=1}^{N}\int_{0}^{T}\big{|}(c^{-1/2}b)(\vartheta_{N}+(N{\mathbb{I}(\vartheta_{N})})^{-1/2}u_{N};t,X^{i}_{t},\mu^{N}_{t})-(c^{-1/2}b)(\vartheta_{N};t,X^{i}_{t},\mu^{N}_{t})\big{|}^{2}dt,$

where the
$B^{i,N,\vartheta_{N}}_{t}=\int_{0}^{t}c^{-1/2}(s,X_{s}^{i})(dX_{s}^{i}-b(\vartheta_{N};s,X_{s}^{i},\mu_{s}^{(N)})ds)$,
$1\leq i\leq N$ are independent Brownian motions on $\mathbb{R}^{d}$ under
$\mathbb{\mathbb{P}}_{\vartheta_{N}}^{N}$. A first-order Taylor’s expansion
therefore yields the representation,

$\displaystyle\zeta_{N}(\vartheta_{N};u_{N})$
$\displaystyle=u_{N}^{\top}(\mathbb{I}(\vartheta_{N})^{-1/2})^{\top}\Delta_{N,\vartheta_{N}}(u_{N})-\tfrac{1}{2}u_{N}^{\top}(\mathbb{I}(\vartheta_{N})^{-1/2})^{\top}\,\widetilde{\mathbb{I}}_{N,\vartheta_{N}}(u_{N})\mathbb{I}(\vartheta_{N})^{-1/2}u_{N}$

where

$\displaystyle\Delta_{N,\vartheta_{N}}(u)=N^{-1/2}\sum_{i=1}^{N}\sum_{j=1}^{d}\int_{0}^{T}\nabla_{\vartheta}(c^{-1/2}b)^{j}([\vartheta_{N},\vartheta_{N}+(N\mathbb{I}(\vartheta_{N}))^{-1/2}u];t,X_{t}^{i},\mu_{t}^{N})d(B^{i,N,\vartheta_{N}}_{t})^{j}$

and

$\displaystyle\widetilde{\mathbb{I}}_{N,\vartheta_{N}}(u)$
$\displaystyle=N^{-1}\sum_{i=1}^{N}\sum_{j=1}^{d}\int_{0}^{T}\nabla_{\vartheta}(c^{-1/2}b)^{j}([\vartheta_{N},\vartheta_{N}+(N\mathbb{I}(\vartheta_{N}))^{-1/2}u];t,X_{t}^{i},\mu_{t}^{N})$
$\displaystyle\;\;\;\;\times\big{(}\nabla_{\vartheta}(c^{-1/2}b)^{j}([\vartheta_{N},\vartheta_{N}+(N\mathbb{I}(\vartheta_{N}))^{-1/2}u];t,X_{t}^{i},\mu_{t}^{N})\big{)}^{\top}dt,$

with the notation
$\phi([\vartheta,\vartheta^{\prime}])=\int_{0}^{1}\phi(\vartheta+\lambda(\vartheta^{\prime}-\vartheta))d\lambda$
that we introduced before. We rewrite the above expansion as

$\displaystyle\zeta_{N}(\vartheta_{N};u_{N})$
$\displaystyle=u_{N}^{\top}(\mathbb{I}(\vartheta_{N})^{-1/2})^{\top}\Delta_{N,\vartheta_{N}}(0)-\tfrac{1}{2}|u|^{2}$
$\displaystyle+u_{N}^{\top}(\mathbb{I}(\vartheta_{N})^{-1/2})^{\top}\Delta_{N,\vartheta_{N}}(u_{N})-u^{\top}(\mathbb{I}(\vartheta_{N})^{-1/2})^{\top}\Delta_{N,\vartheta_{N}}(0)$
$\displaystyle-\tfrac{1}{2}\big{(}u_{N}^{\top}(\mathbb{I}(\vartheta_{N})^{-1/2})^{\top}\,\widetilde{\mathbb{I}}_{N,\vartheta_{N}}(u_{N})\mathbb{I}(\vartheta_{N})^{-1/2}u_{N}-|u|^{2}\big{)}$

and thus (43) follows from

(44)
$(\mathbb{I}(\vartheta_{N})^{-1/2})^{\top}\Delta_{N,\vartheta_{N}}(0)\rightarrow\mathcal{N}(0,\mathrm{Id}_{\mathbb{R}^{p}})$

under $\mathbb{P}_{\vartheta_{N}}^{N}$ in distribution together with the
convergence to $0$ of the last two components.

Step 2. (Convergence of the Gaussian part.) We prove (44) or equivalently, the
convergence

$\xi^{\top}(\mathbb{I}(\vartheta_{N})^{-1/2})^{\top}\Delta_{N,\vartheta_{N}}(0)=\sum_{q,q^{\prime}=1}^{d}\xi_{q}(\mathbb{I}(\vartheta_{N})^{-1/2})_{q^{\prime}q}\Delta_{N,\vartheta_{N}}(0)_{q^{\prime}}\rightarrow\mathcal{N}(0,|\xi|^{2})$

in distribution under $\mathbb{P}_{\vartheta_{N}}^{N}$ for every
$\xi\in\mathbb{R}^{p}$. We apply a classical semimartingale convergence
result, following for instance Jacod and Shiryaev [26] (Corollary 3.24). For
$t\in[0,T]$, the process

$\Delta_{N,\vartheta_{N}}(0)_{t}=N^{-1/2}\sum_{i=1}^{N}\sum_{j=1}^{d}\int_{0}^{t}\nabla_{\vartheta}(c^{-1/2}b)^{j}(\vartheta_{N};s,X_{s}^{i},\mu_{s}^{N})d\lambda\big{)}d(B^{i,N,\vartheta_{N}}_{s})^{j}$

is a continuous local martingale under $\mathbb{P}_{\vartheta_{N}}^{N}$ and so
is
$\big{(}\xi^{\top}(\mathbb{I}(\vartheta_{N})^{-1/2})^{\top}\Delta_{N,\vartheta_{N}}(0)_{t}\big{)}_{t\in[0,T]}$.
It coincides with
$\xi^{\top}(\mathbb{I}(\vartheta_{N})^{-1/2})^{\top}\Delta_{N,\vartheta_{N}}(0)$
at $t=T$ and has predictable compensator

$\displaystyle\big{\langle}\xi^{\top}(\mathbb{I}(\vartheta_{N})^{-1/2})^{\top}$
$\displaystyle\Delta_{N,\vartheta_{N}}(0)_{\cdot}\big{\rangle}_{t}=\sum_{q_{k},q_{k}^{\prime}=1}^{p}\xi_{q_{1}}\xi_{q_{2}}(\mathbb{I}(\vartheta_{N})^{-1/2})_{q_{2}^{\prime}q_{2}}(\mathbb{I}(\vartheta_{N})^{-1/2})_{q_{1}^{\prime}q_{1}}\times$
$\displaystyle
N^{-1}\sum_{i=1}^{N}\sum_{j=1}^{d}\int_{0}^{t}\partial_{\vartheta_{q^{\prime}_{1}}}(c^{-1/2}b)^{j}(\vartheta_{N};s,X_{s}^{i},\mu_{s}^{(N)})\partial_{\vartheta_{q^{\prime}_{2}}}(c^{-1/2}b)^{j}(\vartheta_{N};s,X_{s}^{i},\mu_{s}^{(N)})ds,$

that converges to

$\displaystyle\sum_{q_{k},q_{k}^{\prime}=1}^{p}\xi_{q_{1}}\xi_{q_{2}}(\mathbb{I}(\vartheta)^{-1/2})_{q_{2}^{\prime}q_{2}}(\mathbb{I}(\vartheta)^{-1/2})_{q_{1}^{\prime}q_{1}}\times$
$\displaystyle\sum_{j=1}^{d}\int_{0}^{t}\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}}\big{[}\partial_{\vartheta_{q^{\prime}_{1}}}(c^{-1/2}b)^{j}(\vartheta;s,X_{s}^{i},\mu_{s}^{\vartheta})\partial_{\vartheta_{q^{\prime}_{2}}}(c^{-1/2}b)^{j}(\vartheta;s,X_{s}^{i},\mu_{s}^{\vartheta})\big{]}ds,$

in $\mathbb{P}_{\vartheta_{N}}^{N}$-probability, but that last quantity is
exactly $|\xi|^{2}$ at $t=T$, which proves (44). As for the last convergence
in probability, it is a simple consequence of the continuity of
$\vartheta\mapsto\mathbb{I}(\vartheta)$, see Proposition 11 and the following
lemma

######  Lemma 21.

Let $\beta>0$ and
$\phi:\Theta\times[0,T]\times\mathbb{R}^{d}\times\mathcal{P}_{\beta}\rightarrow\mathbb{R}$
be such that for some $C,\alpha>0$, we have

$\sup_{t\in[0,T]}|\phi(\vartheta^{\prime};t,x^{\prime},\nu^{\prime})-\phi(\vartheta;t,x,\nu)|\leq
C(|\vartheta^{\prime}-\vartheta|+|x^{\prime}-x|+\mathcal{W}_{1}(\nu^{\prime},\nu))(1+|x|^{\alpha}+|x^{\prime}|^{\alpha}+\mathfrak{m}_{\beta}(\nu)+\mathfrak{m}_{\beta}(\nu^{\prime})).$

Then, there exists $0<\delta\leq 1/2$ such that for every $t\in[0,T]$ and
every $m>0$, we have

$\mathbb{E}_{\mathbb{P}_{\vartheta_{N}}^{N}}\Big{[}\Big{|}N^{-1}\sum_{i=1}^{N}\int_{0}^{t}\phi(\vartheta_{N};s,X_{s}^{i},\mu_{s}^{(N)})ds-\int_{0}^{t}\int_{\mathbb{R}^{d}}\phi(\vartheta;s,x,\mu_{s}^{\vartheta})\mu_{s}^{\vartheta}(dx)ds\Big{|}^{m}\Big{]}\leq
C(|\vartheta_{N}-\vartheta|^{m}+N^{-\delta m}).$

We apply Lemma 21 to
$\phi(\vartheta;s,x,\nu)=\partial_{\vartheta_{q_{1}^{\prime}}}(c^{-1/2}b)^{j}(\vartheta;s,x,\nu)\partial_{\vartheta_{q_{2}^{\prime}}}(c^{-1/2}b)^{j}(\vartheta;s,x,\nu)$,
thanks to Assumption 4. The proof is given in Appendix 7.4.

