# Consensus based optimization via jump-diffusion stochastic differential
equations

D. Kalise Department of Mathematics, Imperial College London, South Kensington
Campus, SW7 2AZ London, UK<EMAIL_ADDRESS>A. Sharma School of
Mathematical Sciences, University of Nottingham, UK;
<EMAIL_ADDRESS>M.V. Tretyakov School of Mathematical Sciences,
University of Nottingham, UK<EMAIL_ADDRESS>
###### Abstract

We introduce a new consensus based optimization (CBO) method where interacting
particle system is driven by jump-diffusion stochastic differential equations.
We study well-posedness of the particle system as well as of its mean-field
limit. The major contributions of this paper are proofs of convergence of the
interacting particle system towards the mean-field limit and convergence of a
discretized particle system towards the continuous-time dynamics in the mean-
square sense. We also prove convergence of the mean-field jump-diffusion SDEs
towards global minimizer for a large class of objective functions. We
demonstrate improved performance of the proposed CBO method over earlier CBO
methods in numerical simulations on benchmark objective functions.

##  1 Introduction

Large-scale individual-based models have become a well-established modelling
tool in modern science and engineering, with applications including pedestrian
motion, collective animal behaviour, swarm robotics and molecular dynamics,
among many others. Through the iteration of basic interactions forces such as
attraction, repulsion, and alignment, these complex systems of exhibit a rich
self-organization behaviour (see e.g. [BFM97, CS07, CFRT10, MT14, BRSW15,
ABF+19]).

Over the last decades, individual-based models have also entered the field of
global optimization and its many applications in operations research, control,
engineering, economics, finance, and machine learning. In many applied
problems arising in the aforementioned fields, the objective function to be
optimized can be non-convex and/or non-smooth, disabling the use of
traditional continuous/convex optimization technique. In such scenarios,
individual-based metaheuristic models have been proven surprisingly effective.
Examples include genetic algorithms, ant colony optimization, particle swarm
optimization, simulated annealing, etc. (see [HKS89, DB05, Ken10] and
references therein). These methods are probabilistic in nature which set them
apart from other derivative-free algorithms [CSV09]. Unlike many convex
optimization methods, metaheuristic algorithms, are relatively simple to
implement and easily parallelizable. This combination of simplicity and
effectiveness has fuelled the application of metaheuristic in complex
engineering problems such as shape optimization, scheduling problems, and
hyper-parameter tuning in machine learning models. However, it is often the
case that metaheuristics lack rigorous convergence results, a question which
has become an active area of research [GP21, GHPQ21].

In [PTTM17], the authors introduced a optimization algorithm which employs an
individual-based model to frame a global minimization

$\min\limits_{x\in\mathbb{R}^{d}}f(x),$

where $f(x)$ is a positive function from $\mathbb{R}^{d}$ to $\mathbb{R}$, as
a consensus problem . In this model, each individual particle explores the
energy landscape given by $f(x)$, broadcasting its current value to the rest
of the ensemble through a weighted average. This iterated interaction
generates trajectories which flock towards a consensus point which correspond
to a global minimizer of $f(x)$, hence the name Consensus Based Optimization
(CBO). We refer to [Tot22, GHPQ21] for two recent surveys on the topic. The
dynamics of existing CBO models are governed by stochastic differential
equations with Wiener noise [PTTM17, CCTT18, CJLZ21]. Hence, we can resort to
a toolbox from stochastic calculus and stochastic numerics to perform analysis
of these models. This amenability of CBO models to theoretical as well as
numerical analysis differentiates them from other agent based optimization
algorithms.

In this paper, we propose a new CBO model which is governed by jump-diffusion
stochastic differential equations. This means randomness in the dynamics of
the proposed CBO models comes from Wiener noise as well as compound Poisson
process. The following are the contributions of this paper:

  * (i)

We prove the well-posedness of the interacting-particle system and of its
mean-field limit driven by jump-diffusion SDEs and convergence of the mean-
field SDEs to the global minimum. The approach to study well-posedness and
convergence to the global minimum is similar to [CCTT18] but adapted to the
jump-diffusion case with time-dependent coefficients.

  * (ii)

The major contribution of the paper is that we prove mean-square convergence
of the interacting particle system to the mean-field limit when number of
particles, $N$, tend to $\infty$. This also implies convergence of the
particle system towards the mean-field limit in $2-$Wasserstein metric. Let us
emphasize that we prove this result for quadratically growing objective
function. We also study convergence of the implementable discretized particle
system towards the jump-diffusion SDEs as the discretization step, $h$, goes
to $0$. Our results can be utilized for the earlier CBO models [PTTM17,
CCTT18, CJLZ21].

  * (iii)

As illustrated in the numerical experiments, the addition of a jump-diffusion
process in the particle system leads to a more effective exploration of the
energy landscape. This particularly relevant when a good prior knowledge of
the optimal solution for initialization of the CBO is not available.

As was highlighted in [CCTT18, Remark 3.2], it is not straightforward to prove
convergence of the interacting particle system towards its mean-field limit,
even after proving uniform in $N$ moment bound of the solutions of the SDEs
driving particles system. Convergence results of this type have been proved
for special cases of compact manifolds (see [FHPS20] for compact hypersurfaces
and [HKK+22] for Stiefel manifolds) and globally Lipschitz continuous
objective functions. In this case, not only the objective function is bounded
but also particles are evolving on a compact set. Under the assumptions on the
objective function as in our paper, in the diffusion case weak convergence of
the empirical measure of a particle system to the law of the corresponding
mean field SDEs has been proved in [GHPQ21, HQ21] exploiting Prokhorov’s
theorem. Here we prove convergence of the particle system to the mean-field
SDEs in the mean-square sense for a quadratically growing locally-Lipschitz
objective function defined on $\mathbb{R}^{d}$.

Furthermore, practical implementation of the particle system corresponding to
a CBO model needs a numerical approximation in the mean-square sense. We
utilize an explicit Euler scheme to implement the proposed jump-diffusion CBO
model. This leads to the question whether the Euler scheme converges to the
CBO model taking into account that the coefficients of the particle system are
not globally Lipschitz and the Lipschitz constants grow exponentially when the
objective function is not bounded. At the same time, the coefficients of the
particle system have linear growth at infinity. In the case of jump-diffusion
SDEs, earlier works either showed convergence of the Euler scheme in the case
of globally Lipschitz coefficients [PBL10] or proposed special schemes in the
case of non-globally Lipschitz coefficients with super-linear growth, e.g. a
tamed Euler scheme [DKS16]. Here we prove mean-square convergence of the Euler
scheme and we show that this convergence is uniform in the number of particles
$N$, i.e. the choice of a discretization time-step $h$ is independent of $N$.
Our convergence result also holds for earlier CBO models [PTTM17, CCTT18,
CJLZ21].

In Section 2, we first present a review of existing CBO models and then
describe our CBO model driven by jump-diffusion SDEs. We also formally
introduce mean-field limit of the new CBO model. In Section 3, we focus on
well-posedness of the interacting particle system behind the new CBO model and
its mean-field limit. In Section 4, we discuss convergence of the mean field
limit towards a point in $\mathbb{R}^{d}$ which approximates the global
minimum, convergence of the interacting particle system towards mean field
limit, and convergence of the implementable discretized particle system
towards the particle system. We present results of numerical experiments in
Section 5 to compare performance of our model and the existing CBO models.

Throughout the paper, $C$ is a floating constant which may vary at different
places. We denote $(a\cdot b)$ as dot product between two vectors,
$a,b\in\mathbb{R}^{d}$. We will omit brackets $()$ wherever it does not lead
to any confusion.

##  2  CBO models : existing and new

In Section 2.1, we review the existing CBO models. In Section 2.2, we
introduce a new CBO model driven by jump-diffusion SDEs and and discuss
potential advantages of adding jumps to CBO models which are confirmed by
numerical experiments in Section 5. The numerical experiments of Section 5 are
conducted using the Euler scheme presented in Section 2.2.

###  2.1 Review of the existing CBO models

Let $N\in\mathbb{N}$ denote the number of agents with position vector,
$X^{i}_{N}(t)\in\mathbb{R}^{d}$, $i=1,\dots,N$. The following model was
proposed in [PTTM17]:

$\displaystyle dX^{i}_{N}(t)$
$\displaystyle=-\beta(X^{i}_{N}(t)-\bar{X}^{\alpha,f}_{N}(t))H^{\epsilon}(f(X^{i}_{N}(t))-f(\bar{X}^{\alpha,f}_{N}(t)))dt$
$\displaystyle\;\;\;\;+\sqrt{2}\sigma|X^{i}_{N}(t)-\bar{X}^{\alpha,f}_{N}(t)|dW^{i}(t),\;\;\;\;i=1,\dots,N,$
(2.1)

where $H^{\epsilon}:\mathbb{R}\rightarrow\mathbb{R}$ is a smooth
regularization of the Heaviside function, $W^{i}(t)$, $i=1,\ldots,N,$
represent $N-$independent $d$-dimensional standard Wiener processes,
$\beta>0$, $\sigma>0$, and $\bar{X}^{\alpha,f}_{N}(t)$ is given by

$\bar{X}^{\alpha,f}_{N}(t)=\frac{\sum_{i=1}^{N}X^{i}_{N}(t)w_{f}^{\alpha}(X^{i}_{N}(t))}{\sum_{i=1}^{N}w_{f}^{\alpha}(X^{i}_{N}(t))},$
(2.2)

with $w_{f}^{\alpha}(x)=\exp{(-\alpha f(x))}$, $\alpha>0$.

Each particle $X^{i}_{N}$ at time $t$ is assigned an opinion
$f(X^{i}_{N}(t))$. The lesser the value of $f$ for a particle, the more is the
influence of that particle, i.e. the more weight is assigned to that particle
at that time as can be seen in (2.2) of the instantaneous weighted average. If
the value $f(X^{i}_{N}(t))$ of a particle $X^{i}_{N}$ at time $t$ is greater
than the value $f(\bar{X}_{N}^{\alpha,f}(t))$ at the instantaneous weighted
average $\bar{X}_{N}^{\alpha,f}(t)$ then the regularised Heaviside function
forces the particle $X^{i}_{N}$ to drift towards $\bar{X}_{N}^{\alpha,f}$. If
the opinion of $i$-th particle matters more among the interacting particles,
i.e. the value $f(X^{i}_{N}(t))$ is less than $f(\bar{X}^{i}_{N}(t))$, then it
is not beneficial for it to move towards $\bar{X}_{N}^{\alpha,f}$. The noise
term is added to explore the space $\mathbb{R}^{d}$ and to avoid non-uniform
consensus. The noise intensity induced in the dynamics of the $i-$th particle
at time $t$ takes into account the distance of the particle from the
instantaneous weighted average, $\bar{X}_{N}^{\alpha,f}(t)$. Over a period of
time as the particles start moving towards a consensus opinion, the
coefficients in (2.1) go to zero.

One can observe that the more influential opinion a particular particle has,
the higher is the weight assigned to that particle in the instantaneous
weighted average (2.2). Based on this logic, in [CCTT18] the authors dropped
the regularised Heaviside function in the drift coefficient and the model
(2.1) was simplified as follows:

$dX^{i}_{N}(t)=-\beta(X^{i}_{N}(t)-\bar{X}_{N}^{\alpha,f}(t))dt+\sigma|X^{i}_{N}(t)-\bar{X}_{N}^{\alpha,f}(t)|dW^{i}(t),\;\;\;i=1,\dots,N,$
(2.3)

with $\beta$, $\sigma$, $\bar{X}_{N}^{\alpha,f}$ as in (2.1)-(2.2).

The major drawback of the consensus based models (2.1) and (2.3) is that the
parameters $\beta$ and $\sigma$ are dependent on the dimension $d$. To
illustrate this fact, we replace $\bar{X}_{N}^{\alpha,f}$ in (2.3) by a fixed
vector $V\in\mathbb{R}^{d}$. Then, using Ito’s formula, we have

$\frac{d}{dt}\mathbb{E}|X^{i}_{N}(t)-V|^{2}=(-2\beta+\sigma^{2}d)\mathbb{E}|X^{i}_{N}(t)-V|^{2},\;\;\;\;i=1,\dots,N.$
(2.4)

As one can notice, for particles to reach the consensus point whose position
vector is $V$, one needs $2\beta>d\sigma^{2}$. To overcome this deficiency,
the authors of [CJLZ21] proposed the following model which is based on
component-wise noise intensity instead of isotropic noise used in (2.1) and
(2.3):

$dX^{i}_{N}(t)=-\beta(X^{i}_{N}(t)-\bar{X}_{N}^{\alpha,f}(t))dt+\sqrt{2}\sigma\operatorname{Diag}(X^{i}_{N}(t)-\bar{X}_{N}^{\alpha,f}(t))dW^{i}(t),\;\;\;\;i=1,\dots,N,$
(2.5)

where $\beta,\sigma$, and $\bar{X}_{N}^{\alpha,f}$ are as in (2.1)-(2.2), and
$\operatorname{Diag}(U)$ is a diagonal matrix whose diagonal is a vector
$U\in\mathbb{R}^{d}$. Now, if we replace $\bar{X}_{N}^{\alpha,f}$ by a fixed
vector $V$ and then use Ito’s formula for (2.5), we get

$\displaystyle\frac{d}{dt}\mathbb{E}|X^{i}_{N}(t)-V|^{2}$
$\displaystyle=-2\beta\mathbb{E}|X^{i}_{N}(t)-V|^{2}+\sigma^{2}\mathbb{E}\sum\limits_{j=1}^{d}(X^{i}_{N}(t)-V)_{j}^{2}$
$\displaystyle=(-2\beta+\sigma^{2})\mathbb{E}|X^{i}_{N}(t)-V|^{2},\;\;\;\;i=1,\dots,N,$
(2.6)

where $(X_{N}^{i}(t)-V)_{j}$ denotes the $j-$th component of
$(X_{N}^{i}(t)-V)$. It is clear that in this model there is no dimensional
restriction on $\beta$ and $\sigma$.

Other CBO models [HJK20, HJK21] are based on interacting particles driven by
common noise. Since the same noise drives all the particles, the exploration
is not effective. Therefore, they are not scalable with respect to dimension
and do not perform well in contrast to the CBO models (2.1), (2.3), (2.5) and
model introduced in Section 2.2. This fact is demonstrated in experiments in
Section 5.

###  2.2 Jump-diffusion CBO models

Let us consider the following jump-diffusion model:

$\displaystyle dX^{i}_{N}(t)$
$\displaystyle=-\beta(t)(X^{i}_{N}(t)-\bar{X}_{N}(t))dt+\sqrt{2}\sigma(t)\operatorname{Diag}(X^{i}_{N}(t)-\bar{X}_{N}(t))dW^{i}(t)$
$\displaystyle\;\;\;\;+\gamma(t)\operatorname{Diag}(X^{i}_{N}(t^{-})-\bar{X}_{N}(t^{-}))dJ^{i}(t),\;\;i=1,\dots,N,$
(2.7)

with

$J^{i}(t)=\sum\limits_{j=1}^{N^{i}(t)}Z^{i}_{j},$ (2.8)

where $N^{i}(t)$, $i=1\dots,N$ are $N-$independent Poisson processes with jump
intensity $\lambda$ and $Z_{j}^{i}=(Z_{j,1}^{i},\dots,Z_{j,d}^{i})^{\top}$ are
i.i.d. $d$-dimensional random variables denoting $j-$th jump by $i-$th
particle and $Z_{j}^{i}\sim Z$. The distribution of $Z$ is called as jump size
distribution. For the sake of convenience, we write $Z_{l}$ as the $l$-th
component of vector $Z$. We assume that each component $Z_{l}$ of $Z$ is also
i.i.d. random variable and distributed as

$Z_{l}\sim\text{\ooalign{\raisebox{0.86108pt}{--}\cr$Z$\cr}},$ (2.9)

where  – $Z$ is an $\mathbb{R}-$valued random variable whose probability
density is given by
$\rho_{\text{\ooalign{-\kern-2.10002pt-\cr$z$\cr}}}(\text{\ooalign{-\kern-3.00003pt-\cr$z$\cr}})$
such that
$\mathbb{E}(\text{\ooalign{\raisebox{0.86108pt}{--}\cr$Z$\cr}})=\int_{\mathbb{R}}\text{\ooalign{-\kern-3.00003pt-\cr$z$\cr}}\rho_{\text{\ooalign{-\kern-2.10002pt-\cr$z$\cr}}}(\text{\ooalign{-\kern-3.00003pt-\cr$z$\cr}})d\text{\ooalign{-\kern-3.00003pt-\cr$z$\cr}}=0$.
We also denote the probability density of $Z$ as
$\rho_{z}(z)=\prod_{l=1}^{d}\rho_{\text{\ooalign{-\kern-2.10002pt-\cr$z$\cr}}}(z_{l})$.
Note that $\mathbb{E}(Z)$ is a $d-$dimensional zero vector, since each $Z_{l}$
is distributed as  – $Z$ . The Wiener processes $W^{i}(t)$, the Poisson
processes $N^{i}(t)$, $i=1\dots,N$ and the jump sizes $Z$ are assumed to be
mutually independent (see further theoretical details concerning Lévy-driven
SDEs in [App04]). Also, $\beta(t)$, $\sigma(t),\gamma(t)$ are continuous
functions and

$\bar{X}_{N}(t)=(\bar{X}^{1}_{N}(t),\dots,\bar{X}^{d}_{N}(t)):=\frac{\sum_{i=1}^{N}X^{i}_{N}(t)e^{-\alpha
f(X^{i}_{N}(t))}}{\sum_{i=1}^{N}e^{-\alpha f(X^{i}_{N}(t))}},$ (2.10)

with $\alpha>0$. Note that we have omitted $\alpha$ and $f$ of
$\bar{X}_{N}^{\alpha,f}$ in the notation used in (2.2) for the simplicity of
writing.

We recall the meaning of the jump term

$\int_{0}^{t}\gamma(s)\operatorname{Diag}(X^{i}(s^{-})-\bar{X}_{N}(s^{-}))dJ^{i}(s)=\sum_{j=1}^{N^{i}(t)}\gamma(\tau_{j})\operatorname{Diag}(X^{i}(\tau_{j}^{-})-\bar{X}_{N}(\tau_{j}^{-}))Z^{i}_{j},$

where $\tau_{j}$ denotes the time of $j$-th jump of the Poisson process
$N^{i}(t)$. Thanks to the assumption that
$\mathbb{E}(\text{\ooalign{\raisebox{0.86108pt}{--}\cr$Z$\cr}})=0$ (which in
turn implies $\mathbb{E}(Z^{i}_{j,l})=0$, $j=1,\dots,N^{i}(t)$, $i=1,\dots,N$,
$l=1,\dots,d$), the above integral is a martingale, and hence (similar to
Ito’s integral term in (2.2)) it does not bias trajectories of $X_{N}^{i}(t)$,
$i=1,\dots,N$.

The jump diffusion SDEs (2.2) are different from (2.5) in the two ways:

  * •

The SDEs (2.2) are a consequence of interlacing of Ito’s diffusion by jumps
arriving according to the Poisson process whose jump intensity is given by
$\lambda$.

  * •

We take $\beta(t)$ as a continuous positive non-decreasing function of $t$
such that $\beta(t)\rightarrow\beta>0$ as $t\rightarrow\infty$, $\sigma(t)$ as
a continuous positive non-increasing function of $t$ such that
$\sigma(t)\rightarrow\sigma>0$ as $t\rightarrow\infty$ and $\gamma(t)$ as a
continuous non-negative non-increasing function of $t$ such that
$\gamma(t)\rightarrow\gamma\geq 0$ as $t\rightarrow\infty$.

