# Volterra type McKean-Vlasov SDEs with singular kernels: Well-posedness,
Propagation of Chaos and Euler schemes††thanks: S. Liu is supported by
Postgraduate Research Practice Innovation Program of Jiangsu Province (No.
KYCX23 1668). H. Gao is supported in part by the NSFC Grant Nos. 12171084 and
the fundamental Research Funds for the Central Universities No. RF1028623037.

Shanqi Liu111School of Mathematical Science, Nanjing Normal University,
Nanjing 210023, China and Hongjun Gao222School of Mathematics, Southeast
University, Nanjing 211189, China E-mail<EMAIL_ADDRESS>author. E-mail<EMAIL_ADDRESS>
Abstract In this paper, our work is devoted to studying Volterra type McKean-
Vlasov stochastic differential equations with singular kernels. Firstly, the
well-posedness of Volterra type McKean-Vlasov stochastic differential
equations are established. And then propagation of chaos is proved with
explicit estimate of the convergence rate. Finally, We also propose an
explicit Euler scheme for an interacting particle system associated with the
Volterra type McKean-Vlasov equation.

Key Words: Stochastic Volterra equations, McKean-Vlasov SDEs, Well-posedness,
Propagation of chaos, Euler schemes.

AMS Subject Classification (2020): 60H20; 45D05; 60H35.

##  1 Introduction

Stochastic differential equations (SDEs) are mathematical models that
characterize complex phenomena including physical, chemical, and biological
sciences. Among them, Volterra-type stochastic differential equations are
important and interesting class of research objects with a large number of
applications in mathematics finance, biology, etc. In particular, when the
kernel $K(t,s)=K^{H}(t,s)$, where $K^{H}(t,s)$ is defined in Remark 3.2,
Volterra-type SDEs is reduced to the classical SDEs driven by fractional
Brownian motion. If the kernel $K(t,s)=(t-s)^{-\alpha}$ with suitable index
$\alpha$, Volterra-type SDEs is reduced to Caputo fractional SDEs, see e.g.
[3].

Stochastic Volterra integral equations are more complex models compared with
usual SDEs. On the one hand, these equations are in general neither Markovian
nor semi-martingales, and include fractional Brownian motion with Hurst index
smaller than $1/2$ as a special case. On the other hand, possibly singular
kernels need to be controlled carefully, which causes fundamental technical
difficulties. In the past decade, stochastic Volterra integral equations have
been intensively investigated, e.g. the well-posedness [1, 2, 5, 20, 21, 22,
23, 28, 30, 31, 32, 33, 35], the asymptotic properties such as the large
deviation [15, 10, 19, 34, 35], moderate deviation [16, 10] and central limit
theorem [27], and numerical methods [17, 26, 34].

McKean-Vlasov SDEs (also called distribution dependent SDEs [24] or mean-field
SDEs), it was first discussed by Kac in literatures [12, 13], which describe
limiting behaviors of individual particles in an interacting particle system
of mean-field type when the number of particles goes to infinity (so called
propagation of chaos) and widely used in various fields. One can refer to [18]
and the references therein for more details. Volterra-type McKean-Vlasov SDEs
first appeared in the paper [28], which studied the well-posedness and
Pontryagin’s type maximum principle of a class mean-field backward stochastic
Volterra integral equations. Recently, there has been an growing interest in
the study of mean-field forward (backward) SDEs. In [31], the authors extended
the results in [28] by considering the well-posedness and Pontryagin’s type
maximum principle of the mean-field backward doubly stochastic Volterra
integral equation. Fan and Jiang [7] studied a class of distribution-dependent
SDEs driven by Volterra processes and the associated optimal control problem.

To the best of our knowledge, it seems that there are few works on forward
Volterra type McKean-Valsov SDEs. For the regular kernel, Jie et al. [11]
studied one-dimensional McKean–Vlasov stochastic Volterra equations with
Hölder diffusion coefficients. One can also refer Theorem $2.10$ in [23]. The
aim of the paper is to study basic properties of Volterra type McKean-Vlasov
SDEs with singular kernels. More precisely, we first consider the well-
posedness and propagation of chaos for Volterra type McKean-Vlasov SDEs under
Lipschitz condition in the sense of distribution measure. Very recently, we
remark that Prömel and Scheffels [23] also consider the same problems for
Volterra type McKean-Vlasov SDEs with singular kernels. Compared with Theorem
$3.2$ in [23], our assumption $\mathbf{(H1^{\prime})}$ is weaker than
assumption $2.1$ in [23]. Volterra type Gronwall’s inequality play the key
role, which provides a clear estimation framework. As an application, we
propose an explicit Euler scheme for an interacting particle system associated
with the Volterra type McKean-Vlasov equation with singular kernels inspired
by [34].

The rest of the paper is organized as follows. In Section 2, we review some
basic notations, and a Gronwall’s inequality of Volterra type is introduced.
In Section 3, under some suitable conditions, the existence, uniqueness and
path continuity of solution to Volterra type McKean-Vlasov stochastic
differential equations are established. Later in Section 4, we discuss
Propagation of chaos for Volterra type McKean-Vlasov SDEs. Finally in Section
5, an explicit Euler scheme for an interacting particle system associated with
the Volterra type McKean-Vlasov equation with singular kernels is constructed.

##  2 Preliminary

###  2.1 Notations

Throughout this paper, we denote $\mathbb{R}^{d}$ the $n$-dimensional
Euclidean space. Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq
0},\mathbb{P})$ to be a given complete filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})$, where
$\mathcal{F}_{t}$ is a nondecreasing family of sub-$\sigma$ fields of
$\mathcal{F}$ satisfying the usual conditions. Let $\mathcal{P}$ be the space
of all probability measures $\mu$ on $\mathbb{R}^{d}$, and let

$\mathcal{P}_{2}(\mathbb{R}^{d})=\left\\{\mu\in\mathcal{P}\left(\mathbb{R}^{d}\right):\mu\left(|\cdot|^{2}\right):=\int_{\mathbb{R}^{d}}|x|^{2}\mu(\mathrm{d}x)<\infty\right\\}.$

It is well known that $\mathcal{P}_{2}$ is a Polish space under the
Wasserstein distance

$\mathbb{W}_{2}(\mu,\nu):=\inf_{\pi\in\mathcal{C}(\mu,\nu)}\left(\int_{\mathbb{R}^{m}\times\mathbb{R}^{m}}|x-y|^{2}\pi(\mathrm{d}x,\mathrm{d}y)\right)^{\frac{1}{2}},\
\mu,\nu\in\mathcal{P}_{2}\left(\mathbb{R}^{d}\right),$

where $\mathcal{C}(\mu,\nu)$ is the set of couplings for $\mu$ and $\nu$; that
is, $\pi\in\mathcal{C}(\mu,\nu)$ is a probability measure on
$\mathbb{R}^{d}\times\mathbb{R}^{d}$ such that
$\pi\left(\cdot\times\mathbb{R}^{d}\right)=\mu$ and
$\pi\left(\mathbb{R}^{d}\times\cdot\right)=\nu$.

For a measurable function $K$ on $\mathbb{R}_{+}\times\mathbb{R}_{+}$ and
measurable function $L$ on $\mathbb{R}_{+}\times\mathbb{R}_{+}$, the
convolutions $K*L$ and $L*K$ are defined by

$(K*L)(t,s)=\int_{s}^{t}K(t,u)L(u,s){\mathrm{d}}u,$
$(L*K)(t,s)=\int_{s}^{t}L(t,u)K(u,s){\mathrm{d}}u,$

for almost all $(t,s)\in\Delta$, where
$\Delta:=\\{t,s\in\mathbb{R}_{+}\times\mathbb{R}_{+}:s\leq t\\}.$

###  2.2 Growall’s inequality of Volterra type

We first recall the following lemma due to Girpenberg [9] (Theorem $1$).

######  Lemma 2.1.

Let $K:\Delta\to\mathbb{R}_{+}$ be a measurable function. Assume that for any
$T>0$,

$t\mapsto\int_{0}^{t}K(t,s){\mathrm{d}}s\in L^{\infty}(0,T)$

and

$\limsup_{\varepsilon\downarrow
0}\Big{\|}\int_{\cdot+\varepsilon}^{\cdot}K(\cdot+\varepsilon,s){\mathrm{d}}s\Big{\|}_{L^{\infty}(0,T)}<1.$

Define

$\displaystyle R_{1}(t,s):=K(t,s),\quad
R_{n+1}(t,s):=\int_{s}^{t}K(t,u)R_{n}(u,s){\mathrm{d}}u,n\in\mathbb{N}.$ (2.1)

Then for any $T>0$, there exist constants $C_{T}>0$ and $\gamma\in(0,1)$ such
that

$\Big{\|}\int_{0}^{\cdot}R_{n}(\cdot,s)ds\Big{\|}_{L^{\infty}(0,T)}\leq
C_{T}n\gamma^{n},\forall n\in\mathbb{N}.$

In particular, the series

$\displaystyle R(t,s):=\sum_{n=1}^{\infty}R_{n}(t,s)$ (2.2)

converges for almost all $(t,s)\in\Delta$, and

$\displaystyle R(t,s)-K(t,s)=(K*R)(t,s)-(R*K)(t,s)$ (2.3)

and for any $T>0$,

$\displaystyle t\mapsto\int_{0}^{t}R(t,s){\mathrm{d}}s\in L^{\infty}(0,T).$
(2.4)

The function $R$ defined by (2.2) is called the resolvent of the kernel $K$.
All the functions $K$ in Lemma 2.1 will be denoted by $\mathcal{K}$. We also
denote by $\mathcal{K}_{>1}$ the set of all nonnegative measurable functions
$K$ on $\Delta$ with the property that for any $T>0$ and some $\beta>1$,

$t\mapsto\int_{0}^{t}K^{\beta}(t,s){\mathrm{d}}s\in L^{\infty}(0,T).$

It is clear that $\mathcal{K}_{>1}\subset\mathcal{K}.$ We now present
Gronwall’s lemma of Volterra type due to Zhang [35] (Lemma $2.2$).

######  Lemma 2.2.

Let $K\in\mathcal{K}$ and $R_{n}$ and $R$ be defined by (2.1) and (2.2),
respectively. Let $f,g:\mathbb{R}_{+}\to\mathbb{R}_{+}$ be two measurable
functions satisfying that for any $T>0$ and some $n\in\mathbb{N}$,

$t\mapsto\int_{0}^{t}R_{n}(t,s)f(s){\mathrm{d}}s\in L^{\infty}(0,T)$

and for almost all $t\in(0,\infty)$,

$\int_{0}^{t}R(t,s)g(s){\mathrm{d}}s<\infty.$

If for almost all $t\in(0,\infty)$,

$f(t)\leq g(t)+\int_{0}^{t}K(t,s)f(s){\mathrm{d}}s,$

then for almost all $t\in(0,\infty)$,

$f(t)\leq g(t)+\int_{0}^{t}R(t,s)g(s){\mathrm{d}}s.$

##  3 Well-posedness for Volterra type McKean-Vlasov SDEs

###  3.1 Global existence and uniqueness

Consider the following Volterra type McKean-Vlasov SDEs:

$\displaystyle
X_{t}=X_{0}+\int_{0}^{t}b(t,s,X_{s},\mathcal{L}_{X_{s}}){\mathrm{d}}s+\int_{0}^{t}\sigma(t,s,X_{s},\mathcal{L}_{X_{s}}){\mathrm{d}}W_{s},$
(3.1)

where $X_{0}\in\mathbb{R}^{d}$ and
$\sigma:\mathbb{R}_{+}\times\mathbb{R}_{+}\times\mathbb{R}^{d}\times\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}^{d}\times\mathbb{R}^{m}$,
$b:\mathbb{R}_{+}\times\mathbb{R}_{+}\times\mathbb{R}^{d}\times\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}^{d}$
are Borel measurable functions, and $\\{W_{t}\\}_{t\geq 0}$ is an
m-dimensional standard Brownian motion definied on the classical Wiener space.

Now, we introduce the hypothesis under which we will be able to prove the
global existence and uniqueness of solutions to (3.1).

$\mathbf{(H1)}$ Integrability condition:

$\sup_{t\in[0,T]}\int_{0}^{t}\Big{[}K_{1}(t,s)+K_{2}(t,s)+K_{3}(t,s)+K_{4}(t,s)\Big{]}{\mathrm{d}}s\leq
C_{T},\quad t\in[0,T].$

where $K_{i},i=1,2,3,4$ are from $\mathbf{(H2)}$ and $\mathbf{(H3)}$ below.

