Reflected Mckean-Vlasov stochastic differential equations with jumps in time-
dependent domains

Imane Jarni∗††* Corresponding author.111College of Computing, Mohammed VI
Polytechnic University, Morocco. E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>Badr Missaoui222Moroccan Center of Game Theory,
Mohammed VI Polytechnic University, Morocco. E-mail<EMAIL_ADDRESS>and
Youssef Ouknine 333Africa Business School, Mohammed VI Polytechnic University,
Morocco.444Mathematics Department, Faculty of Sciences Semlalia, Cadi Ayyad
University, Marrakesh, Morocco.555Hassan II Academy of Sciences and
Technologies, Rabat, Morocco<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we investigate the deterministic multidimensional Skorokhod
problem with normal reflection in a family of time-dependent convex domains
that are càdlàg with respect to the Hausdorff metric. We then show the
existence and uniqueness of solutions to multidimensional McKean-Vlasov
stochastic differential equations reflected in these time-dependent domains.
Additionally, we derive stability properties with respect to the initial
condition and the coefficients. Finally, we establish a propagation of chaos
result.

Keyword: Reflection; Skorokhod problem; Time-dependent convex region;
Reflected Mckean-Vlasov; Stability.
AMS Subject Classification: 60H20, 60G07.

##  1 Introduction

Takana in [30] was the first to generalize Skorokhod’s work [25] from one-
dimensional to multidimensional reflection problems, by constructing solutions
reflected in time-independent convex domains. These results were subsequently
extended to encompass a broader class of time-independent domains and driving
processes (see, e.g., [21], [24], [26]). It is worth noting that the
literature primarily focuses on two main types of multidimensional reflection
problems. Those with normal reflection and those with oblique reflection, the
latter of which occurs when the cone of reflection differs from the cone of
inward normal.

In the context of time-dependent domains with oblique reflection, many works
address different types of domains or trajectories (see, for instance, [6],
[8], [13], [16]). Of particular note is the recent work by Nyströ and Önsko
[17], which extends many of the previously cited results. The authors
demonstrate the existence and uniqueness of strong càdlàg solutions for
obliquely reflected SDEs in non-smooth time-dependent domains.

To the best of our knowledge, the case of normal reflection is less developed
in the context of reflected SDEs in time-dependent domains. We quote the work
by Bass and Hsu [2], who demonstrated the existence of a strong solution to
multidimensional SDEs in smooth time-dependent domains and established weak
uniqueness for this equation. However, the seminal reference in the context of
normal reflection is the work by Costantini, Gobet, and El Karoui [7]. In
their work, the authors proved the existence and uniqueness of solutions to
the Skorokhod problem in smooth time-dependent domains for a more general
equation than that considered by [2].

In all previous works, the assumptions made on the domains are geometric,
particularly concerning the regularity of the domain. The one-dimensional
Skorokhod problem reflected in a time-independent interval has been well
investigated (see, e.g., [4], [11], [12] [20], [27]). Typically, assumptions
are made regarding the trajectory of the barriers.

Klimsiak, Rozkosz, and Słomiński [23] proved the existence and uniqueness of
multidimensional BSDEs with continuous driven process with normal reflection
in a time-dependent domain under conditions similar to the one-dimensional
case. Specifically, they assumed that the trajectory of the domain is right-
continuous and left-limited (càdlàg) with respect to the Hausdorff metric.
They also considered the so-called Mokobodzki condition. Additionally, we
mention the work by Fakhouri, Ouknine, and Ren [18], which investigates BSDEs
driven by a Brownian motion and a Poisson point process when the domain is
continuous with respect to the Hausdorff metric. Furthermore, Eddahbi,
Fakhouri, and Ouknine [19] dealt with the general case of càdlàg driven
processes and càdlàg domains with respect to the Hausdorff metric. Note that
all these works about reflected BSDEs are extension of work by Gegout-Petit
and Pardoux in [10] to time-dependent domains. Also, we mention that in the
context of SDEs in time-dependent domains, such an approach has never been
studied in the manner of [23]. Our goal is to study reflected McKean–Vlasov
SDEs in a time-dependent domain in this manner, i.e., by considering
conditions on the regularity of the trajectory of the domain instead of
geometric assumptions.

The McKean–Vlasov SDEs, also known as mean-field SDEs, are a class of SDEs
whose coefficients depend on the law of the solution. These equations have
attracted attention due to their numerous applications, particularly in the
study of large-population particle systems, which are well-formulated in
various scientific fields such as economics, social science, and finance (see,
e.g., [9], [14], [15]).

In the context of reflected McKean–Vlasov equations, Sznitman in [28] was the
first to study the existence and uniqueness of reflected McKean–Vlasov
equations in smooth bounded time-independent domains. Recently, Adam et. al.
in [1] studied reflected McKean–Vlasov diffusions in unbounded convex time-
independent domains driven by a Brownian motion, considering weaker
assumptions on the coefficients of the reflected SDEs.

Our goal in this paper is to investigate reflected McKean–Vlasov equations of
Wiener–Poisson type in time-dependent domains in a similar form to [23]. More
precisely, we are given a family $\mathcal{D}=\\{D_{t},t\in[0,T]\\}$ of time-
dependent càdlàg domains with respect to Hausdorff metric. We are interested
in investigating the following system:

$\textbf{(}\Gamma)\left\\{\begin{aligned}
&X_{t}=X_{0}+\int_{0}^{t}b\left(s,X_{s},\mathcal{L}_{X_{s}}\right)\mathrm{d}s+\int_{0}^{t}\sigma\left(s,X_{s},\mathcal{L}_{X_{s}}\right)\mathrm{d}W_{s}\\\
&\quad\quad+\int_{0}^{t}\int_{\mathbb{R}^{d}\backslash\\{0\\}}\beta\left(s,X_{s^{-}},\mathcal{L}_{X_{s^{-}}},z\right)\tilde{N}(\mathrm{d}s,\mathrm{d}z)+K_{t},\quad
t\in[0,T],\\\ &X_{t}\in D_{t},\quad\text{for every}\;t\in[0,T],\\\ &\text{for
every c\\`{a}dl\\`{a}g and adapted process such that}\;Z_{t}\in
D_{t},\;\int_{0}^{t}\langle X_{s}-Z_{s},\mathrm{d}K_{s}\rangle\leq
0\;\text{for every}\;t\geq 0\end{aligned}\right.,$

where $W$ is a Brownian motion and $\tilde{N}$ is a martingale measure in
$\mathbb{R}^{d}$. The quantities $b,\sigma,\beta$ are measurable random maps
(for more details on the assumptions regarding these quantities, see Section
3), and $\mathcal{L}_{X_{t}}$ means the law of $X_{t}$, $t\geq 0$. By a
solution to reflected SDEs, we mean an adapted càdlàg couple $(X,K)$ that
satisfies the system $(\Gamma)$. The process $K$ is with finite total
variation $|K|$ that acts only when $X$ reaches the boundary.

In [18], [19], and [23], the authors show that the reflected BSDEs in
multidimensional domains are solved under the so-called Mokobodzki condition,
which assumes the existence of a special semimartingale $A$ such that
$A\in\mathring{\mathcal{D}}$. The quadratic variation of the local martingale
part of $A$ is integrable, and the total variation of its finite variation
part is square integrable, with $\inf_{t\in[0,T]}d(A_{t},\partial D_{t})>0$.
In our case, we will consider a weaker assumptions by taking $A$ to be a
càdlàg optional process in $\mathcal{S}_{0,T}^{2}$ (see the next subsection
for notations).

All the papers mentioned above, about reflected SDEs, the construction of
solutions is based on the properties of deterministic mapping on the space of
continuous or right continuous left limited functions. Also in this work, our
approach will be based on the deterministic Skorokhod problem, therefore, our
first goal of this paper is to introduce and solve reflection problem in
deterministic case that suits the type of reflection in the system $(\Gamma)$.

An interesting aspect in the study of Mckean-Vlasov equations is the study of
Propagation of chaos property. Indeed, we know that McKean-Vlasov equations
naturally arise in N-particle systems, where particles interact in a mean-
field way. This interaction is reflected in the dependence of coefficients on
the empirical measure of the system. For a sufficiently large $N$ and under
certain conditions, particle trajectories become asymptotically independent,
even in the presence of initial correlations. This phenomenon can be seen as
the law of large numbers what we commonly refer to as the Propagation of Chaos
property. Another interesting aspect in stochastic dynamical systems theory is
the stability of solutions. The property of stability implies that
trajectories remain relatively unchanged under minor disturbances. Therefore,
one of the goals of this paper is to study the properties of solution
stability and the Propagation of Chaos.

Our main contribution and organization of the paper:

In section 2, we introduce a deterministic Skorokhod problem for càdlàg inputs
in a time-dependent domain that is càdlàg with respect to the Hausdorff
metric. We establish the existence and uniqueness of the solution to this
problem.

Section 3 is divided into four subsections. In the first subsection, we study
the problem of reflection for càdlàg optional processes, then provide some
estimations for solutions, which will be useful for demonstrating the
existence and uniqueness of solutions to the system $(\Gamma)$. In the second
subsection, we demonstrate the existence of solutions to the system $(\Gamma)$
under conditions weaker than the Mokobodzki condition. In the third
subsection, we study the stability of the system $(\Gamma)$ with respect to
both the initial condition and the coefficients. In the last subsection, we
examine the propagation of chaos property of $(\Gamma)$.

It is important to note that our results extend the theory of reflected SDEs
and remain novel even without the McKean–Vlasov component.

###  1.1 Notation and Preliminaries

We introduce the following notations, which will be used throughout this
paper.

  * $\bullet$

$\mathcal{C}:$ The space of all bounded closed convex subsets of
$\mathbb{R}^{d}$ with nonempty interiors endowed with the Hausdorff metric
$d_{H}$, i.e. any $D,D^{\prime}\in\mathcal{C}$,

$d_{H}\left(D,D^{\prime}\right)=\max\left(\sup_{x\in
D}\operatorname{d}\left(x,D^{\prime}\right),\sup_{x\in
D^{\prime}}\operatorname{d}(x,D)\right),$

where $\operatorname{d}(x,D)=\inf_{y\in D}|x-y|$, with |.| is the Euclidean
norm on $\mathbb{R}^{d}.$

  * $\bullet$

$\mathbb{D}([0,T],\mathbb{R}^{d})$: The space of all mapping
$y:[0,T]\to\mathbb{R}^{d}$ with càdlàg trajectories such that
$y_{T}=y_{T^{-}}$.

  * $\bullet$

Let $x,y\in\mathbb{D}([0,T],\mathbb{R}^{d})$. We set
$|x|_{T}:=\sup_{s\in[0,T]}|x_{s}|$. By $d_{\mathbb{S}}$, we mean the Skorokhod
metric on $\mathbb{D}([0,T],\mathbb{R}^{d})$.

  * $\bullet$

Let $a\in\mathbb{D}([0,T],\mathbb{R}^{d})$, $|a|_{T}$ denotes the total
variation on $[0,T]$.

  * $\bullet$

Let $D\in\mathcal{C}$, we denote by $\partial D$ the boundary of $D$. For
$x\in\partial D$, $\mathcal{N}_{x}$ denotes the set of inward normal unit
vectors at $x$ i-e $\eta\in\mathcal{N}_{x}$ iff for any $y\in D$, $\langle
x-y,\eta\rangle\leq 0$.

  * $\bullet$

Let $D\in\mathcal{C}$. For every $y\in\mathbb{R}^{d}$, we denote by
$\Pi_{D}(y)$ the projection of $y$ on $D$.