Step 3. (Convergence of the remainder terms.) We first prove

(45)
$u_{N}^{\top}(\mathbb{I}(\vartheta_{N})^{-1/2})^{\top}\Delta_{N,\vartheta_{N}}(u_{N})-u^{\top}(\mathbb{I}(\vartheta_{N})^{-1/2})^{\top}\Delta_{N,\vartheta_{N}}(0)\rightarrow
0$

in $\mathbb{P}_{\vartheta_{N}}^{N}$-probability. Since
$\mathbb{I}(\vartheta_{N})^{-1/2}$ is well defined for large enough $N$ and
converges to $\mathbb{I}(\vartheta)^{-1/2}$ and $u_{N}\rightarrow u$, it is
sufficient to prove
$\Delta_{N,\vartheta_{N}}(u_{N})-\Delta_{N,\vartheta_{N}}(0)\rightarrow 0$ in
$\mathbb{P}_{\vartheta_{N}}^{N}$-probability. Introduce the process

$G_{N}^{i,r}(\vartheta,u)_{t}=\int_{0}^{1}\big{(}\partial_{\vartheta_{r}}(c^{-1/2}b)(\vartheta+\lambda(N\mathbb{I}(\vartheta))^{-1/2}u;t,X_{t}^{i},\mu_{t}^{(N)})-\partial_{\vartheta_{r}}(c^{-1/2}b)(\vartheta;t,X_{t}^{i},\mu_{t}^{(N)})\big{)}d\lambda$

for $1\leq r\leq p$ and $t\in[0,T]$. By Itô’s isometry

$\displaystyle\mathbb{E}_{\mathbb{P}_{\vartheta_{N}}^{N}}\big{[}|\Delta_{N,\vartheta_{N}}(u_{N})-\Delta_{N,\vartheta_{N}}(0)|^{2}\big{]}$
$\displaystyle=\sum_{r=1}^{p}\mathbb{E}_{\mathbb{P}_{\vartheta_{N}}^{N}}\big{[}|N^{-1/2}\sum_{i=1}^{N}\int_{0}^{T}G_{N}^{i,r}(\vartheta_{N},u_{N})_{t}^{\top}dB_{t}^{i,N,\vartheta_{N}}\big{|}^{2}\big{]}$
$\displaystyle=\sum_{r=1}^{p}N^{-1}\sum_{i=1}^{N}\int_{0}^{T}\mathbb{E}_{\mathbb{P}_{\vartheta_{N}}^{N}}\big{[}|G_{N}^{i,r}(\vartheta_{N},u_{N})_{t}|^{2}\big{]}dt.$

Moreover

$\displaystyle|G_{N}^{i,r}(\vartheta_{N},u_{N})_{t}|^{2}$
$\displaystyle\leq\sum_{j=1}^{d}\sup_{\vartheta\in\Theta}\big{|}(\nabla_{\vartheta}\partial_{\vartheta_{r}}c^{-1/2}b)^{j}(\vartheta;t,X_{t}^{i},\mu_{t}^{(N)})\lambda(N\mathbb{I}(\vartheta_{N}))^{-1/2}u_{N}\big{|}^{2}$
$\displaystyle\leq
CN^{-1}(1+|X_{t}^{i}|^{2r_{1}}+{\mathfrak{m}}_{r_{2}}(\mu_{t}^{(N)})^{2}),$

for large enough $N$, thanks to Assumption 4. We conclude

$\sup_{t\in[0,T]}\mathbb{E}_{\mathbb{P}_{\vartheta_{N}}^{N}}\big{[}|G_{N}^{i,r}(\vartheta_{N},u_{N})_{t}|^{2}\big{]}\leq
CN^{-1}$

for large enough $N$ by Lemma 7 and (45) follows.

The convergence of the second remainder term is a simple consequence of
$\widetilde{\mathbb{I}}_{N,\vartheta_{N}}(u_{N})\rightarrow\mathbb{I}(\vartheta)$
thanks to Lemma 21 together with the continuity of
$\vartheta\mapsto\mathbb{I}(\vartheta)$ and Proposition 11. The Proof of
Theorem 17 is complete for the experiment $\mathcal{E}^{N}$.

Step 4. (The case of the experiment $\mathcal{G}^{\otimes N}$.) We now easily
extend the previous results to the experiment $\mathcal{G}^{\otimes N}$. Since
it is a product of the same experiment $\mathcal{G}$, it is tempting to use
classical criterions for IID data. However, from a simple glance at the
structure of the previous computations, it suffices to retrace Steps 1 to 3
replacing $\mu_{t}^{(N)}$ by $\mu_{t}^{\vartheta_{N}}$ and
$\mathbb{P}_{\vartheta_{N}}^{N}$ by
$\overline{\mathbb{P}}_{\vartheta_{N}}^{\otimes N}$ that actually turn out to
be simpler. We omit the details.

###  5.3. Proof of Theorem 19

We plan to apply the classical theory of Ibragimov-Hasminski, and more
specifically Theorem III.1.1 of [24].We introduce the notation

$\mathcal{Z}_{N}(\vartheta;u)=\frac{d\mathbb{P}_{\vartheta+(N\mathbb{I}_{\mathcal{G}}(\vartheta))^{-1/2}u}^{N}}{d\mathbb{P}_{\vartheta}^{N}}.$

We first establish two key regularity properties of the likelihood process.

Step 1. (A regularity property for the likelihood process.) Here, we prove
that for any $r\geq 2$

(46)
$\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\big{|}\mathcal{Z}_{N}(\vartheta;u)^{1/r}-\mathcal{Z}_{N}(\vartheta;v)^{1/r}\big{|}^{r}\big{]}\leq
C(1+\kappa^{\gamma})|u-v|^{\gamma^{\prime}},$

for some positive $\gamma,\gamma^{\prime}$, uniformly in $u,v$ such that
$\mathcal{Z}_{N}(\vartheta;u)$ and $\mathcal{Z}_{N}(\vartheta;v)$ are well-
defined and $|u|,|v|$ are bounded by $\kappa>0$. Pick any $r\geq 2$. By a
first-order expansion

$\displaystyle\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\big{|}\mathcal{Z}_{N}(\vartheta;u)^{1/r}-\mathcal{Z}_{N}(\vartheta;v)^{1/r}\big{|}^{r}\big{]}$
$\displaystyle\leq|u-v|^{r}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\big{|}\int_{0}^{1}\mathcal{\nabla}_{u}(\mathcal{Z}_{N})^{1/r}(\vartheta;u+\lambda(v-u))d\lambda\big{|}^{r}\big{]}$
(47) $\displaystyle\leq
C|u-v|^{r}\int_{0}^{1}\sum_{q=1}^{p}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\big{|}\partial_{u_{q}}((\mathcal{Z}_{N})^{1/r})(\vartheta;u+\lambda(v-u))\big{|}^{r}\big{]}d\lambda.$

Define, for $t\in[0,T]$, the random process

$\phi_{t}(\vartheta;X^{i},\mu^{(N)})=\exp\Big{(}\int_{0}^{t}(c^{-1}b)(\vartheta;s,X_{s}^{i},\mu_{s}^{(N)})dX_{s}^{i}-\tfrac{1}{2}\int_{0}^{t}|(c^{-1/2}b)(\vartheta;s,X_{s}^{i},\mu_{s}^{(N)})|^{2}ds\Big{)}.$

Since

$\mathcal{Z}_{N}(\vartheta;u)=\prod_{i=1}^{N}\frac{\phi_{T}^{1/r}(\vartheta+(N\mathbb{I}_{\mathcal{G}}(\vartheta))^{-1/2}u,X^{i},\mu^{(N)})}{\phi_{T}^{1/r}(\vartheta,X^{i},\mu^{(N)})},$

we have

$\displaystyle\partial_{u_{q}}((\mathcal{Z}_{N})^{1/r})(\vartheta;u)$
$\displaystyle=\sum_{i=1}^{N}\frac{\partial_{u_{q}}(\phi_{T}^{1/r}(\vartheta+(N\mathbb{I}_{\mathcal{G}}(\vartheta))^{-1/2}u;X^{i},\mu^{(N)}))}{\phi_{T}^{1/r}(\vartheta;X^{i},\mu^{(N)})}$
$\displaystyle\;\;\;\;\times\prod_{i^{\prime}\neq
i}\frac{\phi_{T}^{1/r}(\vartheta+(N\mathbb{I}_{\mathcal{G}}(\vartheta))^{-1/2}u;X^{i^{\prime}},\mu^{(N)})}{\phi_{T}^{1/r}(\vartheta;X^{i^{\prime}},\mu^{(N)})}$
$\displaystyle=\sum_{i=1}^{N}\frac{\nabla_{\vartheta}(\phi_{T}^{1/r})(\vartheta+(N\mathbb{I}_{\mathcal{G}}(\vartheta))^{-1/2}u;X^{i},\mu^{(N)})^{\top}\big{(}(N\mathbb{I}_{\mathcal{G}}(\vartheta))^{-1/2}\big{)}_{\cdot
q}}{\phi_{T}^{1/r}(\vartheta+(N\mathbb{I}_{\mathcal{G}}(\vartheta))^{-1/2}u;X^{i},\mu^{(N)})}$
$\displaystyle\;\;\;\;\times\prod_{i^{\prime}=1}^{N}\frac{\phi_{T}^{1/r}(\vartheta+(N\mathbb{I}_{\mathcal{G}}(\vartheta))^{-1/2}u;X^{i^{\prime}},\mu^{(N)})}{\phi_{T}^{1/r}(\vartheta;X^{i^{\prime}},\mu^{(N)})}$
$\displaystyle=(\mathcal{Z}_{N})^{1/r}(\vartheta;u)\big{(}(N\mathbb{I}_{\mathcal{G}}(\vartheta))^{-1/2}\big{)}_{\cdot
q}\times$
$\displaystyle\;\;\;\;\;\sum_{i=1}^{N}\nabla_{\vartheta}\big{(}\log(\phi_{T}^{1/r})\big{)}(\vartheta+(N\mathbb{I}_{\mathcal{G}}(\vartheta))^{-1/2}u,X^{i},\mu^{(N)}).$