$\mathbf{(H2)}$ Lipschitz condition: There exist positive functions
$K_{1}(t,s)\in\mathcal{K}$ and $K_{2}(t,s)\in\mathcal{K}$ such that for all
$x,y\in\mathbb{R}^{d}$, $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and
$t,s\in[0,T]$

$\displaystyle|b(t,s,x,\mu)-b(t,s,y,\nu)|$ $\displaystyle\leq
K_{1}(t,s)\Big{(}|x-y|+\mathbb{W}_{2}(\mu,\nu)\Big{)},$
$\displaystyle||\sigma(t,s,x,\mu)-\sigma(t,s,y,\nu)||^{2}$ $\displaystyle\leq
K_{2}(t,s)\Big{(}|x-y|^{2}+\mathbb{W}_{2}^{2}(\mu,\nu)\Big{)}.$

$\mathbf{(H3)}$ Linear growth condition: There exist positive functions
$K_{3}(t,s)\in\mathcal{K}$ and $K_{4}(t,s)\in\mathcal{K}$ such that for all
$x\in\mathbb{R}^{d}$, $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and
$t,s\in[0,T]$

$\displaystyle|b(t,s,x,\mu)|$ $\displaystyle\leq
K_{3}(t,s)\Big{(}1+|x|+\mathbb{W}_{2}(\mu,\delta_{0})\Big{)},$
$\displaystyle||\sigma(t,s,x,\mu)||^{2}$ $\displaystyle\leq
K_{4}(t,s)\Big{(}1+|x|^{2}+\mathbb{W}_{2}^{2}(\mu,\delta_{0})\Big{)}.$

######  Theorem 3.1.

Assume that $X_{0}\in L^{p}(\Omega,\mathcal{F}_{0},P)$ and assumptions
$\mathbf{(H1)}$-$\mathbf{(H3)}$ hold, Then there exists a unique measurable
$(\mathcal{F}_{t})$-adapted process $X_{t}$ such that for almost all $t\geq
0$,

$X_{t}=X_{0}+\int_{0}^{t}b(t,s,X_{s},\mathcal{L}_{X_{s}}){\mathrm{d}}s+\int_{0}^{t}\sigma(t,s,X_{s},\mathcal{L}_{X_{s}}){\mathrm{d}}W_{s},t\in[0,T],P\text{-}a.s.,$

and for any $T>0$ some constant $C_{p,T}$

$\displaystyle E|X_{t}|^{p}\leq C_{p,T}(E|X_{0}|^{p}+1).$ (3.2)

for almost all $t\in[0,T]$.

###### Proof.

We utilize the successive approximation scheme to prove Theorem 3.1. Define
recursively $(X^{n})_{n\geq 1}$ as follows: $X^{1}_{t}:=X_{0}$, $t\in[0,T]$
and for each $n\in\mathbb{N}$,

$\displaystyle
X^{n+1}_{t}:=X_{0}+\int_{0}^{t}b(t,s,X^{n}_{s},\mathcal{L}_{X^{n}_{s}}){\mathrm{d}}s+\int_{0}^{t}\sigma(t,s,X^{n}_{s},\mathcal{L}_{X^{n}_{s}}){\mathrm{d}}W_{s},t\in[0,T].$
(3.3)

By Burkholder-Davis-Gundy (BDG) inequality, H$\mathrm{\ddot{o}}$lder’s
inequality,
$\mathbb{W}_{2}(\mathscr{L}_{X},\mathscr{L}_{Y})\leq[E|X-Y|^{2}]^{\frac{1}{2}}$
and $\mathbf{(H1)}$, $\mathbf{(H3)}$ we have

$\displaystyle E\Big{(}|X_{t}^{n+1}|^{p}\Big{)}$ $\displaystyle\leq
3^{p-1}E|X_{0}|^{p}+3^{p-1}E\Big{(}\Big{|}\int_{0}^{t}b(t,s,X^{n}_{s},\mathcal{L}_{X^{n}_{s}}){\mathrm{d}}s\Big{|}^{p}\Big{)}$
$\displaystyle+3^{p-1}E\Big{(}\Big{|}\int_{0}^{t}\sigma(t,s,X^{n}_{s},\mathcal{L}_{X^{n}_{s}}){\mathrm{d}}W_{s}\Big{|}^{p}\Big{)}$
$\displaystyle\leq
3^{p-1}E|X_{0}|^{p}+3^{p-1}E\Big{(}\int_{0}^{t}K_{3}(t,s)\big{|}1+|X^{n}_{s}|+\mathbb{W}_{2}(\mathcal{L}_{X^{n}_{s}},\delta_{0})\big{|}{\mathrm{d}}s\Big{)}^{p}$
$\displaystyle+C_{p}E\Big{(}\int_{0}^{t}||\sigma(t,s,X^{n}_{s},\mathcal{L}_{X^{n}_{s}})||^{2}{\mathrm{d}}s\Big{)}^{\frac{p}{2}}$
$\displaystyle\leq 3^{p-1}E|X_{0}|^{p}+C_{p}\int_{0}^{t}K_{3}(t,s)\cdot
E\Big{(}1+|X^{n}_{s}|^{p}+\mathbb{W}^{p}_{2}(\mathcal{L}_{X^{n}_{s}},\delta_{0})\Big{)}{\mathrm{d}}s\cdot\Big{(}\int_{0}^{t}K_{3}(t,s){\mathrm{d}}s\Big{)}^{p-1}$
$\displaystyle+C_{p}\int_{0}^{t}K_{4}(t,s)\cdot
E\Big{(}1+|X^{n}_{s}|^{p}+\mathbb{W}^{p}_{2}(\mathcal{L}_{X^{n}_{s}},\delta_{0})\Big{)}{\mathrm{d}}s\cdot\Big{(}\int_{0}^{t}K_{3}(t,s){\mathrm{d}}s\Big{)}^{\frac{p-2}{2}}$
$\displaystyle\leq 3^{p-1}E|X_{0}|^{p}+C_{p,T}\int_{0}^{t}K_{3}(t,s)\cdot
E(1+|X_{s}^{n}|^{p}){\mathrm{d}}s+C_{p,T}\int_{0}^{t}K_{4}(t,s)\cdot
E(1+|X_{s}^{n}|^{p}){\mathrm{d}}s$ $\displaystyle\leq
3^{p-1}E|X_{0}|^{p}+C_{p,T}+C_{p,T}\int_{0}^{t}\Big{(}K_{3}(t,s)+K_{4}(t,s)\Big{)}\cdot
E|X_{s}^{n}|^{p}{\mathrm{d}}s$ $\displaystyle\leq
C_{p,T}E|X_{0}|^{p}+C_{p,T}\int_{0}^{t}\Big{(}K_{3}(t,s)+K_{4}(t,s)\Big{)}\cdot
E|X_{s}^{n}|^{p}{\mathrm{d}}s.$ (3.4)

Let

$f_{m}(t):=\sup_{n=1,\cdots,m}E|X_{s}^{n}|^{p},$

then we get

$f_{m}(t)\leq C_{p,T}(E|X_{0}|^{p}+1)+\int_{0}^{t}K_{3,4}(t,s)\cdot
f_{m}(s){\mathrm{d}}s,\quad\forall m\geq 1,$

where $K_{3,4}:=C_{p,T}\cdot(K_{3}(t,s)+K_{4}(t,s))$ and the constant
$C_{p,T}$ is independent of $m$.

Let $R^{K_{3,4}}$ be defined by (2.2) in terms of $K_{3,4}$. Note that by
(2.1), (2.3), (2.4) and $\mathbf{(H1)}$, when $n=1$ we have

$\displaystyle\int_{0}^{T}R_{1}^{K_{3,4}}(t,s)f_{m}(s){\mathrm{d}}s=\int_{0}^{T}K_{3,4}(t,s)f_{m}(s){\mathrm{d}}s<\infty,$

and

$\displaystyle\int_{0}^{t}R^{K_{3,4}}(t,s)\cdot E|X_{0}|^{p}{\mathrm{d}}s$
$\displaystyle=\int_{0}^{t}K_{3,4}(t,s)\cdot
E|X_{0}|^{p}{\mathrm{d}}s+\int_{0}^{t}\Big{(}\int_{s}^{t}R^{K_{3,4}}(t,u)K_{3,4}(u,s){\mathrm{d}}u\Big{)}\cdot
E|X_{0}|^{p}{\mathrm{d}}s$
$\displaystyle=\int_{0}^{t}K_{3,4}(t,s){\mathrm{d}}s\cdot
E|X_{0}|^{p}+\int_{0}^{t}R^{K_{3,4}}(t,u)\Big{(}\int_{0}^{u}K_{3,4}(u,s){\mathrm{d}}s\Big{)}{\mathrm{d}}u\cdot
E|X_{0}|^{p}<\infty.$

Hence, Combining (2.3), (2.4), $\mathbf{(H1)}$ and Lemma 2.2, we obtain that
for almost all $t\in[0,T]$,

$\displaystyle\sup_{n\in\mathbb{N}}E|X^{n}_{t}|^{p}$
$\displaystyle=\lim_{m\to\infty}f_{m}(t)\leq
C_{p,T}(E|X_{0}|^{p}+1)+\int_{0}^{t}R^{K_{3,4}}(t,s)\cdot(E|X_{0}|^{p}+1){\mathrm{d}}s\Big{)}$
$\displaystyle\leq C_{p,T}(E|X_{0}|^{p}+1).$ (3.5)

On the other hand, we set

$Z_{n,m}(t):=X^{n}_{t}-X^{m}_{t}$

and

$f(t):=\limsup_{n,m\to\infty}E|Z_{n,m}(t)|^{p}.$

Applying BDG’s inequality, H$\mathrm{\ddot{o}}$lder’s inequality,
$\mathbf{(H1)}$ and $\mathbf{(H2)}$ we obtain

$\displaystyle E|Z_{n+1,m+1}(t)|^{p}$
$\displaystyle=E|X^{n+1}_{t}-X^{m+1}_{t}|^{p}$ $\displaystyle\leq
2^{p-1}E\Big{(}\Big{|}\int_{0}^{t}|b(t,s,X^{n}_{s},\mathcal{L}_{X^{n}_{s}})-b(t,s,X^{m}_{s},\mathcal{L}_{X^{m}_{s}})|{\mathrm{d}}s\Big{|}^{p}\Big{)}$
$\displaystyle+2^{p-1}E\Big{(}\Big{|}\int_{0}^{t}\big{(}\sigma(t,s,X^{n}_{s},\mathcal{L}_{X^{n}_{s}})-\sigma(t,s,X^{m}_{s},\mathcal{L}_{X^{m}_{s}})\big{)}{\mathrm{d}}W_{s}\Big{|}^{p}\Big{)}$
$\displaystyle\leq
2^{p-1}E\Big{(}\int_{0}^{t}K_{1}(t,s)\cdot\big{(}\big{|}X^{n}_{s}-X^{m}_{s}|+\mathbb{W}_{2}(\mathcal{L}_{X^{n}_{s}},\mathcal{L}_{X^{m}_{s}})\big{)}{\mathrm{d}}s\Big{)}^{p}$
$\displaystyle+C_{p}E\Big{(}\int_{0}^{t}||\sigma(t,s,X^{n}_{s},\mathcal{L}_{X^{n}_{s}})-\sigma(t,s,X^{m}_{s},\mathcal{L}_{X^{m}_{s}})||^{2}{\mathrm{d}}s\Big{)}^{\frac{p}{2}}$
$\displaystyle\leq C_{p}\int_{0}^{t}K_{1}(t,s)\cdot
E\Big{(}\big{|}X^{n}_{s}-X^{m}_{s}\big{|}^{p}+\mathbb{W}^{p}_{2}(\mathcal{L}_{X^{n}_{s}},\mathcal{L}_{X^{m}_{s}})\Big{)}{\mathrm{d}}s\cdot\Big{(}\int_{0}^{t}K_{1}(t,s){\mathrm{d}}s\Big{)}^{p-1}$
$\displaystyle+C_{p}\int_{0}^{t}K_{2}(t,s)\cdot
E\Big{(}\big{|}X^{n}_{s}-X^{m}_{s}\big{|}^{p}+\mathbb{W}^{p}_{2}(\mathcal{L}_{X^{n}_{s}},\mathcal{L}_{X^{m}_{s}})\Big{)}{\mathrm{d}}s\cdot\Big{(}\int_{0}^{t}K_{2}(t,s){\mathrm{d}}s\Big{)}^{\frac{p-1}{2}}$
$\displaystyle\leq C_{p,T}\int_{0}^{t}K_{1}(t,s)\cdot
E|X_{s}^{n}-X_{s}^{m}|^{p}{\mathrm{d}}s+C_{p,T}\int_{0}^{t}K_{2}(t,s)\cdot
E|X_{s}^{n}-X_{s}^{m}|^{p}{\mathrm{d}}s$ $\displaystyle\leq
C_{p,T}\int_{0}^{t}\Big{(}K_{1}(t,s)+K_{2}(t,s)\Big{)}\cdot
E|Z_{n,m}(t)|^{p}{\mathrm{d}}s.$