  * $\bullet$

If $\mathcal{D}=\\{D_{t},\;t\in[0,T]\\}$ is a family of time-dependent sets
such that $D_{t}\in\mathcal{C}$ for every $t\geq 0$, then $D_{t}\subset
B(0,r_{t})$, where $r_{t}$ denotes the radius of the ball $B(0,r_{t})$. For
$z\in\mathbb{D}([0,T],\mathbb{R}^{d})$, we say that $z\in\mathcal{D}$ if
$z_{t}\in D_{t}$ for every $t\geq 0$. We define $\mathring{\mathcal{D}}$ as
the set $\\{\mathring{D}_{t},\;t\in[0,T]\\}$, where $\mathring{D}_{t}$ is the
interior of $D_{t}$.

  * $\bullet$

For a topological space $E$, we denote by $\mathcal{B}(E)$ its Borel
$\sigma$-algebra, and by $\mathcal{P}(E)$, the set of probability measures on
$(E,\mathcal{B}(E))$. We define $\mathcal{P}_{2}(E)$ as the set of all
probability measures on ($E,\mathcal{B}(E)$) with finite second-order moments.
For $\mu,\nu\in\mathcal{P}_{2}(E)$, the second-order Wasserstein distance is
defined by the formula:

$W_{d}(\mu,\nu)=\inf\left\\{\left[\int
d(x,y)^{2}\pi(dx,dy)\right]^{1/2};\pi\in\mathcal{P}(E\times E)\text{ with
marginals }\mu\text{ and }\nu\right\\}\text{. }$

Note that the distance considered in $\mathbb{R}^{d}$ is the Euclidean norm
i-e., $d(x,y)=|x-y|$ for $(x,y)\in\mathbb{R}^{2d}$.

Let $t\in[0,T]$. On the space $\mathbb{D}([0,t],\mathbb{R}^{d})$, we define
the distance
$d(\omega_{1},\omega_{2}):=d_{t}(\omega_{1},\omega_{2})=\sup_{s\in[0,t]}|\omega_{1}(s)-\omega_{2}(s)|$
for every $\omega_{1},\omega_{2}\in\mathbb{D}([0,t],\mathbb{R}^{d})$.

In the following, $W_{t}$ denotes $W_{d_{t}}$ on the space
$\mathcal{P}_{2}(\mathbb{D}([0,t],\mathbb{R}^{d}))$, and $W$ denotes $W_{|.|}$
on the space $\mathcal{P}_{2}(\mathbb{R}^{d})$.

  * •

For $X\in\mathbb{D}([0,T],\mathbb{R}^{d})$, $\mathcal{L}_{X}$ denotes the law
of the process $X$ on $\mathcal{P}(\mathbb{D}([0,T],\mathbb{R}^{d})$.
Similarly, $\mathcal{L}_{X_{t}}$ denotes the law of $X_{t}$ on
$\mathcal{P}(\mathbb{R}^{d})$, where $t\geq 0$.

  * •

If $\mu\in\mathcal{P}(\mathbb{D}([0,T],\mathbb{R}^{d}))$, we denote by
$\mu_{t}$ the push-forward image of the measure $\mu$ under $\Pi_{t}$, and by
$\mu_{t^{-}}$ the push-forward image of the measure $\mu$ under $\Pi_{t^{-}}$.
Here,
$(\Pi_{t},\Pi_{t^{-}}):\mathbb{D}([0,T],\mathbb{R}^{d})\rightarrow\mathbb{R}^{d}\times\mathbb{R}^{d}\text{
is defined as
}\Pi_{t}(\omega)=\omega_{t}\;\text{and}\;\;\Pi_{t^{-}}(\omega)=\omega_{t^{-}}\text{
for }\omega\in\mathbb{D}([0,T],\mathbb{R}^{d})$.

  * •

For $a\in\mathbb{R}^{d},\delta_{a}$ denotes the Dirac measure at $a$.

  * •

We denote the predictable $\sigma$-algebra by $\mathcal{P}$.

  * •

For $k\geq 1$, we denote by $\mathbb{H}^{2,k}$ the set of
$\mathbb{R}^{d}$-valued, càdlàg, and adapted processes $(X_{t})_{t\geq 0}$
such that $\mathbb{E}\left(\int_{0}^{T}|X_{s}|^{2}\mathrm{d}s\right)<\infty.$

  * •

For two stopping times $\tau$ and $\tau^{\prime}$ such that
$\tau\leq\tau^{\prime}$, we denote by $\mathcal{S}_{\tau,\tau^{\prime}}^{2}$
the complete space of $\mathbb{R}^{d}$-valued, càdlàg, and adapted processes
$(X_{t})_{t\geq 0}$ such that
$||X||_{\mathcal{S}^{2}_{\tau,\tau^{\prime}}}:=\mathbb{E}(\sup_{t\in[\tau,\tau^{\prime}]}|X_{t}|^{2})^{\frac{1}{2}}<\infty$.

######  Remark 1.1.

([22]) Note that for $D\in\mathcal{C}$, we have:

  * i)

$\forall y\in D\quad\left\langle\Pi_{D}(x)-x,\Pi_{D}(x)-y\right\rangle\leq 0.$

  * ii)

If $a\in\mathring{D}$, then for every $\eta\in\mathcal{N}_{x}$,

$\langle x-a,\eta\rangle\leq-\operatorname{dist}(a,\partial D).$

######  Proposition 1.1.

([3]) For each $y$ in $\mathbb{D}([0,T],\mathbb{R}^{d})$ and each positive
$\epsilon$, there exist points $t_{0},t_{1},\ldots,t_{v}$ such that

$0=t_{0}<t_{1}<\cdots<t_{v}=T,$

and

$\sup_{t,s\in[t_{i-1},t_{i}[}|y_{t}-y_{s}|<\epsilon,\quad i=1,2,\ldots,v.$

######  Remark 1.2.

The following results can be found in [3], and [5].

  * •

Note that $(\mathbb{D}([0,T],\mathbb{R}^{d}),d_{\mathbb{S}})$ is a Polish
space. It follows that
$(\mathcal{P}_{2}(\mathbb{D}([0,T],\mathbb{R}^{d}),d_{\mathbb{S}}))$ is also a
Polish space.

  * •

The metric $d_{\mathbb{S}}$ satisfies the following: for every
$x,y\in\mathbb{D}([0,T],\mathbb{R}^{d})$, we have
$d_{\mathbb{S}}(x,y)\leq|x-y|_{T}$, which implies $W_{d_{\mathbb{S}}}\leq
W_{T}$.

##  2 Skorokhod reflection problem in a time-dependent convex domains:
Deterministic case

The Skorokhod reflection problem in the deterministic case is stated as
follows:

######  Definition 2.1.

Let $\mathcal{D}=\\{D_{t},t\in[0,T]\\}$ be a family of time-dependent sets
such that for every $t$, $D_{t}\in\mathcal{C}$ and $t\to D_{t}$ is càdlàg with
respect to Hausdorff metric. Let $y\in\mathbb{D}([0,T],\mathbb{R}^{d})$ such
that $y_{0}\in\mathrm{D}_{0}$. We say that a pair of functions
$(x,k)\in\mathbb{D}([0,T],\mathbb{R}^{2d})$ is a solution to the reflection
problem associated with $y$ if:

  * (i)

$x_{t}=y_{t}+k_{t},\;\text{for every}\;t\in[0,T]$,

  * (ii)

$x\in\mathcal{D}$,

  * (iii)

For every càdlàg $z$, such that $z\in\mathcal{D}$, we have for every $t\geq 0$

$\int_{0}^{t}\langle x_{s}-z_{s},\mathrm{d}k_{s}\rangle\leq 0.$

This problem will be denoted by $RP_{\mathcal{D}}(y)$.

######  Remark 2.1 ([23]).

Note that the point (iii) in the definition 2.1 implies the existence of a
unit vector $\eta$ such that $\mathrm{d}k_{s}=\eta_{s}\mathrm{d}|k|_{s}$ with
$\eta_{s}\in\mathcal{N}_{x_{s}}$ if $x_{s}\in\partial D_{s}$.

We establish the following a priori estimate for the solutions of problem
$SP(y)$.

######  Lemma 2.1.

Let $\mathcal{D}=\\{D_{t},t\in[0,T]\\}$ and
$\mathcal{\tilde{D}}=\\{\tilde{D}_{t},t\in[0,T]\\}$ be two families of time-
dependent sets such that for every $t\geq 0$, $D_{t}$ and $\tilde{D}_{t}$
belong to $\mathcal{C}$, $t\to D_{t}$ and $t\to\tilde{D}_{t}$ are càdlàg with
respect to Hausdorff metric, $D_{T}=D_{T^{-}}$, and
$\tilde{D}_{T}=\tilde{D}_{T^{-}}$. Let $y$ and $\tilde{y}$ be elements of
$\mathbb{D}([0,T],\mathbb{R}^{d})$ such that $y_{0}\in D_{0}$ and
$\tilde{y}_{0}\in\tilde{D}_{0}$. Let $(x,k)$ and $(\tilde{x},\tilde{k})$ be
the solutions associated to $SP_{\mathcal{D}}(y)$ and
$SP_{\mathcal{\tilde{D}}}(\tilde{y})$, respectively. Then, for every $t\geq
0$, we have

$|x_{t}-\tilde{x}_{t}|^{2}\leq|y_{t}-\tilde{y}_{t}|^{2}+2\sup_{s\in[0,t]}d_{H}(D_{s},\tilde{D}_{s})(|k|_{t}+|\tilde{k}|_{t})+2\int_{0}^{t}\langle
y_{t}-y_{s}+\tilde{y}_{s}-\tilde{y}_{t},\mathrm{d}(k_{s}-\tilde{k}_{s})\rangle.$
(2.1)

###### Proof.

Let $t\geq 0$, we apply the change of variable formula to $k-\tilde{k}$, we
get:

$|x_{t}-\tilde{x}_{t}|^{2}\leq|y_{t}-\tilde{y}_{t}|^{2}+2\int_{0}^{t}\langle
x_{s}-\tilde{x}_{s},\mathrm{d}(k_{s}-\tilde{k}_{s})\rangle+2\int_{0}^{t}\langle
y_{t}-y_{s}+\tilde{y}_{s}-\tilde{y}_{t},\mathrm{d}(k_{s}-\tilde{k}_{s})\rangle.$

We have,

$\displaystyle\int_{0}^{t}\langle
x_{s}-\tilde{x}_{s},\mathrm{d}(k_{s}-\tilde{k}_{s})\rangle$
$\displaystyle=\int_{0}^{t}\langle
x_{s}-\Pi_{D_{s}}(\tilde{x}_{s})+\Pi_{D_{s}}(\tilde{x}_{s})-\tilde{x}_{s},\eta_{s}\rangle\mathds{1}_{\\{x_{s}\in\partial
D_{s}\\}}\mathrm{d}|k|_{s}$ $\displaystyle-\int_{0}^{t}\langle
x_{s}-\Pi_{\tilde{D}_{s}}(x_{s})+\Pi_{\tilde{D}_{s}}(x_{s})-\tilde{x}_{s},\tilde{\eta}_{s}\rangle\mathds{1}_{\\{\tilde{x}_{s}\in\partial\tilde{D}_{s}\\}}\mathrm{d}|\tilde{k}|_{s}.$