Interpreting $\mathcal{Z}_{N}(\vartheta;u)$ as a Radon-Nikodym derivative
entails

$\displaystyle\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\big{|}\partial_{u_{q}}((\mathcal{Z}_{N})^{1/r})(\vartheta;u)\big{|}^{r}\big{]}$
(48)
$\displaystyle=|\big{(}\mathbb{I}_{\mathcal{G}}(\vartheta))^{-1/2}\big{)}_{\cdot
q}|^{r}\mathbb{E}_{\mathbb{P}_{\vartheta+(N\mathbb{I}_{\mathcal{G}}(\vartheta))^{-1/2}u}^{N}}\big{[}\big{|}N^{-1/2}\sum_{i=1}^{N}\nabla_{\vartheta}\big{(}\log(\phi_{T}^{1/r})\big{)}(\vartheta+(N\mathbb{I}_{\mathcal{G}}(\vartheta))^{-1/2}u)\big{|}^{r}\big{]}$

by a change of probability between $\mathbb{P}_{\vartheta}^{N}$ and
$\mathbb{P}_{\vartheta+(N\mathbb{I}_{\mathcal{G}}(\vartheta))^{-1/2}u}^{N}$.
Next, by definition of $\phi_{t}$, we have, for every
$\vartheta^{\prime}\in\Theta$

$\displaystyle\partial_{\vartheta_{q^{\prime}}}(\log(\phi_{T}^{1/r}))(\vartheta^{\prime};X^{i},\mu^{(N)})$
$\displaystyle=\frac{1}{r}\int_{0}^{T}\partial_{\vartheta_{q^{\prime}}}(c^{-1}b)(\vartheta^{\prime};t;X_{t}^{i},\mu_{t}^{(N)})dX_{t}^{i}$
$\displaystyle\;\;\;-\frac{1}{2r}\int_{0}^{T}2\partial_{\vartheta_{q^{\prime}}}(c^{-1/2}b)(\vartheta^{\prime};t,X_{t}^{i},\mu_{t}^{(N)})^{\top}(c^{-1/2}b)(\vartheta^{\prime};t,X_{t}^{i},\mu_{t}^{(N)})dt$
$\displaystyle=\frac{1}{r}\int_{0}^{T}\partial_{\vartheta_{q^{\prime}}}(c^{-1/2}b)(\vartheta^{\prime};t;X_{t}^{i},\mu_{t}^{(N)})dB_{t}^{i,N,\vartheta^{\prime}},$

where the
$(B^{i,N,\vartheta^{\prime}}_{t})_{t\in[0,T]}=(\int_{0}^{t}c^{-1/2}(s,X_{s}^{i})(dX_{s}^{i}-b(\vartheta^{\prime};s,X_{s}^{i},\mu_{s}^{(N)})ds)_{t\in[0,T]}$,
$1\leq i\leq N$ are independent Brownian motions on $\mathbb{R}^{d}$ under
$\mathbb{P}_{\vartheta^{\prime}}^{N}$. Plugging-in this representation in (48)
at
$\vartheta^{\prime}=\vartheta+(N\mathbb{I}_{\mathcal{G}}(\vartheta))^{-1/2}u$,
we infer

$\displaystyle\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\big{|}\partial_{u_{q}}((\mathcal{Z}_{N})^{1/r})(\vartheta;u)\big{|}^{r}\big{]}$
$\displaystyle\leq
C\mathbb{E}_{\mathbb{P}_{\vartheta^{\prime}}^{N}}\big{[}\big{|}N^{-1/2}\sum_{i=1}^{N}\int_{0}^{T}\partial_{\vartheta_{q^{\prime}}}(c^{-1/2}b)(\vartheta^{\prime};t;X_{t}^{i},\mu_{t}^{(N)})dB_{t}^{i,N,\vartheta^{\prime}}\big{|}^{r}\big{]}$
$\displaystyle\leq
C\mathbb{E}_{\mathbb{P}_{\vartheta^{\prime}}^{N}}\big{[}\big{|}N^{-1}\sum_{i=1}^{N}\int_{0}^{T}|\partial_{\vartheta_{q^{\prime}}}(c^{-1/2}b)(\vartheta^{\prime};t;X_{t}^{i},\mu_{t}^{(N)})|^{2}dt\big{|}^{r/2}\big{]}$
$\displaystyle\leq
CN^{-1}\sum_{i=1}^{N}\int_{0}^{T}(1+\mathbb{E}_{\mathbb{P}_{\vartheta^{\prime}}^{N}}\big{[}|X_{t}^{i}|^{r_{1}r}\big{]}+\mathfrak{m}_{r_{2}}(\mu_{t}^{(N)})^{r})dt$
$\displaystyle\leq
C(1+\int_{0}^{T}\mathbb{E}_{\mathbb{P}_{\vartheta^{\prime}}^{N}}\big{[}|X_{t}^{i}|^{r\max(r_{1},r_{2})}\big{]}dt).$

where we succesively used the Burckholder-Davis-Gundy, Assumption 4 and the
fact that $r\geq 2$. Now, we claim that with
$\vartheta^{\prime}=\vartheta+(N\mathbb{I}_{\mathcal{G}}(\vartheta))^{-1/2}(u+\lambda(v-u))$,
we have

(49)
$\int_{0}^{T}\mathbb{E}_{\mathbb{P}_{\vartheta+(N\mathbb{I}_{\mathcal{G}}(\vartheta))^{-1/2}(u+\lambda(v-u))}^{N}}\big{[}|X_{t}^{i}|^{r\max(r_{1},r_{2})}\big{]}dt\leq
C(1+\kappa^{\gamma})$

for some $\gamma>0$, uniformly in $|u|,|v|$ bounded by $\kappa$ and where $C$
depends on $\Theta$ only. Indeed, keeping up with the abbreviation
$\vartheta^{\prime}$, we have

$\mathbb{E}_{\mathbb{P}_{\vartheta^{\prime}}}\big{[}|X_{t}^{i}|^{r\max(r_{1},r_{2})}\big{]}\leq
C\big{(}\mathbb{E}_{\mathbb{P}_{\vartheta^{\prime}}}\big{[}|X_{t}^{i}-X_{t}^{i,\vartheta}|^{r\max(r_{1},r_{2})}\big{]}+\mathbb{E}_{\mathbb{P}_{\vartheta^{\prime}}}\big{[}|X_{t}^{i,\vartheta}|^{r\max(r_{1},r_{2})}\big{]}.$

By (39) of Lemma 20

$\displaystyle\mathbb{E}_{\mathbb{P}_{\vartheta^{\prime}}}\big{[}|X_{t}^{i}-X_{t}^{i,\vartheta}|^{r\max(r_{1},r_{2})}\big{]}$
$\displaystyle\leq C|\vartheta-\vartheta^{\prime}|^{r\max(r_{1},r_{2})}$
$\displaystyle=C|(N\mathbb{I}_{\mathcal{G}}(\vartheta))^{-1/2}(u+\lambda(v-u))|^{r\max(r_{1},r_{2})}$
$\displaystyle\leq CN^{-r\max(r_{1},r_{2})/2}\kappa^{r\max(r_{1},r_{2})}$

and the second term that only depends on $\vartheta$ by coupling is uniformly
bounded by Lemma 7. The estimate (49) follows. Going back to (47), we conclude

$\displaystyle\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\big{|}\mathcal{Z}_{N}(\vartheta;u)^{1/r}-\mathcal{Z}_{N}(\vartheta;u)^{1/r}\big{|}^{r}\big{]}$
$\displaystyle\leq C|u-v|^{r}(1+\kappa^{r\max(r_{1},r_{2})})$

and (46) is established with $\gamma=r\max(r_{1},r_{2})$ and
$\gamma^{\prime}=r$.