Although we analyse CBO model (2.2) with time-dependent parameters, a decision
to take parameters time-dependent or not is problem specific. Note that the
particles driven by SDEs (2.2) jump at different times with different jump
sizes and jumps arrive according to the Poisson process with intensity
$\lambda$.

We can also write the jump-diffusion SDEs (2.2) in terms of Poisson random
measure [App04] as

$\displaystyle dX^{i}_{N}(t)$
$\displaystyle=-\beta(t)(X^{i}_{N}(t)-\bar{X}_{N}(t))dt+\sqrt{2}\sigma(t)\operatorname{Diag}(X^{i}_{N}(t)-\bar{X}_{N}(t))dW^{i}(t)$
$\displaystyle\;\;\;\;+\int_{\mathbb{R}^{d}}\gamma(t)\operatorname{Diag}(X^{i}_{N}(t^{-})-\bar{X}_{N}(t^{-}))z\mathcal{N}^{i}(dt,dz),$
(2.11)

where $\mathcal{N}^{i}(dt,dz)$, $i=1,\dots,N$, represent the independent
Poisson random measures with intensity measure $\nu(dz)dt$ and $\nu(dz)$ is a
Lévy measure which is finite in our case (2.2). Although for simplicity we
introduced our model as (2.2), in proving well-posedness and convergence
results we will make use of (2.2).

We can formally write the mean field limit of the model (2.2) as the following
McKean-Vlasov SDEs:

$\displaystyle dX(t)$
$\displaystyle=-\beta(t)(X(t)-\bar{X}(t))dt+\sqrt{2}\sigma(t)\operatorname{Diag}(X(t)-\bar{X}(t))dW(t)$
$\displaystyle\;\;\;\;+\gamma(t)\operatorname{Diag}(X(t^{-})-\bar{X}(t^{-}))dJ(t),$
(2.12)

where $J(t)=\sum_{j=1}^{N(t)}Z_{j}$, $N(t)$ is a Poisson process with
intensity $\lambda$, and

$\displaystyle\bar{X}(t):=\bar{X}^{\mathcal{L}_{X(t)}}=\frac{\int_{\mathbb{R}^{d}}xe^{-\alpha
f(x)}\mathcal{L}_{X(t)}(dx)}{\int_{\mathbb{R}^{d}}e^{-\alpha
f(x)}\mathcal{L}_{X(t)}(dx)}=\frac{\mathbb{E}\big{(}X(t)e^{-\alpha
f(X(t))}\big{)}}{\mathbb{E}\big{(}e^{-\alpha f(X(t))}\big{)}},$ (2.13)

with $\mathcal{L}_{X(t)}:=\text{Law}(X(t))$. We can rewrite the mean field
jump diffusion SDEs (2.2) in terms of Poisson random measure as

$\displaystyle dX(t)$
$\displaystyle=-\beta(t)(X(t)-\bar{X}(t))dt+\sqrt{2}\sigma(t)\operatorname{Diag}(X(t)-\bar{X}(t))dW(t)$
$\displaystyle\;\;\;\;+\gamma(t)\int_{\mathbb{R}^{d}}\operatorname{Diag}(X(t^{-})-\bar{X}(t^{-}))z\mathcal{N}(dt,dz).$
(2.14)

####  2.2.1 Other jump-diffusion CBO models

Although the aim of the paper is it to analyse the CBO model (2.2), we discuss
three other jump-diffusion CBO models of interest.

Additional Model 1 : Writing (2.2) in terms of Poisson random measure suggests
that we can also consider an infinite activity Lévy process, e.g. an
$\alpha-$stable process, to introduce jumps in dynamics of particles. We can
write the CBO model as

$\displaystyle dX^{i}_{N}(t)$
$\displaystyle=-\beta(t)(X^{i}_{N}(t)-\bar{X}_{N}(t))dt+\sqrt{2}\sigma(t)\operatorname{Diag}(X^{i}_{N}(t)-\bar{X}_{N}(t))dW^{i}(t)$
$\displaystyle\;\;\;\;+\int_{\mathbb{R}^{d}}\gamma(t)\operatorname{Diag}(X^{i}_{N}(t^{-})-\bar{X}_{N}(t^{-}))z\mathcal{N}^{i}(dt,dz),$
(2.15)

However, numerical approximation of SDEs driven by infinite activity Lévy
processes is computationally more expensive (see e.g. [PBL10, DMT21]), hence
it can be detrimental for the overall CBO performance.

Additional Model 2 : In the SDEs (2.2), the intensity of Poisson process
$\lambda$ is constant. If we take jump intensity as $\lambda(t)$, i.e. a
function of $t$ then the corresponding SDEs will be as follows:

$\displaystyle dX^{i}(t)$
$\displaystyle=-\beta(t)(X^{i}_{N}(t)-\bar{X}_{N}(t))dt+\sqrt{2}\sigma(t)\operatorname{Diag}(X^{i}_{N}(t)-\bar{X}(t))dW^{i}(t)$
$\displaystyle\;\;\;\;+\operatorname{Diag}(X^{i}_{N}(t^{-})-\bar{X}_{N}(t^{-}))dJ^{i}(t),\;\;i=1,\dots,N,$
(2.16)

where all the notation are as in (2.2) and (2.10) except here the intensity of
the Poisson processes $N^{i}(t)$ is a time-dependent function $\lambda(t)$. It
is assumed that $\lambda(t)$ is a decreasing function such that
$\lambda(t)\rightarrow 0$ as $t\rightarrow\infty$. Also, in comparison with
(2.2), there is no $\gamma(t)$ in the jump component of (2.2.1). Note that,
the compound Poisson process with constant jump intensity $\lambda$ is a Lévy
process but with time-dependent jump intensity $\lambda(t)$, it is not a Lévy
process, rather it is an additive process. Additive process is a
generalization of Lévy process which satisfies all conditions of Lévy process
except stationarity of increments [KI99]. The SDEs (2.2.1) present another
jump-diffusion CBO model driven by additive process. The analysis of model
(2.2.1) follows similar arguments since the jump-diffusion SDEs (2.2.1) can
also be written in terms of the Poisson random measure with intensity measure
$\nu_{t}(dz)dt$, where $(\nu_{t})_{t\geq 0}$ is a family of Lévy measures.

Additional Model 3 : In model (2.2), the particles have idiosyncratic noise
which means they are driven by different Wiener processes and different
compound Poisson processes. Instead, we can have a different jump-diffusion
model in which the same Poisson noise drives particle system but jumps sizes
still independently vary for all particles. This means jumps arrive at the
same time for all particles, but particles jump with different jump-sizes. We
can write CBO model as

$\displaystyle dX^{i}_{N}(t)$
$\displaystyle=-\beta(t)(X^{i}_{N}(t)-\bar{X}_{N}(t))dt+\sqrt{2}\sigma(t)\operatorname{Diag}(X^{i}_{N}(t)-\bar{X}_{N}(t))dW^{i}(t)$
$\displaystyle\;\;\;\;+\int_{\mathbb{R}^{d}}\gamma(t)\operatorname{Diag}(X^{i}_{N}(t^{-})-\bar{X}_{N}(t^{-}))z\mathcal{N}(dt,dz).$
(2.17)

We compare performance of the jump-diffusion CBO models (2.2) and (2.2.1) in
Section 5.

####  2.2.2 Discussion

Firstly, we will discuss dependence of the parameters $\beta(t)$, $\sigma(t)$,
$\gamma(t)$ and $\lambda$ on dimension $d$. The independent and identical
distribution of $Z_{l}$, which denotes the $l-$th component of $Z$, result in
the non-dependency of parameters on dimension in the similar manner as for the
model (2.5). We illustrate this fact by fixing a vector $V\in\mathbb{R}^{d}$
and replacing $\bar{X}_{N}$ in (2.2) by $V$ then using Ito’s formula and the
assumption made on
$\rho_{\text{\ooalign{-\kern-2.10002pt-\cr$z$\cr}}}(\text{\ooalign{-\kern-3.00003pt-\cr$z$\cr}})$,
we have

$\displaystyle\frac{d}{dt}\mathbb{E}|X^{i}_{N}(t)-V|^{2}$
$\displaystyle=-2\beta(t)\mathbb{E}|X^{i}_{N}(t)-V|^{2}+\sigma^{2}(t)\sum\limits_{j=1}^{d}\mathbb{E}(X^{i}_{N}(t)-V)_{j}^{2}$
$\displaystyle\;\;\;\;+\lambda\int_{\mathbb{R}^{d}}\big{(}|X^{i}_{N}(t)-V+\gamma(t)\operatorname{Diag}(X^{i}_{N}(t)-V)z|^{2}-|X^{i}_{N}(t)-V|^{2}\big{)}\rho_{z}(z)dz$
$\displaystyle=(-2\beta(t)+\sigma^{2}(t))\mathbb{E}|X^{i}_{N}(t)-V|^{2}+\lambda\int_{\mathbb{R}^{d}}\gamma^{2}(t)|\operatorname{Diag}(X^{i}_{N}(t)-V)z|^{2}\rho_{z}(z)dz$
$\displaystyle=(-2\beta(t)+\sigma^{2}(t))\mathbb{E}|X^{i}_{N}(t)-V|^{2}+\lambda\gamma^{2}(t)\sum\limits_{j=1}^{d}\int_{\mathbb{R}^{d}}(X^{i}_{N}(t)-V)_{j}^{2}z_{j}^{2}\prod_{l=1}^{d}\rho_{\text{\ooalign{-\kern-2.10002pt-\cr$z$\cr}}}(z_{l})dz$
(2.20)
$\displaystyle=\big{(}-2\beta(t)+\sigma^{2}(t)+\lambda\gamma^{2}(t)\mathbb{E}(\text{\ooalign{\raisebox{0.86108pt}{--}\cr$Z$\cr}}^{2})\big{)}\mathbb{E}|X^{i}_{N}(t)-V|^{2}.$
(2.23)

We can choose $\beta(t)$, $\sigma(t)$, $\gamma(t)$, $\lambda$ and distribution
of  – $Z$ guaranteeing that there is a $t_{*}\geq 0$ such that
$-2\beta(t)+\sigma^{2}(t)+\lambda\gamma^{2}(t)\mathbb{E}(\text{\ooalign{\raisebox{0.86108pt}{--}\cr$Z$\cr}}^{2})<0$
for all $t\geq t_{*}$ and such a choice is independent of $d$. It is clear
from (2.23) that with this choice, $\mathbb{E}|X^{i}_{N}(t)-V|^{2}$,
$i=1,\dots,N$, decay in time as $t\rightarrow\infty$.

In the previous CBO models, there were only two terms namely, the drift term
and the diffusion term. The drift tries to take the particles towards their
instantaneous weighted average. The diffusion term helps in exploration of the
state space with the aim to find a state with better weighted average than the
current one. The model (2.2) contains one extra term, which we call the jump
term. Jumps help in intensifying the search in a search space and aids in
avoiding premature convergence or trapping in local minima. This results in
more effective use of the interaction of particles.

Moreover, the effect of jumps decays with time in (2.2) by virtue of
decreasing $\gamma(t)$. The reason for considering the model (2.2) where jumps
affect only the initial period of time is that we want particles to explore
more space faster at the beginning of simulation and, as soon as the weighted
average of particles is in a vicinity of the global minimum, we do not want
jumps to affect convergence of particles towards that consensus point lying in
the close neighbourhood of the global minimum. Therefore, the time-dependent
parameters and degeneracy of the coefficients help in exploiting the searched
space.

As a consequence, the jump-diffusion noise and degenerate time-dependent
coefficients in model (2.2) may help in keeping the balance of exploration and
exploitation by interacting particles over a period of time. We will continue
this discussion on exploration and exploitation in Section 5, where the
proposed CBO method is tested.

####  2.2.3 Implementation

Let $0=t_{0}<\dots<t_{n}=T$ be a uniform partition of the time interval
$[0,T]$ into $n$ sub-intervals such that $h:=t_{k+1}-t_{k}$, $k=0,\dots,n-1$
and $T=nh$. To approximate (2.2), we construct a Markov chain
$(Y_{N}^{i}(t_{k}))$, $k=1,\dots,n$, using the following Euler scheme:

$\displaystyle Y^{i}_{N}(t_{k+1})$
$\displaystyle=Y_{N}^{i}(t_{k})-\beta(t_{k})(Y^{i}_{N}(t_{k})-\bar{Y}_{N}(t_{k}))h+\sigma(t_{k})\operatorname{Diag}(Y^{i}_{N}(t_{k})-\bar{Y}_{N}(t_{k}))\Delta
W(t_{k})$
$\displaystyle\;\;\;\;+\gamma(t_{k})\sum\limits_{j=N^{i}(t_{k})+1}^{N^{i}(t_{k+1})}\operatorname{Diag}(Y^{i}_{N}(t_{k})-\bar{Y}_{N}(t_{k}))Z^{i}_{j},$
(2.24)

where $\Delta W(t_{k})=W(t_{k+1})-W(t_{k})$ has Gaussian distribution with
mean $0$ and variance $h$, $Z^{i}_{j}$ denotes $j-$th jump size of the $i-$th
particle, $N^{i}(t)$ are independent Poisson processes with jump intensity
$\lambda$, and

$\displaystyle\bar{Y}_{N}(t)=\sum\limits_{i=1}^{N}Y^{i}_{N}(t)\frac{e^{-\alpha
f(Y^{i}_{N}(t))}}{\sum_{j=1}^{N}e^{-\alpha f(Y^{i}_{N}(t))}}.$ (2.25)

To implement the discretization scheme we initialize the $N\times d$ matrix
$Y$ at time $t_{0}=0$, and update it for $n$ iterations using (2.2.3) by
calculating (2.25) at each iteration. The code to implement above numerical
scheme utilizing $N\times d$ matrix, which allows to save memory and time in
computations, is available on github. We will discuss the convergence of
scheme (2.2.3) in Subsection 4.3.

##  3 Well-posedness results

In Section 3.1, we discuss well-posedness of the interacting particle system
(2.2) and prove moment bound for this system. In Section 3.2, we prove well-
posedness and moment bound of the mean field limit (2.2) of the particle
system (2.2).

###  3.1 Well-posedness of the jump-diffusion particle system

This section is focused on showing existence and uniqueness of the solution of
(2.2). We first introduce the notation which are required in this section.

Let us denote
$\textbf{x}_{N}:=(x_{N}^{1},\dots,x_{N}^{N})^{\top}\in\mathbb{R}^{Nd}$,
$\bar{\textbf{x}}_{N}=\sum_{i=1}^{N}x^{i}_{N}e^{-\alpha
f(x^{i}_{N})}/\sum_{j=1}^{N}e^{-\alpha f(x^{j}_{N})}$,
$\textbf{W}(t):=(W^{1}(t),\dots,W_{N}(t))^{\top}$,
$\textbf{F}_{N}(\textbf{x}_{N}):=\big{(}F^{1}_{N}(\textbf{x}_{N}),\dots,F^{N}_{N}(\textbf{x}_{N})\big{)}^{\top}\in\mathbb{R}^{Nd}$
with $F_{N}^{i}(\textbf{x}_{N})=(x_{N}^{i}-\bar{x}_{N})\in\mathbb{R}^{d}$ for
all $i=1,\dots,N$,
$\textbf{G}_{N}(\textbf{x}_{N}):=\operatorname{Diag}(\textbf{F}_{N}(\textbf{x}_{N}))\in\mathbb{R}^{Nd\times
Nd}$ and $\textbf{J}(t)=({J}^{1}(t),\dots,{J}^{N}(t))$, where $J^{i}(t)$ is
from (2.8) which implies
$\int_{0}^{t}\gamma(t)\operatorname{Diag}(F^{i}_{N}(\textbf{x}_{N}^{i}))d{J}^{i}(t)=\int_{0}^{t}\int_{\mathbb{R}^{d}}\operatorname{Diag}(F^{i}_{N}(\textbf{x}_{N}))z\mathcal{N}^{i}(dt,dz)$.
Let us represent $\ell(dz)$ as the Lebesgue measure of $dz$, and for the sake
of convenience we will use $dz$ in place of $\ell(dz)$ whenever there is no
confusion. We can write the particle system (2.2) using the above notation as

$\displaystyle
d\textbf{X}_{N}(t)=\beta(t)\textbf{F}_{N}(\textbf{X}_{N}(t^{-}))dt+\sqrt{2}\sigma(t)\textbf{G}_{N}(\textbf{X}_{N}(t^{-}))d\textbf{W}(t)+\gamma(t)\textbf{G}_{N}(\textbf{X}_{N}(t^{-}))d\textbf{J}(t).$
(3.1)

In order to show well-posedness of (3.1), we need the following natural
assumptions on the objective function $f$. Let

$f_{m}:=\inf f.$ (3.2)

######  Assumption 3.1.

$f_{m}>0$.

######  Assumption 3.2.

$f:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is locally Lipschtiz continuous, i.e.
there exists a positive function $L(R)$ such that

$|f(x)-f(y)|\leq L(R)|x-y|,$

whenever $|x|$, $|y|\leq R$, $x$, $y\in\mathbb{R}^{d}$, $R>0$.

Assumption 3.2 is used for proving local Lipschitz continuity and linear
growth of $F^{i}_{N}$ and $G^{i}_{N}$, $i=1,\dots,N$. Let
$B(R)=\\{x\in\mathbb{R}^{d}\;;\;|x|\leq R\\}$.

######  Lemma 3.1.

Under Assumptions 3.1-3.2, the following inequalities hold for any
$\textbf{x}_{N}$, $\textbf{y}_{N}\in\mathbb{R}^{Nd}$ satisfying
$\sup_{i=1,\dots,N}|x^{i}_{N}|,\sup_{i=1,\dots,N}|y^{i}_{N}|\leq R$ and for
all $i=1,\dots,N$:

  1. 1.

$|F^{i}_{N}(\textbf{x}_{N})-F^{i}_{N}(\textbf{y}_{N})|\leq|x^{i}_{N}-y^{i}_{N}|+\frac{C(R)}{N^{1/2}}|\textbf{x}_{N}-\textbf{y}_{N}|,$

  2. 2.

$|F^{i}_{N}(\textbf{x}_{N})|^{2}\leq 2(|x_{N}^{i}|^{2}+|\textbf{x}_{N}|^{2}),$

where $C(R)=e^{\alpha(|f|_{L_{\infty}(B(R))}-f_{m}})\big{(}1+\alpha
RL(R)+\alpha RL(R)e^{\alpha(|f|_{L_{\infty}(B(R))}-f_{m})})$.