By (iii) in Definition 2.1,

$\int_{0}^{t}\langle
x_{s}-\Pi_{D_{s}}(\tilde{x}_{s}),\eta_{s}\rangle\mathds{1}_{\\{x_{s}\in\partial
D_{s}\\}}\mathrm{d}|k|_{s}\leq
0\quad\text{and}\quad\int_{0}^{t}\langle\tilde{x}_{s}-\Pi_{\tilde{D}_{s}}(x_{s}),\tilde{\eta}_{s}\rangle\mathds{1}_{\\{\tilde{x}_{s}\in\partial\tilde{D}_{s}\\}}\mathrm{d}|\tilde{k}|_{s}\leq
0.$

It follows that,

$\displaystyle\int_{0}^{t}\langle
x_{s}-\tilde{x}_{s},\mathrm{d}(k_{s}-\tilde{k}_{s})\rangle$
$\displaystyle\leq\int_{0}^{t}\langle\Pi_{D_{s}}(\tilde{x}_{s})-\tilde{x}_{s},\eta_{s}\rangle\mathds{1}_{\\{x_{s}\in\partial
D_{s}\\}}\mathrm{d}|k|_{s}+\int_{0}^{t}\langle\Pi_{\tilde{D}_{s}}(x_{s})-x_{s},\tilde{\eta}_{s}\rangle\mathds{1}_{\\{\tilde{x}_{s}\in\partial\tilde{D}_{s}\\}}\mathrm{d}|\tilde{k}|_{s}$
$\displaystyle\leq\int_{0}^{t}|\Pi_{D_{s}}(\tilde{x}_{s})-\tilde{x}_{s}|\mathds{1}_{\\{x_{s}\in\partial
D_{s}\\}}\mathrm{d}|k|_{s}+\int_{0}^{t}|\Pi_{\tilde{D}_{s}}(x_{s})-x_{s}|\mathds{1}_{\\{x_{s}\in\partial\tilde{D}_{s}\\}}\mathrm{d}|\tilde{k}|_{s}$
$\displaystyle\leq\int_{0}^{t}d(\tilde{x}_{s},D_{s})\mathrm{d}|k|_{s}+\int_{0}^{t}d(x_{s},\tilde{D}_{s})\mathrm{d}|\tilde{k}|_{s}$
$\displaystyle\leq\sup_{s\in[0,t]}d_{H}(D_{s},\tilde{D}_{s})(|k|_{t}+|\tilde{k}|_{t}).$

Subsequently, we obtain:

$|x_{t}-\tilde{x}_{t}|^{2}\leq|y_{t}-\tilde{y}_{t}|^{2}+2\sup_{s\in[0,t]}d_{H}(D_{s},\tilde{D}_{s})(|k|_{t}+|\tilde{k}|_{t})+2\int_{0}^{t}\langle
y_{t}-y_{s}+\tilde{y}_{s}-\tilde{y}_{t},\mathrm{d}(k_{s}-\tilde{k}_{s})\rangle.$

This completes the proof. ∎

######  Proposition 2.1.

Let $\mathcal{D}^{n}=\\{D_{t}^{n},t\in[0,T]\\}$ and
$\mathcal{D}=\\{D_{t},t\in[0,T]\\}$ be families of time-dependent sets such
that for every $t\geq 0$ and $n\in\mathbb{N}$,
$D_{t}^{n},D_{t}\in\mathcal{C}$, and $t\to D_{t}^{n}$, $t\to D_{t}$ are càdlàg
with respect to Hausdorff metric, $D_{T}^{n}=D_{T^{-}}^{n}$,
$D_{T}=D_{T^{-}}$. Let $\left\\{y^{n}\right\\}$ and $\left\\{a^{n}\right\\}$
be elements of $\mathbb{D}\left([0,T],\mathbb{R}^{d}\right)$ such that
$y_{0}^{n}\in D_{0}^{n}$ and $a^{n}\in\mathring{\mathcal{D}^{n}}$. Let
$\left\\{\left(x^{n},k^{n}\right)\right\\}$ be a sequence of solutions of the
Skorokhod problem $SP_{\mathcal{D}^{n}}(y^{n})$. We assume that
$\inf_{t\in[0,T]}d(a^{n}_{t},\partial D_{t}^{n})\geq m$, where $m$ is a
positive constant independent of $n$, and
$\lim\limits_{n}(\sup_{t\in[0,T]}d_{H}(D_{t}^{n},D_{t})+\sup_{t\in[0,T]}|y_{t}^{n}-y_{t}|+\sup_{t\in[0,T]}|a_{t}^{n}-a_{t}|)=0$.
Then, there exists $C>0$ and $N\in\mathbb{N}$ such that for every $n\geq N$,
we have

$|k^{n}|_{T}\leq
C(\sup_{t\in[0,T]}|y_{t}|^{2}+\sup_{t\in[0,T]}|a_{t}|^{2}+1).$ (2.2)

###### Proof.

Since $y^{n}$ and $a^{n}$ converge uniformly to $y$ and $a$, respectively, and
$a^{n}\in\mathcal{D}$, it follows that $y$ and $a$ are càdlàg, and
$a\in\mathcal{D}$. By proposition 1.1, there exist points
$t_{0},t_{1},\ldots,t_{v}$ such that $0=t_{0}<t_{1}<\cdots<t_{v}=T$ and

$\sup_{t,s\in[t_{i-1},t_{i}[}|a_{t}-a_{s}|+\sup_{t,s\in[t_{i-1},t_{i}[}|y_{t}-y_{s}|<\frac{m}{4},\quad
i=1,2,\ldots,v.$ (2.3)

Let $i\in\\{1\dots v\\}$ and $t\in[t_{i-1},t_{i}]$. We apply the change of
variable formula to the function $k_{.}^{n}-k_{t_{i-1}}^{n}$, we get

$\displaystyle|k_{t}^{n}-k_{t_{i-1}}^{n}|^{2}$ $\displaystyle\leq
2\int_{t_{i-1}}^{t}\langle
x_{u}^{n}-x_{t_{i-1}}^{n},\mathrm{d}k_{u}^{n}\rangle-2\int_{t_{i-1}}^{t}\langle
y_{u}^{n}-y_{t_{i-1}}^{n},\mathrm{d}k_{u}^{n}\rangle$ $\displaystyle\leq
2\int_{t_{i-1}}^{t}\langle
x_{u}^{n}-a_{u}^{n},\mathrm{d}k_{u}^{n}\rangle+2\int_{t_{i-1}}^{t}\langle
a_{u}^{n}-a_{u},\mathrm{d}k_{u}^{n}\rangle+2\int_{t_{i-1}}^{t}\langle
a_{u}-a_{t_{i-1}},\mathrm{d}k_{u}^{n}\rangle$
$\displaystyle+2\int_{t_{i-1}}^{t}\langle
a_{t_{i-1}}-x_{t_{i-1}}^{n},\mathrm{d}k_{u}^{n}\rangle-2\int_{t_{i-1}}^{t}\langle
y_{u}^{n}-y_{u},\mathrm{d}k_{u}^{n}\rangle-2\int_{t_{i-1}}^{t}\langle
y_{u}-y_{t_{i-1}},\mathrm{d}k_{u}^{n}\rangle$
$\displaystyle-2\int_{t_{i-1}}^{t}\langle
y_{t_{i-1}}-y_{t_{i-1}}^{n},\mathrm{d}k_{u}^{n}\rangle,$

where in the last inequality, we have add and subtract $a^{n}$,
$a-a_{t_{i-1}}$ and $y-y_{t_{i-1}}$.

Now, we use (ii) of Remark 1.1, and we apply the Cauchy-Schwartz inequality to
the second, fifth, and seventh integrals. We split the third and sixth
integrals into integrals over the intervals $[t_{i-1},t[$ and $\\{t\\}$. In
the fourth integral, we add and subtract $\Pi_{D_{t_{i-1}}}(x^{n}_{t_{i-1}})$.
By doing so, we obtain

$\begin{split}|k_{t}^{n}-k_{t_{i-1}}^{n}|^{2}&\leq-2\int_{t_{i-1}}^{t}d(a_{u}^{n},\partial
D_{u}^{n})\mathrm{d}|k^{n}|_{u}+2\sup_{s\in[0,T]}|a_{s}^{n}-a_{s}||k^{n}|_{[t_{i-1},t_{i}]}+2\int_{[t_{i-1},t[}\langle
a_{u}-a_{t_{i-1}},\mathrm{d}k_{u}^{n}\rangle\\\ &+2\langle
a_{t}-a_{t_{i-1}},\Delta k_{t}^{n}\rangle+2\langle
a_{t_{i-1}}-\Pi_{D_{t_{i-1}}}(x^{n}_{t_{i-1}}),k_{t}^{n}-k_{t_{i-1}}^{n}\rangle+2\langle\Pi_{D_{t_{i-1}}}(x^{n}_{t_{i-1}})-x_{t_{i-1}}^{n},k_{t}^{n}-k_{t_{i-1}}^{n}\rangle\\\
&+4\sup_{s\in[0,T]}|y_{s}^{n}-y_{s}||k^{n}|_{[t_{i-1},t_{i}]}-2\int_{[t_{i-1},t[}\langle
y_{u}-y_{t_{i-1}},\mathrm{d}k_{u}^{n}\rangle-2\langle y_{t}-y_{t_{i-1}},\Delta
k_{t}^{n}\rangle\\\ &\leq-2\int_{t_{i-1}}^{t}d(a_{u}^{n},\partial
D_{u}^{n})\mathrm{d}|k^{n}|_{u}+2\sup_{s\in[0,T]}|a_{s}^{n}-a_{s}||k^{n}|_{[t_{i-1},t_{i}]}+2\sup_{t\in[t_{i-1,t_{i}}[}|a_{t}-a_{t_{i-1}}||k^{n}|_{[t_{i-1},t_{i}]}\\\
&+8\sup_{s\in[0,T]}|a_{s}|\sup_{t\in[t_{i-1},t_{i}]}|k_{t}^{n}-k_{t_{i-1}}^{n}|+4r_{t_{i-1}}\sup_{t\in[t_{i-1},t_{i}]}|k_{t}^{n}-k_{t_{i-1}}^{n}|+2\sup_{t\in[0,T]}d_{H}(D_{t}^{n},D_{t})|k^{n}|_{[t_{i-1},t_{i}]}\\\
&+4\sup_{s\in[0,T]}|y_{s}^{n}-y_{s}||k^{n}|_{[t_{i-1},t_{i}]}+2\sup_{t\in[t_{i-1,t_{i}}[}|y_{t}-y_{t_{i-1}}||k^{n}|_{[t_{i-1},t_{i}]}+8\sup_{s\in[0,T]}|y_{s}|\sup_{t\in[t_{i-1},t_{i}]}|k_{t}^{n}-k_{t_{i-1}}^{n}|.\end{split}$

According to the assumptions, there exists $N$ such that for every $n\geq N$,

$2\sup_{s\in[0,T]}|a_{s}^{n}-a_{s}|+2\sup_{t\in[0,T]}d_{H}(D_{t}^{n},D_{t})+4\sup_{s\in[0,T]}|y_{s}^{n}-y_{s}|\leq\frac{m}{2}.$

Combining this with (2.3), we obtain,

$2\int_{t_{i-1}}^{t}d(a_{u}^{n},\partial
D_{u}^{n})\mathrm{d}|k^{n}|_{u}+|k_{t}^{n}-k_{t_{i-1}}^{n}|^{2}\leq\\\
m|k^{n}|_{[t_{i-1},t_{i}]}+8(\sup_{s\in[0,T]}|a_{s}|+\sup_{s\in[0,T]}|y_{s}|++\frac{1}{2}r_{t_{i-1}})\sup_{t\in[t_{i-1},t_{i}]}|k_{t}^{n}-k_{t_{i-1}}^{n}|.$

We use the inequality $2|ab|\leq(\frac{1}{2}|a|^{2}+2|b|^{2})$, we get

$\begin{split}m|k^{n}|_{[t_{i-1},t_{i}]}+\sup_{t\in[t_{i-1},t_{i}]}|k_{t}^{n}-k_{t_{i-1}}^{n}|^{2}&\leq\frac{1}{2}\sup_{t\in[t_{i-1},t_{i}]}|k_{t}^{n}-k_{t_{i-1}}^{n}|^{2}+64(\sup_{s\in[0,T]}|a_{s}|^{2}+\sup_{s\in[0,T]}|y_{s}|^{2}+\frac{1}{4}r_{t_{i-1}}^{2}).\end{split}$

By setting $c=\max\Big{(}64,\frac{1}{4}\max_{\\{i=1,\dots
v+1\\}}(r_{t_{i-1}}^{2})\Big{)}$, we obtain

$|k^{n}|_{[t_{i-1},t_{i}]}\leq
c(\sup_{s\in[0,T]}|y_{s}|^{2}+\sup_{s\in[0,T]}|a_{s}|^{2}+1).$

We sum on $i$, which is a finite sum, we get

$|k^{n}|_{T}\leq
C(\sup_{s\in[0,T]}|y_{s}|^{2}+\sup_{s\in[0,T]}|a_{s}|^{2}+1),$

where $C=c(v+1)$.