Step 2. Here we prove a moment bound for the likelihood ratio process, namely,
for every $r>0$

(50)
$\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\mathcal{Z}_{N}(\vartheta;u)^{1/2}\big{]}\leq
C|u|^{-r},$

uniformly in $\vartheta\in\Theta$ and
$u=(N\mathbb{I}_{\mathcal{G}}(\vartheta))^{1/2}(\vartheta^{\prime}-\vartheta)$
with $\vartheta^{\prime}\in\Theta$. Introducing for $t\in[0,T]$ the
$\mathbb{P}_{\vartheta}^{N}$-martingale

$\mathcal{M}_{t}^{N}(\vartheta;u)=\sum_{i=1}^{N}\int_{0}^{t}\big{(}(c^{-1/2}b)(\vartheta+(N\mathbb{I}_{\mathcal{G}}(\vartheta))^{-1/2}u;s,X_{s}^{i};\mu_{s}^{(N)})-(c^{-1/2}b)(\vartheta;s,X_{s}^{i},\mu_{s}^{(N)})\big{)}^{\top}dB_{s}^{i,N,\vartheta},$

we have

$\mathcal{Z}_{N}(\vartheta;u)=\exp\big{(}\mathcal{M}_{T}^{N}(\vartheta;u)-\tfrac{1}{2}\langle\mathcal{M}_{\cdot}^{N}(\vartheta;u)\rangle_{T}\big{)}.$

It follows that

$\displaystyle\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\mathcal{Z}_{N}(\vartheta;u)^{1/2}\big{]}$
$\displaystyle=\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\Big{[}\exp(-\tfrac{1}{2}\mathcal{M}_{T}^{N}(\vartheta;u)-\tfrac{3}{16}\langle\mathcal{M}_{\cdot}^{N}(\vartheta;u)\rangle_{T})\exp\big{(}-\tfrac{1}{16}\langle\mathcal{M}_{\cdot}^{N}(\vartheta;u)\rangle_{T}\big{)}\Big{]}$
$\displaystyle\leq\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\exp(-\tfrac{3}{4}
\mathcal{M}_{T}^{N}(\vartheta;u)-\tfrac{1}{2}\langle\tfrac{3}{4}\mathcal{M}_{\cdot}^{N}(\vartheta;u)\rangle_{T})\big{]}^{\frac{2}{3}}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\exp\big{(}-\tfrac{3}{16}\langle\mathcal{M}_{\cdot}^{N}(\vartheta;u)\rangle_{T}\big{)}\big{]}^{\frac{1}{3}}$
$\displaystyle\leq\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\exp\big{(}-\tfrac{3}{16}\langle\mathcal{M}_{\cdot}^{N}(\vartheta;u)\rangle_{T}\big{)}\big{]}^{1/3},$

thanks to Hölder’s inequality and the martingale property of
$(\tfrac{3}{4}\mathcal{M}_{t}^{N}(\vartheta;u))_{t\in[0,T]}$. With the help of
the parametrisation
$u=(N\mathbb{I}_{\mathcal{G}}(\vartheta))^{1/2}(\vartheta^{\prime}-\vartheta)$,
we rewrite $\langle\mathcal{M}_{\cdot}^{N}(\vartheta;u)\rangle_{T}$ as

$\displaystyle\sum_{i=1}^{N}\int_{0}^{T}\big{|}(c^{-1/2}b)(\vartheta^{\prime};t,X_{t}^{i},\mu_{t}^{(N)})-(c^{-1/2}b)(\vartheta;t;X_{t}^{i},\mu_{t}^{(N)})\big{|}^{2}dt$
$\displaystyle=\sum_{i=1}^{N}\sum_{j=1}^{d}\int_{0}^{T}\big{(}((N\mathbb{I}_{\mathcal{G}}(\vartheta))^{-1/2}u)^{\top}\int_{0}^{1}\nabla_{\vartheta}(c^{-1/2}b)^{j}(\vartheta+\lambda(\vartheta^{\prime}-\vartheta);t,X_{t}^{i},\mu_{t}^{(N)})d\lambda\big{)}^{2}dt$
$\displaystyle=u^{\top}\mathbb{I}_{\mathcal{G}}(\vartheta)^{-1/2}\Sigma^{N}([\vartheta,\vartheta^{\prime}];u)\mathbb{I}_{\mathcal{G}}(\vartheta)^{-1/2}u,$

with

$\displaystyle\Sigma^{N}([\vartheta,\vartheta^{\prime}];u)=N^{-1}\sum_{i=1}^{N}\sum_{j=1}^{d}\int_{0}^{T}$
$\displaystyle\nabla_{\vartheta}(c^{-1/2}b)^{j}([\vartheta,\vartheta^{\prime}];t,X_{t}^{i},\mu_{t}^{(N)})(\nabla_{\vartheta}(c^{-1/2}b)^{j}([\vartheta,\vartheta^{\prime}];t,X_{t}^{i},\mu_{t}^{(N)}))^{\top}dt$

which converges by Lemma 21 to

$\displaystyle\Sigma([\vartheta,\vartheta^{\prime}];u)=\sum_{j=1}^{d}\int_{0}^{T}\int_{\mathbb{R}^{d}}$
$\displaystyle\nabla_{\vartheta}(c^{-1/2}b)^{j}([\vartheta,\vartheta^{\prime}];t,x,\mu_{t}^{\vartheta})(\nabla_{\vartheta}(c^{-1/2}b)^{j}([\vartheta,\vartheta^{\prime}];t,x,\mu_{t}^{\vartheta}))^{\top}\mu_{t}(dx)dt$

in $\mathbb{P}_{\vartheta}^{N}$-probability. Abbreviating further
$\widetilde{\Sigma}^{N}(\vartheta;u)=\mathbb{I}_{\mathcal{G}}(\vartheta)^{-1/2}\Sigma^{N}([\vartheta,\vartheta^{\prime}];u)\mathbb{I}_{\mathcal{G}}(\vartheta)^{-1/2}$
and
$\widetilde{\Sigma}(\vartheta;u)=\mathbb{I}_{\mathcal{G}}(\vartheta)^{-1/2}\Sigma([\vartheta,\vartheta^{\prime}];u)\mathbb{I}_{\mathcal{G}}(\vartheta)^{-1/2}$,
we have

$\displaystyle\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\mathcal{Z}_{N}(\vartheta;u)^{1/2}\big{]}$
$\displaystyle\leq\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\exp\big{(}-\tfrac{3}{16}u^{\top}\widetilde{\Sigma}^{N}(\vartheta;u)u\big{)}\big{]}^{1/3}$
(51)
$\displaystyle\leq\mathbb{P}_{\vartheta}^{N}\big{(}u^{\top}\widetilde{\Sigma}^{N}(\vartheta;u)u\leq\tfrac{1}{2}u^{\top}\widetilde{\Sigma}(\vartheta;u)u\big{)}^{1/3}+\exp\big{(}-\tfrac{1}{32}u^{\top}\widetilde{\Sigma}(\vartheta;u)u\big{)}.$

The non-degeneracy assumption (recall Definition 14) ensures
$u^{\top}\widetilde{\Sigma}(\vartheta;u)u\geq\mathfrak{c}|u|^{2}$ for some
$\mathfrak{c}>0$ that does not depend on $\vartheta$ nor $u$ (but that depends
on $\Theta$), hence the remainder term decays faster than any power of $|u|$.
The first term in the right-hand side of (51) is bounded above by

$\displaystyle\mathbb{P}_{\vartheta}^{N}\big{(}\big{|}u^{\top}\big{(}\widetilde{\Sigma}^{N}(\vartheta;u)-\widetilde{\Sigma}(\vartheta;u)\big{)}u\big{|}\geq\tfrac{1}{2}u^{\top}\widetilde{\Sigma}(\vartheta;u)u\big{)}^{\tfrac{1}{3}}$
$\displaystyle\leq\mathbb{P}_{\vartheta}^{N}\Big{(}\big{|}u^{\top}\big{(}\widetilde{\Sigma}^{N}(\vartheta;u)-\widetilde{\Sigma}(\vartheta;u)\big{)}u\big{|}\geq\tfrac{1}{2}\mathfrak{c}|u|^{2}\Big{)}^{\tfrac{1}{3}}$
$\displaystyle\leq
C|u|^{-\tfrac{2m}{3}}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\big{|}u^{\top}\big{(}\widetilde{\Sigma}^{N}(\vartheta;u)-\widetilde{\Sigma}(\vartheta;u)\big{)}u\big{|}^{m}\big{]}^{\tfrac{1}{3}}$

for every $m>0$ by Markov’s inequality and where we used the non-degeneracy
assumption again. For $1\leq\ell,\ell^{\prime}\leq p$, introduce

$\displaystyle\phi_{\ell,\ell^{\prime}}([\vartheta,\vartheta^{\prime}];t,x,\nu)$
$\displaystyle=(\partial_{\vartheta_{\ell}}(c^{-1/2}b)([\vartheta,\vartheta^{\prime}];t,x,\nu))^{\top}\partial_{\vartheta_{\ell}}(c^{-1/2}b)([\vartheta,\vartheta^{\prime}];t,x,\nu),$

so that
$\big{(}\Sigma^{N}(\vartheta;u)-\Sigma(\vartheta;u)\big{)}_{\ell,\ell^{\prime}}$
is simply

$\displaystyle
N^{-1}\sum_{i=1}^{N}\int_{0}^{T}\phi_{\ell,\ell^{\prime}}([\vartheta,\vartheta^{\prime}];t,X_{t}^{i},\mu_{t}^{(N)})dt-\int_{0}^{T}\int_{\mathbb{R}^{d}}\phi_{\ell,\ell^{\prime}}([\vartheta,\vartheta^{\prime}];t,x,\mu_{t}^{\vartheta})\mu_{t}(dx)dt.$

By Lemma 21, we derive

$\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\big{|}\big{(}\Sigma^{N}(\vartheta;u)-\Sigma(\vartheta;u)\big{)}_{\ell,\ell^{\prime}}\big{|}^{m}\big{]}\leq
CN^{-\delta m},$

therefore

$\displaystyle\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\big{|}u^{\top}\big{(}\widetilde{\Sigma}^{N}(\vartheta;u)-\widetilde{\Sigma}(\vartheta;u)\big{)}u\big{|}^{m}\big{]}^{1/3}\leq
C|u|^{2m/3}N^{-\delta m/3},$

and finally

(52)
$\mathbb{P}_{\vartheta}^{N}\big{(}\big{|}u^{\top}\big{(}\widetilde{\Sigma}^{N}(\vartheta;u)-\widetilde{\Sigma}(\vartheta;u)\big{)}u\big{|}\geq\tfrac{1}{2}u^{\top}\widetilde{\Sigma}(\vartheta;u)u\big{)}^{\tfrac{1}{3}}\leq
CN^{-\delta m/3}$

Pick $r\geq 1$. Combining (51) and (52), we infer

$\displaystyle|u|^{r}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\mathcal{Z}_{N}(\vartheta;u)^{1/2}\big{]}$
$\displaystyle\leq C|u|^{r}N^{-\delta
m/3}+|u|^{r}\exp(-\tfrac{\mathfrak{c}}{32}|u|^{2}).$

For
$u=(N\mathbb{I}_{\mathcal{G}}(\vartheta))^{1/2}(\vartheta^{\prime}-\vartheta)$
with $\vartheta^{\prime}\in\Theta$, we have $|u|\leq CN^{1/2}$. The first term
in the right-hand side of the previous estimate is thus bounded as soon a
$m\geq 3r/(2\delta)$. The second term is bounded uniformly in $|u|$. We thus
have established (50).