###### Proof.

Let us deal with the first inequality above. We have

$\displaystyle|F^{i}_{N}(\textbf{x}_{N})$
$\displaystyle-F^{i}_{N}(\textbf{y}_{N})|\leq|x^{i}_{N}-y^{i}_{N}|+\Bigg{|}\frac{\sum_{i=1}^{N}x^{i}_{N}e^{-\alpha
f(x^{i}_{N})}}{\sum_{i=1}^{N}e^{-\alpha
f(x^{i}_{N})}}-\frac{\sum_{i=1}^{N}y^{i}_{N}e^{-\alpha
f(y^{i}_{N})}}{\sum_{i=1}^{N}e^{-\alpha f(y^{i}_{N})}}\Bigg{|}$
$\displaystyle\leq|x^{i}_{N}-y^{i}_{N}|+\frac{1}{\sum_{j=1}^{N}e^{-\alpha
f(x^{j}_{N})}}\Bigg{|}\sum\limits_{i=1}^{N}\bigg{(}x^{i}_{N}e^{-\alpha
f(x^{i}_{N})}-y^{i}_{N}e^{-\alpha f(y^{i}_{N})}\bigg{)}\Bigg{|}$
$\displaystyle\;\;\;\;+\sum\limits_{i=1}^{N}|y^{i}_{N}|e^{-\alpha
f(y^{i}_{N})}\Bigg{|}\frac{1}{\sum_{j=1}^{N}e^{-\alpha
f(x^{j}_{N})}}-\frac{1}{\sum_{j=1}^{N}e^{-\alpha f(y^{j}_{N})}}\Bigg{|}$
$\displaystyle\leq|x^{i}_{N}-y^{i}_{N}|+\frac{1}{\sum_{j=1}^{N}e^{-\alpha
f(x^{j}_{N})}}\Bigg{(}\Bigg{|}\sum\limits_{i=1}^{N}(x^{i}_{N}-y^{i}_{N})e^{-\alpha
f(x^{i}_{N})}\Bigg{|}+\Bigg{|}\sum\limits_{i=1}^{N}y^{i}_{N}(e^{-\alpha
f(x^{i}_{N})}-e^{-\alpha f(y^{i}_{N})})\Bigg{|}\Bigg{)}$
$\displaystyle\;\;\;\;+\sum\limits_{i=1}^{N}|y^{i}_{N}|e^{-\alpha
f(y^{i}_{N})}\Bigg{|}\frac{1}{\sum_{j=1}^{N}e^{-\alpha
f(x^{j}_{N})}}-\frac{1}{\sum_{j=1}^{N}e^{-\alpha f(y^{j}_{N})}}\Bigg{|}.$

Using Jensen’s inequality, we have

$\displaystyle\frac{1}{\frac{1}{N}\sum_{i=1}^{N}e^{-\alpha f(x^{i}_{N})}}$
$\displaystyle\leq e^{\alpha\frac{1}{N}\sum_{i=1}^{N}f(x^{i}_{N})}.$

Using the Cauchy-Bunyakowsky-Shwartz inequality, we get

$\displaystyle|F^{i}_{N}(\textbf{x}_{N})-F^{i}_{N}(\textbf{y}_{N})|\leq|x^{i}_{N}-y^{i}_{N}|+e^{\alpha|f|_{L_{\infty}(B(R))}}e^{-\alpha
f_{m}}\frac{1}{N}\sum_{i=1}^{N}\big{|}x^{i}_{N}-y^{i}_{N}\big{|}+\alpha
e^{-\alpha f_{m}}e^{\alpha|f|_{L_{\infty}(B(R))}}L(R)$
$\displaystyle\times\bigg{(}\frac{1}{N}\sum\limits_{i=1}^{N}|y^{i}_{N}|^{2}\bigg{)}^{1/2}\bigg{(}\frac{1}{N}\sum\limits_{i=1}^{N}|x^{i}_{N}-y^{i}_{N}|^{2}\bigg{)}^{1/2}+\alpha
e^{-2\alpha
f_{m}}e^{2\alpha|f|_{L_{\infty}(B(R))}}\frac{L(R)}{N}\sum\limits_{i=1}^{N}|y^{i}_{N}|\sum\limits_{i=1}^{N}|x^{i}_{N}-y^{i}_{N}|$
$\displaystyle\leq|x^{i}_{N}-y^{i}_{N}|+e^{\alpha|f|_{L_{\infty}(B(R))}}e^{-\alpha
f_{m}}\frac{1}{N}\sum_{i=1}^{N}\big{|}x^{i}_{N}-y^{i}_{N}\big{|}+\alpha
e^{-\alpha
f_{m}}e^{\alpha|f|_{L_{\infty}(B(R))}}RL(R)\bigg{(}\frac{1}{N}\sum\limits_{i=1}^{N}|x^{i}_{N}-y^{i}_{N}|^{2}\bigg{)}^{1/2}$
$\displaystyle+\alpha e^{-2\alpha
f_{m}}e^{2\alpha|f|_{L_{\infty}(B(R))}}RL(R)\bigg{(}\frac{1}{N}\sum\limits_{i=1}^{N}|x^{i}_{N}-y^{i}_{N}|^{2}\bigg{)}^{1/2}$
$\displaystyle\leq|x^{i}_{N}-y^{i}_{N}|+e^{\alpha(|f|_{L_{\infty}(B(R))}-f_{m})})\big{(}1+\alpha
RL(R)+\alpha
RL(R)e^{\alpha(|f|_{L_{\infty}(B(R))}-f_{m})})\frac{1}{N^{1/2}}|\textbf{x}_{N}-\textbf{y}_{N}|.$

The second inequality directly follows from

$\displaystyle|F^{i}_{N}(\textbf{x}_{N})|\leq|x^{i}_{N}|+|\textbf{x}_{N}|.$

∎

######  Theorem 3.2.

Let the initial condition $\textbf{X}_{N}(0)$ of the jump-diffusion SDE (2.2)
satisfy $\mathbb{E}|\textbf{X}_{N}(0)|^{2}<\infty$ and
$\mathbb{E}|\text{\ooalign{\raisebox{0.86108pt}{--}\cr$Z$\cr}}|^{2}<\infty$,
then the $Nd-$dimensional system (2.2) has a unique strong solution
$\textbf{X}_{N}(t)$ under Assumptions 3.1-3.2.

###### Proof.

Note that
$|G^{i}_{N}(\textbf{x}_{N})-G^{i}_{N}(\textbf{y}_{N})|=|F^{i}_{N}(\textbf{x}_{N})-F^{i}_{N}(\textbf{y}_{N})|$
and for all $i=1\dots,N$,

$\displaystyle\int_{\mathbb{R}^{d}}|{F}^{i}_{N}(\textbf{x}_{N}){z}|^{2}\rho_{{z}}({z})d{z}$
$\displaystyle=\int_{\mathbb{R}^{d}}\sum\limits_{l=1}^{d}|(x_{N}^{i})_{l}-(y_{N}^{i})_{l}|^{2}|z_{l}|^{2}\prod\limits_{k=1}^{d}\rho_{\text{\ooalign{-\kern-2.10002pt-\cr$z$\cr}}}(z_{k})d{z}$
$\displaystyle=\sum\limits_{l=1}^{d}|(x_{N}^{i})_{l}-(y_{N}^{i})_{l}|^{2}\int_{\mathbb{R}^{d}}|z_{l}|^{2}\prod\limits_{k=1}^{d}\rho_{\text{\ooalign{-\kern-2.10002pt-\cr$z$\cr}}}(z_{k})d{z}=|{F}^{i}_{N}(\textbf{x}_{N})|^{2}\mathbb{E}(\text{\ooalign{\raisebox{0.86108pt}{--}\cr$Z$\cr}})^{2},$

where $(x^{i}_{N})_{l}$ means the $l-$th component of $d$-dimensional vector
$x^{i}_{N}$ and $z_{l}$ means the $l-$th component of $d-$dimensional vector
$z$. Therefore, from Lemma 3.1, we can say that we have a positive function
$K(R)$ of $R>0$ such that

$\displaystyle|\textbf{F}_{N}(\textbf{x}_{N})-\textbf{F}_{N}(\textbf{y}_{N})|^{2}+|\textbf{G}_{N}(\textbf{x}_{N})-\textbf{G}_{N}(\textbf{y}_{N})|^{2}$
$\displaystyle+\sum_{i=1}^{N}\int_{\mathbb{R}^{d}}|\operatorname{Diag}({F}^{i}_{N}(\textbf{x}_{N})-{F}^{i}_{N}(\textbf{y}_{N})){z}|^{2}\rho_{{z}}({z})d{z}$
$\displaystyle\leq K(R)|\textbf{x}_{N}-\textbf{y}_{N}|,$

whenever $|\textbf{x}_{N}|$, $|\textbf{y}_{N}|\leq R$. Moreover,

$\displaystyle|\textbf{F}_{N}(\textbf{x}_{N})|^{2}+|\textbf{G}_{N}(\textbf{x}_{N})|^{2}+\sum_{i=1}^{N}\int_{\mathbb{R}^{d}}|\operatorname{Diag}({F}^{i}_{N}({x}_{N})){z}|^{2}\rho_{{z}}({z})d{z}\leq
C|\textbf{x}_{N}|^{2},$

where $C$ is some positive constant independent of $|\textbf{x}_{N}|$. Then
the proof immediately follows from [GK80, Theorem 1].

Consequently, by [DKS16, Lemma 2.3], the following moment bound, provided
$\mathbb{E}|\textbf{X}_{N}(0)|^{2p}<\infty$ and
$\mathbb{E}|\textbf{Z}|^{2p}<\infty$, holds:

$\displaystyle\mathbb{E}\sup_{0\leq t\leq T}|\textbf{X}_{N}(t)|^{2p}\leq
C_{N},$ (3.3)

where $C_{N}$ may depend on $N$ and $p\geq 1$.∎

In the last step of proof above, we highlighted that $C_{N}$ may depend on
$N$. However, for convergence analysis in later sections we need an uniform in
$N$ bound for
$\sup_{i=1,\dots,N}\mathbb{E}\big{(}\sup_{t\in[0,T]}|X^{i}_{N}(t)|^{2p}\big{)}$,
$p\geq 1$ which we prove under the following assumptions as in [CCTT18].

######  Assumption 3.3.

There exists a positive constant $K_{f}$ such that

$\displaystyle|f(x)-f(y)|$ $\displaystyle\leq
K_{f}(1+|x|+|y|)|x-y|,\;\;\text{for all}\;x,y,\in\mathbb{R}^{d}.$

######  Assumption 3.4.

There is a constant $K_{u}>0$

$\displaystyle f(x)-f_{m}$ $\displaystyle\leq K_{u}(1+|x|^{2}),\;\;\text{for
all}\;x\in\mathbb{R}^{d}.$

######  Assumption 3.5.

There exists constants $R>0$ and $K_{l}>0$ such that

$f(x)-f_{m}\geq K_{l}|x|^{2},\;\;|x|\geq R.$

As one can see, we need a stronger Assumption 3.3 as compared to Assumption
3.2 to obtain a moment bound uniform in $N$. The Assumptions 3.4-3.5 are to
make sure that objective function $f$ has quadratic growth at infinity.

From [CCTT18, Lemma 3.3], we have the following result under Assumptions 3.1,
3.3-3.5:

$\displaystyle\sum_{i=1}^{N}|x_{N}^{i}|^{2}\frac{e^{-\alpha
f(x_{N}^{i})}}{\sum_{j=1}^{N}e^{-\alpha f(x_{N}^{j})}}\leq
L_{1}+L_{2}\frac{1}{N}\sum_{i=1}^{N}|x_{N}^{i}|^{2},$ (3.4)

where $L_{1}=R^{2}+L_{2}$ and
$L_{2}=2\frac{K_{u}}{K_{l}}\Big{(}1+\frac{1}{\alpha K_{l}R^{2}}\Big{)}$, $R$
is from Assumption 3.5.

######  Lemma 3.3.

Let Assumptions 3.1, 3.3-3.5 be satisfied. Let $p\geq 1$,
$\sup_{i=1,\dots,N}\mathbb{E}|X^{i}_{N}(0)|^{2p}<\infty$ and
$\mathbb{E}|Z|^{2p}<\infty$. Then

$\sup_{i\in\\{1,\dots,N\\}}\mathbb{E}\sup_{0\leq t\leq
T}|X^{i}_{N}(t)|^{2p}\leq K_{m},$

where $X_{N}^{i}(t)$ is from (2.2) and $K_{m}$ is a positive constant
independent of $N$.

###### Proof.

Let $p$ be a positive integer. Using Ito’s formula, we have

$\displaystyle|X_{N}^{i}(t)|^{2p}$
$\displaystyle=|X^{i}_{N}(0)|^{2p}-2p\mathbb{E}\int_{0}^{t}\beta(s)|X_{N}^{i}(s)|^{2p-2}\big{(}X_{N}^{i}(s)\cdot(X_{N}^{i}(s)-\bar{X}_{N}(s))\big{)}ds$
$\displaystyle\;\;\;\;+2\sqrt{2}p\int_{0}^{t}\sigma(s)|X^{i}_{N}(s)|^{2p-2}\big{(}X_{N}^{i}(s)\cdot\operatorname{Diag}(X_{N}^{i}(s)-\bar{X}_{N}(s))dW^{i}(s)\big{)}$
$\displaystyle\;\;\;\;+4p(p-1)\int_{0}^{t}\sigma^{2}(s)|X_{N}^{i}(s)|^{2p-4}|\operatorname{Diag}(X_{N}^{i}(s)-\bar{X}_{N}(s))X_{N}^{i}(s)|^{2}ds$
$\displaystyle\;\;\;\;+2p\int_{0}^{t}\sigma^{2}(s)|X_{N}^{i}(s)|^{2p-2}|\operatorname{Diag}(X_{N}^{i}(s)-\bar{X}_{N}(s)|^{2}ds$
$\displaystyle\;\;\;\;+\int_{0}^{t}\int_{\mathbb{R}^{d}}\big{(}|X_{N}^{i}(s^{-})+\gamma(s)\operatorname{Diag}(X_{N}^{i}(s^{-})-\bar{X}_{N}(s^{-}))z|^{2p}-|X_{N}^{i}(s^{-})|^{2p}\big{)}\mathcal{N}^{i}(ds,dz).$

First taking supremum over $0\leq t\leq T$ and then taking expectation, we get

$\displaystyle\mathbb{E}\sup_{0\leq t\leq
T}|X^{i}_{N}(t)|^{2p}\leq\mathbb{E}|X^{i}_{N}(0)|^{2p}+C\mathbb{E}\int_{0}^{T}|X_{N}^{i}(s)|^{2p-2}\big{|}X_{N}^{i}(s)\cdot(X_{N}^{i}(s)-\bar{X}_{N}(s))\big{|}ds$
$\displaystyle\;\;\;\;+C\mathbb{E}\sup_{0\leq t\leq
T}\bigg{|}\int_{0}^{t}|X^{i}_{N}(s)|^{2p-2}\big{(}X_{N}^{i}(s)\cdot\operatorname{Diag}(X_{N}^{i}(s)-\bar{X}_{N}(s))dW^{i}(s)\big{)}\bigg{|}$
$\displaystyle\;\;\;\;+C\mathbb{E}\int_{0}^{T}|X_{N}^{i}(s)|^{2p-4}|\operatorname{Diag}(X_{N}^{i}(s)-\bar{X}_{N}(s))X_{N}^{i}(s)|^{2}ds$
$\displaystyle\;\;\;\;+C\mathbb{E}\int_{0}^{T}|X_{N}^{i}(s)|^{2p-2}|\operatorname{Diag}(X_{N}^{i}(s)-\bar{X}_{N}(s)|^{2}ds$
$\displaystyle\;\;\;\;+C\mathbb{E}\sup_{0\leq t\leq
T}\int_{0}^{t}\int_{\mathbb{R}^{d}}\big{(}|X_{N}^{i}(s^{-})+\gamma(s)\operatorname{Diag}(X_{N}^{i}(s^{-})-\bar{X}_{N}(s^{-}))z|^{2p}-|X_{N}^{i}(s^{-})|^{2p}\big{)}\mathcal{N}^{i}(ds,dz).$
(3.5)

To deal with the second term in (3.1), we use Young’s inequality and obtain

$\displaystyle|X_{N}^{i}(s)|^{2p-2}\big{|}X_{N}^{i}(s)\cdot(X_{N}^{i}(s)-\bar{X}_{N}(s))\big{|}$
$\displaystyle\leq|X_{N}^{i}(s)|^{2p}+|X_{N}^{i}(s)|^{2p-1}|\bar{X}_{N}(s)|$
$\displaystyle\leq\frac{4p-1}{2p}|X_{N}^{i}(s)|^{2p}+\frac{1}{2p}|\bar{X}_{N}(s)|^{2p}.$

To ascertain a bound on $|\bar{X}_{N}(s)|^{2p}$, we first apply Jensen’s
inequality to $|\bar{X}_{N}(s)|^{2}$ to get

$|\bar{X}_{N}(s)|^{2}=\Bigg{|}\sum_{i=1}^{N}X_{N}^{i}(s)\frac{e^{-\alpha
f(X_{N}^{i}(s))}}{\sum_{j=1}^{N}e^{-\alpha
f(X_{N}^{j}(s))}}\Bigg{|}^{2}\leq\sum_{i=1}^{N}|X_{N}^{i}(s)|^{2}\frac{e^{-\alpha
f(X_{N}^{i}(s))}}{\sum_{j=1}^{N}e^{-\alpha f(X_{N}^{j}(s))}},$

then using (3.4), we obtain $|\bar{X}_{N}(s)|^{2}\leq
L_{1}+L_{2}\frac{1}{N}\sum\limits_{i=1}^{N}|X_{N}^{i}(s)|^{2},$ which on
applying the elementary inequality, $(a+b)^{p}\leq
2^{p-1}(a^{p}+b^{p}),\;a,b\in\mathbb{R}_{+}$ and Jensen’s inequality, gives

$\displaystyle|\bar{X}_{N}(s)|^{2p}\leq
2^{p-1}\Big{(}L_{1}^{p}+L_{2}^{p}\frac{1}{N}\sum\limits_{i=1}^{N}|X_{N}^{i}(s)|^{2p}\Big{)}.$

As a consequence of the above calculations, we get

$\displaystyle|X_{N}^{i}(s)|^{2p-2}\big{|}X_{N}^{i}(s)\cdot(X_{N}^{i}(s)-\bar{X}_{N}(s))\big{|}\leq
C\Big{(}1+|X^{i}_{N}(s)|^{2p}+\frac{1}{N}\sum\limits_{i=1}^{N}|X_{N}^{i}(s)|^{2p}\Big{)},$
(3.6)

where $C$ is a positive constant independent of $N$.

Using the Burkholder-Davis-Gundy inequality, we get

$\displaystyle\mathbb{E}$ $\displaystyle\sup_{0\leq t\leq
T}\bigg{|}\int_{0}^{t}|X^{i}_{N}(s)|^{2p-2}\big{(}X_{N}^{i}(s)\cdot\operatorname{Diag}(X_{N}^{i}(s)-\bar{X}_{N}(s))dW^{i}(s)\big{)}\bigg{|}$
$\displaystyle\leq\mathbb{E}\bigg{(}\int_{0}^{T}\big{(}|X^{i}_{N}(s)|^{2p-2}\big{(}X_{N}^{i}(s)\cdot\operatorname{Diag}(X_{N}^{i}(s)-\bar{X}_{N}(s))\big{)}\big{)}^{2}ds\bigg{)}^{1/2}$
$\displaystyle\leq\mathbb{E}\Bigg{(}\sup_{0\leq t\leq
T}|X_{N}^{i}(t)|^{2p-1}\bigg{(}\int_{0}^{T}|X_{N}^{i}(s)-\bar{X}_{N}(s))|^{2}ds\bigg{)}^{1/2}\Bigg{)},$

which on applying generalized Young’s inequality ($ab\leq(\epsilon
a^{q_{1}})/q_{1}+b^{q_{2}}/(\epsilon^{q_{2}/q_{1}}q_{2}),\;\epsilon,q_{1},q_{2}>0,1/q_{1}+1/q_{2}=1$)
yields

$\displaystyle\mathbb{E}$ $\displaystyle\sup_{0\leq t\leq
T}\bigg{|}\int_{0}^{t}|X^{i}_{N}(s)|^{2p-2}\big{(}X_{N}^{i}(s)\cdot\operatorname{Diag}(X_{N}^{i}(s)-\bar{X}_{N}(s))dW^{i}(s)\big{)}\bigg{|}$
$\displaystyle\leq\frac{1}{2}\mathbb{E}\sup_{0\leq t\leq
T}|X^{i}_{N}(t)|^{2p}+C\mathbb{E}\bigg{(}\int_{0}^{T}|X_{N}^{i}(s)-\bar{X}_{N}(s))|^{2}ds\bigg{)}^{p}$
$\displaystyle\leq\frac{1}{2}\mathbb{E}\sup_{0\leq t\leq
T}|X^{i}_{N}(t)|^{2p}+C\mathbb{E}\bigg{(}\int_{0}^{T}|X_{N}^{i}(s)-\bar{X}_{N}(s))|^{2p}ds\bigg{)},$
(3.7)

where in the last step we have utilized Holder’s inequality.