This completes the proof. ∎

######  Lemma 2.2.

Let $\mathcal{D}=\\{D_{t},t\in[0,T]\\}$ be a family of time-dependent sets
such that for every $t$, $D_{t}\in\mathcal{C}$ and $t\to D_{t}$ is càdlàg with
respect to Hausdorff metric such that $D_{T}=D_{T^{-}}$. Let $y$ and $a$ be
elements of $\mathbb{D}([0,T],\mathbb{R}^{d})$. For $n\geq 0$, we consider the
following recursive sequence of times defined by: $t_{0}^{n}=0$ and for $i\geq
1$,

$t_{i}^{n}=(t_{i-1}^{n}+1/n)\land\inf\\{t>t_{i-1}^{n}\quad
d_{H}(D_{t_{i-1}^{n}},D_{t})\geq\frac{1}{n}\;\;\text{or}\;\;|y_{t_{i-1}^{n}}-y_{t}|\geq\frac{1}{n}\;\;\text{or}\;\;|a_{t_{i-1}^{n}}-a_{t}|\geq\frac{1}{n}\\}\land
T.$

Then, we have

For every $n\geq 1$, there exists $i_{n}$ such that for every $i>i_{n}$
$t_{i}^{n}=T$.

In addition, if we consider the discretization of $y$, $a$ and $\mathcal{D}$
given by the following:

$D_{t}^{n}=\begin{cases}D_{t_{i-1}^{n}},&t\in[t_{i-1}^{n},t_{i}^{n}[,i=1,\ldots,i_{n},\\\
D_{t_{i_{n}}^{n}},&t\in\left[t_{i_{n}}^{n},T\right].\end{cases}$
$y^{n}_{t}=\begin{cases}y_{t_{i-1}^{n}},&t\in[t_{i-1}^{n},t_{i}^{n}[,i=1,\ldots,i_{n},\\\
y_{t_{i_{n}}^{n}},&t\in\left[t_{i_{n}}^{n},T\right].\end{cases}$
$a^{n}_{t}=\begin{cases}a_{t_{i-1}^{n}},&t\in[t_{i-1}^{n},t_{i}^{n}[,i=1,\ldots,i_{n},\\\
a_{t_{i_{n}}^{n}},&t\in\left[t_{i_{n}}^{n},T\right].\end{cases}$

We have,

$\lim\limits_{n}\sup_{0\leq t\leq
T}d_{H}(D_{t}^{n},D_{t})=0,\;\;\lim\limits_{n}\sup_{0\leq t\leq
T}|y_{t}^{n}-y_{t}|=0,\;\;\text{and}\;\;\lim\limits_{n}\sup_{0\leq t\leq
T}|a_{t}^{n}-a_{t}|=0.$

###### Proof.

  * i)

The first point follows immediately from the fact that $y,a$ and $\mathcal{D}$
are càdlàg.

  * ii)

Assume that there exists a sequence $(t_{n})$ in $[0,T]$ such that
$\lim\limits_{n}t_{n}=t$ and
$\lim\limits_{n}d_{H}(D_{t_{n}}^{n},D_{t_{n}})\neq 0$. Setting
$s_{n}=\max\\{t_{i}^{n},t_{i}^{n}\leq t_{n}\\}$ implies that, for every $n\geq
1$, there exists $j_{n}$ such that $s_{n}=t_{j_{n}}^{n}$,
$D_{t_{n}}^{n}=D_{s_{n}}$ and $s_{n}\leq t_{n}<t_{{j_{n}}+1}^{n}\leq
s_{n}+\frac{1}{n}$, hence $\lim\limits_{n}s_{n}=t$. We distinguish the
following cases:

    * case 1:

If $\\{n:t<s_{n}\\}$ is infinite, we have

$0\leq d_{H}(D_{t_{n}}^{n},D_{t_{n}})=d_{H}(D_{s_{n}},D_{t_{n}})\leq
d_{H}(D_{s_{n}},D_{t})+d_{H}(D_{t},D_{t_{n}}).$

The last terms in the previous inequality tend to 0, which is a contradiction.

    * case 2:

If $\\{n:s_{n}\leq t\leq t_{n}\\}$ is infinite, then

$0\leq d_{H}(D_{t_{n}}^{n},D_{t_{n}})\leq
d_{H}(D_{s_{n}},D_{t})+d_{H}(D_{t},D_{t_{n}})\leq\frac{1}{n}+d_{H}(D_{t},D_{t_{n}}),$

which also tend to 0. It is also a contradiction.

    * case 3:

If $\\{n:t_{n}<t\\}$ is infinite, then

$0\leq d_{H}(D_{t_{n}}^{n},D_{t_{n}})\leq
d_{H}(D_{s_{n}},D_{t^{-}})+d_{H}(D_{t^{-}},D_{t_{n}}),$

which is also a contradiction.

    * iii)

Using the fact that $y$ and $a$ are càdlàg, the convergence of $y^{n}$ and
$a^{n}$ follows similarly as (ii).

∎

######  Theorem 2.1.

Let $\mathcal{D}=\\{D_{t},t\in[0,T]\\}$ be a family of time-dependent sets
such that for every $t$, $D_{t}\in\mathcal{C}$, and $t\to D_{t}$ is càdlàg
with respect to Hausdorff metric. Let $y$ and $a$ be elements of
$\mathbb{D}([0,T],\mathbb{R}^{d})$ such that $y_{0}\in D_{0}$,
$a\in\mathring{\mathcal{D}}$, and $\inf_{t\in[0,T]}d(a_{t},\partial D_{t})>0$.
Then, there exists a unique solution of the problem $SP_{\mathcal{D}}(y)$. In
addition,

$|k|_{T}\leq C(\sup_{s\in[0,T]}|y_{s}|^{2}+\sup_{s\in[0,T]}|a_{s}|^{2}+1).$
(2.4)

###### Proof.

  * i)

Uniqueness: It follows immediately from Lemma 2.1.

  * ii)

Existence: We consider the same discretization of $y$, $a$ and $\mathcal{D}$
as in Lemma 2.2 along with the following scheme.

$\displaystyle x_{t}^{n}=\begin{cases}y_{0},&\text{if}\quad
t\in\left[0,t_{1}^{n}\right[,\\\
\Pi_{D_{t_{i-1}^{n}}^{n}}(x_{t_{i-2}^{n}}^{n}+y_{t}^{n}-y_{t_{i-2}^{n}}),&\text{if}\quad
t\in[t_{i-1}^{n},t_{i}^{n}[,i=1,\ldots,i_{n},\\\
\Pi_{D_{t_{i_{n}-1}^{n}}^{n}}(x_{t_{i_{n}-1}^{n}}^{n}+y_{t}^{n}-y_{t_{i_{n}-1}^{n}}),&\text{if}\quad
t\in\left[t_{i_{n}}^{n},T\right]\end{cases}$ $\displaystyle
k_{t}^{n}=\begin{cases}0,&\text{if}\quad t\in\left[0,t_{1}^{n}[\right.\\\
k_{t_{i-2}^{n}}^{n}+x_{t}^{n}-x_{t_{i-2}^{n}}^{n}-y_{t}^{n}+y_{t_{i-2}^{n}},&\text{if}\quad
t\in\left[t_{i-1}^{n},t_{i}^{n}[\right.\,i=1,\ldots,i_{n},\\\
k_{t_{i_{n}-1}^{n}}^{n}+x_{t}^{n}-x_{t_{i_{n}-1}^{n}}^{n}-y_{t}^{n}+y_{t_{i_{n}-1}^{n}},&\text{if}\quad
t\in[t_{i_{n}}^{n},T].\end{cases}$

We will complete the proof in two steps.

    * Step 1:

We show that $(x^{n},k^{n})=SP_{\mathcal{D}^{n}}(y^{n})$, where
$\mathcal{D}^{n}=\\{D^{n}_{t},\quad t\in[0,T]\\}$.

      * $\bullet$

By the construction of the scheme above, we have for every $t\in[0,T]$,

$x^{n}_{t}=y^{n}_{t}+k^{n}_{t}\in D_{t}^{n}.$

      * $\bullet$

Let $(v^{n})_{n\geq 1}$ be a sequence of elements of
$\mathbb{D}([0,T],\mathbb{R}^{d})$ such that for every $n\geq 0$,
$v^{n}\in\mathcal{D}^{n}$, then

$\displaystyle\int_{0}^{t}\langle
x^{n}_{s}-v_{s}^{n},\mathrm{d}k^{n}_{s}\rangle=\sum_{i=1}^{i_{n}}\langle
x_{{t_{i}^{n}}\land t}^{n}-v_{{t_{i}^{n}}\land t}^{n},\Delta
k^{n}_{{t_{i}^{n}}\land t}\rangle$
$\displaystyle=\sum_{i=1}^{i_{n}}\langle\Pi_{D_{{t_{i}^{n}}\land
t}^{n}}(x_{{{t_{i-1}^{n}}\land t}}^{n}+\Delta y_{{t_{i}^{n}}\land t}^{n})-$
$\displaystyle v_{{t_{i}^{n}}\land t}^{n},\Pi_{D_{{t_{i}^{n}}\land
t}^{n}}(x_{{{t_{i-1}^{n}}\land t}}^{n}+\Delta y_{{t_{i}^{n}}\land
t}^{n})-(x_{{{t_{i-1}^{n}}\land t}}^{n}+\Delta y_{{t_{i-1}^{n}}\land
t}^{n})\rangle.$

By (i) in Remark (1.1), the previous sum is non positive.

    * Step 2:

We show that $x^{n},k^{n}$ converge to the solution of the reflection problem
$RP_{\mathcal{D}}(y)$ in the space
$\big{(}\mathbb{D}([0,T],\mathbb{R}^{d}),|.|_{T}\big{)}$.