Step 3. We are now ready to apply Theorem III.1.1 of [24] and gather several
properties of the maximum likelihood estimator. Note that the continuity of
the likelihood function $\vartheta\mapsto\mathcal{L}^{N}(\vartheta;X^{(N)})$
and the compactness of $\Theta$ ensures that a solution
$\widehat{\vartheta}_{N}^{\,\tt{mle}}$ to (23) exists.

The uniform LAN condition given in the proof of Theorem 17 is the Condition N1
of Chapter III [24]. The non-degeneracy assumption (according to Definition
(14)) is related to the uniform use of Condition N2 of [24]. Step 1 and Step 2
are respectively Condition N3 and N4 of [24]. We may thus apply Theorem
III.1.1 of [24] and we readily obtain Statement (i) of Theorem 19. Statement
(ii) is a consequence of Corollary III.1.1 of [24] while Statement (iii) is a
consequence of Theorem III.1.3 of [24].

The proof of Theorem 19 is complete.

##  6\. Remaining proofs

###  6.1. Proof of Proposition 9

Anticipating the proof of Lemma 20, we prove a slightly stronger result,
namely

(53)
$\big{(}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|\overline{X}_{t}^{i,\vartheta}-\overline{X}_{t}^{i,\vartheta^{\prime}}|^{r}\big{]}\big{)}^{1/r}\leq
C|\vartheta-\vartheta^{\prime}|$

for $r\geq 1$. Indeed, Proposition 9 is then a consequence of

$\mathcal{W}_{1}(\mu_{t}^{\vartheta},\mu_{t}^{\vartheta^{\prime}})\leq\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|\overline{X}_{t}^{i,\vartheta}-\overline{X}_{t}^{i,\vartheta^{\prime}}|\big{]}\leq\big{(}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|\overline{X}_{t}^{i,\vartheta}-\overline{X}_{t}^{i,\vartheta^{\prime}}|^{2r}\big{]}\big{)}^{1/2r},$

for any $r\geq 1$. From
$\overline{X}_{0}^{i,\vartheta}=\overline{X}_{0}^{i,\vartheta^{\prime}}$, we
have

$\overline{X}_{t}^{i,\vartheta}-\overline{X}_{t}^{i,\vartheta^{\prime}}=\int_{0}^{t}\big{(}b(\vartheta;s,\overline{X}_{s}^{i,\vartheta},\mu_{s}^{\vartheta})-b(\vartheta^{\prime};s,\overline{X}_{s}^{i,\vartheta^{\prime}},\mu_{s}^{\vartheta^{\prime}})\big{)}ds+\int_{0}^{t}\big{(}\sigma(s,\overline{X}_{s}^{i,\vartheta})-\sigma(s,\overline{X}_{s}^{i,\vartheta^{\prime}})\big{)}dB_{s}^{i,N,\vartheta}.$

Using Assumption 2 and the Burckholder-Davis-Gundy inequality, we infer

$\displaystyle\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\Big{[}\Big{|}\int_{0}^{t}\big{(}\sigma(s,\overline{X}_{s}^{i,\vartheta})-\sigma(s,\overline{X}_{s}^{i,\vartheta^{\prime}})\big{)}dB_{s}^{i,N,\vartheta}\Big{|}^{2r}\Big{]}$
$\displaystyle\leq
C\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\Big{[}\Big{(}\int_{0}^{t}|\overline{X}_{s}^{i,\vartheta}-\overline{X}_{s}^{i,\vartheta^{\prime}}\big{|}^{2}ds\Big{)}^{r}\Big{]}$
$\displaystyle\leq
C\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\Big{[}\int_{0}^{t}|\overline{X}_{s}^{i,\vartheta}-\overline{X}_{s}^{i,\vartheta^{\prime}}\big{|}^{2r}ds\Big{]}$

since $r\geq 1$. Thanks to the smoothness properties of $b$ granted by
Assumptions 3 and 4 and incorporating the previous estimate, we obtain

$\displaystyle\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|\overline{X}_{t}^{i,\vartheta}-\overline{X}_{t}^{i,\vartheta^{\prime}}|^{2r}\big{]}$
$\displaystyle\leq
C\int_{0}^{t}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|\vartheta-\vartheta^{\prime}|^{2r}(1+|\overline{X}_{s}^{i,\vartheta}|^{2r_{1}r}+\mathfrak{m}_{r_{2}}(\mu_{s}^{\vartheta})^{2r})+|\overline{X}_{s}^{i,\vartheta}-\overline{X}_{s}^{i,\vartheta^{\prime}}|^{2r}+\mathcal{W}_{1}(\mu_{s}^{\vartheta},\mu_{s}^{\vartheta^{\prime}})^{2r}\big{]}ds$
$\displaystyle\leq
C\big{(}|\vartheta-\vartheta^{\prime}|^{2r}+\int_{0}^{t}\big{(}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|\overline{X}_{s}^{i,\vartheta}-\overline{X}_{s}^{i,\vartheta^{\prime}}|^{2r}+\mathcal{W}_{1}(\mu_{s}^{\vartheta},\mu_{s}^{\vartheta^{\prime}})^{2r}\big{]}ds\big{)}$

where we used that
$\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}[|\overline{X}_{t}^{i,\vartheta}|^{r^{\prime}}]=\mathfrak{m}_{r^{\prime}}(\mu_{t}^{\vartheta})$
is bounded uniformly in $t\in[0,T]$ and $\vartheta\in\Theta$ for all values of
$r^{\prime}\geq 1$ by Lemma 8. We obtain (53) for $2r$ by Grönwall’s lemma,
hence for every $r\geq 1$ by Cauchy-Schwarz’s inequality. The proposition
follows.

###  6.2. Proof of Proposition 10

By Girsanov’s theorem,

$\displaystyle\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}^{\otimes
N}}\Big{[}\log\frac{d\overline{\mathbb{P}}_{\vartheta}^{\otimes
N}}{d\mathbb{P}_{\vartheta}^{N}}\Big{]}$
$\displaystyle=\tfrac{1}{2}\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}^{\otimes
N}}\Big{[}\sum_{i=1}^{N}\int_{0}^{T}|b(\vartheta;t,X_{t}^{i},\mu_{t}^{(N)})-b(\vartheta;t,X_{t}^{i},\mu_{t}^{\vartheta})|^{2}dt$
$\displaystyle=\tfrac{1}{2}\sum_{i=1}^{N}\int_{0}^{T}\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}^{\otimes
N}}\big{[}\big{|}N^{-1}\sum_{j=1}^{N}\big{(}\widetilde{b}(\vartheta;t,X_{t}^{i},X_{t}^{j})-\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}}[\widetilde{b}(\vartheta;t,\zeta,X_{t}^{j})]_{\zeta=X_{t}^{i}}\big{)}\big{|}^{2}\big{]}dt.$

We plan to use the following decomposition

$\displaystyle
N^{-1}\sum_{j=1}^{N}\widetilde{b}(\vartheta;t,X_{t}^{i},X_{t}^{j})-\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}}\big{[}\widetilde{b}(\vartheta;t,\zeta,X_{t}^{j})\big{]}_{\zeta=X_{t}^{i}}$
$\displaystyle=N^{-1}\big{(}\widetilde{b}(\vartheta;t,X_{t}^{i},X_{t}^{i})-\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}}[\widetilde{b}(\vartheta;t,\zeta,X_{t}^{i})]_{\zeta=X_{t}^{i}}\big{)}$
$\displaystyle+\tfrac{N-1}{N}(N-1)^{-1}\sum_{j=1,j\neq
i}^{N}\big{(}\widetilde{b}(\vartheta;t,X_{t}^{i},X_{t}^{j})-\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}^{\otimes
2}}[\widetilde{b}(\vartheta;t,X_{t}^{i},X_{t}^{j})\,|X_{t}^{i}]\big{)}.$

Using the elementary inequality
$(a+b)^{2}\leq(1+\rho)a^{2}+(1+\rho^{-1})b^{2}$ valid for every $\rho>0$, we
obtain