By (3.1), $\mathbf{(H1)}$ and applying Fatou’s lemma, we deduce that

$f(t)\leq C_{p,T}\int_{0}^{t}\Big{(}K_{1}(t,s)+K_{2}(t,s)\Big{)}\cdot
f(s){\mathrm{d}}s.$

Using Gronwall’s inequality of Volterra type Lemma 2.2, we get for almost all
$t\in[0,T]$,

$f(t)=\limsup_{n,m\to\infty}E|Z_{n,m}(t)|^{p}=0.$

Hence, there exists a $\mathbb{R}^{d}$-valued $(\mathcal{F}_{t})$-adapted
process $X_{t}$ such that for almost all $t\in[0,T]$,

$\lim_{m\to\infty}E|X^{n}_{t}-X_{t}|^{p}=0.$

Now taking limits for (3.3) gives the existence. The uniqueness follows from a
similar discussion. ∎

###  3.2 Path continuity of solutions

In this subsection, in order to study path continuity of solution to (3.1), in
addition to $(\mathbf{H2})$-$(\mathbf{H3})$, we also assume that:

$\mathbf{(H1^{\prime})}$ There exists a positive constant $\gamma>0$ such that
for any $t,t^{\prime}\in[0,T]$

$\int_{0}^{t}|K_{i}(t^{\prime},s)-K_{i}(t,s)|{\mathrm{d}}s\leq
C_{T}|t-t^{\prime}|^{\gamma},\quad t\in[0,T],$
$\int_{0}^{t}|K_{j}(t^{\prime},s)-K_{j}(t,s)|^{2}{\mathrm{d}}s\leq
C_{T}|t-t^{\prime}|^{2\gamma},\quad t\in[0,T],$
$\int_{t}^{t^{\prime}}|K_{i}(t,s)|^{2}{\mathrm{d}}s\leq
C_{T}|t-t^{\prime}|^{2\gamma},\quad t\in[0,T],$
$\int_{t}^{t^{\prime}}|K_{j}(t,s)|^{2}{\mathrm{d}}s\leq
C_{T}|t-t^{\prime}|^{2\gamma},\quad t\in[0,T].$

where $i=1,3,j=2,4$.

$\mathbf{(H4)}$ For all $t,t^{\prime},s\in[0,T]$,
$\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $x\in\mathbb{R}^{d}$,

$\displaystyle|b(t^{\prime},s,x,\mu)-b(t,s,x,\mu)|$ $\displaystyle\leq
F_{1}(t^{\prime},t,s)\Big{(}1+|x|+\mathbb{W}_{2}(\mu,\delta_{0})\Big{)},$
$\displaystyle||\sigma(t,s,x,\mu)-\sigma(t,s,y,\mu)||^{2}$ $\displaystyle\leq
F_{2}(t^{\prime},t,s)\Big{(}1+|x|^{2}+\mathbb{W}_{2}^{2}(\mu,\delta_{0})\Big{)},$

where $F_{i}(t^{\prime},t,s),i=1,2,$ are positive function on
$[0,T]\times[0,T]\times[0,T]$, and satisfy for some $\gamma>0$

$\int_{0}^{t}F_{1}(t^{\prime},t,s){\mathrm{d}}s\leq
C_{T}|t-t^{\prime}|^{\gamma},\quad\int_{0}^{t}F_{2}(t^{\prime},t,s){\mathrm{d}}s\leq
C_{T}|t-t^{\prime}|^{2\gamma}.$

######  Remark 3.1.

It is clear that $\mathbf{(H1^{\prime})}$ implies
$K_{k}\in\mathcal{K},k=1,2,3,4$. Indeed,

$t\mapsto\int_{0}^{t}K_{k}(t,s){\mathrm{d}}s\in L^{\infty}(0,T)$

and

$\limsup_{\varepsilon\downarrow
0}\Big{\|}\int_{\cdot+\varepsilon}^{\cdot}K_{k}(\cdot+\varepsilon,s){\mathrm{d}}s\Big{\|}_{L^{\infty}(0,T)}<1.$

So under stronger assumptions $\mathbf{(H1^{\prime})}$, $\mathbf{(H2)}$ and
$\mathbf{(H3)}$, we still can prove the results as Theorem 3.1. Moreover,
compared with Theorem $3.2$ in [23], our assumption $\mathbf{(H1^{\prime})}$
is weaker than assumption $2.1$ in [23].

######  Remark 3.2.

Assume $K_{1}=K_{2}=K_{3}=K_{4}=K$, let us list some examples of kernels that
satisfy $\mathbf{(H1^{\prime})}$:

$\bullet$ fractional Brownian motion kernel. The fractional Brownian motion
has the integral representation in law:

$\displaystyle B_{t}^{H}=\int_{0}^{1}K^{H}(t,s)dW_{s},$

where $W$ is a standard Brownian motion and $K^{H}$ is the square integral
kernel:

$\displaystyle
K^{H}(t,s)=K(t,s)=c_{H}(t-s)^{H-\frac{1}{2}}+c_{H}(\frac{1}{2}-H)\int_{s}^{t}(\theta-s)^{H-\frac{3}{2}}\Big{(}1-(\frac{s}{\theta})^{\frac{1}{2}-H}\Big{)}\mathrm{d}\theta,$

with

$c_{H}=\Big{(}\frac{2H{\Gamma}(\frac{3}{2}-H)}{{\Gamma}(H+\frac{1}{2}){\Gamma}(2-2H)}\Big{)}^{\frac{1}{2}}.$

It was proved in [3] that for singular case $(H<\frac{1}{2})$ and regular case
$(H>\frac{1}{2})$ we have $(\alpha=|H-\frac{1}{2}|)$

$\displaystyle\text{Singular case}\quad\big{|}\frac{\partial K^{H}}{\partial
t}(t,s)\big{|}\leq C_{\alpha}(t-s)^{-\alpha-1}\
\text{and}\int_{s}^{t}K^{H}(t,u)^{2}\mathrm{d}u\leq
C_{\alpha}(t-s)^{1-2\alpha},\ s<t.$

and

$\displaystyle\text{Regular case}\quad\big{|}\frac{\partial K^{H}}{\partial
t}(t,s)\big{|}\leq C_{\alpha}(t-s)^{\alpha-1}s^{-\alpha}\
\text{and}\int_{s}^{t}K^{H}(t,u)^{2}\mathrm{d}u\leq
C_{\alpha}(t-s)^{1+2\alpha},\ s<t.$

So for the singular case and regular case we have that $(t<t^{\prime})$

$\int_{t}^{t^{\prime}}(K^{H}(t,s))^{2}\mathrm{d}s\leq
C_{\alpha}(t^{\prime}-t)^{1-2\alpha}=C_{\alpha}(t^{\prime}-t)^{2H},$

and

$\int_{t}^{t^{\prime}}(K^{H}(t,s))^{2}\mathrm{d}s\leq
C_{\alpha}(t^{\prime}-t)^{1+2\alpha}=C_{\alpha}(t^{\prime}-t)^{2H}.$

Moreover, it was proved in [34] that for $H\in(0,1)$

$\int_{0}^{t}\big{|}K^{H}(t^{\prime},s)-K^{H}(t,s)\big{|}^{2}\mathrm{d}s\leq|t^{\prime}-t|^{2H}.$

Therefore, it is clear to verify $\mathbf{(H1^{\prime})}$ hold with
$\gamma=2H$.

$\bullet$ Convolutional kernel, for the details see Example $2.3$ in [1]. Note
that our assumption $\mathbf{(H1^{\prime})}$ covers the conditions $(2.5)$ in
[1].

######  Theorem 3.2.

Assume that $X_{0}\in L^{p}(\Omega,\mathcal{F}_{0},P)$,
$p>2\vee\frac{2}{\gamma}$ and assumptions $\mathbf{(H1^{\prime})}$,
$\mathbf{(H2)}$-$\mathbf{(H4)}$ hold. Then $t\mapsto X(t)$ admits a
${\alpha}$-order H$\mathrm{\ddot{o}}$lder continuous modification for any
$\delta\in[0,\frac{\gamma}{2}-\frac{1}{p})$.

###### Proof.

For any $0\leq t<t^{\prime}\leq T$, we first rewrite $X_{t^{\prime}}-X_{t}$ as

$\displaystyle X_{t^{\prime}}-X_{t}$
$\displaystyle=\int_{t}^{t^{\prime}}b(t^{\prime},t,X_{s},\mathcal{L}_{X_{s}}){\mathrm{d}}s+\int_{0}^{t}\big{(}b(t^{\prime},s,X_{s},\mathcal{L}_{X_{s}})-b(t,s,X_{s},\mathcal{L}_{X_{s}})\big{)}{\mathrm{d}}s$
$\displaystyle+\int_{t}^{t^{\prime}}\sigma(t^{\prime},t,X_{s},\mathcal{L}_{X_{s}}){\mathrm{d}}W_{s}+\int_{0}^{t}\big{(}\sigma(t^{\prime},s,X_{s},\mathcal{L}_{X_{s}})-\sigma(t,s,X_{s},\mathcal{L}_{X_{s}})\big{)}{\mathrm{d}}W_{s}$
$\displaystyle:=I_{1}(t^{\prime},t)+I_{2}(t^{\prime},t)+I_{3}(t^{\prime},t)+I_{4}(t^{\prime},t).$

As for the term $I_{1}(t^{\prime},t)$, applying H$\mathrm{\ddot{o}}$lder’s
inequality, $\mathbf{(H1^{\prime})}$, $\mathbf{(H3)}$ and (3.2), we get

$\displaystyle E|I_{1}(t^{\prime},t)|^{p}$ $\displaystyle\leq
E\Big{|}\int_{t}^{t^{\prime}}K_{3}(t,s)\cdot\Big{(}1+|X_{s}|+\mathbb{W}_{2}(\mathcal{L}_{X_{s}},\delta_{0})\Big{)}{\mathrm{d}}s\Big{|}^{p}$
$\displaystyle\leq
E\int_{t}^{t^{\prime}}K_{3}(t,s)\cdot\Big{|}1+|X_{s}|+\mathbb{W}_{2}(\mathcal{L}_{X_{s}},\delta_{0})\Big{|}^{p}{\mathrm{d}}s\cdot\Big{(}\int_{t}^{t^{\prime}}K_{3}(t,s){\mathrm{d}}s\Big{)}^{p-1}$
$\displaystyle\leq
C_{p,T}|t-t^{\prime}|^{\gamma(p-1)}\int_{t}^{t^{\prime}}K_{3}(t,s)\cdot\big{(}1+E|X_{s}|^{p}\big{)}{\mathrm{d}}s$
$\displaystyle\leq C_{p,T}\big{(}1+E|X_{0}|^{p}\big{)}|t-t^{\prime}|^{\gamma
p}.$

As for the term $I_{2}(t^{\prime},t)$, by H$\mathrm{\ddot{o}}$lder’s
inequality, $\mathbf{(H1^{\prime})}$, $\mathbf{(H4)}$ and (3.2), we have

$\displaystyle E|I_{2}(t^{\prime},t)|^{p}$ $\displaystyle\leq
E\Big{|}\int_{t}^{t^{\prime}}F_{1}(t^{\prime},t,s)\cdot\Big{(}1+|X_{s}|+\mathbb{W}_{2}(\mathcal{L}_{X_{s}},\delta_{0})\Big{)}{\mathrm{d}}s\Big{|}^{p}$
$\displaystyle\leq
E\int_{t}^{t^{\prime}}F_{1}(t^{\prime},t,s)\cdot\Big{|}1+|X_{s}|+\mathbb{W}_{2}(\mathcal{L}_{X_{s}},\delta_{0})\Big{|}^{p}{\mathrm{d}}s\cdot\Big{(}\int_{t}^{t^{\prime}}F_{1}(t^{\prime},t,s){\mathrm{d}}s\Big{)}^{p-1}$
$\displaystyle\leq
C_{p,T}|t-t^{\prime}|^{\gamma(p-1)}\int_{t}^{t^{\prime}}F_{1}(t^{\prime},t,s)\cdot\big{(}1+E|X_{s}|^{p}\big{)}{\mathrm{d}}s$
$\displaystyle\leq C_{p,T}\big{(}1+E|X_{0}|^{p}\big{)}|t-t^{\prime}|^{\gamma
p}.$