On the one hand, we have

$\inf_{t\in[0,T]}d(a_{t}^{n},\partial
D_{t}^{n})=\min_{i=0,\dots,i_{n}-1}\left(\inf_{t\in[t_{i}^{n},t_{i+1}^{n}[}d(a_{t_{i}^{n}},\partial
D_{t_{i}^{n}})\right)\land d(a_{T},\partial
D_{T})\geq\inf_{t\in[0,T]}d(a_{t},\partial D_{t}),$

which is positive and independent of $n$. Combined with Lemma 2.2, the
assumptions of Proposition 2.1 are satisfied. Therefore, $|k^{n}|$ satisfies
inequality (2.2). On the other hand, combining (2.2) with the inequality (2.1)
in Lemma 2.1, we obtain $x^{n}$ and $k^{n}$ are Cauchy sequences in
$\big{(}\mathbb{D}([0,T],\mathbb{R}^{d}),|.|_{T}\big{)}$. Again from
Proposition 2.1, $k^{n}$ converges uniformly to a function with finite
variation. Denote by $x$ and $k$ the limits of $x^{n}$ and $k^{n}$,
respectively.

      * $\bullet$

Let $t\in[0,T]$, we have

$d(x_{t},D_{t})\leq|x_{t}-x_{t}^{n}|+d(x_{t}^{n},D_{t})\leq\sup_{t\in[0,T]}|x_{t}-x_{t}^{n}|+\sup_{t\in[0,T]}d_{H}(D_{t}^{n},D_{t}).$

We tend n to $\infty$, we get $d(x_{t},D_{t})=0$, this implies $x_{t}\in
D_{t}$.

      * $\bullet$

We show (iii) of Definition 2.1.

Let $v\in\mathbb{D}([0,T],\mathbb{R}^{d})$ such that $v\in\mathcal{D}$. We
have $x^{n}$ and $k^{n}$ converge uniformly to $x$ and $k$. On the one hand,
using Lemma 3.9 in [18], for every $t\geq 0$, we have

$\lim\limits_{n}\int_{0}^{t}\langle
x^{n}_{s}-v_{s},\mathrm{d}k^{n}_{s}\rangle=\int_{0}^{t}\langle
x_{s}-v_{s},\mathrm{d}k_{s}\rangle,$

on the other hand,

$\displaystyle\int_{0}^{t}\langle x^{n}_{s}-v_{s},\mathrm{d}k^{n}_{s}\rangle$
$\displaystyle=\sum_{i=0}^{i_{n}}\langle x_{{t_{i}^{n}}\land
t}^{n}-v_{{t_{i}^{n}}\land t},\Delta k_{{t_{i}^{n}}\land t}^{n}\rangle\leq 0.$

It is clear that $(x,k)$ satisfy the assumptions of Proposition 2.2, hence
(2.4) holds true.

This completes the proof.

∎

##  3 Reflected McKean-Vlasov stochastic differential equation

In this section, we consider a probability space
$\left(\Omega,\mathbb{F},\left(\mathcal{F}_{t}\right)_{t\leq
T},\mathbb{P}\right)$ satisfying the usual conditions, with an
$\mathbb{F}$-adapted and $\mathbb{R}^{m}$-valued Brownian motion
$(W_{t})_{0\leq t\leq T}$ and $(\tilde{N}_{t})_{0\leq t\leq T}$ is a
martingale measure in $\mathbb{R}^{d}\backslash\\{0\\}$ with Lévy measure
$\lambda$ independent of $(W_{t})_{0\leq t\leq T}$, corresponding to a
standard Poisson random measure $N(t,A)$. For any measurable subset of
$\mathbb{R}^{d}\backslash\\{0\\}$ such that $\lambda(A)<\infty$, we have:

$\tilde{N}_{t}(A)=N(t,A)-t\lambda(A),$

where $N(t,A)$ satisfies that

$\mathbb{E}[N(t,A)]=t\lambda(A).$

The measure $\lambda(\cdot)$ is supposed to be a $\sigma$-finite measure on
$\mathbb{R}^{d}\backslash\\{0\\}$ and satisfies the following integrability
condition

$\int_{\mathbb{R}^{d}\backslash\\{0\\}}\left(1\wedge|z|^{2}\right)\lambda(\mathrm{d}z)<\infty.$

###  3.1 Application to stochastic processes

Now we give the definition of Skorokhod problem for processes.

######  Definition 3.1.

Let $\mathcal{D}=\\{D_{t},t\in[0,T]\\}$ be a family of time-dependent sets
such that for every $t$, $D_{t}\in\mathcal{C}$ and $t\to D_{t}$ is càdlàg with
respect to Hausdorff metric, and $D_{T}=D_{T^{-}}$. Let $Y$ be a càdlàg
$\mathbb{F}$-adapted process such that $Y_{0}\in D_{0}$. We say that a pair of
processes $(X,K)$ is a solution of the reflection problem associated with $Y$
and $\mathcal{D}$ if $X$ and $K$ are càdlàg and $\mathbb{F}$-adapted
processes, such that

  * (i)

$X_{t}=Y_{t}+K_{t},\;\text{for every}\;t\in[0,T]$,

  * (ii)

$X_{t}\in\mathrm{D}_{t},\;\text{for every}\;t\in[0,T]$,

  * (iii)

For every càdlàg and $\mathbb{F}$-adapted process $Z$ such that
$Z\in\mathcal{D}$, we have

$\text{for every }t\geq 0\;\;\;\int_{0}^{t}\langle
X_{s}-Z_{s},\mathrm{d}K_{s}\rangle\leq 0.$

We write $(X,K)=SP_{\mathcal{D}}(Y)$.

######  Proposition 3.1.

Let $\mathcal{D}=\\{D_{t},t\in[0,T]\\}$ be a family of time-dependent sets
such that for every $t$, $D_{t}\in\mathcal{C}$ and $t\to D_{t}$ is càdlàg with
respect to Hausdorff metric. Let $Y$ be a càdlàg $\mathbb{F}$-adapted process
such that $Y_{0}\in D_{0}$. We assume that there exists a càdlàg and
$\mathbb{F}$-adapted process $A$ such $A\in\mathring{\mathcal{D}}$ and
$\inf_{t\in[0,T]}d(A_{t},\partial D_{t})>0$. Then, there exists a unique
solution to the problem $SP_{\mathcal{D}}(Y)$.

###### Proof.

The proof is similar to the one given in the previous section, as the integral
in the reflection problem is in Stieltjes sense. The proof in the previous
section shows that if $Y$ is càdlàg and $\mathbb{F}$-adapted, then $X$ and $K$
are also càdlàg and $\mathbb{F}$-adapted processes. ∎

In the following, we provide an estimate for the solution for inputs that are
semimartingales.

######  Proposition 3.2.

Let $\mathcal{D}=\\{D_{t},t\in[0,T]\\}$ be a family of time-dependent sets
such that for every $t$, $D_{t}\in\mathcal{C}$ and $t\to D_{t}$ is càdlàg with
respect to Hausdorff metric. Let $Y$ and $\tilde{Y}$ be two semimartingales
with the following decompositions: $Y=M+V$ and
$\tilde{Y}=\tilde{M}+\tilde{V}$, such that $Y_{0},\tilde{Y}_{0}\in D_{0}$.
Assume that $M$ and $\tilde{M}$ are martingales in $\mathcal{S}_{0,T}$, $V$
and $\tilde{V}$ are càdlàg adapted processes with finite variation such that
$|V|$ and $|\tilde{V}|$ are square-integrable. Let $(X,K)$ and
$(\tilde{X},\tilde{K})$ be the solutions associated with $RP_{\mathcal{D}}(Y)$
and $RP_{\mathcal{D}}(\tilde{Y})$, respectively. Then, there exists a constant
$C$ such that for all stopping times $\tau$ in $[0,T]$, we have:

$\mathbb{E}(\sup_{0\leq t\leq\tau}|X_{t}-\tilde{X}_{t}|^{2})\leq
C\mathbb{E}\left(|X_{0}-\tilde{X}_{0}|^{2}+[M-\tilde{M}]_{\tau}+|V-\tilde{V}|_{\tau}^{2}\right),$

where $[M-\tilde{M}]$ denotes the quadratic variation of $M-\tilde{M}$.

###### Proof.

By Lemma 2.1, we have

$|X_{t}-\tilde{X}_{t}|^{2}\leq|Y_{t}-\tilde{Y}_{t}|^{2}+2\int_{0}^{t}\langle
Y_{t}-Y_{s}+\tilde{Y}_{s}-\tilde{Y}_{t},\mathrm{d}(K_{s}-\tilde{K}_{s})\rangle.$

We apply Itô’s formula, we get

$\displaystyle\int_{0}^{t}\langle
Y_{t}-Y_{s}+\tilde{Y}_{s}-\tilde{Y}_{t},\mathrm{d}(K_{s}-\tilde{K}_{s})$
$\displaystyle=\int_{0}^{t}\langle
K_{s^{-}}-\tilde{K}_{s^{-}},\mathrm{d}(Y_{s}-\tilde{Y}_{s})\rangle$
$\displaystyle=\int_{0}^{t}\langle
X_{s^{-}}-\tilde{X}_{s^{-}},\mathrm{d}(Y_{s}-\tilde{Y}_{s})\rangle-\int_{0}^{t}\langle
Y_{s^{-}}-\tilde{Y}_{s^{-}},\mathrm{d}(Y_{s}-\tilde{Y}_{s})\rangle$
$\displaystyle=\int_{0}^{t}\langle
X_{s^{-}}-\tilde{X}_{s^{-}},\mathrm{d}(Y_{s}-\tilde{Y}_{s})\rangle+[Y-\tilde{Y}]_{t}+|Y_{0}-\tilde{Y}_{0}|^{2}-|Y_{t}-\tilde{Y}_{t}|^{2}$

We use the decomposition of $Y$ and $\tilde{Y}$, apply Burkholder-Davis-
Gundy’s inequality, and perform a simple computation to obtain

$\displaystyle\mathbb{E}(\sup_{0\leq t\leq\tau}|X_{t}-\tilde{X}_{t}|^{2})$
$\displaystyle\leq\frac{1}{2}\mathbb{E}(\sup_{s\leq\tau}|X_{s}-\hat{X}_{s}|^{2})+C\mathbb{E}\left(|Y_{0}-\tilde{Y}_{0}|^{2}+[M-\tilde{M}]_{\tau}+|V-\tilde{V}|_{\tau}^{2}\right).$

This completes the proof. ∎

######  Remark 3.1.

Note that if for some stopping time $\tau$ in $[0,T]$, we have

$\mathbb{E}(\sup_{t\in[0,\tau]}|A_{t}|^{2})+\mathbb{E}(\sup_{t\in[0,\tau]}|Y_{t}|^{2})<\infty.$

From inequality (2.4), we also have

$\mathbb{E}(|K|_{\tau}+\sup_{t\in[0,\tau]}|X_{t}|^{2})<\infty.$

###  3.2 Existence and uniqueness of strong solution for reflected Mckean-
Vlasov equations

We first recall the existence and uniqueness result in the case of SDEs.