Now, we move on to obtain estimates which are required to deal with fourth and
fifth term in (3.1). Using Young’s inequality, we have

$\displaystyle A_{1}:=|X_{N}^{i}(s)|^{2p-4}(|X_{N}^{i}(s)|^{2}$
$\displaystyle-(X_{N}^{i}(s)\cdot\bar{X}_{N}(s)))^{2}\leq
2|X_{N}^{i}(s)|^{2p}+2|X_{N}^{i}(s)|^{2p-2}|\bar{X}_{N}(s)|^{2}$
$\displaystyle\leq\frac{4p-2}{p}|X_{N}^{i}(s)|^{2p}+\frac{2}{p}|\bar{X}_{N}(s)|^{2p}.$
(3.8)

In the same way, applying Young’s inequality, we obtain

$\displaystyle A_{2}:=|X_{N}^{i}(s)|^{2p-2}|\operatorname{Diag}(X_{N}^{i}(s)$
$\displaystyle-\bar{X}_{N}(s))|^{2}\leq
2|X_{N}^{i}(s)|^{2p}+2|X_{N}^{i}(s)|^{2p-2}|\bar{X}_{N}(s)|^{2}$
$\displaystyle\leq\frac{4p-2}{p}|X_{N}^{i}(s)|^{2p}+\frac{2}{p}|\bar{X}_{N}(s)|^{2p}.$
(3.9)

Following the same procedure based on (3.4), which we followed to obtain bound
(3.6), we also get

$\displaystyle A_{1}+A_{2}\leq
C\Big{(}1+|X_{N}^{i}(s)|^{2p}+\frac{1}{N}\sum\limits_{i=1}^{N}|X_{N}^{i}(s)|^{2p}\Big{)},$
(3.10)

where $C$ is a positive constant independent of $N$.

It is left to deal with the last term in (3.1). Using the Cauchy-Bunyakowsky-
Schwartz inequality, we get

$\displaystyle\mathbb{E}\sup_{0\leq t\leq
T}\int_{0}^{t}\int_{\mathbb{R}^{d}}\big{(}|X_{N}^{i}(s^{-})+\gamma(s)\operatorname{Diag}(X_{N}^{i}(s^{-})-\bar{X}_{N}(s^{-}))z|^{2p}-|X_{N}^{i}(s^{-})|^{2p}\big{)}\mathcal{N}^{i}(ds,dz)$
$\displaystyle\leq\mathbb{E}\sup_{0\leq t\leq
T}\int_{0}^{t}\int_{\mathbb{R}^{d}}\bigg{(}2^{2p-1}\big{(}|X_{N}^{i}(s^{-})|^{2p}+|\gamma(s)\operatorname{Diag}(X_{N}^{i}(s^{-})-\bar{X}_{N}(s^{-}))z|^{2p}\big{)}-|X_{N}^{i}(s^{-})|^{2p}\bigg{)}\mathcal{N}^{i}(ds,dz)$
$\displaystyle\leq
C\mathbb{E}\int_{0}^{T}\int_{\mathbb{R}^{d}}\big{(}|X_{N}^{i}(s^{-})|^{2p}+|\gamma(s)\operatorname{Diag}(X_{N}^{i}(s^{-})-\bar{X}_{N}(s^{-}))z|^{2p}\big{)}\mathcal{N}^{i}(ds,dz)$
$\displaystyle\leq
C\mathbb{E}\int_{0}^{T}\int_{\mathbb{R}^{d}}(|X_{N}^{i}(s)|^{2p}+|\gamma(s)\operatorname{Diag}(X_{N}^{i}(s)-\bar{X}_{N}(s))z|^{2p}\big{)}\rho_{z}(z)dz$
$\displaystyle\leq
C\mathbb{E}\int_{0}^{T}\Big{(}|X_{N}^{i}(s)|^{2p}+|X_{N}^{i}(s)-\bar{X}_{N}(s)|^{2p}\int_{\mathbb{R}^{d}}|z|^{2p}\rho_{z}(z)dz\Big{)}ds.$

We have

$\displaystyle|X_{N}^{i}(s)-\bar{X}_{N}(s)|^{2p}$ $\displaystyle\leq
2^{2p-1}\big{(}|X_{N}^{i}(s)|^{2p}+|\bar{X}_{N}^{i}(s)|^{2p}\big{)}\leq
C\Big{(}1+|X_{N}^{i}(s)|^{2p}+\frac{1}{N}\sum\limits_{i=1}^{N}|X_{N}^{i}(s)|^{2p}\Big{)},$

and hence

$\displaystyle\mathbb{E}\sup_{0\leq t\leq
T}\int_{0}^{t}\int_{\mathbb{R}^{d}}\big{(}|X_{N}^{i}(s^{-})+\gamma(s)\operatorname{Diag}(X_{N}^{i}(s^{-})-\bar{X}_{N}(s^{-}))z|^{2p}-|X_{N}^{i}(s^{-})|^{2p}\big{)}\mathcal{N}^{i}(ds,dz)$
$\displaystyle\leq
C\mathbb{E}\int_{0}^{T}\Big{(}1+|X_{N}^{i}(s)|^{2p}+\frac{1}{N}\sum\limits_{i=1}^{N}|X_{N}^{i}(s)|^{2p}\Big{)}ds,$
(3.11)

where $C>0$ does not depend on $N$.

Using (3.6), (3.7), (3.10) and (3.11) in (3.1), we get

$\displaystyle\frac{1}{2}\mathbb{E}\sup_{0\leq t\leq T}|X_{N}^{i}(t)|^{2p}$
$\displaystyle\leq\mathbb{E}|X_{N}^{i}(0)|^{2p}+C\mathbb{E}\int_{0}^{T}\Big{(}1+|X_{N}^{i}(s)|^{2p}+\frac{1}{N}\sum\limits_{i=1}^{N}|X_{N}^{i}(s)|^{2p}\Big{)}ds$

and

$\displaystyle\mathbb{E}\sup_{0\leq t\leq T}|X_{N}^{i}(t)|^{2p}$
$\displaystyle\leq
2\mathbb{E}|X_{N}^{i}(0)|^{2p}+C\mathbb{E}\int_{0}^{T}\Big{(}1+\sup_{0\leq
u\leq s}|X_{N}^{i}(u)|^{2p}+\frac{1}{N}\sum\limits_{i=1}^{N}\sup_{0\leq u\leq
s}|X_{N}^{i}(u)|^{2p}\Big{)}ds.$

Taking supremum over $\\{1,\dots,N\\}$, we obtain

$\displaystyle\sup_{i=1,\dots,N}\mathbb{E}\sup_{0\leq t\leq
T}|X_{N}^{i}(t)|^{2p}$ $\displaystyle\leq
2\sup_{i=\\{1,\dots,N\\}}\mathbb{E}|X_{N}^{i}(0)|^{2p}+C\bigg{(}1+\int_{0}^{T}\sup_{i=1,\dots,N}\mathbb{E}\sup_{0\leq
u\leq s}|X_{N}^{i}(u)|^{2p}ds\bigg{)},$

which gives our targeted result for positive integer valued $p$ by applying
Grönwall’s lemma (note that we can apply Grönwall’s lemma due to (3.3)). We
can extend the result to non-integer values of $p\geq 1$ using Holder’s
inequality. ∎

###  3.2 Well-posedness of mean-field jump-diffusion SDEs

In this section, we first introduce Wasserstein metric and state Lemma 3.4
which is crucial for establishing well-posedness of the mean-field limit.
Then, we prove existence and uniqueness of the McKean-Vlasov jump-diffusion
SDEs (2.2) in Theorem 3.5.

Let $\mathbb{D}([0,T];\mathbb{R}^{d})$ be the space of $\mathbb{R}^{d}$ valued
cádlág functions and $\mathcal{P}_{p}(\mathbb{R}^{d}),\;p\geq 1$, be the space
of probability measures on the measurable space
$(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$ such that for any
$\mu\in\mathcal{P}_{p}(\mathbb{R}^{d})$,
$\int_{\mathbb{R}^{d}}|x|^{p}\mu(dx)<\infty$, and which is equipped with the
$p$-Wasserstein metric

$\mathcal{W}_{p}(\mu,\vartheta):=\inf_{\pi\in\prod(\mu,\vartheta)}\Big{(}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|x-y|^{p}\pi(dx,dy)\Big{)}^{\frac{1}{p}},$

where $\prod(\mu,\vartheta)$ is the set of couplings of
$\mu,\vartheta\in\mathcal{P}_{p}(\mathbb{R}^{d})$ [Vil03].

Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ with
$\int_{\mathbb{R}^{d}}|x|^{2}\mu(dx)\leq K$. Then, using Jensen’s inequality,
we have

$\displaystyle
e^{-\alpha\int_{\mathbb{R}^{d}}f(x)\mu(dx)}\leq\int_{\mathbb{R}^{d}}e^{-\alpha
f(x)}\mu(dx),$

and the simple rearrangement together with Assumption 3.4, gives

$\displaystyle\frac{e^{-\alpha f_{m}}}{\int_{\mathbb{R}^{d}}e^{-\alpha
f(x)}\mu(dx)}\leq e^{\alpha(\int_{\mathbb{R}^{d}}f(x)\mu(dx)-f_{m})}\leq
e^{\alpha K_{u}\int_{\mathbb{R}^{d}}(1+|x|^{2})\mu(dx)}\leq C_{K},$ (3.12)

where $C_{K}>0$ is a constant. We will also need the following notation:

$\displaystyle\bar{X}^{\mu}=\frac{\int_{\mathbb{R}^{d}}xe^{-\alpha
f(x)}\mu(dx)}{\int_{\mathbb{R}^{d}}e^{-\alpha f(x)}\mu(dx)},$

where $\mu\in\mathcal{P}_{4}(\mathbb{R}^{d})$.

The next lemma is required for proving well-posedness of the McKean-Vlasov
SDEs (2.2). Its proof is available in [CCTT18, Lemma 3.2].

######  Lemma 3.4.

Let Assumptions 3.1, 3.3-3.5 hold and there exists a constant $K>0$ such that
$\int|x|^{4}\mu(dx)\leq K$ and $\int|y|^{4}\vartheta(dy)\leq K$ for all
$\mu,\vartheta\in\mathcal{P}_{4}(\mathbb{R}^{d})$, then the following
inequality is satisfied:

$|\bar{X}^{\mu}-\bar{X}^{\vartheta}|\leq C\mathcal{W}_{2}(\mu,\vartheta),$

where $C>0$ is independent of $\mu$ and $\vartheta$.

######  Theorem 3.5.

Let Assumptions 3.1, 3.3-3.5 hold, and let $\mathbb{E}|X(0)|^{4}<\infty$ and
$\int_{\mathbb{R}^{d}}|z|^{4}\rho_{z}(z)dz<\infty$. Then, there exists a
unique nonlinear process $X\in\mathbb{D}([0,T];\mathbb{R}^{d})$, $T>0$ which
satisfies the McKean-Vlasov SDEs (2.2) in the strong sense.

###### Proof.

Let $v\in C([0,T];\mathbb{R}^{d})$. Consider the following SDEs:

$\displaystyle dX_{v}(t)$
$\displaystyle=-\beta(t)(X_{v}(t)-v(t))dt+\sigma(t)\operatorname{Diag}(X_{v}(t)-v(t))dW(t)$
$\displaystyle\;\;\;\;+\gamma(t)\int_{\mathbb{R}^{d}}\operatorname{Diag}(X_{v}(t^{-})-v(t)))z\mathcal{N}(dt,dz)$
(3.13)

for any $t\in[0,T]$.

Note that $v(t)$ is a deterministic function of $t$, therefore the
coefficients of SDEs (3.13) only depend on $x$ and $t$. The coefficients are
globally Lipschitz continuous and have linear growth in $x$. The existence and
uniqueness of a process $X_{v}\in\mathbb{D}([0,T];\mathbb{R}^{d})$ satisfying
SDEs with Lévy noise (3.13) follows from [App04, pp. 311-312]. We also have
$\int_{\mathbb{R}^{d}}|x|^{4}\mathcal{L}_{X_{v}(t)}(dx)=\mathbb{E}|X_{v}(t)|^{4}\leq\sup_{t\in[0,T]}\mathbb{E}|X_{v}(t)|^{4}\leq
K$, where $K$ is a positive constant depending on $v$ and $T$, and
$\mathcal{L}_{X_{v}(t)}$ represents the law of $X_{v}(t)$.

We define a mapping

$\displaystyle\mathbb{T}:C([0,T];\mathbb{R}^{d})\rightarrow
C([0,T];\mathbb{R}^{d}),\;\;\mathbb{T}(v)=\bar{X}_{v},$ (3.14)

where

$\displaystyle\mathbb{T}v(t)$
$\displaystyle=\bar{X}_{v}(t)=\mathbb{E}(X_{v}(t)e^{-\alpha
f(X_{v}(t))})\Big{/}\mathbb{E}(e^{-\alpha f(X_{v}(t))})$
$\displaystyle=\int_{\mathbb{R}^{d}}xe^{-\alpha
f(x)}\mathcal{L}_{X_{v}(t)}(dx)\bigg{/}\int_{\mathbb{R}^{d}}e^{-\alpha
f(x)}\mathcal{L}_{X_{v}(t)}(dx)=\bar{X}^{\mathcal{L}_{X_{v}(t)}}(t).$

Let $\delta\in(0,1)$. For all $t,t+\delta\in(0,T)$, Ito’s isometry provides

$\displaystyle\mathbb{E}|X_{v}(t+\delta)-X_{v}(t)|^{2}$ $\displaystyle\leq
C\int_{t}^{t+\delta}\mathbb{E}|X_{v}(s)-v(s)|^{2}ds$
$\displaystyle\;\;\;\;+\int_{t}^{t+\delta}\int_{\mathbb{R}^{d}}\mathbb{E}|X_{v}(s)-v(s)|^{2}|z|^{2}\rho(z)dzds\leq
C\delta,$ (3.15)

where $C$ is a positive constant independent of $\delta$. Using Lemma 3.4 and
(3.15), we obtain

$\displaystyle|\bar{X}_{v}(t+\delta)-\bar{X}_{v}(t)|$
$\displaystyle=|\bar{X}^{\mathcal{L}_{X_{v}(t+\delta)}}(t+\delta)-\bar{X}^{\mathcal{L}_{X_{v}(t)}}(t)|\leq
C\mathcal{W}_{2}(\mathcal{L}_{X_{v}(t+\delta)},\mathcal{L}_{X_{v}(t)})$
$\displaystyle\leq
C\big{(}\mathbb{E}|X_{v}(t+\delta)-X_{v}(t)|^{2}\big{)}^{1/2}\leq
C|\delta|^{1/2},$

where $C$ is a positive constant independent $\delta$. This implies the Hölder
continuity of the map $t\rightarrow\bar{X}_{v}(t)$. Therefore, the compactness
of $\mathbb{T}$ follows from the compact embedding
$C^{0,\frac{1}{2}}([0,T];\mathbb{R}^{d})\hookrightarrow
C([0,T];\mathbb{R}^{d})$.

Using Ito’s isometry, we have

$\displaystyle\mathbb{E}|X_{v}(t)|^{2}$ $\displaystyle\leq
4\bigg{(}\mathbb{E}|X_{v}(0)|^{2}+\mathbb{E}\bigg{|}\int_{0}^{t}\beta(s)(X_{v}(s)-v(s))ds\bigg{|}^{2}+\mathbb{E}\bigg{|}\int_{0}^{t}\sigma(s)\operatorname{Diag}(X_{v}(s)-v(s))dW(s)\bigg{|}^{2}$
$\displaystyle\;\;\;\;+\mathbb{E}\bigg{|}\int_{0}^{t}\gamma(s)\operatorname{Diag}(X_{v}(s^{-})-v(s))z\mathcal{N}(ds,dz)\bigg{|}^{2}\bigg{)}$
$\displaystyle\leq
C\bigg{(}1+\int_{0}^{t}\mathbb{E}|X_{v}(s)-v(s)|^{2}ds\bigg{)}\leq
C\bigg{(}1+\int_{0}^{t}(\mathbb{E}|X_{v}(s)|^{2}+|v(s)|^{2})ds\bigg{)},$
(3.16)

where $C$ is a positive constant independent of $v$. Moreover, we have the
following result under Assumptions 3.1, 3.3-3.5 [CCTT18, Lemma 3.3]:

$\displaystyle|\bar{X}_{v}(t)|^{2}\leq L_{1}+L_{2}\mathbb{E}|X_{v}(t)|^{2},$
(3.17)

where $L_{1}$ and $L_{2}$ are from (3.4). Consider a set $\mathcal{S}=\\{v\in
C([0,T];\mathbb{R}^{d}):v=\epsilon\mathbb{T}v,\;0\leq\epsilon\leq 1\\}$. The
set $\mathcal{S}$ is non-empty due to the fact that $\mathbb{T}$ is compact
(see the remark after Theorem 10.3 in [GT83]). Therefore, for any
$v\in\mathcal{S}$, we have the corresponding unique process
$X_{v}(t)\in\mathbb{D}([0,T];\mathbb{R}^{d})$ satisfying (3.13), and
$\mathcal{L}_{X_{v}(t)}$ represents the law of $X_{v}(t)$, such that the
following holds due to (3.17):

$\displaystyle|v(s)|^{2}=\epsilon^{2}|\mathbb{T}v(s)|^{2}=\epsilon^{2}|\bar{X}_{v}(s)|^{2}\leq\epsilon^{2}\big{(}L_{1}+L_{2}\mathbb{E}|X(s)|^{2})$
(3.18)

for all $s\in[0,T]$. Substituting (3.18) in (3.16), we get

$\displaystyle\mathbb{E}|X_{v}(t)|^{2}\leq
C\bigg{(}1+\int_{0}^{t}\mathbb{E}|X_{v}(s)|^{2}ds\bigg{)},$

which on applying Grönwall’s lemma gives

$\displaystyle\mathbb{E}|X_{v}(t)|^{2}\leq C,$ (3.19)

where $C$ is independent of $v$. Due to (3.18) and (3.19), we can claim the
boundedness of the set $\mathcal{S}$. Therefore, from the Leray-Schauder
theorem [GT83, Theorem 10.3] there exists a fixed point of the mapping
$\mathbb{T}$. This proves existence of the solution of (2.2).