As for the term $I_{3}(t^{\prime},t)$, by BDG’s inequality,
H$\mathrm{\ddot{o}}$lder’s inequality, $\mathbf{(H1^{\prime})}$,
$\mathbf{(H3)}$ and (3.2), for $p>2$ we have

$\displaystyle E|I_{3}(t^{\prime},t)|^{p}$ $\displaystyle\leq
C_{p}E\Big{(}\int_{t}^{t^{\prime}}\|\sigma(t^{\prime},s,X_{s},\mathcal{L}_{X_{s}})\|^{2}{\mathrm{d}}s\Big{)}^{\frac{p}{2}}$
$\displaystyle\leq
C_{p}E\Big{(}\int_{t}^{t^{\prime}}K_{4}(t^{\prime},s)\cdot\Big{(}1+|X_{s}|^{2}+\mathbb{W}_{2}^{2}(\mathcal{L}_{X_{s}},\delta_{0})\Big{)}{\mathrm{d}}s\Big{)}^{\frac{p}{2}}$
$\displaystyle\leq
C_{p}E\int_{t}^{t^{\prime}}K_{4}(t^{\prime},s)\cdot\Big{(}1+|X_{s}|^{2}+\mathbb{W}_{2}^{2}(\mathcal{L}_{X_{s}},\delta_{0})\Big{)}^{p}{\mathrm{d}}s\cdot\Big{(}\int_{t}^{t^{\prime}}K_{4}(t^{\prime},s){\mathrm{d}}s\Big{)}^{\frac{p}{2}-1}$
$\displaystyle\leq
C_{p,T}|t-t^{\prime}|^{p-2}\int_{t}^{t^{\prime}}K_{4}(t^{\prime},s)\cdot\Big{(}1+E|X_{s}|^{p}\Big{)}{\mathrm{d}}s$
$\displaystyle\leq C_{p,T}\big{(}1+E|X_{0}|^{p}\big{)}|t-t^{\prime}|^{\gamma
p}.$

As for the term $I_{4}(t^{\prime},t)$, by BDG’s inequality,
H$\mathrm{\ddot{o}}$lder’s inequality, extended Minkowski’s inequality
(Corollary $1.30$ in [14]), $\mathbf{(H4)}$ and (3.2), we obtain

$\displaystyle E|I_{4}(t^{\prime},t)|^{p}$ $\displaystyle\leq
C_{p}E\Big{(}\int_{0}^{t}\|\sigma(t^{\prime},s,X_{s},\mathcal{L}_{X_{s}})-\sigma(t,s,X_{s},\mathcal{L}_{X_{s}})\|^{2}{\mathrm{d}}s\Big{)}^{\frac{p}{2}}$
$\displaystyle\leq
C_{p}E\Big{(}\int_{0}^{t}F_{2}(t^{\prime},t,s)\cdot\Big{(}1+|X_{s}|^{2}+\mathbb{W}_{2}^{2}(\mathcal{L}_{X_{s}},\delta_{0})\Big{)}{\mathrm{d}}s\Big{)}^{\frac{p}{2}}$
$\displaystyle\leq
C_{p}\Big{(}\int_{0}^{t}F_{2}(t^{\prime},t,s)\cdot\big{(}E\big{[}1+|X_{s}|^{p}+\mathbb{W}_{2}^{p}(\mathcal{L}_{X_{s}},\delta_{0})\big{]}\big{)}^{\frac{2}{p}}{\mathrm{d}}s\Big{)}^{\frac{p}{2}}$
$\displaystyle\leq
C_{p}\Big{(}\int_{0}^{t}F_{2}(t^{\prime},t,s)\cdot\big{(}1+(E|X_{s}|^{p})^{\frac{2}{p}}\big{)}{\mathrm{d}}s\Big{)}^{\frac{p}{2}}$
$\displaystyle\leq
C_{p,T}\big{(}1+E|X_{0}|^{p}\big{)}\Big{(}\int_{0}^{t}F_{2}(t^{\prime},t,s){\mathrm{d}}s\Big{)}^{\frac{p}{2}}$
$\displaystyle\leq C_{p,T}\big{(}1+E|X_{0}|^{p}\big{)}|t^{\prime}-t|^{\gamma
p}.$

Hence, for all $0\leq t<t^{\prime}\leq T$,

$E|X_{t^{\prime}}-X_{t}|^{p}\leq
C_{p,T}\big{(}1+E|X_{0}|^{p}\big{)}|t^{\prime}-t|^{\gamma p}.$

Existence of a continuous version as well as the bouud now follows from the
Kolmogorov’s continuity theorem, see Theorem I. $2.1$ in [25]. ∎

##  4 Propagation of chaos

In this section, let us now introduce the interacting particle system of (3.1)
in order to study propagation of chaos property (see, e.g., [29]). More
precisely, we consider the following interacting particle system with singular
kernels:

$\displaystyle
X_{t}^{i,N}=X_{0}^{i}+\int_{0}^{t}b(t,s,X_{s}^{i,N},\mu^{X,N}_{s}){\mathrm{d}}s+\int_{0}^{t}\sigma(t,s,X_{s}^{i,N},\mu^{X,N}_{s}){\mathrm{d}}W_{s},$
(4.1)

almost surely, where the empirical measures for any $t\in[0,T]$ is defined by

$\mu^{X,N}_{t}(\cdot):=\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{t}^{i,N}}(\cdot),$

and $\delta_{x}$ denotes the Dirac measure at point $x$. Also, we consider the
following Volterra system of non-interacting particles,

$\displaystyle
X_{t}^{i}=X_{0}^{i}+\int_{0}^{t}b(t,s,X_{s}^{i},\mathcal{L}_{X_{s}^{i}}){\mathrm{d}}s+\int_{0}^{t}\sigma(t,s,X_{s}^{i},\mathcal{L}_{X_{s}^{i}}){\mathrm{d}}W_{s},$
(4.2)

almost surely for any $t\in[0,T]$ and $i\in\\{1,\cdots,N\\}$.

######  Lemma 4.1.

Assume that $X_{0}\in L^{q}(\Omega,\mathcal{F}_{0},P)$ with some $q>p$ and
assumptions $\mathbf{(H1)}$-$\mathbf{(H3)}$ hold, then there exists a constant
depending only on $q,T,X_{0}^{i}$ such that for all $i\in\\{1,\cdots,N\\}$,

$E|X^{i}_{t}|^{q}\leq C_{q,T}\big{(}E|X_{0}|^{q}+1\big{)}.$

###### Proof.

Applying BDG’s inequality, H$\mathrm{\ddot{o}}$lder’s inequality,
$\mathbf{(H1)}$ and $\mathbf{(H3)}$ yields

$\displaystyle E|X_{t}^{i}|^{q}$
$\displaystyle=E\Big{|}X_{0}^{i}+\int_{0}^{t}b(t,s,X_{s}^{i},\mathcal{L}_{X_{s}^{i}}){\mathrm{d}}s+\int_{0}^{t}\sigma(t,s,X_{s}^{i},\mathcal{L}_{X_{s}^{i}}){\mathrm{d}}W_{s}\Big{|}^{q}$
$\displaystyle\leq
3^{q-1}E|X_{0}^{i}|^{q}+3^{q-1}E\Big{|}\int_{0}^{t}b(t,s,X_{s}^{i},\mathcal{L}_{X_{s}^{i}}){\mathrm{d}}s\Big{|}^{q}+3^{q-1}E\Big{|}\int_{0}^{t}\sigma(t,s,X_{s}^{i},\mathcal{L}_{X_{s}^{i}}){\mathrm{d}}W_{s}\Big{|}^{q}$
$\displaystyle\leq
3^{q-1}E|X_{0}^{i}|^{q}+3^{q-1}E\Big{|}\int_{0}^{t}K_{3}(t,s)\cdot(1+|X_{s}^{i}|+\mathbb{W}_{2}(\mathcal{L}_{X_{s}^{i}},\delta_{0})){\mathrm{d}}s\Big{|}^{q}+C_{q}E\Big{|}\int_{0}^{t}\|\sigma(t,s,X_{s}^{i},\mathcal{L}_{X_{s}^{i}})\|^{2}{\mathrm{d}}s\Big{|}^{\frac{q}{2}}$
$\displaystyle\leq 3^{q-1}E|X^{i}_{0}|^{q}+C_{q}\int_{0}^{t}K_{3}(t,s)\cdot
E\Big{(}1+|X^{i}_{s}|^{p}+\mathbb{W}^{q}_{2}(\mathcal{L}_{X^{i}_{s}},\delta_{0})\Big{)}{\mathrm{d}}s\cdot\Big{(}\int_{0}^{t}K_{3}(t,s){\mathrm{d}}s\Big{)}^{q-1}$
$\displaystyle+C_{q}\int_{0}^{t}K_{4}(t,s)\cdot
E\Big{(}1+|X^{i}_{s}|^{p}+\mathbb{W}^{q}_{2}(\mathcal{L}_{X^{i}_{s}},\delta_{0})\Big{)}{\mathrm{d}}s\cdot\Big{(}\int_{0}^{t}K_{3}(t,s){\mathrm{d}}s\Big{)}^{\frac{q-1}{2}}$
$\displaystyle\leq 3^{q-1}E|X_{0}|^{p}+C_{p,T}\int_{0}^{t}K_{3}(t,s)\cdot
E(1+|X_{s}^{i}|^{q}){\mathrm{d}}s+C_{q,T}\int_{0}^{t}K_{4}(t,s)\cdot
E(1+|X_{s}^{i}|^{q}){\mathrm{d}}s$ $\displaystyle\leq
3^{q-1}E|X_{0}|^{p}+C_{q,T}+C_{q,T}\int_{0}^{t}\Big{(}K_{3}(t,s)+K_{4}(t,s)\Big{)}\cdot
E|X_{s}^{i}|^{q}{\mathrm{d}}s$ $\displaystyle\leq
C_{q,T}\big{(}E|X_{0}|^{q}+1\big{)}+C_{q,T}\int_{0}^{t}\Big{(}K_{3}(t,s)+K_{4}(t,s)\Big{)}\cdot
E|X_{s}^{i}|^{q}{\mathrm{d}}s,$

then by Lemma 2.2, we have that

$\displaystyle E|X^{i}_{t}|^{q}\leq C_{q,T}\big{(}E|X_{0}|^{q}+1\big{)}.$

∎

######  Theorem 4.1.

Assume that $X_{0}\in L^{q}(\Omega,\mathcal{F}_{0},P)$ with some $q>p$ and
assumptions $\mathbf{(H1)}$-$\mathbf{(H3)}$ hold, then there exists a constant
C depending only on $d,p,q,T,X_{0}^{i}$ such that for all $N\geq 1$,

$\sup_{i=1,\cdots,N}\sup_{t\in[0,T]}E|X_{t}^{i,N}-X_{t}^{i}|^{p}\leq
C_{p,d,q,T,X_{0}^{i}}\times\begin{cases}N^{-\frac{1}{2}}+N^{-\frac{p-q}{q}}&\quad\text{if}\quad
p>\frac{d}{2}\ \text{and}\ q\neq 2p,\\\
N^{-\frac{1}{2}}\log(1+N)+N^{-\frac{p-q}{q}}&\quad\text{if}\quad
p=\frac{d}{2}\ \text{and}\ q\neq 2p,\\\
N^{-\frac{p}{d}}+N^{-\frac{p-q}{q}}&\quad\text{if}\quad p\in(0,\frac{d}{2})\
\text{and}\ q\neq\frac{d}{d-p},\end{cases}$

###### Proof.

By BDG’s inequality, H$\mathrm{\ddot{o}}$lder’s inequality, $\mathbf{(H1)}$
and $\mathbf{(H2)}$, we obtain