$\displaystyle\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}^{\otimes
N}}\big{[}\big{|}N^{-1}\sum_{j=1}^{N}\widetilde{b}(\vartheta;t,X_{t}^{i},X_{t}^{j})-\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}}[\widetilde{b}(\vartheta;t,\zeta,X_{t}^{j})]_{\zeta=X_{t}^{i}}\big{|}^{2}\big{]}$
$\displaystyle\leq(1+\rho^{-1})N^{-2}\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}}\big{[}|\widetilde{b}(\vartheta;t,X_{t}^{1},X_{t}^{1})|^{2}\big{]}+(1+\rho)\frac{N-1}{N^{2}}\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}^{\otimes
2}}\big{[}|\widetilde{b}(\vartheta;t,X_{t}^{1},X_{t}^{2})|^{2}\big{]},$

and therefore

$\limsup_{N\rightarrow\infty}\sup_{\vartheta\in\Theta}\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}^{\otimes
N}}\Big{[}\log\frac{d\overline{\mathbb{P}}_{\vartheta}^{\otimes
N}}{d\mathbb{P}_{\vartheta}^{N}}\Big{]}\leq\frac{1+\rho}{2}\sup_{\vartheta\in\Theta}\int_{0}^{T}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|\widetilde{b}(\vartheta;t,x,y)|^{2}(\mu_{t}^{\vartheta}\otimes\mu_{t}^{\vartheta})(dx,dy)dt.$

By assumption,

$\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|\widetilde{b}(\vartheta;t,x,y)|^{2}(\mu_{t}^{\vartheta}\otimes\mu_{t}^{\vartheta})(dx,dy)\leq
C\big{(}1+\sup_{t\in[0,T],\vartheta\in\Theta}(\mathfrak{m}_{r_{1}}(\mu_{t}^{\vartheta})+\mathfrak{m}_{r_{2}}(\mu_{t}^{\vartheta}))\big{)}$

which is finite by Lemma 8. We thus obtain (15). In order to obtain (16), we
simply apply Pinsker’s inequality:

$\displaystyle\limsup_{N\rightarrow\infty}\sup_{\vartheta\in\Theta}\|\mathbb{P}_{\vartheta}^{N}-\overline{\mathbb{P}}_{\vartheta}^{\otimes
N}\|_{TV}^{2}$
$\displaystyle\leq\tfrac{1}{2}\limsup_{N\rightarrow\infty}\sup_{\vartheta\in\Theta}\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}^{\otimes
N}}\Big{[}\log\frac{d\overline{\mathbb{P}}_{\vartheta}^{\otimes
N}}{d\mathbb{P}_{\vartheta}^{N}}\Big{]}$
$\displaystyle\leq\frac{1+\rho}{4}\sup_{\vartheta\in\Theta}\int_{0}^{T}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|\widetilde{b}(\vartheta;t,x,y)|^{2}(\mu_{t}^{\vartheta}\otimes\mu_{t}^{\vartheta})(dx,dy).$

The conclusion follows by taking $\rho$ sufficiently small.

###  6.3. Proof of Proposition 11

Let

$\big{(}\phi_{\ell,\ell^{\prime}}(\vartheta;t,x,\nu)\big{)}_{1\leq\ell,\ell^{\prime}\leq
p}=\big{(}\partial_{\vartheta_{\ell}}(c^{-1/2}b)(\vartheta;t,x,\mu^{\vartheta}_{t})^{\top}\partial_{\vartheta_{\ell^{\prime}}}(c^{-1/2}b)(\vartheta;t,x,\mu^{\vartheta}_{t})\big{)}_{1\leq\ell,\ell^{\prime}\leq
p}.$

We have

$\displaystyle\big{(}\mathbb{I}_{\mathcal{G}}(\vartheta)\big{)}_{\ell,\ell^{\prime}}-\big{(}\mathbb{I}_{\mathcal{G}}(\vartheta^{\prime})\big{)}_{\ell,\ell^{\prime}}$
$\displaystyle=\int_{0}^{T}\big{(}\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}}\big{[}\phi_{\ell,\ell^{\prime}}(\vartheta;t,X_{t},\mu_{t}^{\vartheta})\big{]}-\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta^{\prime}}}\big{[}\phi_{\ell,\ell^{\prime}}(\vartheta^{\prime};t,X_{t},\mu_{t}^{\vartheta^{\prime}})\big{]}\big{)}dt$
$\displaystyle=\int_{0}^{T}\big{(}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\phi_{\ell,\ell^{\prime}}(\vartheta;t,\overline{X}_{t}^{1,\vartheta},\mu_{t}^{\vartheta})-\phi_{\ell,\ell^{\prime}}(\vartheta^{\prime};t,\overline{X}_{t}^{1,\vartheta^{\prime}},\mu_{t}^{\vartheta^{\prime}})\big{]}\big{)}dt.$

Thanks to the smoothness properties of $b$ and $\sigma$ granted by Assumptions
2, 3 and 4, we have

$\displaystyle\big{|}\phi_{\ell,\ell^{\prime}}(\vartheta;t,\overline{X}_{t}^{1,\vartheta},\mu_{t}^{\vartheta})-\phi_{\ell,\ell^{\prime}}(\vartheta^{\prime};t,\overline{X}_{t}^{1,\vartheta^{\prime}},\mu_{t}^{\vartheta^{\prime}})\big{|}$
$\displaystyle\leq
C\big{(}|\vartheta-\vartheta^{\prime}|+|\overline{X}_{t}^{1,\vartheta}-\overline{X}_{t}^{1,\vartheta^{\prime}}|+\mathcal{W}_{1}(\mu_{t}^{\vartheta^{\prime}},\mu_{t}^{\vartheta})\big{)}\big{(}1+|\overline{X}_{t}^{1,\vartheta}|^{\alpha}+|\overline{X}_{t}^{1,\vartheta^{\prime}}|^{\alpha}+\mathfrak{m}_{\beta}(\mu_{t}^{\vartheta})+\mathfrak{m}_{\beta}(\mu_{t}^{\vartheta^{\prime}})\big{)}.$

We have
$\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}[|\overline{X}_{t}^{1,\vartheta}|^{r}]=\mathfrak{m}_{r}(\mu_{t}^{\vartheta})$,
which is uniformly bounded in $t\in[0,T],\vartheta\in\Theta$ for every $r\geq
1$ by Lemma 8. Likewise
$\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}[|\overline{X}_{t}^{1,\vartheta^{\prime}}|^{r}]\leq
C(\mathfrak{m}_{r}(\mu_{t}^{\vartheta})+\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}[|\overline{X}_{t}^{1,\vartheta}-\overline{X}_{t}^{1,\vartheta^{\prime}}|^{r}])$,
therefore, by Cauchy-Schwarz’s inequality

$\displaystyle\big{|}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\phi_{\ell,\ell^{\prime}}(\vartheta,t,\overline{X}_{t}^{1,\vartheta},\mu_{t}^{\vartheta})-\phi_{\ell,\ell^{\prime}}(\vartheta^{\prime},t,\overline{X}_{t}^{1,\vartheta^{\prime}},\mu_{t}^{\vartheta^{\prime}})\big{]}\big{|}$
$\displaystyle\leq
C\big{(}|\vartheta-\vartheta^{\prime}|+(\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}[|\overline{X}_{t}^{1,\vartheta}-\overline{X}_{t}^{1,\vartheta^{\prime}}|^{2}])^{1/2}+\mathcal{W}_{1}(\mu_{t}^{\vartheta},\mu_{t}^{\vartheta^{\prime}})\big{)}(1+(\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}[|\overline{X}_{t}^{1,\vartheta}-\overline{X}_{t}^{1,\vartheta^{\prime}}|^{2}])^{1/2})$

and the Lipschitz smoothness follows by applying (40) of Lemma 20 and
Proposition 9. The convergence
$N^{-1}\mathbb{I}_{\mathcal{E}^{N}}(\vartheta)\rightarrow\mathbb{I}_{\mathcal{G}}(\vartheta)$
is a simple consequence of Lemma 21.

###  6.4. Proof of Proposition 16

Note that for every $1\leq\ell\leq p$:

$\partial_{\vartheta_{\ell}}\ell^{N}(\vartheta;X^{N})=G^{N}(X^{(N)})_{\ell}+2\sum_{\ell^{\prime}=1}^{p}\vartheta_{\ell^{\prime}}H^{N}(X^{(N)})_{\ell,\ell^{\prime}}.$

Let $\vartheta,\vartheta^{\prime}$ in $\Theta_{0}$ be such that
$\mathbb{P}_{\vartheta}^{N}=\mathbb{P}_{\vartheta^{\prime}}^{N}$ for every
$N\geq 1$. (This also implies
$\overline{\mathbb{P}}_{\vartheta}=\overline{\mathbb{P}}_{\vartheta^{\prime}}$
by Lemma 21 for instance.) Using the symmetry of $H^{N}(X^{(N)})$, we note
that

$\vartheta^{\top}H^{N}(X^{(N)})\vartheta-(\vartheta^{\prime})^{\top}H^{N}(X^{(N)})\vartheta^{\prime}=\sum_{\ell=1}^{p}(\vartheta_{\ell}-\vartheta^{\prime}_{\ell})\sum_{\ell^{\prime}=1}^{p}H^{N}(X^{(N)})_{\ell,\ell^{\prime}}(\vartheta_{\ell^{\prime}}+\vartheta^{\prime}_{\ell^{\prime}}).$