Let $v_{1}$ and $v_{2}$ be two fixed points of the mapping $\mathbb{T}$ and
let us denote the corresponding solutions of (3.13) as $X_{v_{1}}$ and
$X_{v_{2}}$. Using Ito’s isometry, we can get

$\displaystyle\mathbb{E}|X_{v_{1}}(t)-X_{v_{2}}(t)|^{2}\leq\mathbb{E}|X_{v_{1}}(0)-X_{v_{2}}(0)|^{2}+C\int_{0}^{t}\big{(}\mathbb{E}|X_{v_{1}}(s)-X_{v_{2}}(s)|^{2}+|v_{1}(s)-v_{2}(s)|^{2}\big{)}ds.$
(3.20)

Note that $\mathcal{S}$ is a bounded set and by definiiton $v_{1}$ and $v_{2}$
belong to $\mathcal{S}$. Then, we can apply Lemma 3.4 to ascertain

$\displaystyle|v_{1}(s)-v_{2}(s)|^{2}=|\bar{X}_{v_{1}}(s)-\bar{X}_{v_{2}}(s)|^{2}\leq
C\mathcal{W}_{2}(\mathcal{L}_{X_{v_{1}}(s)},\mathcal{L}_{X_{v_{2}}(s)})\leq
C\mathbb{E}|X_{v_{1}}(s)-X_{v_{2}}(s)|^{2}.$

Using the above estimate, Grönwall’s lemma and the fact
$X_{v_{1}}(0)=X_{v_{2}}(0)$ in (3.20), we get uniqueness of the solution of
(2.2). ∎

######  Theorem 3.6.

Let Assumptions 3.1, 3.3-3.5 are satisfied. Let $p\geq 1$,
$\mathbb{E}|X(0)|^{2p}<\infty$ and $\mathbb{E}|Z|^{2p}<\infty$, then the
following holds:

$\displaystyle\mathbb{E}\sup_{0\leq t\leq T}|X(t)|^{2p}\leq K_{p},$

where $X(t)$ satisfies (2.2) and $K_{p}$ is a positive constant.

###### Proof.

Recall that under the assumptions of this theorem, Theorem 3.5 guarantees
existence of a strong solution of (2.2).

Let $p$ be a positive integer. Let us denote $\theta_{R}=\inf\\{s\geq
0\;;\;|X(s)|\geq R\\}$. Using Ito’s formula, we obtain

$\displaystyle|X(t)|^{2p}$
$\displaystyle=|X(0)|^{2p}-2p\int_{0}^{t}\beta(s)|X(s)|^{2p-2}\big{(}X(s)\cdot(X(s)-\bar{X}(s))\big{)}ds$
$\displaystyle\;\;\;\;+2\sqrt{2}p\int_{0}^{t}\sigma(s)|X(s)|^{2p-2}\big{(}X(s)\cdot(\operatorname{Diag}(X(s)-\bar{X}(s))dW(s))\big{)}$
$\displaystyle\;\;\;\;+4p(p-1)\int_{0}^{t}\sigma^{2}(s)|X(s)|^{2p-4}|\operatorname{Diag}(X(s)-\bar{X}(s))X(s)|^{2}ds$
$\displaystyle\;\;\;\;+2p\int_{0}^{t}\sigma^{2}(s)|X(s)|^{2p-2}|\operatorname{Diag}(X(s)-\bar{X}(s))|^{2}ds$
$\displaystyle\;\;\;\;+\int_{0}^{t}\int_{\mathbb{R}^{d}}(|X(s^{-})+\gamma(s)\operatorname{Diag}(X(s^{-})-\bar{X}(s^{-}))z|^{2p}-|X(s^{-})|^{2p})\mathcal{N}(ds,dz).$

First taking suprema over $0\leq t\leq T\wedge\theta_{R}$ and then taking
expectation on both sides, we get

$\displaystyle\mathbb{E}$ $\displaystyle\sup_{0\leq t\leq
T\wedge\theta_{R}}|X(t)|^{2p}\leq\mathbb{E}|X(0)|^{2p}+C\mathbb{E}\int_{0}^{T\wedge\theta_{R}}|X(s)|^{2p-2}\big{|}X(s)\cdot(X(s)-\bar{X}(s))\big{|}ds$
$\displaystyle\;\;\;\;+C\mathbb{E}\sup_{0\leq t\leq
T\wedge\theta_{R}}\bigg{|}\int_{0}^{t}|X(s)|^{2p-2}\big{(}X(s)\cdot(\operatorname{Diag}(X(s)-\bar{X}(s))dW(s))\big{)}\bigg{|}$
$\displaystyle\;\;\;\;+C\mathbb{E}\int_{0}^{T\wedge\theta_{R}}|X(s)|^{2p-4}|\operatorname{Diag}(X(s)-\bar{X}(s))X(s)|^{2}ds$
$\displaystyle\;\;\;\;+C\mathbb{E}\int_{0}^{T\wedge\theta_{R}}|X(s)|^{2p-2}|\operatorname{Diag}(X(s)-\bar{X}(s))|^{2}ds$
$\displaystyle\;\;\;\;+\mathbb{E}\sup_{0\leq t\leq
T\wedge\theta_{R}}\int_{0}^{t}\int_{\mathbb{R}^{d}}(|X(s^{-})+\gamma(s)\operatorname{Diag}(X(s^{-})-\bar{X}(s^{-}))z|^{2p}-|X(s^{-})|^{2p})\mathcal{N}(ds,dz).$
(3.21)

To deal with the second term in (3.21), we use Young’s inequality and
ascertain

$\displaystyle|X(s)|^{2p-2}\big{|}X(s)\cdot(X(s)-\bar{X}(s))\big{|}\leq|X(s)|^{2p}+|X(s)|^{2p-1}|\bar{X}(s)|$
$\displaystyle\leq\frac{4p-1}{2p}|X(s)|^{2p}+\frac{1}{2p}|\bar{X}(s)|^{2p}\leq
C(|X(s)|^{2p}+|\bar{X}(s)|^{2p}).$ (3.22)

Using Burkholder-Davis-Gundy inequality, we have

$\displaystyle\mathbb{E}$ $\displaystyle\sup_{0\leq t\leq
T\wedge\theta_{R}}\bigg{|}\int_{0}^{t}|X(s)|^{2p-2}\big{(}X(s)\cdot(\operatorname{Diag}(X(s)-\bar{X}(s))dW(s))\big{)}\bigg{|}$
$\displaystyle\leq\mathbb{E}\bigg{(}\int_{0}^{T\wedge\theta_{R}}|X(s)|^{4p-2}|X(s)-\bar{X}(s)|^{2}ds\bigg{)}^{1/2}$
$\displaystyle\leq\mathbb{E}\Bigg{(}\sup_{0\leq t\leq
T\wedge\theta_{R}}|X(t)|^{2p-1}\bigg{(}\int_{0}^{T\wedge\theta_{R}}|X(s)-\bar{X}(s))|^{2}ds\bigg{)}^{1/2}\Bigg{)}.$
(3.23)

We apply generalized Young’s inequality $\big{(}ab\leq(\epsilon
a^{q_{1}})/q_{1}+b^{q_{2}}/(\epsilon^{q_{2}/q_{1}}q_{2}),\;\epsilon,q_{1},q_{2}>0,1/q_{1}+1/q_{2}=1$)
and Holder’s inequality on the right hand side of (3.23) to get

$\displaystyle\mathbb{E}$ $\displaystyle\sup_{0\leq t\leq
T\wedge\theta_{R}}\bigg{|}\int_{0}^{t}|X(s)|^{2p-2}\big{(}X(s)\cdot\operatorname{Diag}(X(s)-\bar{X}(s))dW(s)\big{)}\bigg{|}$
$\displaystyle\leq\frac{1}{2}\mathbb{E}\sup_{0\leq t\leq
T\wedge\theta_{R}}|X(t)|^{2p}+C\mathbb{E}\bigg{(}\int_{0}^{T\wedge\theta_{R}}|X(s)-\bar{X}(s)|^{2}ds\bigg{)}^{p}$
$\displaystyle\leq\frac{1}{2}\mathbb{E}\sup_{0\leq t\leq
T\wedge\theta_{R}}|X(t)|^{2p}+C\mathbb{E}\bigg{(}\int_{0}^{T\wedge\theta_{R}}|X(s)-\bar{X}(s)|^{2p}ds\bigg{)}$
$\displaystyle\leq\frac{1}{2}\mathbb{E}\sup_{0\leq t\leq
T\wedge\theta_{R}}|X(t)|^{2p}+C\mathbb{E}\bigg{(}\int_{0}^{T\wedge\theta_{R}}\big{(}|X(s)|^{2p}+|\bar{X}(s)|^{2p}\big{)}ds\bigg{)}.$
(3.24)

We have the following estimate to use in the fourth term in (3.21):

$\displaystyle|X(s)|^{2p-4}$
$\displaystyle|\operatorname{Diag}(X(s)-\bar{X}(s))X(s)|^{2}\leq|X(s)|^{2p-4}(|X(s)|^{2}+(X(s)\cdot\bar{X}(s)))^{2}$
$\displaystyle\leq 2|X(s)|^{2p}+2|X(s)|^{2p-2}|\bar{X}(s)|^{2}\leq
C\big{(}|X(s)|^{2p}+|\bar{X}(s)|^{2p}\big{)}.$ (3.25)

We make use of Minkowski’s inequality to get

$\displaystyle|X(s)|^{2p-2}|\operatorname{Diag}(X(s)-\bar{X}(s))|^{2}=|X(s)|^{2p-2}|X(s)-\bar{X}(s)|^{2}\leq
2|X(s)|^{2p}+2|X(s)|^{2p-2}|\bar{X}(s)|^{2},$

then Young’s inequality implies

$\displaystyle|X(s)|^{2p-2}|X(s)-\bar{X}(s)|^{2}\leq
C(|X(s)|^{2p}+|\bar{X}(s)|^{2p}).$ (3.26)

Now, we find an estimate for the last term in (3.21). Using the Cauchy-
Bunyakowsky-Schwartz inequality, we obtain

$\displaystyle\mathbb{E}$ $\displaystyle\sup_{0\leq t\leq
T\wedge\theta_{R}}\int_{0}^{t}\int_{\mathbb{R}^{d}}(|X(s^{-})+\gamma(s)\operatorname{Diag}(X(s^{-})-\bar{X}(s^{-}))z|^{2p}-|X(s^{-})|^{2p})\mathcal{N}(ds,dz)$
$\displaystyle\leq\mathbb{E}\sup_{0\leq t\leq
T\wedge\theta_{R}}\int_{0}^{t}\int_{\mathbb{R}^{d}}2^{2p-1}(|X(s^{-})|^{2p}+|\gamma(s)\operatorname{Diag}(X(s^{-})-\bar{X}(s^{-}))z|^{2p})-|X(s^{-})|^{2p}\mathcal{N}(ds,dz)$
$\displaystyle\leq
C\mathbb{E}\int_{0}^{T\wedge\theta_{R}}\int_{\mathbb{R}^{d}}(|X(s^{-})|^{2p}+|\gamma(s)\operatorname{Diag}(X(s^{-})-\bar{X}(s^{-}))z|^{2p})\mathcal{N}(ds,dz).$

Using Doob’s optional stopping theorem [App04, Theorem 2.2.1], we get

$\displaystyle\mathbb{E}$ $\displaystyle\sup_{0\leq t\leq
T\wedge\theta_{R}}\int_{0}^{t}\int_{\mathbb{R}^{d}}(|X(s^{-})+\gamma(s)\operatorname{Diag}(X(s^{-})-\bar{X}(s^{-}))z|^{2p}-|X(s^{-})|^{2p})\mathcal{N}(ds,dz)$
$\displaystyle\leq
C\mathbb{E}\int_{0}^{T\wedge\theta_{R}}\int_{\mathbb{R}^{d}}(|X(s)|^{2p}+|\gamma(s)\operatorname{Diag}(X(s)-\bar{X}(s))z|^{2p})\rho_{z}(z)dzds$
$\displaystyle\leq
C\mathbb{E}\int_{0}^{T\wedge\theta_{R}}\Big{(}|X(s)|^{2p}+|\bar{X}(s)|^{2p}\Big{)}\Big{(}1+\int_{\mathbb{R}^{d}}|z|^{2p}\rho_{z}(z)dz\Big{)}ds$
$\displaystyle\leq
C\mathbb{E}\int_{0}^{T\wedge\theta_{R}}\big{(}|X(s)|^{2p}+|\bar{X}(s)|^{2p}\big{)}ds.$
(3.27)

We have the following result under Assumptions 3.1, 3.3-3.5 [CCTT18, Lemma
3.3]:

$\displaystyle|\bar{X}(s)|^{2}\leq L_{1}+L_{2}\mathbb{E}|X(s)|^{2},$ (3.28)

where $L_{1}$ and $L_{2}$ are from (3.4).

Substituting (3.22), (3.24)-(3.28) in (3.21), using Holder’s inequality, we
arrive at the following bound:

$\displaystyle\mathbb{E}\sup_{0\leq t\leq T\wedge\theta_{R}}|X(t)|^{2p}$
$\displaystyle\leq
2\mathbb{E}|X(0)|^{2p}+C\mathbb{E}\int_{0}^{T\wedge\theta_{R}}(|X(s)|^{2p}+|\bar{X}(s)|^{2p})ds$
$\displaystyle\leq
C+C\mathbb{E}\int_{0}^{T\wedge\theta_{R}}(1+|X(s)|^{2p}+\mathbb{E}|X(s)|^{2p})ds$
$\displaystyle\leq C+C\int_{0}^{T}\mathbb{E}\sup_{0\leq u\leq
s\wedge\theta_{R}}|X(u)|^{2p}ds,$

which on using Grönwall’s lemma gives

$\displaystyle\mathbb{E}\sup_{0\leq t\leq T\wedge\theta_{R}}|X(t)|^{2p}\leq
C,$

where $C$ is independent of $R$. Then, tending $R\rightarrow\infty$ and
applying Fatau’s lemma give the desired result. ∎

##  4 Convergence results

In Section 4.1, we prove the convergence of $X(t)$, which is the mean field
limit of the particle system (2.2), towards global minimizer. This convergence
proof is based on the Laplace principle. Our approach in Section 4.1 is
similar to [CJLZ21, Appendix A]. The main result (Theorem 4.3) of Section 4.1
differs from [CJLZ21] in three respects. First, in our model (2.2), the
parameters are time-dependent. Second, we need to treat the jump part of
(2.2). Third, the analysis in [CJLZ21] is done for quadratic loss function but
the assumptions that we impose on the objective function here are less
restrictive. In Section 4.2, we prove convergence of the interacting particle
system (2.2) towards the mean-field limit (2.2) as $N\rightarrow\infty$. In
Section 4.3, we prove uniform in $N$ convergence of the Euler scheme (2.2.3)
to (2.2) as $h\rightarrow 0$, where $h$ is the discretization step.

###  4.1 Convergence towards the global minimum

The aim of this section is to show that the non-linear process $X(t)$ driven
by the distribution dependent SDEs (2.2) converges to a point $x^{*}$ which
lies in a close vicinity of the global minimum which we denote as $x_{\min}$.
To this end, we will first prove that
$\operatorname{Var}(t):=\mathbb{E}|X(t)-\mathbb{E}(X(t))|^{2}$ satisfies a
differential inequality which, with particular choice of parameters, implies
exponential decay of $\operatorname{Var}(t)$ as $t\rightarrow\infty$. We also
obtain a differential inequality for $M(t):=\mathbb{E}\big{(}e^{-\alpha
f(X(t))}\big{)}$.

The approach that we follow in this section is along the lines of [CCTT18,
CJLZ21] but with necessary adjustments for the jump term in (2.2).

######  Lemma 4.1.

Under Assumptions 3.1, 3.3-3.5, the following inequality is satisfied for
$\operatorname{Var}(t)$:

$\displaystyle\frac{d}{dt}\operatorname{Var}(t)$
$\displaystyle\leq-\bigg{(}2\beta(t)-\big{(}2\sigma^{2}(t)+\lambda\gamma^{2}(t)\mathbb{E}|\text{\ooalign{\raisebox{0.86108pt}{--}\cr$Z$\cr}}|^{2}\big{)}\Big{(}1+\frac{e^{-\alpha
f_{m}}}{M(t)}\Big{)}\bigg{)}\operatorname{Var}(t).$ (4.3)

###### Proof.

Using Ito’s formula, we have

$\displaystyle|X(t)$
$\displaystyle-\mathbb{E}X(t)|^{2}=|X(0)-\mathbb{E}X(0)|^{2}-2\int_{0}^{t}\beta(s)(X(s)-\mathbb{E}X(s))\cdot(X(s)-\bar{X}(s))ds$
$\displaystyle-2\int_{0}^{t}(X(s)-\mathbb{E}X(s))\cdot
d\mathbb{E}X(s)+2\int_{0}^{t}\sigma^{2}(s)|X(s)-\bar{X}(s)|^{2}ds$
$\displaystyle+2\sqrt{2}\int_{0}^{t}\sigma(s)(X(s)-\mathbb{E}X(s))\cdot\big{(}\operatorname{Diag}(X(s)-\bar{X}(s))dW(s)\big{)}$
$\displaystyle+\int_{0}^{t}\int_{\mathbb{R}^{d}}\big{\\{}|X(s^{-})-\mathbb{E}X(s^{-})+\gamma(s)\operatorname{Diag}(X(s^{-})-\bar{X}(s^{-}))z|^{2}-|X(s^{-})-\mathbb{E}(X(s^{-}))|^{2}\big{\\}}\mathcal{N}(ds,dz).$

Taking expectation on both sides, we get

$\displaystyle\operatorname{Var}(t)=\operatorname{Var}(0)-2\mathbb{E}\int_{0}^{t}\beta(s)\mathbb{E}\big{(}(X(s)-\mathbb{E}X(s))\cdot(X(s)-\bar{X}(s))\big{)}dt+2\int_{0}^{t}\sigma^{2}(s)\mathbb{E}|X(s)-\bar{X}(s)|^{2}ds$
$\displaystyle\;\;\;\;+\lambda\gamma^{2}(s)\int_{0}^{t}\int_{\mathbb{R}^{d}}\mathbb{E}|\operatorname{Diag}(X(s)-\bar{X}(s))z|^{2}\rho_{z}(z)dzds$
$\displaystyle=\operatorname{Var}(0)-2\int_{0}^{t}\big{(}\beta(s)\operatorname{Var}(s)+2\sigma^{2}(s)\mathbb{E}|X(s)-\bar{X}(s)|^{2}+\lambda\gamma^{2}(s)\mathbb{E}|\text{\ooalign{\raisebox{0.86108pt}{--}\cr$Z$\cr}}|^{2}\mathbb{E}|X(s)-\bar{X}(s)|^{2}\big{)}ds,$
(4.6)

since

$\displaystyle\mathbb{E}\big{(}(X(t)-\mathbb{E}X(t))\cdot(\mathbb{E}X(t)-\bar{X}(t))\big{)}=0,$
$\displaystyle|X(t)-\mathbb{E}X(t)+\operatorname{Diag}(X(t)-\bar{X}(t))z|^{2}=|X(t)-\mathbb{E}X(t)|^{2}+|\operatorname{Diag}(X(t)-\bar{X}(t))z|^{2}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+2\big{(}(X(t)-\mathbb{E}X(t))\cdot\operatorname{Diag}(X(t)-\bar{X}(t))z\big{)},$
$\displaystyle\int_{\mathbb{R}^{d}}\big{(}(X(t)-\mathbb{E}X(t))\cdot\operatorname{Diag}(X(t)-\bar{X}(t))z\big{)}\rho_{z}(z)dz=0.$

Moreover,
$\int_{\mathbb{R}^{d}}\sum_{l=1}^{d}(X(t)-\bar{X}(t))_{l}^{2}z_{l}^{2}\rho_{z}(z)dz=\sum_{l=1}^{d}(X(t)-\bar{X}(t))_{l}^{2}\int_{\mathbb{R}{{}^{d}}}z_{l}^{2}\prod_{i=1}^{d}\rho_{\text{\ooalign{-\kern-2.10002pt-\cr$z$\cr}}}(z_{i})dz=|X(t)-\bar{X}(t)|^{2}\mathbb{E}|\text{\ooalign{\raisebox{0.86108pt}{--}\cr$Z$\cr}}|^{2}$,
since each component $Z_{l}$ of $Z$ is distributed as  – $Z$ .