$\displaystyle E|X_{t}^{i,N}-X_{t}^{i}|^{p}$
$\displaystyle=E\big{|}\int_{0}^{t}\big{(}b(t,s,X_{s}^{i,N},\mu^{X,N}_{s})-b(t,s,X_{s}^{i},\mathcal{L}_{X_{s}^{i}})\big{)}{\mathrm{d}}s+\int_{0}^{t}\big{(}\sigma(t,s,X_{s}^{i,N},\mu^{X,N}_{s})-\sigma(t,s,X_{s}^{i},\mathcal{L}_{X_{s}^{i}})\big{)}{\mathrm{d}}W_{s}\big{|}^{p}$
$\displaystyle\leq
2^{p-1}E\Big{(}\Big{|}\int_{0}^{t}|b(t,s,X^{i,N}_{s},\mu^{X,N}_{s})-b(t,s,X^{i}_{s},\mathcal{L}_{X^{i}_{s}})|{\mathrm{d}}s\Big{|}^{p}\Big{)}$
$\displaystyle+2^{p-1}E\Big{(}\Big{|}\int_{0}^{t}\big{(}\sigma(t,s,X^{i,N}_{s},\mu^{X,N}_{s})-\sigma(t,s,X^{i}_{s},\mathcal{L}_{X^{i}_{s}})\big{)}{\mathrm{d}}W_{s}\Big{|}^{p}\Big{)}$
$\displaystyle\leq
2^{p-1}E\Big{(}\int_{0}^{t}K_{1}(t,s)\big{(}\big{|}X^{i,N}_{s}-X^{i}_{s}|+\mathbb{W}_{2}(\mu^{X,N}_{s},\mathcal{L}_{X^{i}_{s}})\big{)}{\mathrm{d}}s\Big{)}^{p}$
$\displaystyle+C_{p}E\Big{(}\int_{0}^{t}\big{\|}\sigma(t,s,X^{i,N}_{s},\mu^{X,N}_{s})-\sigma(t,s,X^{i}_{s},\mathcal{L}_{X^{i}_{s}})\big{\|}^{2}{\mathrm{d}}s\Big{)}^{\frac{p}{2}}$
$\displaystyle\leq C_{p}\int_{0}^{t}K_{1}(t,s)\cdot
E\Big{(}\big{|}X^{i,N}_{s}-X^{i}_{s}\big{|}^{p}+\mathbb{W}^{p}_{2}(\mu^{X,N}_{s},\mathcal{L}_{X^{i}_{s}})\Big{)}{\mathrm{d}}s\cdot\Big{(}\int_{0}^{t}K_{1}(t,s){\mathrm{d}}s\Big{)}^{p-1}$
$\displaystyle+C_{p}\int_{0}^{t}K_{2}(t,s)\cdot
E\Big{(}\big{|}X^{i,N}_{s}-X^{i}_{s}\big{|}^{p}+\mathbb{W}^{p}_{2}(\mu^{X,N}_{s},\mathcal{L}_{X^{i}_{s}})\Big{)}{\mathrm{d}}s\cdot\Big{(}\int_{0}^{t}K_{2}(t,s){\mathrm{d}}s\Big{)}^{\frac{p-1}{2}}$
$\displaystyle\leq C_{p,T}\int_{0}^{t}K_{1}(t,s)\cdot
E|X_{s}^{i,N}-X_{s}^{i}|^{p}{\mathrm{d}}s+C_{p,T}\int_{0}^{t}K_{2}(t,s)\cdot
E|X_{s}^{i,N}-X_{s}^{i}|^{p}{\mathrm{d}}s$
$\displaystyle+C_{p,T}\int_{0}^{t}\Big{(}K_{1}(t,s)+K_{2}(t,s)\Big{)}\cdot
E\mathbb{W}^{p}_{2}(\mu^{X,N}_{s},\mathcal{L}_{X^{i}_{s}}){\mathrm{d}}s.$

Note that

$\displaystyle\mathbb{W}^{p}_{2}(\mu^{X,N}_{s},\mathcal{L}_{X^{i}_{s}})$
$\displaystyle\leq\Big{(}\mathbb{W}^{2}_{2}(\mu^{X,N}_{s},\mathcal{L}_{X^{i}_{s}})\Big{)}^{\frac{p}{2}}\leq\Big{(}\frac{2}{N}\sum_{i=1}^{N}|X_{s}^{i}-X_{s}^{i,N}|^{2}+2\mathbb{W}^{2}_{2}(\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{t}^{i}},\mathcal{L}_{X^{i}_{s}})\Big{)}^{\frac{p}{2}}$
$\displaystyle\leq
C_{p}\Big{(}\frac{1}{N}\sum_{i=1}^{N}|X_{s}^{i}-X_{s}^{i,N}|^{2}\Big{)}^{\frac{p}{2}}+C_{p}\mathbb{W}^{p}_{2}(\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{t}^{i}},\mathcal{L}_{X^{i}_{s}}).$

Therefore, we have that

$\displaystyle E\mathbb{W}^{p}_{2}(\mu^{X,N}_{s},\mathcal{L}_{X^{i}_{s}})$
$\displaystyle\leq
C_{p}E\Big{(}\frac{1}{N}\sum_{i=1}^{N}|X_{s}^{i}-X_{s}^{i,N}|^{2}\Big{)}^{\frac{p}{2}}+C_{p}E\mathbb{W}^{p}_{2}(\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{t}^{i}},\mathcal{L}_{X^{i}_{s}}).$

As a consequence, by extended Minkowski’s inequality and
H$\mathrm{\ddot{o}}$lder’s inequality we have

$\displaystyle E|X_{t}^{i,N}-X_{t}^{i}|^{p}$ $\displaystyle\leq
C_{p,T}\int_{0}^{t}K_{1}(t,s)\cdot
E|X_{s}^{i,N}-X_{s}^{i}|^{p}{\mathrm{d}}s+C_{p,T}\int_{0}^{t}K_{2}(t,s)\cdot
E|X_{s}^{i,N}-X_{s}^{i}|^{p}{\mathrm{d}}s$
$\displaystyle+C_{p,T}\int_{0}^{t}\Big{(}K_{1}(t,s)+K_{2}(t,s)\Big{)}\cdot
E\mathbb{W}^{p}_{2}(\mu^{X,N}_{s},\mathcal{L}_{X^{i}_{s}}){\mathrm{d}}s$
$\displaystyle\leq C_{p,T}\int_{0}^{t}K_{1}(t,s)\cdot
E|X_{s}^{i,N}-X_{s}^{i}|^{p}{\mathrm{d}}s+C_{p,T}\int_{0}^{t}K_{2}(t,s)\cdot
E|X_{s}^{i,N}-X_{s}^{i}|^{p}{\mathrm{d}}s$
$\displaystyle+C_{p,T}\int_{0}^{t}\Big{(}K_{1}(t,s)+K_{2}(t,s)\Big{)}\cdot\Big{[}E\Big{(}\frac{1}{N}\sum_{i=1}^{N}|X_{s}^{i}-X_{s}^{i,N}|^{2}\Big{)}^{\frac{p}{2}}+E\mathbb{W}^{p}_{2}(\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{t}^{i}},\mathcal{L}_{X^{i}_{s}})\Big{]}{\mathrm{d}}s$
$\displaystyle\leq C_{p,T}\int_{0}^{t}K_{1}(t,s)\cdot
E|X_{s}^{i,N}-X_{s}^{i}|^{p}{\mathrm{d}}s+C_{p,T}\int_{0}^{t}K_{2}(t,s)\cdot
E|X_{s}^{i,N}-X_{s}^{i}|^{p}{\mathrm{d}}s$
$\displaystyle+C_{p,T}\int_{0}^{t}\Big{(}K_{1}(t,s)+K_{2}(t,s)\Big{)}\cdot\Big{[}\Big{(}\frac{1}{N}\sum_{i=1}^{N}\big{(}E|X_{s}^{i}-X_{s}^{i,N}|^{p}\big{)}^{\frac{2}{p}}\Big{)}^{\frac{p}{2}}+E\mathbb{W}^{p}_{2}(\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{t}^{i}},\mathcal{L}_{X^{i}_{s}})\Big{]}{\mathrm{d}}s$
$\displaystyle\leq C_{p,T}\int_{0}^{t}\Big{(}K_{1}(t,s)+K_{2}(t,s)\Big{)}\cdot
E|X_{s}^{i,N}-X_{s}^{i}|^{p}{\mathrm{d}}s+C_{p,T}\int_{0}^{t}\Big{(}K_{1}(t,s)+K_{2}(t,s)\Big{)}\cdot
E\mathbb{W}^{p}_{p}(\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{t}^{i}},\mathcal{L}_{X^{i}_{s}}){\mathrm{d}}s,$

then applying Theorem $1$ of [8], we obtain

$\displaystyle
E\mathbb{W}^{p}_{p}(\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{t}^{i}},\mathcal{L}_{X^{i}_{s}})\leq
C_{p,d,q}M_{q}^{\frac{q}{p}}(\mathcal{L}_{X_{s}^{i}})\times\begin{cases}N^{-\frac{1}{2}}+N^{-\frac{p-q}{q}}&\quad\text{if}\quad
p>\frac{d}{2}\ \text{and}\ q\neq 2p,\\\
N^{-\frac{1}{2}}\log(1+N)+N^{-\frac{p-q}{q}}&\quad\text{if}\quad
p=\frac{d}{2}\ \text{and}\ q\neq 2p,\\\
N^{-\frac{p}{d}}+N^{-\frac{p-q}{q}}&\quad\text{if}\quad p\in(0,\frac{d}{2})\
\text{and}\ q\neq\frac{d}{d-p},\end{cases}$

where $M_{q}(\mathcal{L}_{X_{s}^{i}})$ is defined by

$M_{q}(\mathcal{L}_{X_{s}^{i}})=\int_{\mathbb{R}^{d}}|X_{s}^{i}|^{q}\mathcal{L}_{X_{s}^{i}}({\mathrm{d}}X_{s}^{i}).$

By Lemma 4.1, $M_{q}(\mathcal{L}_{X_{s}^{i}})<\infty$, using Lemma 2.2, we can
get desired result. ∎

##  5 Euler schemes for Volterra type McKean-Vlasov SDEs

###  5.1 Discretized scheme for interacting particle system

In this subsection, we construct a Euler schemes for the interacting particle
system (4.1) associated with (4.2). When the coefficients $b$ and $\sigma$ are
independent of the law of $X_{t}$, Zhang [34] proposed Euler schemes for
classsical stochastic Volterra equations with singular kernels, where the
author made a delicate design to avoid touching the singular points. We now
propose the Euler schemes for the interacting particle system (4.1) associated
with (4.2). Without loss of generality, we assume $T=1$.

$\displaystyle
X_{t}^{i,N,n}=X_{0}^{i}+\int_{0}^{t}b^{i,N,n}(\tilde{t},\tilde{s}_{n},X_{s_{n}}^{i,N,n},\mu^{X,N,n}_{s_{n}}){\mathrm{d}}s+\int_{0}^{t}\sigma^{i,N,n}(\tilde{t},\tilde{s}_{n},X_{s_{n}}^{i,N,n},\mu^{X,N,n}_{s_{n}}){\mathrm{d}}W_{s},\
t\in[0,1],$ (5.1)

for each $i\in\\{1,\cdots,N\\}$, where
$X_{0}^{i}:=\xi\in\mathbb{R}^{d},s_{n}:=\frac{[2^{n}s]}{2^{n}}$ and

$\mu^{X,N,n}_{t}(\cdot):=\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{t}^{i,N,n}}(\cdot),$

almost surely for $t\in[0,1]$ and $n\in\mathbb{N}$. Here $[a]$ denotes the
integral part of a real number $a$. And $0<\tilde{s}_{n}<\tilde{t}$ are
defined by

$\displaystyle\tilde{t}:=t\cdot\mathbb{I}_{\\{t\geq
2^{-n}\\}}+2^{-n}\cdot\mathbb{I}_{\\{t\in[0,2^{-n}]\\}},$ (5.2)
$\displaystyle\tilde{s}_{n}:=s_{n}\cdot\mathbb{I}_{\\{s\in[2^{-n},1)\\}}+2^{-n-1}\cdot\mathbb{I}_{\\{s\in(0,2^{-n})\\}}.$
(5.3)

In order to index the dependence of $\xi$, we denote the solution of (4.2) and
(5.1) by $X^{i}_{t,\xi}$ and $X^{i,N,n}_{t,\xi}$. Moreover, we also need to
add more assumptions below.

$\mathbf{(H1^{\prime\prime})}$ High-order integrability condition:

$\sup_{t\in[0,1]}\int_{0}^{t}\Big{[}K^{{\alpha}}_{1}(t,s)+K^{{\alpha}}_{2}(t,s)+K^{{\alpha}}_{3}(t,s)+K^{{\alpha}}_{4}(t,s)\Big{]}{\mathrm{d}}s\leq
C,\quad t\in[0,1],$
$\sup_{t\in[0,1]}\int_{0}^{t}\Big{[}K^{{\alpha}}_{1}(\tilde{t},\tilde{s}_{n})+K^{{\alpha}}_{2}(\tilde{t},\tilde{s}_{n})+K^{{\alpha}}_{3}(\tilde{t},\tilde{s}_{n})+K^{{\alpha}}_{4}(\tilde{t},\tilde{s}_{n})\Big{]}{\mathrm{d}}s\leq
C,\quad t\in[0,1],$

for some ${\alpha}>1$, where $K_{i},i=1,2,3,4$ are from $\mathbf{(H2)}$ and
$\mathbf{(H3)}$.