It follows that

$\displaystyle
0=\ell^{N}(\vartheta;X^{(N)})-\ell^{N}(\vartheta^{\prime};X^{(N)})$
$\displaystyle=\sum_{\ell=1}^{p}(\vartheta_{\ell}-\vartheta^{\prime}_{\ell})\Big{(}G^{N}(X^{(N)})_{\ell}+2\sum_{\ell^{\prime}=1}^{p}H^{N}(X^{(N)})_{\ell,\ell^{\prime}}\tfrac{\vartheta_{\ell^{\prime}}+\vartheta^{\prime}_{\ell^{\prime}}}{2}\Big{)}$
$\displaystyle=\sum_{\ell=1}^{p}\big{(}G^{N}(X^{(N)})_{\ell}+2\sum_{\ell^{\prime}=1}^{p}\tfrac{\vartheta_{\ell^{\prime}}+\vartheta^{\prime}_{\ell^{\prime}}}{2}H^{N}(X^{(N)})_{\ell,\ell^{\prime}}\big{)}(\vartheta_{\ell}-\vartheta_{\ell}^{\prime})$
$\displaystyle=\sum_{\ell=1}^{p}\partial_{\vartheta_{\ell}}\ell^{N}(\vartheta^{\star};X^{(N)})\xi_{\ell}=\nabla_{\vartheta}\ell^{N}(\vartheta^{\star};X^{(N)})^{\top}\xi,$

with $\xi=\vartheta-\vartheta^{\prime}$ and
$\vartheta^{\star}=\tfrac{1}{2}(\vartheta+\vartheta^{\prime})\in\Theta_{0}$ by
the convexity of $\Theta_{0}$ and that does not depend on $X^{(N)}$. Assume
now that $\vartheta\neq\vartheta^{\prime}$. This implies that for some
$\xi\neq 0$, we have

$0=(\nabla_{\vartheta}\ell^{N}(\vartheta^{\star};X^{(N)})^{\top}\xi)^{\top}\nabla_{\vartheta}\ell^{N}(\vartheta^{\star};X^{(N)})^{\top}\xi=\xi^{\top}\mathbb{I}_{\mathcal{E}^{N}}(\vartheta^{\star})\xi.$

Thus $\mathbb{I}_{\mathcal{E}^{N}}(\vartheta^{\star})$ is degenerate for every
$N\geq 1$. Letting $N\rightarrow\infty$ and applying Proposition 11, we infer
that $\mathbb{I}_{\mathcal{G}}(\vartheta^{\star})$ is degenerate as well, a
contradiction. The conclusion follows for $\mathcal{G}$ likewise.

##  7\. Appendix

###  7.1. Proof of Lemma 7

By Assumption 3 we have
$b_{0}=\sup_{t\in[0,T]}|b(\vartheta_{0};t,0,\delta_{0})|<\infty$ for some
$\vartheta_{0}\in\Theta$. Combined with Assumption 4, since $\Theta$ is
compact, we infer

$|b(\vartheta;s,X_{s}^{i},\mu_{s}^{(N)})|\leq
C(1+|X_{s}^{i}|+N^{-1}\sum_{i=1}^{N}|X_{s}^{i}|)$

uniformly in $s\in[0,T]$ and $\vartheta\in\Theta$. For $M>0$, define
$\tau_{M}=\inf\\{s\geq 0,\max_{1\leq i\leq N}|X_{s}^{i}|\geq M\\}\wedge T$ and
note that $\tau_{M}$ is a $(\mathcal{F}_{t})$-stopping time. We have

$\displaystyle|X_{t\wedge\tau_{M}}^{i}|$
$\displaystyle\leq|X_{0}^{i}|+\int_{0}^{t\wedge\tau_{M}}|b(\vartheta,s,X_{s}^{i},\mu_{s}^{(N)})|ds+\big{|}\int_{0}^{t\wedge\tau_{M}}\sigma(s,X_{s}^{i})dB_{s}^{i,N,\vartheta}\big{|}$
$\displaystyle\leq|X_{0}^{i}|+C\int_{0}^{t\wedge\tau_{M}}(1+|X_{s}^{i}|+N^{-1}\sum_{i=1}^{N}|X_{s}^{i}|)ds+\big{|}\int_{0}^{t\wedge\tau_{M}}\sigma(s,X_{s}^{i})dB_{s}^{i,N,\vartheta}\big{|}$
$\displaystyle\leq|X_{0}^{i}|+C\int_{0}^{t}(1+|X_{s\wedge\tau_{M}}^{i}|+N^{-1}\sum_{i=1}^{N}|X_{s\wedge\tau_{M}}^{i}|)ds+\sup_{0\leq
t\leq T}\big{|}\int_{0}^{t}\sigma(s,X_{s}^{i})dB_{s}^{i,N,\vartheta}\big{|}$

Taking $\mathbb{P}_{\vartheta}^{N}$-expectation of order $r\geq 1$, we obtain

$\displaystyle\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|X_{t\wedge\tau_{M}}^{i}|^{r}\big{]}$
$\displaystyle\leq
C\big{(}1+\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|X_{0}^{i}|^{r}\big{]}+\int_{0}^{t}(1+\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|X_{s\wedge\tau_{M}}^{i}|^{r}\big{]}ds\big{)},$

using Jensen’s inequality, the exchangeability of $\mathbb{P}_{\vartheta}^{N}$
and the Burckholder-Davis-Gundy inequality together with Assumption 2 to
obtain

$\mathbb{E}\big{[}\sup_{0\leq t\leq
T}\big{|}\int_{0}^{t}\sigma(s,X_{s}^{i})dB_{s}^{i,N,\vartheta}\big{|}^{r}\big{]}\leq
C\mathbb{E}\big{[}\big{(}\int_{0}^{t}|\sigma(s,X_{s}^{i})|^{2}s\big{)}^{r/2}\big{]}\leq
C^{\prime}.$

By Grönwall’s lemma, we infer

$\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|X_{t\wedge\tau_{M}}^{i}|^{r}\big{]}\leq(\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|X_{0}^{i}|^{r}\big{]}+C^{\prime})\exp(Ct).$

Letting $M\rightarrow\infty$, we conclude by Fatou’s lemma.

###  7.2. Proof of Lemma 8

It is a slight variation of the proof of Lemma 7. By Theorem 4.21 in [12], we
have $\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}}[\sup_{0\leq t\leq
T}|X_{t}^{i}|^{r}]<\infty$ for every $r\geq 1$. (Their argument is developed
for $r=2$ but the extension to any $r\geq 1$ is straightforward.) Therefore,
only the uniformity in $\vartheta$ requires a proof. From

$\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|\overline{X}_{t}^{i,\vartheta}|^{r}\big{]}\leq
C\big{(}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|X_{t}^{i}-\overline{X}_{t}^{i,\vartheta}|^{r}\big{]}+\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|X_{t}^{i}|^{r}\big{]}\big{)},$

and the fact that
$\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|\overline{X}_{t}^{i,\vartheta}|^{r}\big{]}=\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}}\big{[}|X_{t}^{i,\vartheta}|^{r}\big{]}$,
Lemma 8 is now a simple consequence of (42) in Lemma 20 together with Lemma 7.

###  7.3. Proof of Lemma 20

#### Proof of (39)

The first inequality is obvious. Then, since
$X_{0}^{i}=X_{0}^{i,\vartheta^{\prime}}$, we have

$X_{t}^{i}-X_{t}^{i,\vartheta^{\prime}}=\int_{0}^{t}\big{(}b(\vartheta;s,X_{s}^{i},\mu_{s}^{(N)})-b(\vartheta^{\prime};s,X_{s}^{i,\vartheta^{\prime}},\mu_{s}^{(N),\vartheta^{\prime}})\big{)}ds+\int_{0}^{t}\big{(}\sigma(s,X_{s}^{i})-\sigma(s,X_{s}^{i,\vartheta^{\prime}})\big{)}dB_{s}^{i,N,\vartheta}.$

Thanks to the smoothness properties of $b$ and $\sigma$ granted by Assumptions
2, 3 and 4, taking expectation to the power $r$ on both side and applying the
Burckholder-Davis-Gundy inequality, we infer

$\displaystyle\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|X_{t}^{i}-X_{t}^{i,\vartheta^{\prime}}|^{r}\big{]}$
$\displaystyle\leq
C\int_{0}^{t}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|\vartheta-\vartheta^{\prime}|^{r}(1+|X_{s}^{i}|^{r_{1}}+\mathfrak{m}_{r_{2}}(\mu_{s}^{(N)}))^{r}+|X_{s}^{i}-X_{s}^{i,\vartheta^{\prime}}|^{r}+\mathcal{W}_{1}(\mu_{s}^{(N),\vartheta^{\prime}},\mu_{s}^{(N)})^{r}\big{]}ds$
$\displaystyle\leq
C\big{(}|\vartheta-\vartheta^{\prime}|^{r}+\int_{0}^{t}\big{(}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|X_{s}^{i}-X_{s}^{i,\vartheta^{\prime}}|^{r}+\mathcal{W}_{1}(\mu_{s}^{(N),\vartheta^{\prime}},\mu_{s}^{(N)})^{r}\big{]}\big{)}ds\big{)}$

where we used that
$\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\mathfrak{m}_{r_{2}}(\mu_{t}^{(N)})^{r}\big{]}\leq\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}[|X_{t}^{i}|^{r_{2}r}]$
which is bounded uniformly in $t\in[0,T]$ and $\vartheta\in\Theta$ by Lemma 7.
Also, using the first part of (39), namely

$\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\mathcal{W}_{1}(\mu_{s}^{(N),\vartheta^{\prime}},\mu_{s}^{(N)})^{r}\big{]}\leq
CN^{-1}\sum_{i=1}^{N}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|X_{s}^{i}-X_{s}^{i,\vartheta^{\prime}}|^{r}\big{]}$

and taking averages over $i=1,\ldots,N$ on both sides, we infer

$\displaystyle
N^{-1}\sum_{i=1}^{N}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|X_{t}^{i}-X_{t}^{i,\vartheta^{\prime}}|^{r}\big{]}\leq
C\big{(}|\vartheta-\vartheta^{\prime}|^{r}+\int_{0}^{t}N^{-1}\sum_{i=1}^{N}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|X_{s}^{i}-X_{s}^{i,\vartheta^{\prime}}|^{r}\big{]}ds\big{)}.$

We obtain the second part of (39) by Grönwall’s lemma.

#### Proof of (40)

The first inequality is obvious. The second part is simply (53) from the proof
of Proposition 9.