We also have

$\displaystyle\mathbb{E}|X(t)-\bar{X}(t)|^{2}=\operatorname{Var}(t)+|\mathbb{E}X(t)-\bar{X}(t)|^{2}.$
(4.7)

We estimate the term $|\mathbb{E}(X(t))-\bar{X}(t)|^{2}$ using Jensen’s
inequality as

$\displaystyle|\mathbb{E}X(t)-\bar{X}(t)|^{2}$
$\displaystyle=\bigg{|}\mathbb{E}X(t)-\frac{\mathbb{E}X(t)e^{-\alpha
f(X(t))}}{\mathbb{E}e^{-\alpha
f(X(t))}}\bigg{|}^{2}=\bigg{|}\mathbb{E}\bigg{(}\Big{(}\mathbb{E}X(t)-X(t)\Big{)}\frac{e^{-\alpha
f(X(t))}}{\mathbb{E}e^{-\alpha f(X(t))}}\bigg{)}\bigg{|}^{2}$
$\displaystyle=\bigg{|}\int_{\mathbb{R}^{d}}\big{(}\mathbb{E}X(t)-x\big{)}\vartheta_{X(t)}(dx)\bigg{|}^{2}\leq\int_{\mathbb{R}^{d}}\big{|}\mathbb{E}X(t)-x\big{|}^{2}\vartheta_{X(t)}(dx)$
$\displaystyle=\mathbb{E}\bigg{(}|X(t)-\mathbb{E}(X(t))|^{2}\frac{e^{-\alpha
f(X(t))}}{\mathbb{E}e^{-\alpha f(X(t))}}\bigg{)}\leq\frac{e^{-\alpha
f_{m}}}{\operatorname{M}(t)}\operatorname{Var}(t),$ (4.8)

where $\vartheta_{X(t)}(dx)=e^{-\alpha f(x)}/\mathbb{E}(e^{-\alpha
f(X(t))})\mathcal{L}_{X(t)}(dx)$ which implies
$\int_{\mathbb{R}^{d}}\vartheta_{X(t)}(dx)=1$. Using (4.7) and (4.1) in (4.6)
gives the targeted result. ∎

To prove the main result of this section, we need an additional inequality,
which is proved under the following assumption.

######  Assumption 4.1.

$f\in C^{2}(\mathbb{R}^{d})$ and there exist three constants
$K_{1}$,$K_{2},K_{3}>0$ such that the following inequalities are satisfied for
sufficiently large $\alpha$:

  * (i)

$(\nabla f(x)-\nabla f(y))\cdot(x-y)\geq-K_{1}|x-y|^{2}$ for all $x$,
$y\in\mathbb{R}^{d}$.

  * (ii)

$\alpha\Big{(}\frac{\partial f}{\partial
x_{i}}\Big{)}^{2}-\frac{\partial^{2}f}{\partial x_{i}^{2}}\geq-K_{2}$ for all
$i=1,\dots,d$ and $x\in\mathbb{R}^{d}$.

  * (iii)

$\mathbb{E}f(x+\operatorname{Diag}(x)Z)-f(x)\leq
K_{3}|x|^{2}\mathbb{E}|\text{\ooalign{\raisebox{0.86108pt}{--}\cr$Z$\cr}}|^{2}$,
where $Z$ is a d-dimensional random vector and  – $Z$ is real valued random
variable introduced in Section 2.2.

We note that for $f(x)=1+|x|^{2}$, $x\in\mathbb{R}^{d}$, we have
$\mathbb{E}|x+\operatorname{Diag}(x)Z|^{2}-|x|^{2}=\mathbb{E}|\operatorname{Diag}(x)Z|^{2}=\sum_{l=1}^{d}\mathbb{E}(x_{l}Z_{l})^{2}$.
However, each $Z_{l}$ is distributed as  – $Z$ . Hence,
$\mathbb{E}|x+\operatorname{Diag}(x)Z|^{2}-|x|^{2}=|x|^{2}\mathbb{E}|\text{\ooalign{\raisebox{0.86108pt}{--}\cr$Z$\cr}}|^{2}$.
The conditions $(i)$ and $(ii)$ are straightforward to verify for $1+|x|^{2}$.
This implies the existence of a function satisfying the above assumption. This
ensures that the class of functions satisfying the above assumption is not
empty and is consistent with Assumptions 3.1, 3.3-3.5. The most important
implication is that the above assumption allows $f$ to have quadratic growth
which is important for several loss functions in machine learning problems.

In [CCTT18], the authors assumed $f\in C^{2}(\mathbb{R}^{d})$, the norm of
Hessian of $f$ being bounded by a constant, and the norm of gradient and
Laplacian of $f$ satisfying the inequality, $\Delta f\leq c_{0}+c_{1}|\nabla
f|^{2}$, where $c_{0}$ and $c_{1}$ are positive constants. Therefore, in
Assumption 4.1, we have imposed restrictions on $f$ similar to [CCTT18] in the
essence of regularity but adapted to our jump-diffusion case with component-
wise Wiener noise.

######  Lemma 4.2.

The following inequality holds under Assumptions 3.1, 3.3-3.5 and 4.1:

$\displaystyle\frac{d}{dt}\operatorname{M}^{2}(t)$ $\displaystyle\geq-4\alpha
e^{-\alpha
f_{m}}\Big{(}\beta(t)K_{1}+\sigma^{2}(t)K_{2}+\lambda\gamma^{2}(t)K_{3}\mathbb{E}|\text{\ooalign{\raisebox{0.86108pt}{--}\cr$Z$\cr}}|^{2}\Big{)}\operatorname{Var}(t).$
(4.11)

###### Proof.

Using Ito’s formula, we get

$\displaystyle e^{-\alpha f(X(t))}$
$\displaystyle=\int_{0}^{t}\alpha\beta(s)e^{-\alpha f(X(s))}\nabla
f(X(s))\cdot(X(s)-\bar{X}(s))ds$
$\displaystyle\;\;\;\;-\sqrt{2}\int_{0}^{t}\alpha\sigma(s)e^{-\alpha
f(X(s))}\nabla
f(X(s))\cdot\big{(}\operatorname{Diag}(X(s)-\bar{X}(s))dW(s)\big{)}$
$\displaystyle\;\;\;\;+\int_{0}^{t}\sigma^{2}(s)e^{-\alpha
f(X(s))}\sum\limits_{j=1}^{d}\bigg{(}\big{(}X(s)-\bar{X}(s)\big{)}^{2}_{j}\Big{(}\alpha^{2}\Big{(}\frac{\partial
f(X(s))}{\partial x_{j}}\Big{)}^{2}-\alpha\frac{\partial^{2}f(X(s))}{\partial
x_{j}^{2}}\Big{)}\bigg{)}ds$
$\displaystyle\;\;\;\;+\int_{0}^{t}\int_{\mathbb{R}^{d}}\Big{(}e^{-\alpha
f(X(s^{-})+\gamma(s)\operatorname{Diag}(X(s^{-})-\bar{X}(s^{-}))z)}-e^{-\alpha
f(X(s^{-}))}\Big{)}\mathcal{N}(ds,dz).$

Taking expectation on both sides and writing in the differential form yield

$\displaystyle d\mathbb{E}e^{-\alpha f(X(t))}$
$\displaystyle=\alpha\beta(t)\mathbb{E}\big{(}e^{-\alpha f(X(t))}(\nabla
f(X(t))-\nabla f(\bar{X}(t)))\cdot(X(t)-\bar{X}(t))\big{)}dt$
$\displaystyle+\sigma^{2}(t)\mathbb{E}\Bigg{(}e^{-\alpha
f(X(t))}\sum\limits_{j=1}^{d}\bigg{(}\big{(}X(t)-\bar{X}(t)\big{)}^{2}_{j}\Big{(}\alpha^{2}\Big{(}\frac{\partial
f(X(t))}{\partial x_{j}}\Big{)}^{2}-\alpha\frac{\partial^{2}f(X(t))}{\partial
x_{j}^{2}}\Big{)}\bigg{)}\Bigg{)}dt$
$\displaystyle+\lambda\int_{\mathbb{R}^{d}}\mathbb{E}\Big{(}e^{-\alpha
f(X(t)+\gamma(t)\operatorname{Diag}(X(t)-\bar{X}(t))z)}-e^{-\alpha
f(X(t))}\Big{)}\rho_{z}(z)dzdt,$

where we have used the fact $\mathbb{E}\big{[}e^{-\alpha f(X(t))}(\nabla
f(\bar{X}(t))\cdot(X(t)-\bar{X}(t)))\big{]}=0$.

Note that $|e^{-\alpha f(x)}-e^{-\alpha f(y)}|\leq\alpha e^{-\alpha
f_{m}}|f(x)-f(y)|$ which means $e^{-\alpha f(x)}-e^{-\alpha f(y)}\geq-\alpha
e^{-\alpha f_{m}}|f(x)-f(y)|$. Using Assumption 4.1, we get

$\displaystyle d\mathbb{E}e^{-\alpha f(X(t))}\geq-\alpha e^{-\alpha
f_{m}}\big{(}\beta(t)K_{1}+\sigma^{2}(t)K_{2}+\lambda\gamma^{2}(t)K_{3}\mathbb{E}|\text{\ooalign{\raisebox{0.86108pt}{--}\cr$Z$\cr}}|^{2}\big{)}\mathbb{E}|X(t)-\bar{X}(t)|^{2}.$

From (4.7) and (4.1), we have

$\displaystyle\mathbb{E}|X(t)-\bar{X}(t)|^{2}\leq\operatorname{Var}(t)+\frac{e^{-\alpha
f_{m}}}{\operatorname{M}(t)}\operatorname{Var}(t)\leq 2\frac{e^{-\alpha
f_{m}}}{\operatorname{M}(t)}\operatorname{Var}(t).$

This implies

$\displaystyle d\operatorname{M}(t)\geq-2\alpha e^{-\alpha
f_{m}}\big{(}\beta(t)K_{1}+\sigma^{2}(t)K_{2}+\lambda\gamma^{2}(t)K_{3}\mathbb{E}|\text{\ooalign{\raisebox{0.86108pt}{--}\cr$Z$\cr}}|^{2}\big{)}\frac{e^{-\alpha
f_{m}}}{\operatorname{M}(t)}\operatorname{Var}(t)dt,$

which is what we aimed to prove in this lemma. ∎

Our next objective is to show that $\mathbb{E}(X(t))$ converges to $x^{*}$ as
$t\rightarrow\infty$, where $x^{*}$ is close to $x_{\min}$, i.e. the point at
which $f(x)$ attains its minimum value, $f_{m}$. Applying Laplace’s method
(see e.g. [FW12, Chap. 3] and also [PTTM17, CCTT18]), we can calculate the
following asymptotics: for any compactly supported probability measure
$\rho\in\mathcal{P}(\mathbb{R}^{d})$ with $x_{\min}\in\text{supp}(\rho)$, we
have

$\displaystyle\lim\limits_{\alpha\rightarrow\infty}\Bigg{(}-\frac{1}{\alpha}\log\bigg{(}\int_{\mathbb{R}^{d}}e^{-\alpha
f(x)}d\rho(x)\bigg{)}\Bigg{)}=f_{m}>0.$ (4.12)

Based on the above asymptotics, we aim to prove that

$\displaystyle f(x^{*})\leq
f_{m}+\Gamma(\alpha)+\mathcal{O}\bigg{(}\frac{1}{\alpha}\bigg{)},$

where a function $\Gamma(\alpha)\rightarrow 0$ as $\alpha\rightarrow\infty$.

We introduce the following function:

$\displaystyle\chi(t)=2\beta(t)-\big{(}2\sigma^{2}(t)+\lambda\gamma^{2}(t)\mathbb{E}|\text{\ooalign{\raisebox{0.86108pt}{--}\cr$Z$\cr}}|^{2}\big{)}\Big{(}1+\frac{2e^{-\alpha
f_{m}}}{M(0)}\Big{)}.$

We choose $\alpha$, $\beta(t)$, $\sigma(t)$, $\gamma(t)$, $\lambda$,
distribution of  – $Z$ such that

  * (i)

$\chi(t)$ is a continuous function of time $t$,

  * (ii)

$\chi(t)>0$ for all $t\geq 0$, and

  * (iii)

$\chi(t)$ attains its minimum which we denote as $\chi_{\min}$.

We also introduce

$\displaystyle\eta$ $\displaystyle:=4\alpha e^{-\alpha
f_{m}}\operatorname{Var}(0)\frac{K_{1}\beta+K_{2}\sigma^{2}(0)+K_{3}\lambda\gamma^{2}(0)\mathbb{E}|\text{\ooalign{\raisebox{0.86108pt}{--}\cr$Z$\cr}}|^{2}}{\operatorname{M}^{2}(0)\chi_{\min}},$

where $\beta$ is introduced in Section 2.2, and $K_{1}$, $K_{2}$ and $K_{3}$
are from Assumption 4.1.

The next theorem is the main result of this section. We will be assuming that
$\eta\leq 3/4$ which can always be achieved by choosing sufficiently small
$\operatorname{Var}(0)$.

######  Theorem 4.3.

Let Assumptions 3.1, 3.3-3.5 and 4.1 hold. Let us also assume that
$\mathcal{L}_{X(0)}$ is compactly supported and
$x_{\min}\in\text{supp}(\mathcal{L}_{X(0)})$. If $\eta\leq 3/4$, then
$\operatorname{Var}(t)$ exponentially decays to zero as $t\rightarrow\infty$.
Further, there exists an $x^{*}\in\mathbb{R}^{d}$ such that $X(t)\rightarrow
x^{*}$ a.s., $\mathbb{E}(X(t))\rightarrow x^{*}$, $\bar{X}(t)\rightarrow
x^{*}$ as $t\rightarrow\infty$ and the following inequality holds:

$\displaystyle f(x^{*})\leq f_{m}+\Gamma(\alpha)+\frac{\log{2}}{\alpha},$

where function $\Gamma(\alpha)\rightarrow 0$ as $\alpha\rightarrow\infty$.

###### Proof.

Let
$T^{*}=\sup\big{\\{}t\;;\;\operatorname{M}(s)>\frac{\operatorname{M}(0)}{2},\text{for
all}\;s\in[0,t]\big{\\}}.$ Observe that $T^{*}>0$ by definition.

Let us assume that $T^{*}<\infty$. We can deduce that the following holds by
definition of $T^{*}$ for all $t\in[0,T^{*}]$:

$\displaystyle
2\beta(t)-\big{(}2\sigma^{2}(t)+\lambda\gamma^{2}(t)\mathbb{E}|\text{\ooalign{\raisebox{0.86108pt}{--}\cr$Z$\cr}}|^{2}\big{)}\Big{(}1+\frac{e^{-\alpha
f_{m}}}{M(t)}\Big{)}\geq
2\beta(t)-\big{(}2\sigma^{2}(t)+\lambda\gamma^{2}(t)\mathbb{E}|\text{\ooalign{\raisebox{0.86108pt}{--}\cr$Z$\cr}}|^{2}\big{)}\Big{(}1+\frac{2e^{-\alpha
f_{m}}}{M(0)}\Big{)}=\chi(t),$

where the left hand side of the above inequality is from (4.3). Using Lemma
4.1, the fact that $\chi(t)$ is continuous and $\chi(t)>0$ for all $t\geq 0$,
we get for all $t\in[0,T^{*}]$:

$\displaystyle\operatorname{Var}(t)\leq\operatorname{Var}(0)e^{-\chi(t)t}\leq\operatorname{Var}(0)e^{-\chi_{\min}t}.$

We have from Lemma 4.2 for all $t\in(0,T^{*}]$:

$\displaystyle\operatorname{M}^{2}(t)$
$\displaystyle\geq\operatorname{M}^{2}(0)-4\alpha e^{-\alpha
f_{m}}\int_{0}^{t}\big{(}K_{1}\beta(s)+K_{2}\sigma^{2}(s)+K_{3}\lambda\gamma^{2}(s)\mathbb{E}|\text{\ooalign{\raisebox{0.86108pt}{--}\cr$Z$\cr}}|^{2}\big{)}\operatorname{Var}(s)ds$
$\displaystyle\geq\operatorname{M}^{2}(0)-4\alpha e^{-\alpha
f_{m}}\big{(}K_{1}\beta+K_{2}\sigma^{2}(0)+K_{3}\lambda\gamma^{2}(0)\mathbb{E}|\text{\ooalign{\raisebox{0.86108pt}{--}\cr$Z$\cr}}|^{2}\big{)}\operatorname{Var}(0)\int_{0}^{t}e^{-\chi_{\min}s}ds$
$\displaystyle=\operatorname{M}^{2}(0)-4\alpha e^{-\alpha
f_{m}}\big{(}K_{1}\beta+K_{2}\sigma^{2}(0)+K_{3}\lambda\gamma^{2}(0)\mathbb{E}|\text{\ooalign{\raisebox{0.86108pt}{--}\cr$Z$\cr}}|^{2}\big{)}\frac{\operatorname{Var}(0)}{\chi_{\min}}\big{(}1-e^{-\chi_{\min}t}\big{)}$
$\displaystyle>\operatorname{M}^{2}(0)-4\alpha e^{-\alpha
f_{m}}\big{(}K_{1}\beta+K_{2}\sigma^{2}(0)+K_{3}\lambda\gamma^{2}(0)\mathbb{E}|\text{\ooalign{\raisebox{0.86108pt}{--}\cr$Z$\cr}}|^{2}\big{)}\frac{\operatorname{Var}(0)}{\chi_{\min}}\geq\frac{\operatorname{M}^{2}(0)}{4},$

where in the last step we have used the fact that $\eta\leq 3/4$. This shows
$\operatorname{M}(t)>\operatorname{M}(0)/2$ which implies
$\operatorname{M}(t)-\operatorname{M}(0)/2>0$ on the set $(0,T^{*}]$. Also,
note that $M(t)$ is continuous in $t$, therefore there exists an $\epsilon>0$
such that $\operatorname{M}(t)>\operatorname{M}(0)/2$ for all
$t\in[T^{*},T^{*}+\epsilon)$. This creates a contradiction which implies
$T^{*}=\infty$. Hence,

$\operatorname{Var}(t)\leq\operatorname{Var}(0)e^{-\chi_{\min}t}\;\;\text{and}\;\;\operatorname{M}(t)>\operatorname{M}(0)/2\;\text{
for all}\;t>0.$ (4.13)

This implies $\operatorname{Var}(t)$ exponentially decays to zero as
$t\rightarrow\infty$. From (4.1) and (4.13), we get

$\displaystyle|\mathbb{E}X(t)-\bar{X}(t)|^{2}\leq e^{-\alpha
f_{m}}\frac{\operatorname{Var}(t)}{\operatorname{M}(t)}\leq
Ce^{-\chi_{\min}t},\;\;\;\;t>0,$ (4.14)

where $C$ is a positive constant independent of $t$.