$\mathbf{(H4^{\prime})}$ For all $t,t^{\prime},s\in[0,1]$,
$\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $x\in\mathbb{R}^{d}$,

$\displaystyle|b(t^{\prime},s,x,\mu)-b(t,s,x,\mu)|$ $\displaystyle\leq
F_{1}(t^{\prime},t,s)\Big{(}1+|x|+\mathbb{W}_{2}^{2}(\mu,\delta_{0})\Big{)},$
$\displaystyle||\sigma(t^{\prime},s,x,\mu)-\sigma(t,s,x,\mu)||^{2}$
$\displaystyle\leq
F_{2}(t^{\prime},t,s)\Big{(}1+|x|^{2}+\mathbb{W}_{2}^{2}(\mu,\delta_{0})\Big{)},$

where $F_{i}(t^{\prime},t,s),i=1,2,$ are positive function on
$[0,1]\times[0,1]\times[0,1]$, and satisfy for some $\gamma>0$

$\int_{0}^{t\wedge
t^{\prime}}\big{(}F_{1}(t^{\prime},t,s)+F_{2}(t^{\prime},t,s)\big{)}{\mathrm{d}}s\leq
C|t-t^{\prime}|^{\gamma},$

and

$\int_{0}^{t\wedge
t^{\prime}}\big{(}F_{1}(\tilde{t^{\prime}},\tilde{t},\tilde{s}_{n})+F_{2}(\tilde{t^{\prime}},\tilde{t},\tilde{s}_{n})\big{)}{\mathrm{d}}s\leq
C|\tilde{t}-\tilde{t^{\prime}}|^{\gamma}.$

$\mathbf{(H5)}$ For all $t,s,s^{\prime}\in[0,1]$,
$\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $x\in\mathbb{R}^{d}$,

$\displaystyle|b(t,s,x,\mu)-b(t,s^{\prime},x,\mu)|$ $\displaystyle\leq
F_{3}(t,s,s^{\prime})\Big{(}1+|x|+\mathbb{W}_{2}^{2}(\mu,\delta_{0})\Big{)},$
$\displaystyle||\sigma(t,s,x,\mu)-\sigma(t,s^{\prime},x,\mu)||^{2}$
$\displaystyle\leq
F_{4}(t,s,s^{\prime})\Big{(}1+|x|^{2}+\mathbb{W}_{2}^{2}(\mu,\delta_{0})\Big{)},$

where $F_{3}(t,s,s^{\prime})$ and $F_{4}(t,s,s^{\prime})$ satisfy for some
$\delta>0$ and all $t\in[0,1]$

$\int_{0}^{t}\big{(}F_{3}(\tilde{t},s,\tilde{s}_{n})+F_{4}(\tilde{t},s,\tilde{s}_{n})\big{)}{\mathrm{d}}s\leq
C2^{-\delta n},\ n\in\mathbb{N}.$

###  5.2 The Convergence Rate for the Euler schemes

We first prove two lemmas.

######  Lemma 5.1.

For any $p\geq\frac{2\alpha}{\alpha-1}$, suppose that
$\mathbf{(H1^{\prime\prime})}$ and $\mathbf{(H2)}$-$\mathbf{(H3)}$ hold, for
all $n,N\in\mathbb{N}$, $t\in[0,1]$ and $\xi,\eta\in\mathbb{R}^{d}$, we have

$\sup_{i=1,\cdots,N}\sup_{t\in[0,1]}E|X_{t,\xi}^{i,N,n}|^{p}\leq
C_{p}(1+|\xi|^{p}),$

and

$\sup_{i=1,\cdots,N}\sup_{t\in[0,1]}E|X_{t,\xi}^{i,N,n}-X_{t,\eta}^{i,N,n}|^{p}\leq
C_{p}|\xi-\eta|^{p},$

where $C_{p}>0$ does not depend on $n,N\in\mathbb{N}$.

###### Proof.

Recall the discretized scheme (5.1), by H$\mathrm{\ddot{o}}$lder’s inequality,
BDG’s inequality, $\mathbf{(H1^{\prime\prime})}$ and $\mathbf{(H3)}$, we have
for $p\geq\frac{2\alpha}{\alpha-1}$

$\displaystyle
E|X_{t,\xi}^{i,N,n}|^{p}=E\Bigg{|}\xi+\int_{0}^{t}b^{i,N,n}(\tilde{t},\tilde{s}_{n},X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi}){\mathrm{d}}s+\int_{0}^{t}\sigma^{i,N,n}(\tilde{t},\tilde{s}_{n},X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi}){\mathrm{d}}W_{s}\Bigg{|}^{p}$
$\displaystyle\leq
C_{p}|\xi|^{p}+C_{p}E\Big{|}\int_{0}^{t}b^{i,N,n}(\tilde{t},\tilde{s}_{n},X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi}){\mathrm{d}}s\Big{|}^{p}+C_{p}E\Big{|}\int_{0}^{t}\sigma^{i,N,n}(\tilde{t},\tilde{s}_{n},X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi}){\mathrm{d}}W_{s}\Big{|}^{p}$
$\displaystyle\leq
C_{p}|\xi|^{p}+C_{p}E\Big{|}\int_{0}^{t}K_{3}(\tilde{t},\tilde{s}_{n})\cdot\Big{(}1+X_{s_{n},\xi}^{i,N,n}+\mathbb{W}_{2}(\mu^{X,N,n}_{s_{n},\xi},\delta_{0})\Big{)}{\mathrm{d}}s\Big{|}^{p}$
$\displaystyle+C_{p}E\Big{(}\int_{0}^{t}\big{\|}\sigma^{i,N,n}(\tilde{t},\tilde{s}_{n},X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})\big{\|}^{2}{\mathrm{d}}s\Big{)}^{\frac{p}{2}}$
$\displaystyle\leq
C_{p}|\xi|^{p}+C_{p}E\Bigg{(}\int_{0}^{t}\Big{(}1+|X^{i,N,n}_{s_{n},\xi}|+\mathbb{W}_{2}(\mathcal{L}_{X^{i,N,n}_{s_{n},\xi}},\delta_{0})\Big{)}^{{\alpha}^{*}}ds\Bigg{)}^{\frac{p}{{\alpha}^{*}}}\cdot\Big{(}\int_{0}^{t}K^{{\alpha}}_{3}(\tilde{t},\tilde{s}_{n})ds\Big{)}^{\frac{p}{{\alpha}}}$
$\displaystyle+C_{p}E\Bigg{(}\int_{0}^{t}\Big{(}1+|X^{i,N,n}_{s_{n},\xi}|^{2}+\mathbb{W}^{2}_{2}(\mathcal{L}_{X^{i,N,n}_{s_{n},\xi}},\delta_{0})\Big{)}^{{\alpha}^{*}}{\mathrm{d}}s\Bigg{)}^{\frac{p}{2{\alpha}^{*}}}\cdot\Big{(}\int_{0}^{t}K^{{\alpha}}_{4}(\tilde{t},\tilde{s}){\mathrm{d}}s\Big{)}^{\frac{p}{2{\alpha}}}$
$\displaystyle\leq
C_{p}|\xi|^{p}+C_{p}E\Bigg{(}\int_{0}^{t}\Big{(}1+|X^{i,N,n}_{s_{n},\xi}|+\mathbb{W}_{2}(\mathcal{L}_{X^{i,N,n}_{s_{n},\xi}},\delta_{0})\Big{)}^{{\alpha}^{*}}{\mathrm{d}}s\Bigg{)}^{\frac{p}{{\alpha}^{*}}}$
$\displaystyle+C_{p}E\Bigg{(}\int_{0}^{t}\Big{(}1+|X^{i,N,n}_{s_{n},\xi}|^{2}+\mathbb{W}^{2}_{2}(\mathcal{L}_{X^{i,N,n}_{s_{n},\xi}},\delta_{0})\Big{)}^{{\alpha}^{*}}{\mathrm{d}}s\Bigg{)}^{\frac{p}{2{\alpha}^{*}}}$
$\displaystyle\leq
C_{p}|\xi|^{p}+C_{p}\int_{0}^{t}E\Big{(}1+|X^{i,N,n}_{s_{n},\xi}|+\mathbb{W}_{2}(\mathcal{L}_{X^{i,N,n}_{s_{n},\xi}},\delta_{0})\Big{)}^{p}{\mathrm{d}}s+C_{p}\int_{0}^{t}E\Big{(}1+|X^{i,N,n}_{s_{n},\xi}|^{2}+\mathbb{W}^{2}_{2}(\mathcal{L}_{X^{i,N,n}_{s_{n},\xi}},\delta_{0})\Big{)}^{\frac{p}{2}}{\mathrm{d}}s$
$\displaystyle\leq
C_{p}(1+|\xi|^{p})+C_{p}\int_{0}^{t}\Big{(}E|X^{i,N,n}_{s_{n},\xi}|^{p}+E\mathbb{W}^{p}_{2}(\mathcal{L}_{X^{i,N,n}_{s_{n},\xi}},\delta_{0})\Big{)}{\mathrm{d}}s,$

for $\alpha>1,\alpha^{*}:=\frac{p}{p-1}$, any $t\in[0,1],i\in\\{1,\cdots,N\\}$
and $n,N\in\mathbb{N}$. By a simple calculation (see e.g., Lemma $2.3$ in
[6]), one can observe that

$\mathbb{W}^{2}_{2}(\mathcal{L}_{X^{i,N,n}_{s_{n},\xi}},\delta_{0})=\frac{1}{N}\sum_{i=1}^{N}|X_{s_{n},\xi}^{i,N,n}|^{2},$

which yields that

$\displaystyle\sup_{i=1,\cdots,N}\sup_{s\in[0,t]}E|X_{s,\xi}^{i,N,n}|^{p}$
$\displaystyle\leq
C_{p}(1+|\xi|^{p})+C_{p}\int_{0}^{t}\sup_{i=1,\cdots,N}\sup_{u\in[0,s]}E|X_{u,\xi}^{i,N,n}|^{p}{\mathrm{d}}s.$

Applying Gronwall’s inequality, we have

$\sup_{i=1,\cdots,N}\sup_{t\in[0,1]}E|X_{t,\xi}^{i,N,n}|^{p}\leq
C_{p}(1+|\xi|^{p}).$

The second estimate can be obtained by similar estimations. ∎

######  Remark 5.1.

Note that classical Gronwall’s inequality instead of Volterra type Gronwall’s
equality is used to prove above Lemma. High-order integrability condition is
required. We have attempted to apply Volterra type Gronwall’s equality, but it
seems that Volterra type Gronwall’s equality is invaild when we obtain the
following estimate due to the different indexes $t,\tilde{t}:$

$E|X_{t,\xi}^{i,N,n}|^{p}\leq
C_{p}(1+|\xi|^{p})+C_{p}\int_{0}^{t}\Big{(}K_{3}(\tilde{t},s_{n})+K_{4}(\tilde{t},s_{n})\Big{)}\cdot
E|X_{s_{n},\xi}^{i,N,n}|^{p}{\mathrm{d}}s.$

We now prove the following estimate associated with
$X^{i,N,n}_{t^{\prime},\xi}-X^{i,N,n}_{t,\xi}$.

######  Lemma 5.2.

For any $p\geq 2$ sufficiently large, suppose that
$\mathbf{(H1^{\prime\prime})}$, $\mathbf{(H2)}$, $\mathbf{(H3)}$ and
$\mathbf{(H4^{\prime})}$ hold. Then there exists a constant $C_{p}>0$ such
that for all $n,N\in\mathbb{N}$, $t,t^{\prime}\in[0,1]$ and
$\xi\in\mathbb{R}^{d}$, we have

$E|X^{i,N,n}_{t^{\prime},\xi}-X^{i,N,n}_{t,\xi}|\leq
C_{p}(1+|\xi|^{p})|t^{\prime}-t|^{\theta p},$

where $\theta>0$ depends only ${\alpha},\gamma$.

###### Proof.

For any $t<t^{\prime}$, we first divide
$X^{i,N,n}_{t^{\prime},\xi}-X^{i,N,n}_{t,\xi}$ into four parts.