######  Definition 3.2.

Let $\mathcal{D}=\\{D_{t},t\in[0,T]\\}$ be a family of time-dependent sets
such that for every $t$, $D_{t}\in\mathcal{C}$ and $t\to D_{t}$ is càdlàg with
respect to Hausdorff metric. Let $X_{0}$ be an $\mathcal{F}_{0}$ measurable
random variable, with $X_{0}\in D_{0}$,
$(b,\sigma):[0,T]\times\Omega\times\mathbb{R}^{d}\times\mathcal{P}\left(\mathbb{R}^{d}\right)\rightarrow\mathbb{R}^{d}\times\mathbb{R}^{d\times
m}$ and
$\beta:[0,T]\times\Omega\times\mathbb{R}^{d}\times\mathcal{P}\left(\mathbb{R}^{d}\right)\times\mathbb{R}^{d}\backslash\\{0\\}\rightarrow\mathbb{R}^{d}$
be measurable random maps. A couple $(X,K)$ is said to be a solution to the
reflected McKean-Vlasov SDE, which we denote by
$E(\mathcal{D},b,\sigma,\beta)$, if $(X,K)=RP_{\mathcal{D}}(Y)$, where $Y$ is
given by the following:

$Y_{t}=X_{0}+\int_{0}^{t}b\left(s,X_{s},\mathcal{L}_{X_{s}}\right)\mathrm{d}s+\int_{0}^{t}\sigma\left(s,X_{s},\mathcal{L}_{X_{s}}\right)\mathrm{d}W_{s}+\int_{0}^{t}\int_{\mathbb{R}^{d}\backslash\\{0\\}}\beta\left(s,X_{s^{-}},\mathcal{L}_{X_{s^{-}}},z\right)\tilde{N}(\mathrm{d}s,\mathrm{d}z),\,\text{for}\;\;t\geq
0.$

We consider the following set of assumptions:

  * •

$\left(\mathbf{A_{1}}\right)$: $\mathcal{D}=\\{D_{t},t\in[0,T]\\}$ is a family
of time-dependent sets such that for every $t$, $D_{t}\in\mathcal{C}$ and
$t\to D_{t}$ is càdlàg with respect to Hausdorff metric.

  * •

$\left(\mathbf{A_{2}}\right)$: There exists a process $A$ in
$\mathcal{S}_{0,T}$ such that $A\in\mathring{\mathcal{D}}$, and
$\inf_{t\in[0,T]}d(A_{t},\partial D_{t})>0$.

  * •

$\left(\mathbf{A_{3}}\right)$: $X_{0}$ is $\mathcal{F}_{0}$ measurable random
variable with $X_{0}\in D_{0}$ and $\mathbb{E}(|X_{0}|^{2})<\infty$.

  * •

$\left(\mathbf{A_{4}}\right)$: For fixed $\mu\in\mathcal{P}(\mathbb{R}^{d})$,
the processes $b(.,.,0,\mu)$ and $\sigma(.,.,0,\mu)$ are elements of
$\mathbb{H}^{2,d}$ and $\mathbb{H}^{2,dm}$, respectively. In addition, for
$(\omega,t)$ fixed, there exists $\gamma>0$ such that

$\quad\left|(b,\sigma)(t,\omega,x,\mu)-(b,\sigma)\left(t,\omega,x^{\prime},\mu^{\prime}\right)\right|\leq\gamma\left(\left|x-x^{\prime}\right|+W\left(\mu,\mu^{\prime}\right)\right).$

  * •

$\left(\mathbf{A_{5}}\right)$: $\beta$ is
$\mathscr{P}\otimes\mathscr{B}_{\mathbb{R}^{d}}\otimes\mathcal{P}_{2}(\mathbb{R}^{d})\otimes\mathcal{B}_{\mathbb{R}^{d}\backslash\\{0\\}}$-measurable.
For every $(t,\omega)$, we have

$\left(\int_{\mathbb{R}^{d}\backslash\\{0\\}}\left|\beta(s,\omega,x,\mu,z)-\beta\left(s,\omega,x^{\prime},\mu^{\prime},z\right)\right|^{2}\lambda(\mathrm{d}z)\right)^{\frac{1}{2}}\leq\gamma\left(\left|x-x^{\prime}\right|+W\left(\mu,\mu^{\prime}\right)\right),$
$\int_{\mathbb{R}^{d}\backslash\\{0\\}}|\beta(s,\omega,x,\mu,z)|^{2}\lambda(\mathrm{d}z)\leq\gamma(1+|x|^{2}).$

######  Theorem 3.1.

Under assumptions $\left(\mathbf{A}_{1}\right)$-$\left(\mathbf{A}_{5}\right)$,
for every $\mu\in\mathcal{P}_{2}(\mathbb{D}([0,T],\mathbb{R}^{d}))$, there
exists a unique solution $(X^{\mu},K^{\mu})=RP_{\mathcal{D}}(Y^{\mu})$, where
$X^{\mu}\in\mathcal{S}^{2}_{0,T}$, and $Y^{\mu}$ is given by:

$Y_{t}^{\mu}=X_{0}+\int_{0}^{t}b\left(s,X_{s}^{\mu},\mu_{s}\right)\mathrm{d}s+\int_{0}^{t}\sigma\left(s,X_{s}^{\mu},\mu_{s}\right)\mathrm{d}W_{s}+\int_{0}^{t}\int_{\mathbb{R}^{d}\backslash\\{0\\}}\beta\left(s,X_{s^{-}}^{\mu},\mu_{s^{-}},z\right)\tilde{N}(\mathrm{d}s,\mathrm{d}z),\;\;\text{for}\;\;t\geq
0.$ (3.1)

###### Proof.

Note that for $\mu\in\mathcal{P}_{2}(\mathbb{D}([0,T],\mathbb{R}^{d}))$ and
$s,t\geq 0$ we have

$W\left(\mu_{s},\mu_{t}\right)^{2}\leq\int_{\mathbb{D}([0,T],\mathbb{R}^{d})}\left|\pi_{s}(\omega)-\pi_{t}(\omega)\right|^{2}\mu(\mathrm{d}\omega),\;\text{and}\;W\left(\mu_{s},\mu_{t^{-}}\right)^{2}\leq\int_{\mathbb{D}([0,T],\mathbb{R}^{d})}\left|\pi_{s}(\omega)-\pi_{t^{-}}(\omega)\right|^{2}\mu(\mathrm{d}\omega).$

By Lebesgue’s dominated convergence theorem, we conclude that $t\to\mu_{t}$ is
càdlàg with respect to Wasserstein metric. Hence, by
$\left(\mathbf{A}_{4}\right)$ and $\left(\mathbf{A}_{5}\right)$, the
stochastic integrals in (3.1) are well-defined.

Let $\alpha\in]0,1[$, we consider the following stopping time:

$\tau:=\inf\\{t>0\;:\;C\gamma^{2}(2t+t^{2})\geq\alpha\\}\land T,$

with $\tau=\infty\;\text{for}\;\inf(\emptyset)$, and $C$ is the constant in
Proposition 3.2.

We consider the mapping
$\varphi\colon\mathcal{S}^{2}_{0,\tau}\to\mathcal{S}_{0,\tau}^{2}$ that
associates $X\in\mathcal{S}_{0,\tau}^{2}$ with $\varphi(X)$, where
$\varphi(X)$ is defined as the first coordinate of the solution of the
reflection problem $RP_{\mathcal{D}}\Big{(}Y^{\mu}_{.\land\tau}\Big{)}$, which
is justified by Proposition 3.1.

We have for every $X\in\mathcal{S}^{2}_{0,\tau}$,
$\varphi(X)\in\mathcal{S}^{2}_{0,\tau}$. Indeed, from assumptions
$\left(\mathbf{A}_{4}\right)$, $\left(\mathbf{A}_{5}\right)$, and through the
application of Burkholder-Davis-Gundy inequality, we get

$\displaystyle||Y^{\mu}||_{\mathcal{S}^{2}_{0,\tau}}^{2}$ $\displaystyle\leq
C_{T,\gamma,\mu}\mathbb{E}\left(|X_{0}|^{2}+\int_{0}^{\tau}|X_{s}|^{2}\mathrm{d}s+\int_{0}^{\tau}|b(s,0,\mu_{s})|^{2}\mathrm{d}s+\int_{0}^{\tau}|\sigma(s,0,\mu_{s})|^{2}\mathrm{d}s\right)$
$\displaystyle+C_{T,\gamma,\mu}\mathbb{E}\left((1+\sup_{0\leq
t\leq\tau}|X_{t}|^{2})\right)$ $\displaystyle\leq
C_{T,\gamma,\mu}\left(\mathbb{E}\left(|X_{0}|^{2}\right)+||X||_{\mathcal{S}^{2}_{0,\tau}}^{2}+1\right).$

The value of the constant $C_{T,\gamma,\mu}$ depend on $T$, $\gamma$, and
$\alpha$. It varies from one line to another. Given that
$X\in\mathcal{S}^{2}_{0,\tau}$ and based on assumptions
$\left(\mathbf{A}_{2}\right)$ and $\left(\mathbf{A}_{3}\right)$, as well as
the definition of the stopping time $\tau$, we can conclude, as indicated by
Remark 3.1, that $||\varphi(X^{\mu})||_{\mathcal{S}^{2}_{0,\tau}}<\infty$.

Now, we show that $\varphi$ is a contraction. Let
$X,\tilde{X}\in\mathcal{S}^{2}_{0,\tau}$, by proposition 3.2, we have

$\displaystyle||\varphi(X)-\varphi(\tilde{X})||_{\mathcal{S}^{2}_{0,\tau}}^{2}$
$\displaystyle\leq
C\Big{(}\mathbb{E}(\int_{0}^{\tau}|\sigma\left(s,X_{s},\mu_{s}\right)-\sigma(s,\tilde{X}_{s},\mu_{s})|^{2}\mathrm{d}s)$
$\displaystyle+\mathbb{E}(\int_{0}^{\tau}\int_{\mathbb{R}^{d}\backslash\\{0\\}}|\beta(s,X_{s^{-}},\mu_{s^{-}},z)-\beta(s,\tilde{X}_{s},\mu_{s^{-}},z)|^{2}\lambda(\mathrm{d}z)\mathrm{d}s)$
$\displaystyle+\mathbb{E}(\tau\int_{0}^{\tau}|b(s,X_{s},\mu_{s})-b(s,\tilde{X}_{s},\mu_{s})|^{2}\mathrm{d}s\Big{)}$
$\displaystyle\leq C\gamma^{2}\mathbb{E}\left((2\tau+\tau^{2})\sup_{0\leq
t\leq\tau}|X_{t}-\tilde{X}_{t}|^{2}\right)$
$\displaystyle\leq\alpha||X-\tilde{X}||_{\mathcal{S}^{2}_{0,\tau}}^{2}.$

Therefore, by the Banach fixed point theorem, we have existence and uniqueness
of solution on $[0,\tau]$.

By induction, we define the following sequence of stopping times:
$\tau_{0}=0,\;\text{ and }\;n\geq 1$

$\tau_{n+1}:=\inf\\{t>\tau_{n}:\;C\gamma^{2}(2(t-\tau_{n})+t^{2}-\tau_{n}^{2})\geq\alpha\\}\land
T.$

Using a similar argument as previously discussed, one can demonstrate the
existence and uniqueness of the solution on each interval
$[\tau_{n},\tau_{n+1}]$. Since $t\to
C\gamma^{2}(2(t-\tau_{n})+t^{2}-\tau_{n}^{2})$ is continuous, we obtain
$\mathbb{P}(\cap_{n\geq 1}\\{\tau_{n}=T\\})=1$, which implies the existence
and uniqueness of a solution on $[0,T]$.

We show that $X^{\mu}\in\mathcal{S}^{2}_{0,T}$. Given assumptions
$\left(\mathbf{A}_{4}\right)$ and $\left(\mathbf{A}_{5}\right)$, along with
inequality 2.4, we find

$\displaystyle||X^{\mu}||_{\mathcal{S}^{2}_{0,T}}^{2}$ $\displaystyle\leq
C_{T,\gamma}\mathbb{E}\left(|X_{0}|^{2}+\int_{0}^{T}(1+\sup_{0\leq s\leq
u}|X_{u}|^{2})\mathrm{d}s+\int_{0}^{T}|b(s,0,\mu_{s})|^{2}\mathrm{d}s+\int_{0}^{T}|\sigma(s,0,\mu_{s})|^{2}\mathrm{d}s+||A||_{\mathcal{S}^{2}_{0,T}}\right).$

We utilize Gronwall’s Lemma, and we conclude based on assumptions
$\left(\mathbf{A}_{2}\right)$ and $\left(\mathbf{A}_{3}\right)$. This
completes the proof. ∎

######  Theorem 3.2.

Under assumptions $\left(\mathbf{A}_{1}\right)$-$\left(\mathbf{A}_{5}\right)$,
a unique solution exists for the system $E(\mathcal{D},b,\sigma,\beta)$.