#### Proof of (41) and (42)

By triangle inequality,

(54)
$\mathcal{W}_{1}(\mu_{t}^{(N)},\mu_{t}^{\vartheta})\leq\mathcal{W}_{1}(\mu_{t}^{(N)},\overline{\mu}_{t}^{(N),\vartheta})+\mathcal{W}_{1}(\overline{\mu}_{t}^{(N),\vartheta},\mu_{t}^{\vartheta})\leq
N^{-1}\sum_{i=1}^{N}|X_{t}^{i}-\overline{X}_{t}^{i,\vartheta}|+\mathcal{W}_{1}(\overline{\mu}_{t}^{(N),\vartheta},\mu_{t}^{\vartheta}).$

By Theorem 2 of [18], we have
$\sup_{t\in[0,T],\vartheta\in\Theta}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\mathcal{W}_{1}(\overline{\mu}_{t}^{(N),\vartheta},\mu_{t}^{\vartheta})^{r}\big{]}\leq
CN^{-\delta r}$ for every $r\geq 1$ and some $\delta>0$. The value of $\delta$
depends on the dimension $d$ of the state space. The uniformity in
$(t,\vartheta)$ follows in particular from the uniform moment bounds of Lemma
8 (see the conditions of Theorem 2 of [18]). Therefore (41) is a consequence
of (42).

In order to establish (42), since $X_{0}^{i}=\overline{X}_{0}^{i,\vartheta}$,
we write

$X_{t}^{i}-\overline{X}_{t}^{i,\vartheta}=\int_{0}^{t}\big{(}b(\vartheta;X_{s}^{i},\mu_{s}^{(N)})-b(\vartheta;s,\overline{X}_{s}^{i,\vartheta},\mu_{s}^{\vartheta})\big{)}ds+\int_{0}^{t}\big{(}\sigma(s,X_{s}^{i})-\sigma(s,\overline{X}_{s}^{i,\vartheta})\big{)}dB_{s}^{i,N,\vartheta}.$

Taking expectation to the power $r$ on both side and applying the Burckholder-
Davis-Gundy inequality, we infer

$\displaystyle\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|X_{t}^{i}-\overline{X}_{t}^{i,\vartheta}|^{r}\big{]}$
$\displaystyle\leq
C\int_{0}^{t}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|X_{t}^{i}-\overline{X}_{s}^{i,\vartheta}|^{r}+\mathcal{W}_{1}(\mu_{s}^{(N)},\mu_{s}^{\vartheta})^{r}\big{]}ds$
$\displaystyle\leq
C\int_{0}^{t}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|X_{t}^{i}-\overline{X}_{s}^{i,\vartheta}|^{r}+\mathcal{W}_{1}(\overline{\mu}_{s}^{(N),\vartheta},\mu_{s}^{\vartheta})^{r}\big{]}ds$
$\displaystyle\leq\varepsilon_{N}+C\int_{0}^{t}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}|X_{t}^{i}-\overline{X}_{s}^{i,\vartheta}|^{r}\big{]}ds,$

arguing as in (54), and where
$\varepsilon_{N}=CT\sup_{t\in[0,T],\vartheta\in\Theta}\mathbb{E}_{\mathbb{P}_{\vartheta}^{N}}\big{[}\mathcal{W}_{1}(\overline{\mu}_{s}^{(N),\vartheta},\mu_{s}^{\vartheta})^{r}\big{]}\leq
CN^{-\delta r}$. Note that the constants in each line are uniformly bounded in
$\vartheta\in\Theta$. We obtain (42) by Grönwall’s lemma.

###  7.4. Proof of Lemma 21

We plan to use the following decomposition

$N^{-1}\sum_{i=1}^{N}\int_{0}^{t}\phi(\vartheta_{N};s,X_{s}^{i},\mu_{s}^{(N)})ds-\int_{0}^{t}\int_{\mathbb{R}^{d}}\phi(\vartheta;s,x,\mu_{s}^{\vartheta})\mu_{s}^{\vartheta}(dx)ds=I+II+III,$

with

$\displaystyle I$
$\displaystyle=N^{-1}\sum_{i=1}^{N}\int_{0}^{t}\big{(}\phi(\vartheta_{N};s,X_{s}^{i},\mu_{s}^{(N)})-\phi(\vartheta_{N};s,\overline{X}_{s}^{i,\vartheta_{N}},\mu_{s}^{\vartheta_{N}})\big{)}ds$
$\displaystyle II$
$\displaystyle=N^{-1}\sum_{i=1}^{N}\int_{0}^{t}\big{(}\phi(\vartheta_{N};s,\overline{X}_{s}^{i,\vartheta_{N}},\mu_{s}^{\vartheta_{N}})-\phi(\vartheta;s,\overline{X}_{s}^{i,\vartheta},\mu_{s}^{\vartheta})\big{)}ds$
$\displaystyle III$
$\displaystyle=N^{-1}\sum_{i=1}^{N}\int_{0}^{t}\phi(\vartheta;s,\overline{X}_{s}^{i,\vartheta},\mu_{s}^{\vartheta})ds-\mathbb{E}_{\overline{\mathbb{P}}_{\vartheta}}\big{[}\int_{0}^{t}\phi(\vartheta;s,\overline{X}_{s}^{i,\vartheta},\mu_{s}^{\vartheta})ds\big{]}.$

Thanks to the properties of $\phi$ the term $I$ is bounded by a constant times

$\displaystyle
N^{-1}\sum_{i=1}^{N}\int_{0}^{t}\big{(}|X_{s}^{i}-\overline{X}_{s}^{i,\vartheta_{N}}|+\mathcal{W}_{1}(\mu_{s}^{(N)},\mu_{s}^{\vartheta_{N}})\big{)}\big{(}1+|\overline{X}_{s}^{i,\vartheta_{N}}|^{\alpha}+|X_{s}^{i}|^{\alpha}+\mathfrak{m}_{\beta}(\mu_{s}^{\vartheta_{N}})+\mathfrak{m}_{\beta}(\mu_{s}^{(N)})\big{)}ds$
$\displaystyle\leq\Big{(}N^{-1}\sum_{i=1}^{N}\int_{0}^{t}\big{(}|X_{s}^{i}-\overline{X}_{s}^{i,\vartheta_{N}}|+\mathcal{W}_{1}(\mu_{s}^{(N)},\mu_{s}^{\vartheta_{N}})\big{)}^{2}ds\Big{)}^{1/2}$
$\displaystyle\;\;\;\times
C\Big{(}N^{-1}\sum_{i=1}^{N}\int_{0}^{t}\big{(}1+|\overline{X}_{s}^{i,\vartheta_{N}}|^{2\alpha}+|X_{s}^{i}|^{2\alpha}+\mathfrak{m}_{2\beta}(\mu_{s}^{\vartheta_{N}})+\mathfrak{m}_{2\beta}(\mu_{s}^{(N)})\big{)}ds\Big{)}^{1/2}.$

by Cauchy-Schwarz’s inequality. Applying Cauchy-Schwarz’s inequality again
together with Jensen’s inequality, the
$\mathbb{P}_{\vartheta_{N}}^{N}$-expectation to the power $m$ of the first
term is then bounded by a constant times

$\displaystyle\mathbb{E}_{\mathbb{P}_{\vartheta_{N}}^{N}}\Big{[}\Big{|}N^{-1}\sum_{i=1}^{N}\int_{0}^{t}\big{(}|X_{s}^{i}-\overline{X}_{s}^{i,\vartheta_{N}}|^{2}+\mathcal{W}_{1}(\mu_{s}^{(N)},\mu_{s}^{\vartheta_{N}})^{2}\big{)}ds\Big{|}^{m}\Big{]}^{1/2}$
$\displaystyle\;\;\times\Big{(}\int_{0}^{t}\big{(}1+\mathbb{E}_{\mathbb{P}_{\vartheta_{N}}^{N}}\big{[}|\overline{X}_{s}^{i,\vartheta_{N}}|^{2m\alpha}+|X_{s}^{i}|^{2m\alpha}\big{]}+\mathfrak{m}_{2m\beta}(\mu_{s}^{\vartheta_{N}})+\mathfrak{m}_{2m\beta}(\mu_{s}^{(N)})\big{)}ds\Big{)}^{1/2}.$

The first term is bounded by a constant times $N^{-m\delta}$ by (41) and (42)
of Lemma 20. Also the $\mathbb{P}_{\vartheta_{N}}^{N}$-expectation of
$|\overline{X}_{s}^{i,\vartheta_{N}}|^{2m\alpha}$, $|X_{s}^{i}|^{2m\alpha}$
and $\mathfrak{m}_{2m\beta}(\mu_{s}^{(N)})$ is uniformly bounded in
$s\in[0,T]$ by Lemma 7 and so is
$\mathfrak{m}_{2m\beta}(\mu_{s}^{\vartheta_{N}})$ by Lemma 8. We conclude

$\mathbb{E}_{\mathbb{P}_{\vartheta_{N}}^{N}}\big{[}\big{|}I\big{|}^{m}\big{]}\leq
CN^{-\delta m}.$

The second term $II$ is bounded by a constant times

$N^{-1}\sum_{i=1}^{N}\int_{0}^{t}\big{(}|\vartheta_{N}-\vartheta|+|\overline{X}_{s}^{i,\vartheta_{N}}-\overline{X}_{s}^{i,\vartheta}|+\mathcal{W}_{1}(\mu_{s}^{\vartheta_{N}},\mu_{s}^{\vartheta})\big{)}ds.$

Taking $\mathbb{P}_{\vartheta_{N}}^{N}$-expectation to the power $m$ and
applying successively the first and second part of (40) in Lemma 20, we obtain