$\displaystyle X^{i,N,n}_{t^{\prime},\xi}-X^{i,N,n}_{t,\xi}$
$\displaystyle=\int_{t}^{t^{\prime}}b^{i,N,n}(\tilde{t^{\prime}},\tilde{s}_{n},X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi}){\mathrm{d}}s+\int_{t}^{t^{\prime}}\sigma^{i,N,n}(\tilde{t^{\prime}},\tilde{s}_{n},X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi}){\mathrm{d}}W_{s}$
$\displaystyle+\int_{0}^{t}\Big{(}b^{i,N,n}(\tilde{t^{\prime}},\tilde{s}_{n},X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})-b^{i,N,n}(\tilde{t},\tilde{s}_{n},X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})\Big{)}{\mathrm{d}}s$
$\displaystyle+\int_{0}^{t}\Big{(}\sigma^{i,N,n}(\tilde{t^{\prime}},\tilde{s}_{n},X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})-\sigma^{i,N,n}(\tilde{t},\tilde{s}_{n},X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})\Big{)}{\mathrm{d}}W_{s}$
$\displaystyle:=\Xi_{1}+\Xi_{2}+\Xi_{3}+\Xi_{4}.$

For the term $\Xi_{2}$, by Lemma 5.1, H$\mathrm{\ddot{o}}$lder’s inequality,
BDG’s inequality, $\mathbf{(H1^{\prime\prime})}$ and $\mathbf{(H3)}$, we have

$\displaystyle E|\Xi_{2}|^{p}$
$\displaystyle=E\Big{|}\int_{t}^{t^{\prime}}\sigma^{i,N,n}(\tilde{t^{\prime}},\tilde{s}_{n},X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi}){\mathrm{d}}W_{s}\Big{|}^{p}$
$\displaystyle\leq
C_{p}E\Big{(}\int_{t}^{t^{\prime}}\big{\|}\sigma^{i,N,n}(\tilde{t^{\prime}},\tilde{s}_{n},X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})\big{\|}^{2}{\mathrm{d}}s\Big{)}^{\frac{p}{2}}$
$\displaystyle\leq
C_{p}E\Big{(}\int_{t}^{t^{\prime}}K_{4}(\tilde{t^{\prime}},\tilde{s}_{n},\xi)\cdot\big{(}1+|X_{s_{n},\xi}^{i,N,n}|^{2}+\mathbb{W}^{2}_{2}(\mu^{X,N,n}_{s_{n},\xi},\delta_{0})\big{)}{\mathrm{d}}s\Big{)}^{\frac{p}{2}}$
$\displaystyle\leq
C_{p}\Big{(}\int_{t}^{t^{\prime}}K^{{\alpha}}_{4}(\tilde{t^{\prime}},\tilde{s}_{n}){\mathrm{d}}s\Big{)}^{\frac{p}{2{\alpha}}}E\Big{(}\int_{t}^{t^{\prime}}\big{(}1+|X_{s_{n},\xi}^{i,N,n}|^{2}+\mathbb{W}^{2}_{2}(\mu^{X,N,n}_{s_{n},\xi},\delta_{0})\big{)}^{{\alpha}^{*}}{\mathrm{d}}s\Big{)}^{\frac{p}{2{\alpha}^{*}}}$
$\displaystyle\leq
C_{p}(1+|\xi|^{p})|t^{\prime}-t|^{\frac{p}{2{\alpha}^{*}}}.$

For the term $\Xi_{4}$, combining BDG’s inequality, extended Minkowski’s
inequality, $\mathbf{(H1^{\prime\prime})}$, $\mathbf{(H4)}$ with Lemma 5.1, we
have that

$\displaystyle E|\Xi_{4}|^{p}$
$\displaystyle=E\Big{|}\int_{0}^{t}\Big{(}\sigma^{i,N,n}(\tilde{t^{\prime}},\tilde{s}_{n},X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})-\sigma^{i,N,n}(\tilde{t},\tilde{s}_{n},X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n}})\Big{)}{\mathrm{d}}W_{s}\Big{|}^{p}$
$\displaystyle\leq
C_{p}E\Big{(}\int_{0}^{t}\big{\|}\sigma^{i,N,n}(\tilde{t^{\prime}},\tilde{s}_{n},X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})-\sigma^{i,N,n}(\tilde{t},\tilde{s}_{n},X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})\big{\|}^{2}{\mathrm{d}}s\Big{)}^{\frac{p}{2}}$
$\displaystyle\leq
C_{p}E\Big{(}\int_{0}^{t}F_{2}(\tilde{t^{\prime}},\tilde{t},\tilde{s}_{n})\cdot\big{(}1+|X_{s_{n},\xi}^{i,N,n}|^{2}+\mathbb{W}^{2}_{2}(\mu^{X,N,n}_{s_{n},\xi},\delta_{0})\big{)}{\mathrm{d}}s\Big{)}^{\frac{p}{2}}$
$\displaystyle\leq
C_{p}\Bigg{(}\int_{0}^{t}F_{2}(\tilde{t^{\prime}},\tilde{t},\tilde{s}_{n})\cdot\big{(}E\big{[}1+|X^{i,N,n}_{s_{n},\xi}|^{p}+\mathbb{W}_{2}^{p}(\mathcal{L}_{X^{i,N,n}_{s_{n},\xi}},\delta_{0})\big{]}\big{)}^{\frac{2}{p}}{\mathrm{d}}s\Bigg{)}^{\frac{p}{2}}$
$\displaystyle\leq
C_{p}(1+|\xi|^{p})\Big{(}\int_{0}^{t}F_{2}(\tilde{t^{\prime}},\tilde{t},\tilde{s}_{n}){\mathrm{d}}s\Big{)}^{\frac{p}{2}}\leq
C_{p}(1+|\xi|^{p})|t^{\prime}-t|^{\frac{\gamma p}{2}}.$

Similarly, we may deal with the terms $\Xi_{1}$ and $\Xi_{3}$, and obtain

$\displaystyle E|\Xi_{1}|^{p}\leq
C_{p}(1+|\xi|^{p})|t^{\prime}-t|^{\frac{p}{{\alpha}^{*}}}$ $\displaystyle
E|\Xi_{3}|^{p}\leq C_{p}(1+|\xi|^{p})|t^{\prime}-t|^{\gamma p}.$

Combining the above calculation, we have

$\displaystyle E|X^{i,N,n}_{t^{\prime}}-X^{i,N,n}_{t}|\leq
C_{p}(1+|X_{0}^{i}|^{p})\Big{(}|t^{\prime}-t|^{\frac{p}{{\alpha}^{*}}}+|t^{\prime}-t|^{\frac{p}{2{\alpha}^{*}}}+|t^{\prime}-t|^{\gamma
p}+|t^{\prime}-t|^{\frac{\gamma p}{2}}\Big{)},$

which concludes the proof of Lemma 5.2. ∎

The main results of this section are given in the following Theorems.

######  Theorem 5.1.

Assume that $\mathbf{(H1^{\prime\prime})}$, $\mathbf{(H2)}$, $\mathbf{(H3)}$
,$\mathbf{(H4^{\prime})}$ and $\mathbf{(H5)}$ hold, the Euler scheme (5.1)
converges to the true solution of the interacting particle system (4.1)
associated with (4.2) in a strong sense with $L^{p}$, i.e.,

$\sup_{i=1,\cdots,N}\sup_{t\in[0,1]}E|X_{t,\xi}^{i,N}-X_{t,\xi}^{i,N,n}|^{p}\leq
C_{p}(1+|\xi|^{p})\cdot 2^{-n\eta p},$

for $p$ sufficiently large, where the constant $C_{p}$ does depend on
$n,N\in\mathbb{N}$ and $\eta>0$ depends only ${\alpha},\gamma,\delta$.

###### Proof.

Set

$Z^{i,N,n}_{t,\xi}:=X^{i,N}_{t,\xi}-X^{i,N,n}_{t,\xi}.$

Then for any $t\in[0,1],i\in\\{1,\cdots,N\\}$ and $n,N\in\mathbb{N}$, (where
$b(t,s,X_{s,\xi}^{i,N,n},\mu^{X,N,n}_{s,\xi})=b^{i,N,n}(t,s,X_{s,\xi}^{i,N,n},\mu^{X,N,n}_{s,\xi})$
and
$\sigma(t,s,X_{s,\xi}^{i,N,n},\mu^{X,N,n}_{s,\xi})=\sigma^{i,N,n}(t,s,X_{s,\xi}^{i,N,n},\mu^{X,N,n}_{s,\xi}$)

$\displaystyle Z^{i,N,n}_{t,\xi}$
$\displaystyle=\int_{0}^{t}\big{(}b(t,s,X_{s,\xi}^{i,N},\mu^{X,N}_{s,\xi})-b^{i,N,n}(\tilde{t},\tilde{s}_{n},X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})\big{)}{\mathrm{d}}s$
$\displaystyle+\int_{0}^{t}\big{(}\sigma(t,s,X_{s,\xi}^{i,N},\mu^{X,N}_{s,\xi})-\sigma^{i,N,n}(\tilde{t},\tilde{s}_{n},X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})\big{)}{\mathrm{d}}W_{s}$
$\displaystyle=\int_{0}^{t}\Big{(}b(t,s,X_{s,\xi}^{i,N},\mu^{X,N}_{s,\xi})-b(t,s,X_{s,\xi}^{i,N,n},\mu^{X,N,n}_{s,\xi})\Big{)}{\mathrm{d}}s$
$\displaystyle+\int_{0}^{t}\Big{(}b^{i,N,n}(t,s,X_{s,\xi}^{i,N,n},\mu^{X,N,n}_{s,\xi})-b^{i,N,n}(t,s,X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})\Big{)}{\mathrm{d}}s$
$\displaystyle+\int_{0}^{t}\Big{(}b^{i,N,n}(t,s,X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})-b^{i,N,n}(\tilde{t},s,X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})\Big{)}{\mathrm{d}}s$
$\displaystyle+\int_{0}^{t}\Big{(}b^{i,N,n}(\tilde{t},s,X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})-b^{i,N,n}(\tilde{t},s_{n},X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})\Big{)}{\mathrm{d}}s$
$\displaystyle+\int_{0}^{t}\Big{(}\sigma(t,s,X_{s,\xi}^{i,N},\mu^{X,N}_{s,\xi})-\sigma(t,s,X_{s,\xi}^{i,N,n},\mu^{X,N,n}_{s,\xi})\Big{)}{\mathrm{d}}W_{s}$
$\displaystyle+\int_{0}^{t}\Big{(}\sigma^{i,N,n}(t,s,X_{s,\xi}^{i,N,n},\mu^{X,N,n}_{s,\xi})-\sigma^{i,N,n}(t,s,X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})\Big{)}{\mathrm{d}}W_{s}$
$\displaystyle+\int_{0}^{t}\Big{(}\sigma^{i,N,n}(t,s,X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})-\sigma^{i,N,n}(\tilde{t},s,X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})\Big{)}{\mathrm{d}}W_{s}$
$\displaystyle+\int_{0}^{t}\Big{(}\sigma^{i,N,n}(\tilde{t},s,X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})-\sigma^{i,N,n}(\tilde{t},s_{n},X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})\Big{)}{\mathrm{d}}W_{s}$
$\displaystyle:=\Pi_{1}+\Pi_{2}+\Pi_{3}+\Pi_{4}+\Pi_{5}+\Pi_{6}+\Pi_{7}+\Pi_{8}.$

For the term $\Pi_{5}$, by H$\mathrm{\ddot{o}}$lder’s inequality, BDG’s
inequality, $\mathbf{(H1^{\prime\prime})}$ and $\mathbf{(H2)}$, we have