###### Proof.

By Theorem 3.1, for every
$\mu\in\mathcal{P}_{2}(\mathbb{D}([0,T],\mathbb{R}^{d}))$, there exists a
unique solution $(X^{\mu},K^{\mu})$ satisfying (ii), (iii) of Definition 3.1
and the following equation

$X_{t}^{\mu}=X_{0}+\int_{0}^{t}b\left(s,X_{s}^{\mu},\mu_{s}\right)\mathrm{d}s+\int_{0}^{t}\sigma\left(s,X_{s}^{\mu},\mu_{s}\right)\mathrm{d}W_{s}+\int_{0}^{t}\int_{\mathbb{R}^{d}\backslash\\{0\\}}\beta\left(s,X_{s^{-}}^{\mu},\mu_{s^{-}},z\right)\tilde{N}(\mathrm{d}s,\mathrm{d}z)+K_{t},\;\;\text{for}\;t\geq
0.$

We define the mapping
$\psi\colon\mathcal{P}_{2}(\mathbb{D}([0,T],\mathbb{R}^{d}))\to\mathcal{P}_{2}(\mathbb{D}([0,T],\mathbb{R}^{d}))$
which associates $\mu\in\mathcal{P}_{2}(\mathbb{D}([0,T],\mathbb{R}^{d}))$
with $\psi(\mu)=\mathcal{L}(X^{\mu})$. We show that $\psi$ is a contraction.

Consider $\mu,\mu^{\prime}\in\mathbb{D}([0,T],\mathbb{R}^{d})$. By Proposition
3.2 and under assumptions $\left(\mathbf{A}_{4}\right)$ and
$\left(\mathbf{A}_{5}\right)$, for every $t$, we have

$\displaystyle\mathbb{E}(\sup_{0\leq s\leq
t}|X_{s}^{\mu}-X_{s}^{\mu^{\prime}}|^{2})$ $\displaystyle\leq
C\gamma^{2}(2+t)\left(\int_{0}^{t}\mathbb{E}(\sup_{0\leq u\leq
s}|X_{u}^{\mu}-X_{u}^{\mu^{\prime}}|^{2})+W(\mu_{s},\mu^{\prime}_{s})^{2}\mathrm{d}s\right)$
$\displaystyle+\int_{0}^{t}W(\mu_{s^{-}},\mu_{s^{-}}^{\prime})^{2}\mathrm{d}s$
$\displaystyle\leq 2C\gamma^{2}(2+T)\left(\int_{0}^{t}\mathbb{E}(\sup_{0\leq
u\leq
s}|X_{u}^{\mu}-X_{u}^{\mu^{\prime}}|^{2})\mathrm{d}s+\int_{0}^{T}W_{s}(\mu,\mu^{\prime})^{2}\mathrm{d}s\right).$

In the final line, we have made use of the fact that
$W(\mu_{s},\mu^{\prime}_{s})\lor W(\mu_{s^{-}},\mu^{\prime}_{s^{-}})\leq
W_{s}(\mu,\mu^{\prime})$. By applying Gronwall’s inequality and considering
the definition of the Wasserstein metric, we obtain

$W_{T}\left(\psi(\mu),\psi(\mu^{\prime})\right)^{2}\leq
C_{T,\gamma}\int_{0}^{T}W_{s}(\mu,\mu^{\prime})^{2}\mathrm{d}s.$

Here, $C_{T,\gamma}$ is a constant depend on $T$ and $\gamma$. By iterating
the previous inequality, we get for any $n\geq 1$

$W_{T}\left(\psi^{n}(\mu),\psi^{n}\left(\mu^{\prime}\right)\right)^{2}\leq\frac{C_{T,\gamma}^{n}T^{n}}{n!}W_{T}\left(\mu,\mu^{\prime}\right)^{2},$

for $n$ large enough, $\psi$ is a contraction.

This completes the proof. ∎

###  3.3 Stability of Reflected Mckean-Vlasov equations

In this subsection, we study a stability result for the solution of the system
$E(\mathcal{D},b,\sigma,\beta)$ with respect to both the initial condition and
the coefficients.

######  Theorem 3.3.

Assume that $\left(\mathbf{A}_{1}\right)$-$\left(\mathbf{A}_{2}\right)$ are
satisfied. Let $(X_{0}^{n})_{n\geq 0}$, $X_{0}$ be sequence of random
variables satisfying $\left(\mathbf{A}_{3}\right)$. Let
$b,\sigma,\beta,(b^{n})_{n\geq 0},(\sigma^{n})_{n\geq 0},(\beta^{n})_{n\geq
0}$ be sequence of random maps satisfying
$\left(\mathbf{A}_{4}\right)$-$\left(\mathbf{A}_{5}\right)$ uniformly in $n$.
In addition, we assume the following:

  * i)

$\lim_{n\rightarrow\infty}\mathbb{E}|X^{n}_{0}-X_{0}|^{2}=0$.

  * ii)

There exists a positive constant $\delta$, such that

$\sup_{n\geq
0}\left(\left|b^{n}(t,x,\mu)\right|+\left|\sigma^{n}(t,x,\mu)\right|\right)\leq\delta\left(1+|x|\right).$

  * iii)

On each compact $K$ of $\mathbb{R}^{d}$, we have

$\lim_{n\rightarrow\infty}\sup_{0\leq t\leq T}\sup_{x\in
K}\sup_{\mu\in\mathcal{P}_{2}\left(\mathbb{R}^{d}\right)}\left|b^{n}(t,x,\mu)-b(t,x,\mu)\right|+\left|\sigma^{n}(t,x,\mu)-\sigma(t,x,\mu)\right|=0,\;\;\text{a.s}$
$\lim_{n\rightarrow\infty}\sup_{0\leq t\leq T}\sup_{x\in
K}\sup_{\mu\in\mathcal{P}_{2}\left(\mathbb{R}^{d}\right)}\int_{\mathbb{R}^{d}\backslash\\{0\\}}\left|\beta^{n}(t,x,\mu,z)-\beta(t,x,\mu,z)\right|^{2}\lambda(\rm{d}z)=0,\;\;\text{a.s}$

then

$\lim_{n\rightarrow\infty}E\left(\sup_{0\leq t\leq
T}\left|X_{t}^{n}-X_{t}\right|^{2}\right)=0,$

where $(X^{n},K^{n})$ is sequence of solutions of
$E_{\mathcal{D}}(b^{n},\sigma^{n},\beta^{n})$, and $(X,K)$ is the solution of
$E_{\mathcal{D}}(b,\sigma,\beta)$.

###### Proof.

We use Proposition 3.2, we get

$\displaystyle\mathbb{E}(\sup_{0\leq t\leq T}|X_{t}^{n}-X_{t}|^{2})$
$\displaystyle\leq
C\Big{(}\mathbb{E}(|X_{0}^{n}-X_{0}|^{2}+\int_{0}^{T}|\sigma^{n}\left(s,X_{s}^{n},\mathcal{L}_{X_{s}^{n}}\right)-\sigma\left(s,X_{s},\mathcal{L}_{X_{s}}\right)|^{2}\mathrm{d}s)$
$\displaystyle+\mathbb{E}(\int_{0}^{T}\int_{\mathbb{R}^{d}\backslash\\{0\\}}|\beta^{n}(s,X_{s^{-}}^{n},\mathcal{L}_{X_{s^{-}}^{n}},z)-\beta(s,X_{s},\mathcal{L}_{X_{s^{-}}},z)|^{2}\lambda(\mathrm{d}z)\mathrm{d}s)$
$\displaystyle+\mathbb{E}(T\int_{0}^{T}|b^{n}\left(s,X_{s}^{n},\mathcal{L}_{X_{s}^{n}}\right)-b\left(s,X_{s},\mathcal{L}_{X_{s}}\right)|^{2}\mathrm{d}s)\Big{)}.$

We add and subtract the quantities
$\sigma^{n}\left(s,X_{s},\mathcal{L}_{X_{s}}\right),\beta^{n}\left(s,X_{s},\mathcal{L}_{X_{s^{-}}},z\right)$,
and $b^{n}\left(s,X_{s},\mathcal{L}_{X_{s}}\right)$, by using the fact that
the coefficients are Lipschitz, and from the definition of Wassertein metric,
we get

$\displaystyle\mathbb{E}\left(\sup_{0\leq t\leq
T}|X_{t}^{n}-X_{t}|^{2}\right)$ $\displaystyle\leq
C_{T,\gamma}\Big{(}\int_{0}^{T}\mathbb{E}\left(\sup_{0\leq u\leq
s}\left|X_{u}^{n}-X_{u}\right|^{2}\right)\mathrm{d}s+\mathbb{E}(\left|X_{0}^{n}-X_{0}|^{2}\right)$
$\displaystyle+\mathbb{E}(\int_{0}^{T}|\sigma^{n}\left(s,X_{s},\mathcal{L}_{X_{s}}\right)-\sigma\left(s,X_{s},\mathcal{L}_{X_{s}}\right)|^{2}\mathrm{d}s)$
$\displaystyle+\mathbb{E}(\int_{0}^{T}\int_{\mathbb{R}^{d}\backslash\\{0\\}}|\beta^{n}(s,X_{s^{-}},\mathcal{L}_{X_{s^{-}}},z)-\beta(s,X_{s},\mathcal{L}_{X_{s^{-}}},z)|^{2}\lambda(\mathrm{d}z)\mathrm{d}s)$
$\displaystyle+\mathbb{E}(\int_{0}^{T}|b^{n}\left(s,X_{s},\mathcal{L}_{X_{s}}\right)-b\left(s,X_{s},\mathcal{L}_{X_{s}}\right)|^{2}\mathrm{d}s)\Big{)}.$

By applying Gronwall’s lemma, we obtain

$\displaystyle\mathbb{E}\left(\sup_{0\leq t\leq
T}|X_{t}^{n}-X_{t}|^{2}\right)$ $\displaystyle\leq
C_{T,\gamma}\Big{(}\mathbb{E}(|X_{0}^{n}-X_{0}|^{2})+\mathbb{E}(\int_{0}^{T}|\sigma^{n}\left(s,X_{s},\mathcal{L}_{X_{s}}\right)-\sigma\left(s,X_{s},\mathcal{L}_{X_{s}}\right)|^{2}\mathrm{d}s)$
$\displaystyle+\mathbb{E}(\int_{0}^{T}\int_{\mathbb{R}^{d}\backslash\\{0\\}}|\beta^{n}(s,X_{s^{-}},\mathcal{L}_{X_{s^{-}}},z)-\beta(s,X_{s},\mathcal{L}_{X_{s^{-}}},z)|^{2}\lambda(\mathrm{d}z)\mathrm{d}s)$
$\displaystyle+\mathbb{E}(\int_{0}^{T}|b^{n}\left(s,X_{s},\mathcal{L}_{X_{s}}\right)-b\left(s,X_{s},\mathcal{L}_{X_{s}}\right)|^{2}\mathrm{d}s)\Big{)}.$

The first term of this inequality tend to $0$ by $(i)$. From the assumptions
$\left(\mathbf{A}_{5}\right)$, $(ii)$, and $(iii)$, the Dominated Convergence
Theorem guarantees that the three last terms tend to $0$.