Taking expectation on both sides of (2.2) (recall that
$\mathbb{E}\text{\ooalign{\raisebox{0.86108pt}{--}\cr$Z$\cr}}=0$), applying
Holder’s inequality and using (4.7) gives

$\displaystyle\bigg{|}\frac{d}{dt}\mathbb{E}X(t)\bigg{|}$
$\displaystyle\leq\beta\mathbb{E}|X(t)-\bar{X}(t)|\leq\beta(\mathbb{E}|X(t)-\bar{X}(t)|^{2})^{1/2}\leq\beta\big{(}\operatorname{Var}(t)+|\mathbb{E}X(t)-\bar{X}(t)|^{2}\big{)}^{1/2}$
$\displaystyle\leq Ce^{-\chi_{\min}t/2},\;\;\;\;t>0,$ (4.15)

where $C$ is a positive constant independent of $t$.

It is clear from (4.15) that there exists an $x^{*}\in\mathbb{R}^{d}$ such
that $\mathbb{E}(X(t))\rightarrow x^{*}$ as $t\rightarrow\infty$. Further,
$\bar{X}(t)\rightarrow x^{*}$ as $t\rightarrow\infty$ due to (4.14).

Let $\ell>0$. Using Chebyshev’s inequality, we have

$\displaystyle\mathbb{P}(|X(t)-\mathbb{E}X(t)|\geq e^{-\ell
t})\leq\frac{\operatorname{Var}{(t)}}{e^{-2\ell t}}\leq
Ce^{-(\chi_{\min}-2\ell)t},$

where $C>0$ is independent of $t$. If we choose $\ell<\chi_{\min}/2$, then we
can say $|X(t)-\mathbb{E}X(t)|\rightarrow 0$ as $t\rightarrow 0$ a.s. due to
the Borel-Cantelli lemma. This implies $X(t)\rightarrow x^{*}$ a.s.
Application of the bounded convergence theorem gives the convergence result:
$\mathbb{E}e^{-\alpha f(X(t))}\rightarrow e^{-\alpha f(x^{*})}$ as
$t\rightarrow\infty$. Then, due to (4.13), we obtain

$\displaystyle e^{-2\alpha f(x^{*})}\geq M^{2}(0)/4$

and hence

$\displaystyle
f(x^{*})\leq-\frac{1}{\alpha}\log(\operatorname{M}(0))+\frac{1}{\alpha}\log{2}.$

Then, using the asymptotics (4.12), we get

$\displaystyle f(x^{*})\leq f_{m}+\Gamma(\alpha)+\frac{1}{\alpha}\log{2},$
(4.16)

where the function $\Gamma(\alpha)\rightarrow 0$ as $\alpha\rightarrow\infty$.
∎

###  4.2 Convergence to the mean-field SDEs

In the previous section, we showed convergence of the non-linear process
$X(t)$ from (2.2) towards the global minimizer. However, the CBO method is
based on the system (2.2) of finite particles. This means there is a missing
link in the theoretical analysis which we fill in this section by showing
convergence of the particle system (2.2) to the mean-field limit in mean-
square sense (2.2) as the number of particles tends to infinity. The proof of
this result has some ingredients inspired from [MT05] (see also [MT21]),
precisely where we partition the sample space (cf. Theorem 4.7). Further, it
is clear from the proof that we need stronger moment bound result like in
Lemmas 3.3 and 3.6, as compared to [CCTT18, Lemma 3.4].

We first discuss some concepts necessary for later use in this section. We
introduce the following notation for the empirical measure of i.i.d. particles
driven by the McKean-Vlasov SDEs (2.2):

$\displaystyle\mathcal{E}_{t}:=\frac{1}{N}\sum\limits_{i=1}^{N}\delta_{X^{i}(t)},$
(4.17)

where $\delta_{x}$ is the Dirac measure at $x\in\mathbb{R}^{d}$. We will also
need the following notation:

$\displaystyle\bar{X}^{\mathcal{E}_{t}}(t)=\frac{\int_{\mathbb{R}^{d}}xe^{-\alpha
f(x)}\mathcal{E}_{t}(dx)}{\int_{\mathbb{R}^{d}}e^{-\alpha
f(x)}\mathcal{E}_{t}(dx)}=\frac{\sum_{i=1}^{N}X^{i}(t)e^{-\alpha
f(X^{i}(t))}}{\sum_{i=1}^{N}e^{-\alpha f(X^{i}(t))}}.$ (4.18)

Using discrete Jensen’s inequality, we have

$\displaystyle\exp{\bigg{(}-\alpha\frac{1}{N}\sum\limits_{i=1}^{N}f(X^{i}(t))\bigg{)}}$
$\displaystyle\leq\frac{1}{N}\sum\limits_{i=1}^{N}\exp{\Big{(}-\alpha
f(X^{i}(t))\Big{)}},$

which, on rearrangement and multiplying both sides by $e^{-\alpha f_{m}}$,
gives

$\displaystyle\frac{e^{-\alpha f_{m}}}{\frac{1}{N}\sum_{i=1}^{N}e^{-\alpha
f(X^{i}(t))}}$
$\displaystyle\leq\exp{\bigg{(}\alpha\Big{(}\frac{1}{N}\sum\limits_{i=1}^{N}f(X^{i}(t))-f_{m}\Big{)}\bigg{)}}\leq
e^{\alpha K_{u}}\exp{\Big{(}\frac{\alpha
K_{u}}{N}\sum\limits_{i=1}^{N}|X^{i}(t)|^{2}\Big{)}},$ (4.19)

where we have used Assumption 3.4 for the second inequality.

We recall that a random variable $\zeta(\omega)$ is a.s. finite if there is an
increasing sequence $\\{e_{k}\\}_{k\in\mathbb{N}}$ with
$e_{k}\rightarrow\infty$ as $k\rightarrow\infty$ such that

$\displaystyle\mathbb{P}\big{(}\cup_{k=1}^{\infty}\\{\omega\;:\;|\zeta(\omega)|<e_{k}\\}\big{)}=1,$

which means

$\displaystyle\mathbb{P}\big{(}\cap_{k=1}^{\infty}\\{\omega\;:\;|\zeta(\omega)|\geq
e_{k}\\}\big{)}=0,\;\;\;\;\text{i.e.}\;\;\;\;\;\;\mathbb{P}\big{(}\lim_{k\rightarrow\infty}\\{\omega\;:\;|\zeta(\omega)|\geq
e_{k}\\}\big{)}=0.$

Let $g(x)$ be an increasing continuous function of $x\in\mathbb{R}$ then
$g(\zeta(\omega))$ is a.s. finite random variable as well. Also, if
$\zeta_{1}(\omega)$ and $\zeta_{2}(\omega)$ are a.s. finite random variables
then $\zeta_{1}(\omega)\vee\zeta_{2}(\omega)$ is also an a.s. finite random
variable. If $\zeta(\omega)$ is a.s. finite then by continuity of probability
we have [Shi13]:

$\displaystyle\lim_{k\rightarrow\infty}\mathbb{P}(\\{\omega\;:\;|\zeta(\omega)|\geq
e_{k}\\})=0.$ (4.20)

We know that $X^{i}(t)$, governed by the McKean-Vlasov SDEs (2.2), are i.i.d.
random variables for every $t\geq 0$, therefore using Chebyshev’s inequality,
we get

$\displaystyle\mathbb{P}\Big{(}\frac{1}{N}\sum\limits_{i=1}^{N}|X^{i}(t)|^{2}-\mathbb{E}|X(t)|^{2}\geq
N^{(\epsilon-1)/4}\Big{)}\leq\frac{\mathbb{E}\Big{|}\frac{1}{N}\sum_{i=1}^{N}|X^{i}(t)|^{2}-\mathbb{E}|X(t)|^{2}\Big{|}^{4}}{N^{(\epsilon-1)}}$
$\displaystyle=\frac{\mathbb{E}\Big{|}\sum_{i=1}^{N}\big{(}|X^{i}(t)|^{2}-\mathbb{E}|X(t)|^{2}\big{)}\Big{|}^{4}}{N^{3+\epsilon}}=\frac{\sum_{i=1}^{N}\mathbb{E}U_{i}^{4}}{N^{3+\epsilon}}+\frac{\sum_{i=1}^{N}\mathbb{E}U_{i}^{2}\sum_{j=1}^{N}\mathbb{E}U_{j}^{2}}{N^{3+\epsilon}}$
$\displaystyle\leq\frac{C}{N^{1+\epsilon}},$

where we have used Lemma 3.6, $U_{i}=|X^{i}(t)|^{2}-\mathbb{E}|X(t)|^{2}$ and
$C$ is independent of $N$. We take $\epsilon\in(0,1)$ and define
$E_{N}=\left\\{\frac{1}{N}\sum\limits_{i=1}^{N}|X^{i}(t)|^{2}-\mathbb{E}|X(t)|^{2}>\frac{1}{N^{(1-\epsilon)/4}}\right\\}$
then

$\displaystyle\sum\limits_{N=1}^{\infty}\mathbb{P}(E_{N})<\infty.$

The Borel-Cantelli lemma implies that the random variable

$\displaystyle\zeta_{1}(t):=\sup_{N\in\mathbb{N}}N^{(1-\epsilon)/4}\Big{(}\frac{1}{N}\sum_{i=1}^{N}|X^{i}(t)|^{2}-\mathbb{E}|X(t)|^{2}\Big{)}$

is a.s. finite. Therefore,

$\displaystyle\frac{1}{N}\sum\limits_{i=1}^{N}|X^{i}(t)|^{2}\leq\mathbb{E}|X(t)|^{2}+\zeta_{1}(t,\omega)N^{(-1+\epsilon)/4},\;\;\;\;a.s.,$
(4.21)

for all $t\in[0,T]$. Using (4.21) in (4.19) and Lemma 3.6, we get

$\displaystyle\frac{e^{-\alpha
f_{m}}}{\frac{1}{N}\sum\limits_{i=1}^{N}e^{-\alpha f(X^{i}(t))}}\leq e^{\alpha
K_{u}(1+K_{p}+\zeta_{1}(t,\omega)N^{(-1+\epsilon)/4})},\;\;\;\;a.s.$ (4.22)

This show that

$\displaystyle\lim\limits_{N\rightarrow\infty}\frac{e^{-\alpha
f_{m}}}{\frac{1}{N}\sum\limits_{i=1}^{N}e^{-\alpha f(X^{i}(t))}}\leq e^{\alpha
K_{u}(1+K_{p})},\;\;\;\;a.s.$ (4.23)

######  Lemma 4.4.

Let Assumptions 3.1, 3.3-3.5 be satisfied. Let $\mathbb{E}|X(0)|^{4}<\infty$
and $\mathbb{E}|Z|^{4}<\infty$. Then, the following bound holds for all
$t\in[0,T]$ and sufficiently large $N$:

$\displaystyle|\bar{X}^{\mathcal{E}_{t}}(t)-\bar{X}(t)|\leq\frac{\zeta(t,\omega)}{N^{(1-\epsilon)/4}},\;\;\;\;a.s.,$
(4.24)

where $\bar{X}^{\mathcal{E}_{t}}(t)$ is from (4.18), $\bar{X}(t)$ is from
(2.13), $\zeta(t,\omega)$ is an $a.s.$ finite $\mathscr{F}_{t}-$ measurable
random variable and $\epsilon\in(0,1)$.

###### Proof.

We have

$\displaystyle|\bar{X}^{\mathcal{E}_{t}}(t)$
$\displaystyle-\bar{X}(t)|=\bigg{|}\sum_{i=1}^{N}X^{i}(t)\frac{e^{-\alpha
f(X^{i}(t))}}{\sum_{j=1}^{N}e^{-\alpha
f(X^{j}(t))}}-\int_{\mathbb{R}^{d}}x\frac{e^{-\alpha
f(x)}}{\int_{\mathbb{R}^{d}}e^{-\alpha
f(x)}\mathcal{L}_{X(t)}(dx)}\mathcal{L}_{X(t)}(dx)\bigg{|}$
$\displaystyle\leq\bigg{|}\frac{1}{\sum_{j=1}^{N}e^{-\alpha
f(X^{j}(t))}}\bigg{(}\sum_{i=1}^{N}X^{i}(t)e^{-\alpha
f(X^{i}(t))}-\int_{\mathbb{R}^{d}}xe^{-\alpha
f(x)}\mathcal{L}_{X(t)}(dx)\bigg{)}\bigg{|}$
$\displaystyle\;\;\;\;+\bigg{|}\int_{\mathbb{R}^{d}}xe^{-\alpha
f(x)}\mathcal{L}_{X(t)}(dx)\bigg{(}\frac{1}{\sum_{j=1}^{N}e^{-\alpha
f(X^{j}(t))}}-\frac{1}{\int_{\mathbb{R}^{d}}e^{-\alpha
f(x)}\mathcal{L}_{X(t)}(dx)}\bigg{)}\bigg{|}.$ (4.25)

Let $Y^{i}(t)=X^{i}(t)e^{-\alpha f(X^{i}(t))}-\int_{\mathbb{R}^{d}}xe^{-\alpha
f(x)}\mathcal{L}_{X(t)}(dx)$. Note that $\mathbb{E}Y^{i}(t)$ is a
$d-$dimensional zero vector and $\mathbb{E}(Y^{i}(t)\cdot Y^{j}(t))=0$, $i\neq
j$. Then, using Theorem 3.6, we obtain

$\displaystyle\mathbb{E}\Big{|}\sum_{i=1}^{N}X^{i}(t)e^{-\alpha f(X^{i}(t))}$
$\displaystyle-\int_{\mathbb{R}^{d}}xe^{-\alpha
f(x)}\mathcal{L}_{X(t)}(dx)\Big{|}^{4}=\frac{1}{N^{4}}\mathbb{E}\Big{|}\sum\limits_{i=1}^{N}Y^{i}(t)\Big{|}^{4}$
$\displaystyle=\frac{1}{N^{4}}\mathbb{E}\bigg{(}\sum\limits_{i=1}^{N}|Y^{i}(t)|^{4}+\sum_{i=1}^{N}|Y^{i}(t)|^{2}\sum_{j=1}^{N}|Y^{j}(t)|^{2}\bigg{)}\leq\frac{C}{N^{2}},$
(4.26)

where $C$ is a positive constant independent of $N$. As a consequence of above
estimate and using Chebyshev’s inequality, we get

$\displaystyle\mathbb{P}\bigg{(}\Big{|}\sum_{i=1}^{N}X^{i}(t)e^{-\alpha
f(X^{i}(t))}$ $\displaystyle-\int_{\mathbb{R}^{d}}X(t)e^{-\alpha
f(X(t))}\mathcal{L}_{X(t)}(dx)\Big{|}\geq
N^{(\epsilon-1)/4}\bigg{)}\leq\frac{C}{N^{1+\epsilon}}.$

Therefore, by the Borel-Cantelli lemma there exists an a.s. finite
$\mathcal{F}_{t}$-measurable random variable $\zeta_{2}(t,\omega)$ such that
the following bound holds:

$\displaystyle\Big{|}\sum_{i=1}^{N}X^{i}(t)e^{-\alpha f(X^{i}(t))}$
$\displaystyle-\int_{\mathbb{R}^{d}}X(t)e^{-\alpha
f(X(t))}\mathcal{L}_{X(t)}(dx)\Big{|}\leq\frac{\zeta_{2}(t,\omega)}{N^{(1-\epsilon)/4}},\;\;\;\;a.s.$
(4.27)

In the same manner, we can ascertain

$\displaystyle\Big{|}\sum_{i=1}^{N}e^{-\alpha f(X^{i}(t))}$
$\displaystyle-\int_{\mathbb{R}^{d}}e^{-\alpha
f(X(t))}\mathcal{L}_{X(t)}(dx)\Big{|}\leq\frac{\zeta_{3}(t,\omega)}{N^{(1-\epsilon)/4}},\;\;\;\;a.s.,$
(4.28)

where $\zeta_{3}(t,\omega)$ is an a.s. finite $\mathcal{F}_{t}$-measurable
random variable. Substituting (4.22), (4.27) and (4.28) in (4.25), we conclude
that (4.24) is true for sufficiently large $N$. ∎

######  Remark 4.1.

From (4.21), we have
$\lim_{N\rightarrow\infty}\int_{\mathbb{R}^{d}}|x|^{2}\mathcal{E}_{t}(dx)=\mathbb{E}|X(t)|^{2}$,
$a.s.$, which is the strong law of large numbers for i.i.d. random variables
$|X^{i}(t)|^{2}$. Also, the result of Lemma 4.4 can be treated as a law of
large numbers which shows a.s. convergence of weighted average
$\bar{X}^{\mathcal{E}_{t}}(t)$ (as compared to empirical average of (4.21)) of
i.i.d. particle system towards $\bar{X}(t)$ as $N\rightarrow\infty$. ∎

Let $R>0$ be a sufficiently large real number. Let us fix a $t\in[0,T]$. Let
us denote

$\displaystyle\tau_{1,R}=\inf\Big{\\{}s\geq
0\;;\;\frac{1}{N}\sum\limits_{i=1}^{N}|X^{i}_{N}(s)|^{4}\geq R\Big{\\}},$
$\displaystyle\;\;\;\;\tau_{2,R}=\inf\Big{\\{}s\geq
0\;;\;\frac{1}{N}\sum\limits_{i=1}^{N}|X^{i}(s)|^{4}\geq R\Big{\\}},$ (4.29)
$\displaystyle\tau_{R}$ $\displaystyle=\tau_{1,R}\wedge\tau_{2,R},$ (4.30)

and

$\displaystyle\Omega_{1}(t)$ $\displaystyle=\\{\tau_{1,R}\leq
t\\}\cup\\{\tau_{2,R}\leq t\\},$ (4.31) $\displaystyle\Omega_{2}(t)$
$\displaystyle=\Omega\backslash\Omega_{1}(t)=\\{\tau_{1,R}>t\\}\cap\\{\tau_{2,R}>t\\}.$
(4.32)

######  Lemma 4.5.

Let Assumptions 3.1, 3.3-3.5 be satisfied. Then, the following inequality
holds for all $t\in[0,T]$:

$\displaystyle\mathbb{E}\int_{0}^{t\wedge\tau_{R}}|\bar{X}_{N}(s)$
$\displaystyle-\bar{X}^{\mathcal{E}_{s}}(s)|^{2}ds\leq CRe^{4\alpha
K_{u}\sqrt{R}}\int_{0}^{t}\frac{1}{N}\sum\limits_{i=1}^{N}\mathbb{E}|X^{i}_{N}(s\wedge\tau_{R})-X^{i}(s\wedge\tau_{R})|^{2}ds,$
(4.33)

where $\tau_{R}$ is from (4.30), $\bar{X}_{N}(s)$ is from (2.10),
$\bar{X}^{\mathcal{E}_{s}}(s)$ is from (4.18), $C>0$ is independent of $N$ and
$R$.