$\displaystyle E|\Pi_{5}|^{p}$
$\displaystyle=E\Big{|}\int_{0}^{t}\Big{(}\sigma(t,s,X_{s,\xi}^{i,N},\mu^{X,N}_{s,\xi})-\sigma(t,s,X_{s,\xi}^{i,N,n},\mu^{X,N,n}_{s,\xi})\Big{)}{\mathrm{d}}W_{s}\Big{|}^{p}$
$\displaystyle\leq
C_{p}E\Big{|}\int_{0}^{t}\Big{(}\sigma(t,s,X_{s,\xi}^{i,N},\mu^{X,N}_{s,\xi})-\sigma(t,s,X_{s,\xi}^{i,N,n},\mu^{X,N,n}_{s,\xi})\Big{)}^{2}{\mathrm{d}}s\Big{|}^{\frac{p}{2}}$
$\displaystyle\leq
C_{p}E\Big{|}\int_{0}^{t}K_{2}(t,s)\cdot\Big{(}|X_{s,\xi}^{i,N}-X_{s,\xi}^{i,N,n}|^{2}+\mathbb{W}_{2}^{2}(\mu^{X,N}_{s,\xi},\mu^{X,N,n}_{s,\xi})\Big{)}^{2}{\mathrm{d}}s\Big{|}^{\frac{p}{2}}$
$\displaystyle\leq
C_{p}\Big{(}\int_{0}^{t}K^{{\alpha}}_{2}(t,s){\mathrm{d}}s\Big{)}^{\frac{p}{2{\alpha}}}\cdot
E\Big{|}\int_{0}^{t}\Big{(}|X_{s,\xi}^{i,N}-X_{s,\xi}^{i,N,n}|^{2}+\mathbb{W}_{2}^{2}(\mu^{X,N}_{s,\xi},\mu^{X,N,n}_{s,\xi})\Big{)}^{{\alpha}^{*}}{\mathrm{d}}s\Big{|}^{\frac{p}{2{\alpha}^{*}}}$
$\displaystyle\leq
C_{p}\int_{0}^{t}\Big{(}E|Z_{s,\xi}^{i,N,n}|^{p}+E\Big{(}\frac{1}{N}\sum_{i=1}^{N}|Z^{i,N,n}_{s,\xi}|^{2}\Big{)}^{\frac{p}{2}}\Big{)}{\mathrm{d}}s,$

where we have used the following elementary estimate

$\mathbb{W}_{2}^{2}(\mu^{X,N}_{s,\xi},\mu^{X,N,n}_{s,\xi})\leq\frac{1}{N}\sum_{i=1}^{N}|X^{i,N}_{s,\xi}-X^{i,N,n}_{s,\xi}|^{2}.$

For the term $\Pi_{6}$, applying H$\mathrm{\ddot{o}}$lder’s inequality, BDG’s
inequality, $\mathbf{(H1^{\prime\prime})}$ and $\mathbf{(H2)}$, we get

$\displaystyle E|\Pi_{6}|^{p}$
$\displaystyle=\int_{0}^{t}\Big{(}\sigma^{i,N,n}(t,s,X_{s,\xi}^{i,N,n},\mu^{X,N,n}_{s,\xi})-\sigma^{i,N,n}(t,s,X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})\Big{)}{\mathrm{d}}W_{s}$
$\displaystyle\leq
C_{p}E\Big{|}\int_{0}^{t}\big{\|}\sigma^{i,N,n}(t,s,X_{s,\xi}^{i,N,n},\mu^{X,N,n}_{s,\xi})-\sigma^{i,N,n}(t,s,X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})\big{\|}^{2}{\mathrm{d}}s\Big{|}^{\frac{p}{2}}$
$\displaystyle\leq
C_{p}E\Big{|}\int_{0}^{t}K_{2}(t,s)\cdot\Big{(}|X_{s,\xi}^{i,N,n}-X_{s_{n},\xi}^{i,N,n}|^{2}+\mathbb{W}_{2}^{2}(\mu^{X,N,n}_{s,\xi},\mu^{X,N,n}_{s_{n},\xi})\Big{)}^{2}{\mathrm{d}}s\Big{|}^{\frac{p}{2}}$
$\displaystyle\leq
C_{p}\int_{0}^{t}\Big{(}E|X_{s,\xi}^{i,N,n}-X_{s_{n},\xi}^{i,N,n}|^{p}+E\Big{(}\frac{1}{N}\sum_{i=1}^{N}|X_{s,\xi}^{i,N,n}-X_{s_{n},\xi}^{i,N,n}|^{2}\Big{)}^{\frac{p}{2}}\Big{)}{\mathrm{d}}s$
$\displaystyle\leq C_{p}(1+|\xi|^{p})|s-s_{n}|^{\theta p}\leq
C_{p}(1+|\xi|^{p})\cdot 2^{-n\theta p},$

where the last step we have used Lemma 5.2.

For the term $\Pi_{7}$, by Lemma 5.1, BDG’s inequality, extended Minkowski’s
inequality, $\mathbf{(H1^{\prime})}$ and $\mathbf{(H4^{\prime})}$, we have

$\displaystyle E|\Pi_{7}|^{p}$
$\displaystyle=E\Big{|}\int_{0}^{t}\Big{(}\sigma^{i,N,n}(t,s,X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})-\sigma^{i,N,n}(\tilde{t},s,X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})\Big{)}{\mathrm{d}}W_{s}\Big{|}^{p}$
$\displaystyle\leq
C_{p}E\Big{|}\int_{0}^{t}\big{\|}\sigma^{i,N,n}(t,s,X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})-\sigma^{i,N,n}(\tilde{t},s,X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})\big{\|}^{2}{\mathrm{d}}s\Big{|}^{\frac{p}{2}}$
$\displaystyle\leq
C_{p}E\Big{|}\int_{0}^{t}\big{\|}\sigma^{i,N,n}(t,s,X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})-\sigma^{i,N,n}(\tilde{t},s,X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})\big{\|}^{2}{\mathrm{d}}s\Big{|}^{\frac{p}{2}}$
$\displaystyle\leq
C_{p}E\Big{|}\int_{0}^{t}F_{2}(t,\tilde{t},s)\cdot\big{(}1+|X_{s_{n},\xi}^{i,N,n}|^{2}+\mathbb{W}^{2}_{2}(\mu^{X,N,n}_{s_{n},\xi},\delta_{0})\big{)}{\mathrm{d}}s\Big{|}^{\frac{p}{2}}$
$\displaystyle\leq
C_{p}\Bigg{(}\int_{0}^{t}F_{2}(t,\tilde{t},s)\cdot\big{(}E\big{[}1+|X^{i,N,n}_{s_{n},\xi}|^{p}+\mathbb{W}_{2}^{p}(\mathcal{L}_{X^{i,N,n}_{s_{n},\xi}},\delta_{0})\big{]}\big{)}^{\frac{2}{p}}{\mathrm{d}}s\Bigg{)}^{\frac{p}{2}}$
$\displaystyle\leq
C_{p}(1+|\xi|^{p})\Big{(}\int_{0}^{t}F_{2}(t,\tilde{t},s){\mathrm{d}}s\Big{)}^{\frac{p}{2}}\leq
C_{p}(1+|\xi|^{p})|t^{\prime}-t|^{\frac{\gamma p}{2}}\leq
C_{p}(1+|\xi|^{p})\cdot 2^{-\frac{n\gamma p}{2}}.$

For the term $\Pi_{8}$, by Lemma 5.1, BDG’s inequality, extended Minkowski’s
inequality, $\mathbf{(H1^{\prime})}$ and $\mathbf{(H5)}$, we have

$\displaystyle E|\Pi_{8}|^{p}$
$\displaystyle=E\Big{|}\int_{0}^{t}\Big{(}\sigma^{i,N,n}(\tilde{t},s,X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})-\sigma^{i,N,n}(\tilde{t},s_{n},X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})\Big{)}{\mathrm{d}}W_{s}\Big{|}^{p}$
$\displaystyle\leq
C_{p}E\Big{|}\int_{0}^{t}\big{\|}\sigma^{i,N,n}(\tilde{t},s,X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})-\sigma^{i,N,n}(\tilde{t},s_{n},X_{s_{n},\xi}^{i,N,n},\mu^{X,N,n}_{s_{n},\xi})\big{\|}^{2}{\mathrm{d}}s\Big{|}^{\frac{p}{2}}$
$\displaystyle\leq
C_{p}E\Big{|}\int_{0}^{t}F_{4}(\tilde{t},s,s_{n})\cdot\Big{(}1+|X_{s_{n},\xi}^{i,N,n}|^{2}+\mathbb{W}_{2}^{2}(\mu^{X,N,n}_{s_{n},\xi},\delta_{0})\Big{)}^{2}{\mathrm{d}}s\Big{|}^{\frac{p}{2}}$
$\displaystyle\leq
C_{p}(1+|\xi|^{p})\Big{(}\int_{0}^{t}F_{4}(\tilde{t},s,s_{n}){\mathrm{d}}s\Big{)}^{\frac{p}{2}}\leq
C_{p}(1+|\xi|^{p})\cdot 2^{-\frac{n\delta p}{2}}.$

$\Pi_{1},\Pi_{2},\Pi_{3},\Pi_{4}$ can be similarly dealt with, i.e.

$\displaystyle E|\Pi_{1}|^{p}\leq
C_{p}\int_{0}^{t}\Big{(}E|Z_{s,\xi}^{i,N,n}|^{p}+E\Big{(}\frac{1}{N}\sum_{i=1}^{N}|Z^{i,N,n}_{s,\xi}|^{2}\Big{)}^{\frac{p}{2}}\Big{)}{\mathrm{d}}s,$
$\displaystyle E|\Pi_{2}|^{p}\leq C_{p}(1+|\xi|^{p})\cdot 2^{-n\theta p},$
$\displaystyle E|\Pi_{3}|^{p}\leq C_{p}(1+|\xi|^{p})\cdot 2^{-n\gamma p},$
$\displaystyle E|\Pi_{4}|^{p}\leq C_{p}(1+|\xi|^{p})\cdot 2^{-n\delta p}.$

Therefore, if we define

$g(t):=\sup_{i=1,\cdots,N}\sup_{s\in[0,t]}EZ_{s,\xi}^{i,N,n},$

then we easily obtain

$g(t)\leq C_{p}\int_{0}^{t}g(s){\mathrm{d}}s+C_{p}(1+|\xi|^{p})\cdot 2^{-n\eta
p},$

where $\eta>0$ depends only ${\alpha},\gamma,\delta$. Gronwall’s inequality
gives the desired result. ∎

######  Theorem 5.2.

Assume that $\mathbf{(H1^{\prime\prime})}$, $\mathbf{(H2)}$, $\mathbf{(H3)}$
,$\mathbf{(H4^{\prime})}$ and $\mathbf{(H5)}$ hold, there exists $\lambda>0$
such that for any $R>0$ and $p>2$ sufficiently large,

$E\Big{(}\sup_{t\in[0,T],|\xi|\leq
R}\big{|}X_{t,\xi}^{i,N}-X_{t,\xi}^{i,N,n}\big{|}^{p}\Big{)}\leq
E|A(\omega)|^{p}2^{-np\lambda}.$

In particular,

$P\Big{\\{}\lim_{n\to\infty}\sup_{t\in[0,T],|\xi|\leq
R}\big{|}X_{t,\xi}^{i,N}-X_{t,\xi}^{i,N,n}\big{|}=0\Big{\\}}=1.$

###### Proof.

One constructs the following process:

$Z(r,t,\xi)=\begin{cases}X^{i,N}_{t,\xi},\ r=0,\\\
Z^{i,N,n}_{t,\xi}+2^{n+1}(r-2^{-n})\big{(}X^{i,N,n+1}_{t,\xi}-X^{i,N,n}_{t,\xi}\big{)},\
2^{-(n+1)}<r\leq 2^{-n},\ n\in\mathbb{N}.\end{cases}$

Fix $R>0$, by Lemmas 5.1, 5.2 and Theorem 5.1, there exists a constant
$\beta:=\beta(\gamma)>0$ such that for all $p>2$, $t,t^{\prime}\in[0,T]$,
$\xi,\eta\in D_{R}:=\\{\xi\in\mathbb{R}^{d},|\xi|\leq R\\}$, we have

$E\Big{|}Z(r,t,\xi)-Z(r^{\prime},t^{\prime},\eta)\Big{|}^{p}\leq
C_{p,T,R,L_{3},L_{4}}\Big{(}|r-r^{\prime}|^{\beta p}+|t-t^{\prime}|^{\beta
p}+|\xi-\eta|^{\beta p}\Big{)}.$

Thus, by Kolmogorov’s continuity criterium, there is a $p$-order integrable
random variable $A(\omega)$ such that

$\sup_{t\in[0,T],|\xi|\leq R}\Big{|}Z(r,t,\xi)-Z(r^{\prime},t,\xi)\Big{|}\leq
A(\omega)|r-r^{\prime}|^{\lambda},\quad a.s.,$

where $\lambda\in(0,\beta-\frac{d+2}{p})$. In particular, taking
$r=0,r^{\prime}=2^{-n}$, we have

$E\Big{(}\sup_{t\in[0,T],|\xi|\leq
R}\big{|}X_{t,\xi}^{i,N}-X_{t,\xi}^{i,N,n}\big{|}\Big{)}^{p}\leq
E|A(\omega)|^{p}2^{-np\lambda},$

which yields the desired convergence. Borel-Cantelli’s Lemma gives the second
conclusion. ∎

## Conflicts of interests

The authors declare no conflict of interests.

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