This completes the proof. ∎

###  3.4 Propagation of Chaos

Let $N\geq 1$ and $i\in\\{1,\dots,N\\}$. We consider
$(X_{0}^{i},W^{i},\tilde{N}^{i})$ as a sequence of independent copies of
$(X_{0},W,\tilde{N})$. In this subsection, we study the law of a solution to
the interacting $N$-particle system $(X^{N,i},K^{N,i})_{1\leq i\leq N}$ given
by the following

$\textbf{(}S)\left\\{\begin{aligned}
&X_{t}^{N,i}=X_{0}^{i}+\int_{0}^{t}b\left(s,X_{s}^{N,i},\frac{1}{N}\sum_{j=1}^{N}\delta_{X_{s}^{j,N}}\right)\mathrm{d}s+\int_{0}^{t}\sigma\left(s,X_{s}^{N,i},\frac{1}{N}\sum_{j=1}^{N}\delta_{X_{s}^{j,N}}\right)\mathrm{d}W_{s}^{i}\\\
&\quad\quad+\int_{0}^{t}\int_{\mathbb{R}^{d}\backslash\\{0\\}}\beta\left(s,X_{s^{-}}^{N,i},\frac{1}{N}\sum_{j=1}^{N}\delta_{X_{s^{-}}^{j,N}},z\right)\tilde{N}^{i}(\mathrm{d}s,\mathrm{d}z)+K_{t}^{i,N},\quad
t\in[0,T],\\\ &X_{t}^{i,N}\in D_{t},\quad\text{for every}\;t\in[0,T],\\\
&\text{for every c\\`{a}dl\\`{a}g and adapted
process}\;Z\in\mathcal{D},\;\int_{0}^{t}\langle
X_{s}^{i,N}-Z_{s},\mathrm{d}K_{s}^{i,N}\rangle\leq 0\;\text{for every}\;t\geq
0.\end{aligned}\right.$

######  Proposition 3.3.

Under assumptions $\left(\mathbf{A}_{1}\right)$-$\left(\mathbf{A}_{5}\right)$,
a unique solution exists for the system $(\textbf{S})$.

###### Proof.

Since for every $u=(u_{1},\dots,u_{N})$ and $v=(v_{1},\dots,v_{N})$ in
$\mathbb{R}^{dN}$,

$W\left(\frac{1}{N}\sum_{i=1}^{N}\delta_{u_{i}},\frac{1}{N}\sum_{i=1}^{N}\delta_{v_{i}}\right)\leq\frac{1}{\sqrt{N}}|u-v|,$

the proof follows similarly as the proof of Theorem 3.2. ∎

######  Theorem 3.4.

Let $(X^{i},K^{i})$ be the solution of $RP(Y^{i})_{\mathcal{D}}$, where
$Y^{i}$ is given by:

$Y_{t}^{i}=X_{0}^{i}+\int_{0}^{t}b\left(s,X_{s}^{i},\mathcal{L}_{X_{s}^{i}}\right)\mathrm{d}s+\int_{0}^{t}\sigma\left(s,X_{s}^{i},\mathcal{L}_{X_{s}^{i}}\right)\mathrm{d}W_{s}^{i}+\int_{0}^{t}\int_{\mathbb{R}^{d}\backslash\\{0\\}}\beta\left(s,X_{s^{-}}^{i},\mathcal{L}_{X_{s^{-}}^{i}},z\right)\tilde{N}^{i}(\mathrm{d}s,\mathrm{d}z).$

Then, there exists a constant $C$, which depend on $T$, such that

$\mathbb{E}(\sup_{0\leq t\leq T}|X_{t}^{i,N}-X_{t}^{i}|^{2})\leq
C\begin{cases}N^{-1/2},&d<4\\\ N^{-1/2}\log N,&d=4\\\
N^{-\frac{2}{d}},&d>4\end{cases}$

Consequently, the following convergence hold true:

$\lim_{N\rightarrow\infty}\sup_{1\leq i\leq N}\mathbb{E}\sup_{0\leq t\leq
T}|X_{t}^{i,N}-X_{t}^{i}|^{2}=0.$

And for all $k\geq 1$, the following weak convergence holds true:

$\mathcal{L}_{\left(X^{N,1},X^{N,2},\ldots,X^{N,k}\right)}\underset{N\rightarrow\infty}{\longrightarrow}\mathcal{L}_{\left(X^{1},X^{2},\ldots,X^{k}\right)}.$

###### Proof.

Set

$\mu^{N}=\frac{1}{N}\sum_{j=1}^{N}\delta_{X^{j,N}},\;\text{and}\;\mu=\frac{1}{N}\sum_{j=1}^{N}\delta_{X^{j}}.$

By applying the inequality in Proposition 3.2, and by adding and subtracting
the terms
$\sigma\left(s,X_{s}^{i},\mu_{s}^{N}\right),\;\sigma\left(s,X_{s}^{i},\mu_{s}\right)$,
$\beta\left(s,X_{s^{-}}^{i},\mu_{s^{-}}^{N},z\right),\;\beta\left(s,X_{s^{-}}^{i},\mu_{s^{-}},z\right)$,
$b\left(s,X_{s}^{i},\mu_{s}^{N}\right)$, and
$b\left(s,X_{s}^{i},\mu_{s}\right)$, we obtain

$\displaystyle\mathbb{E}(\sup_{0\leq t\leq T}|X_{t}^{i,N}-X_{t}^{i}|^{2})$
$\displaystyle\leq
C\Big{(}\mathbb{E}(\int_{0}^{T}|\sigma\left(s,X_{s}^{i,N},\mu_{s}^{N}\right)-\sigma\left(s,X_{s}^{i},\mu_{s}^{N}\right)|^{2}\mathrm{d}s)$
$\displaystyle+\mathbb{E}(\int_{0}^{T}|\sigma\left(s,X_{s}^{i},\mu_{s}^{N}\right)-\sigma(s,X_{s}^{i},\mu_{s})|^{2}\mathrm{d}s)$
$\displaystyle+\mathbb{E}(\int_{0}^{T}|\sigma\left(s,X_{s}^{i},\mu_{s}\right)-\sigma(s,X_{s}^{i},\mathcal{L}_{X^{i}_{s}})|^{2}\mathrm{d}s)$
$\displaystyle+\mathbb{E}(\int_{0}^{T}\int_{\mathbb{R}^{d}\backslash\\{0\\}}|\beta(s,X_{s^{-}}^{i,N},\mu_{s^{-}}^{N},z)-\beta(s,X_{s^{-}}^{i},\mu_{s^{-}}^{N},z)|^{2}\lambda(\mathrm{d}z)\mathrm{d}s)$
$\displaystyle+\mathbb{E}(\int_{0}^{T}\int_{\mathbb{R}^{d}\backslash\\{0\\}}|\beta(s,X_{s^{-}}^{i},\mu_{s^{-}}^{N},z)-\beta(s,X_{s^{-}}^{i},\mu_{s^{-}},z)|^{2}\lambda(\mathrm{d}z)\mathrm{d}s)$
$\displaystyle+\mathbb{E}(\int_{0}^{T}\int_{\mathbb{R}^{d}\backslash\\{0\\}}|\beta(s,X_{s^{-}}^{i},\mu_{s^{-}},z)-\beta(s,X_{s^{-}}^{i},\mathcal{L}_{X^{i}_{s^{-}}},z)|^{2}\lambda(\mathrm{d}z)\mathrm{d}s)$
$\displaystyle+\mathbb{E}(T\int_{0}^{T}|b\left(s,X_{s}^{i,N},\mu_{s}^{N}\right)-b\left(s,X_{s}^{i},\mu_{s}^{N}\right)|^{2}\mathrm{d}s)$
$\displaystyle+\mathbb{E}(T\int_{0}^{T}|b\left(s,X_{s}^{i},\mu_{s}^{N}\right)-b(s,X_{s}^{i},\mu_{s})|^{2}\mathrm{d}s)$
$\displaystyle+\mathbb{E}(T\int_{0}^{T}|b\left(s,X_{s}^{i},\mu_{s}\right)-b(s,X_{s}^{i},\mathcal{L}_{X^{i}_{s}})|^{2}\mathrm{d}s)\Big{)}.$

Using assumptions $\left(\mathbf{A}_{4}\right)$ and
$\left(\mathbf{A}_{5}\right)$, along with Gronwall’s Lemma and the
exchangeability of $(X^{i,N},X^{i})$, we find:

$\displaystyle\mathbb{E}(\sup_{0\leq t\leq T}|X_{t}^{i,N}-X_{t}^{i}|^{2})$
$\displaystyle\leq
C_{T,\gamma}\left(\mathbb{E}\left(\int_{0}^{T}W(\mu_{s},\mathcal{L}_{X_{s}^{i}})^{2}\rm{d}s\right)+\mathbb{E}\left(\int_{0}^{T}W(\mu_{s^{-}},\mathcal{L}_{X_{s^{-}}^{i}})^{2}\rm{d}s\right)\right).$

By Theorem 3.1, $X^{i}\in\mathcal{S}^{2}_{0,T}$, then $\mu$ and
$\mathcal{L}_{X^{i}}$ belong to
$\mathcal{P}_{2}(\mathbb{D}([0,T],\mathbb{R}^{d}))$. Similar to the proof of
Theorem 3.1, it follows that $t\to\mu_{t}$ and $t\to\mathcal{L}_{X_{t}}$ are
càdlàg with respect to the Wasserstein metric, hence

$\mathbb{E}(\sup_{0\leq t\leq T}|X_{t}^{i,N}-X_{t}^{i}|^{2})\leq
2TC_{T,\gamma}\sup_{0\leq t\leq
T}\mathbb{E}\left(W(\mu_{s},\mathcal{L}_{X_{s}^{i}})^{2}\right).$ (3.2)

By Theorem 5.8 in [5], there exists a positive constant $C$ such that

$\mathbb{E}\left[W\left(\mu_{s},\mathcal{L}_{X_{s}^{i}}\right)^{2}\right]\leq
C\begin{cases}N^{-1/2},&d<4,\\\ N^{-1/2}\log N,&d=4,\\\
N^{-\frac{2}{d}},&d>4.\end{cases}$ (3.3)

The second point follows immediately from this inequality.

For the last point, we use the second point of Remark 1.2, the definition of
Wasserstein metric and the exchangeability of $(X^{i,N},X^{i})$, we get

$\displaystyle
W_{d_{\mathbb{S}}}\left(\mathcal{L}_{\left(X^{N,1},X^{N,2},\ldots,X^{N,k}\right)},\mathcal{L}_{\left(X^{1},X^{2},\ldots,X^{k}\right)}\right)^{2}$
$\displaystyle\leq
W_{T}\left(\mathcal{L}_{\left(X^{N,1},X^{N,2},\ldots,X^{N,k}\right)},\mathcal{L}_{\left(X^{1},X^{2},\ldots,X^{k}\right)}\right)^{2}$
$\displaystyle\leq k\mathbb{E}(\sup_{0\leq i\leq
k}\left|X^{i,n}-X^{i}\right|^{2}).$

Again by remark 1.2, and Theorem 6.9 in [29], we get
$\mathcal{L}_{\left(X^{N,1},X^{N,2},\ldots,X^{N,k}\right)}$ converges weakly
to $\mathcal{L}_{\left(X^{1},X^{2},\ldots,X^{k}\right)}$.

This completes the proof. ∎

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