###### Proof.

We have

$\displaystyle|\bar{X}_{N}(s)-\bar{X}^{\mathcal{E}_{s}}(s)|=\bigg{|}\sum\limits_{i=1}^{N}X^{i}_{N}(s)\frac{e^{-\alpha
f(X^{i}_{N}(s))}}{\sum_{j=1}^{N}e^{-\alpha
f(X_{N}^{j}(s))}}-\sum\limits_{i=1}^{N}X^{i}(s)\frac{e^{-\alpha
f(X^{i}(s))}}{\sum_{j=1}^{N}e^{-\alpha f(X^{j}(s))}}\bigg{|}$
$\displaystyle\leq\Bigg{|}\frac{1}{N}\sum\limits_{i=1}^{N}\big{(}X_{N}^{i}(s)-X^{i}(s)\big{)}\frac{e^{-\alpha
f(X_{N}^{i}(s))}}{\frac{1}{N}\sum_{j=1}^{N}e^{-\alpha
f(X^{j}_{N}(s))}}\Bigg{|}+\Bigg{|}\frac{\frac{1}{N}\sum_{i=1}^{N}X^{i}(s)\big{(}e^{-\alpha
f(X_{N}^{i}(s))}-e^{-\alpha
f(X^{i}(s))}\big{)}}{\frac{1}{N}\sum_{j=1}^{N}e^{-\alpha
f(X^{j}_{N}(s))}}\Bigg{|}$
$\displaystyle\;\;\;\;+\Bigg{|}\frac{1}{N}\sum_{i=1}^{N}X^{i}(s)e^{-\alpha
f(X^{i}(s))}\bigg{(}\frac{1}{\frac{1}{N}\sum_{j=1}^{N}e^{-\alpha
f(X^{j}_{N}(s))}}-\frac{1}{\frac{1}{N}\sum_{j=1}^{N}e^{-\alpha
f(X^{j}(s))}}\bigg{)}\Bigg{|}.$

Using the discrete Jensen inequality, we get

$\displaystyle|\bar{X}_{N}(s)-\bar{X}^{\mathcal{E}_{s}}(s)|\leq
C\Bigg{(}e^{\frac{\alpha}{N}\sum_{j=1}^{N}f(X^{j}_{N}(s))}\frac{1}{N}\sum_{i=1}^{N}|X^{i}_{N}(s)-X^{i}(s)|$
$\displaystyle\;\;\;\;+e^{\frac{\alpha}{N}\sum_{j=1}^{N}f(X^{j}_{N}(s))}\frac{1}{N}\sum_{i=1}^{N}|X^{i}(s)||e^{-\alpha
f(X_{N}^{i}(s))}-e^{-\alpha f(X^{i}(s))}|$
$\displaystyle\;\;\;\;+e^{\frac{\alpha}{N}\sum_{j=1}^{N}(f(X^{j}_{N}(s))+f(X^{j}(s)))}\frac{1}{N}\sum_{i=1}^{N}|X^{i}(s)|\frac{1}{N}\sum_{j=1}^{N}|e^{-\alpha
f(X_{N}^{j}(s))}-e^{-\alpha f(X^{j}(s))}|\Bigg{)},$ (4.34)

where $C$ is a positive constant independent of $N$. Applying Assumptions
3.3-3.4, the Cauchy-Bunyakowsky-Schwartz inequality and Young’s inequality,
$ab\leq a^{2}/2+b^{2}/2$, $a,b>0$, we obtain

$\displaystyle|\bar{X}_{N}(s)-\bar{X}^{\mathcal{E}_{s}}(s)|\leq
C\Bigg{(}e^{\frac{\alpha
K_{u}}{N}\sum_{j=1}^{N}|X^{j}_{N}(s)|^{2}}\frac{1}{N}\sum_{i=1}^{N}|X^{i}_{N}(s)-X^{i}(s)|$
$\displaystyle\;\;\;\;+e^{\frac{\alpha
K_{u}}{N}\sum_{j=1}^{N}|X^{j}_{N}(s)|^{2}}\frac{1}{N}\sum_{i=1}^{N}|X^{i}(s)|\big{(}1+|X^{i}_{N}(s)|+|X^{i}(s)|\big{)}|X_{N}^{i}(s)-X^{i}(s)|$
$\displaystyle\;\;\;\;+e^{\frac{\alpha
K_{u}}{N}\sum_{j=1}^{N}(|X^{j}_{N}(s)|^{2}+|X^{j}(s)|^{2})}\frac{1}{N}\sum_{i=1}^{N}|X^{i}(s)|\frac{1}{N}\sum_{j=1}^{N}\big{(}1+|X^{j}_{N}(s)|+|X^{j}(s)|\big{)}|X_{N}^{j}(s)-X^{j}(s)|\Bigg{)}$
$\displaystyle\leq C\Bigg{(}e^{\frac{\alpha
K_{u}}{N}\sum_{j=1}^{N}|X^{j}_{N}(s)|^{2}}\frac{1}{N}\sum_{i=1}^{N}|X^{i}_{N}(s)-X^{i}(s)|$
$\displaystyle\;\;\;\;+e^{\frac{\alpha
K_{u}}{N}\sum_{j=1}^{N}(|X^{j}_{N}(s)|^{2}+|X^{j}(s)|^{2})}\frac{1}{N}\sum_{i=1}^{N}\big{(}1+|X^{i}_{N}(s)|^{2}+|X^{i}(s)|^{2}\big{)}|X_{N}^{i}(s)-X^{i}(s)|$
$\displaystyle\;\;\;\;+e^{\frac{\alpha
K_{u}}{N}\sum_{j=1}^{N}(|X^{j}_{N}(s)|^{2}+|X^{j}(s)|^{2})}\frac{1}{N}\sum_{i=1}^{N}|X^{i}(s)|^{2}\frac{1}{N}\sum\limits_{j=1}^{N}|X^{j}_{N}(s)-X^{j}(s)|\Bigg{)}$
$\displaystyle\leq C\Bigg{(}e^{\frac{\alpha
K_{u}}{N}\sum_{j=1}^{N}|X^{j}_{N}(s)|^{2}}\frac{1}{N}\sum_{i=1}^{N}|X^{i}_{N}(s)-X^{i}(s)|+e^{\frac{\alpha
K_{u}}{N}\sum_{j=1}^{N}(|X^{j}_{N}(s)|^{2}+|X^{j}(s)|^{2})}$
$\displaystyle\;\;\;\;\times\bigg{(}\frac{1}{N}\sum\limits_{i=1}^{N}\big{(}1+|X_{N}^{i}(s)|^{2}+|X^{i}(s)|^{2}\big{)}^{2}\bigg{)}^{1/2}\bigg{(}\frac{1}{N}\sum\limits_{i=1}^{N}|X_{N}^{i}(s)-X^{i}(s)|^{2}\bigg{)}^{1/2}\Bigg{)}.$
(4.35)

On squaring both sides, we ascertain

$\displaystyle|\bar{X}_{N}(s)-\bar{X}^{\mathcal{E}_{s}}(s)|^{2}\leq
C\Bigg{(}e^{\frac{2\alpha
K_{u}}{N}\sum_{j=1}^{N}|X^{j}_{N}(s)|^{2}}\frac{1}{N}\sum_{i=1}^{N}|X^{i}_{N}(s)-X^{i}(s)|^{2}+e^{\frac{2\alpha
K_{u}}{N}\sum_{j=1}^{N}(|X^{j}_{N}(s)|^{2}+|X^{j}(s)|^{2})}$
$\displaystyle\;\;\;\;\times\bigg{(}\frac{1}{N}\sum\limits_{i=1}^{N}\big{(}1+|X_{N}^{i}(s)|^{2}+|X^{i}(s)|^{2}\big{)}^{2}\bigg{)}\bigg{(}\frac{1}{N}\sum\limits_{i=1}^{N}|X_{N}^{i}(s)-X^{i}(s)|^{2}\bigg{)}\Bigg{)}.$

Using Holder’s inequality, we have

$\displaystyle\frac{1}{N}\sum_{j=1}^{N}(|X^{j}_{N}(s)|^{2}+|X^{j}(s)|^{2})\leq\frac{2}{N^{1/2}}\bigg{(}\sum_{j=1}^{N}(|X^{j}_{N}(s)|^{4}+|X^{j}(s)|^{4})\bigg{)}^{1/2}.$

Therefore,

$\displaystyle\mathbb{E}\int_{0}^{t\wedge\tau_{R}}|\bar{X}_{N}(s)$
$\displaystyle-\bar{X}^{\mathcal{E}_{s}}(s)|^{2}ds\leq CRe^{4\alpha
K_{u}\sqrt{R}}\int_{0}^{t}\frac{1}{N}\sum\limits_{i=1}^{N}\mathbb{E}|X^{i}_{N}(s\wedge\tau_{R})-X^{i}(s\wedge\tau_{R})|^{2}ds,$

where $C>0$ is independent of $N$ and $R$. ∎

######  Lemma 4.6.

Let Assumptions 3.1, 3.3-3.5 be satisfied. Then, the following inequality
holds for all $t\in[0,T]$:

$\displaystyle\mathbb{E}\int_{0}^{t\wedge\tau_{R}}|\bar{X}^{\mathcal{E}_{s}}(s)$
$\displaystyle-\bar{X}(s)|^{2}ds\leq C\frac{e^{2\alpha K_{u}\sqrt{R}}}{N},$
(4.36)

where $\tau_{R}$ is from (4.30), $\bar{X}^{\mathcal{E}_{s}}(s)$ is from
(4.18), $\bar{X}(s)$ is from (2.13), $C>0$ is independent of $N$ and $R$.

###### Proof.

We have

$\displaystyle|\bar{X}^{\mathcal{E}_{s}}(s)$
$\displaystyle-\bar{X}(s)|=\bigg{|}\sum_{i=1}^{N}X^{i}(s)\frac{e^{-\alpha
f(X^{i}(s))}}{\sum_{j=1}^{N}e^{-\alpha
f(X^{j}(s))}}-\int_{\mathbb{R}^{d}}x\frac{e^{-\alpha
f(x)}}{\int_{\mathbb{R}^{d}}e^{-\alpha
f(x)}\mathcal{L}_{X(s)}(dx)}\mathcal{L}_{X(s)}(dx)\bigg{|}$
$\displaystyle\leq\frac{1}{\frac{1}{N}\sum_{j=1}^{N}e^{-\alpha
f(X^{j}(s))}}\Bigg{|}\frac{1}{N}\sum_{i=1}^{N}\bigg{(}X^{i}(s)e^{-\alpha
f(X^{i}(s))}-\int_{\mathbb{R}^{d}}xe^{-\alpha
f(x)}\mathcal{L}_{X(s)}(dx)\bigg{)}\Bigg{|}$
$\displaystyle\;\;\;\;+\Bigg{|}\int_{\mathbb{R}^{d}}xe^{-\alpha
f(x)}\mathcal{L}_{X(s)}(dx)\frac{\frac{1}{N}\sum_{j=1}^{N}\Big{(}e^{-\alpha
f(X^{j}(s))}-\int_{\mathbb{R}^{d}}e^{-\alpha
f(x)}\mathcal{L}_{X(s)}(dx)\Big{)}}{\frac{1}{N}\sum_{j=1}^{N}e^{-\alpha
f(X^{j}(s))}\int_{\mathbb{R}^{d}}e^{-\alpha
f(x)}\mathcal{L}_{X(s)}(dx)}\Bigg{|}.$

Using Jensen’s inequality and squaring both sides, we get

$\displaystyle|\bar{X}^{\mathcal{E}_{s}}(s)$
$\displaystyle-\bar{X}(s)|^{2}\leq
Ce^{\frac{2\alpha}{N}\sum_{j=1}^{N}f(X^{j}(s))}\bigg{|}\frac{1}{N}\sum_{i=1}^{N}\Big{(}X^{i}(s)e^{-\alpha
f(X^{i}(s))}-\mathbb{E}\big{(}X(s)e^{-\alpha
f(X(s))}\big{)}\Big{)}\bigg{|}^{2}$
$\displaystyle\;\;\;\;+Ce^{\frac{2\alpha}{N}\sum_{j=1}^{N}f(X^{j}(s))}e^{2\alpha\mathbb{E}f(X(s))}(\mathbb{E}|X(s)|)^{2}\bigg{|}\frac{1}{N}\sum_{j=1}^{N}\Big{(}e^{-\alpha
f(X^{j}(s))}-\mathbb{E}\big{(}e^{-\alpha f(X(s))}\big{)}\Big{)}\bigg{|}^{2},$

where $C$ is a positive constant independent of $N$. Applying Assumption 3.4,
we ascertain

$\displaystyle|\bar{X}^{\mathcal{E}_{s}}(s)$
$\displaystyle-\bar{X}(s)|^{2}\leq Ce^{\frac{2\alpha
K_{u}}{N}\sum_{j=1}^{N}|X^{j}(s)|^{2}}\bigg{|}\frac{1}{N}\sum_{i=1}^{N}\Big{(}X^{i}(s)e^{-\alpha
f(X^{i}(s))}-\mathbb{E}\big{(}X(s)e^{-\alpha
f(X(s))}\big{)}\Big{)}\bigg{|}^{2}$ $\displaystyle\;\;\;\;+Ce^{\frac{2\alpha
K_{u}}{N}\sum_{j=1}^{N}|X^{j}(s)|^{2}}e^{2\alpha
K_{u}\mathbb{E}|X(s)|^{2}}(\mathbb{E}|X(s)|)^{2}\bigg{|}\frac{1}{N}\sum_{j=1}^{N}\Big{(}e^{-\alpha
f(X^{j}(s))}-\mathbb{E}\big{(}e^{-\alpha f(X(s))}\big{)}\Big{)}\bigg{|}^{2}.$

Hence, using Theorem 3.6, we obtain

$\displaystyle\mathbb{E}\int_{0}^{t\wedge\tau_{R}}|\bar{X}^{\mathcal{E}_{s}}(s)$
$\displaystyle-\bar{X}(s)|^{2}ds\leq Ce^{2\alpha
K_{u}\sqrt{R}}\mathbb{E}\int_{0}^{t\wedge\tau_{R}}\bigg{|}\frac{1}{N}\sum_{i=1}^{N}\Big{(}X^{i}(s)e^{-\alpha
f(X^{i}(s))}-\mathbb{E}\big{(}X(s)e^{-\alpha
f(X(s))}\big{)}\Big{)}\bigg{|}^{2}ds$ $\displaystyle\;\;\;\;+Ce^{2\alpha
K_{u}\sqrt{R}}\mathbb{E}\int_{0}^{t\wedge\tau_{R}}\bigg{|}\frac{1}{N}\sum_{j=1}^{N}\Big{(}e^{-\alpha
f(X^{j}(s))}-\mathbb{E}\big{(}e^{-\alpha f(X(s))}\big{)}\Big{)}\bigg{|}^{2}ds$
$\displaystyle\leq Ce^{2\alpha
K_{u}\sqrt{R}}\int_{0}^{t}\mathbb{E}\bigg{|}\frac{1}{N}\sum_{i=1}^{N}U^{i}_{1}(s\wedge\tau_{R})\bigg{|}^{2}ds+Ce^{2\alpha
K_{u}\sqrt{R}}\int_{0}^{t}\mathbb{E}\bigg{|}\frac{1}{N}\sum_{i=1}^{N}U^{i}_{2}(s\wedge\tau_{R})\bigg{|}^{2}ds,$

where $U_{1}^{i}(s\wedge\tau_{R})=X^{i}(s\wedge\tau_{R})e^{-\alpha
f(X^{i}(s\wedge\tau_{R}))}-\mathbb{E}\big{(}X(s\wedge\tau_{R})e^{-\alpha
f(X(s\wedge\tau_{R}))}\big{)}$, $U_{2}^{i}(s\wedge\tau_{R})=e^{-\alpha
f(X^{i}(s))}-\mathbb{E}\big{(}e^{-\alpha f(X(s))}\big{)}$, and $C$ is
independent of $N$ and $R$. We have

$\displaystyle\mathbb{E}\bigg{|}\frac{1}{N}\sum_{i=1}^{N}U^{i}_{1}(s\wedge\tau_{R})\bigg{|}^{2}=\frac{1}{N^{2}}\sum\limits_{i=1}^{N}\mathbb{E}|U_{1}^{i}(s\wedge\tau_{R})|^{2}+\frac{1}{N^{2}}\sum_{\begin{subarray}{c}i,j=1,\;i\neq
j\end{subarray}}^{N}\mathbb{E}\big{(}U^{i}_{1}(s\wedge\tau_{R})\cdot
U_{1}^{j}(s\wedge\tau_{R})\big{)}.$

Note that $\mathbb{E}\big{(}U^{i}_{1}(s)\cdot U_{1}^{j}(s)\big{)}=0$ for
$i\neq j$ and $s\wedge\tau_{R}$ is a bounded stopping time then
$\mathbb{E}\big{(}U^{i}_{1}(s\wedge\tau_{R})\cdot
U_{1}^{j}(s\wedge\tau_{R})\big{)}=0$ for $i\neq j$ because of Doob’s optional
stopping theorem [App04, Theorem 2.2.1]. Using Theorem 3.6, we deduce

$\displaystyle\mathbb{E}\bigg{|}\frac{1}{N}\sum_{i=1}^{N}U^{i}_{1}(s\wedge\tau_{R})\bigg{|}^{2}\leq\frac{C}{N},$
(4.37)

where $C$ is independent of $N$. In the similar manner, we can obtain

$\displaystyle\mathbb{E}\bigg{|}\frac{1}{N}\sum_{i=1}^{N}U^{i}_{2}(s\wedge\tau_{R})\bigg{|}^{2}\leq\frac{C}{N},$
(4.38)

where $C$ is independent of $N$. Using (4.37) and (4.38), we get the following
estimate:

$\displaystyle\mathbb{E}\int_{0}^{t\wedge\tau_{R}}|\bar{X}^{\mathcal{E}_{s}}(s)$
$\displaystyle-\bar{X}(s)|^{2}ds\leq C\frac{e^{2\alpha K_{u}\sqrt{R}}}{N},$

where $C$ is independent of $N$ and $R$. ∎

######  Theorem 4.7.

Let Assumptions 3.1, 3.3-3.5 be satisfied. Let $X_{N}^{i}(t)$ solve (2.2). Let
$X^{i}(t)$ represent independent processes which solve (2.2). Let us assume
that $X^{i}_{N}(0)=X^{i}(0)$, a.s., $i=1\dots,N$. Let $\mathbb{E}|Z|^{4}\leq
C$, $\sup_{i=1,\dots,N}\mathbb{E}|X^{i}(0)|^{4}\leq C$, and
$\sup_{i=1,\dots,N}\mathbb{E}|X^{i}_{N}(0)|^{4}\leq C$. Then, the following
mean-square convergence result holds for all $t\in[0,T]$:

$\displaystyle\lim\limits_{N\rightarrow\infty}\sup_{i=1,\dots,N}\mathbb{E}|X_{N}^{i}(t)-X^{i}(t)|^{2}=0.$
(4.39)

###### Proof.

Let $t\in(0,T]$. We can write

$\displaystyle\mathbb{E}|X_{N}^{i}(t)-X^{i}(t)|^{2}$
$\displaystyle=\mathbb{E}\big{(}|X_{N}^{i}(t)-X^{i}(t)|^{2}I_{\Omega_{1}(t)}\big{)}+\mathbb{E}\big{(}|X_{N}^{i}(t)-X^{i}(t)|^{2}I_{\Omega_{2}(t)}\big{)}$
$\displaystyle=:E_{1}(t)+E_{2}(t),$

where $\Omega_{1}(t)$ and $\Omega_{2}(t)$ are from (4.31) and (4.32),
respectively. Using the Cauchy-Bunyakowsky-Shwartz inequality and Chebyshev’s
inequality, we obtain

$\displaystyle E_{1}(t):$
$\displaystyle=\mathbb{E}\big{(}|X_{N}^{i}(t)-X^{i}(t)|^{2}I_{\Omega_{1}(t)}\big{)}\leq\big{(}\mathbb{E}|X_{N}^{i}(t)-X^{i}(t)|^{4}\big{)}^{1/2}\big{(}\mathbb{E}I_{\Omega_{1}(t)}\big{)}^{1/2}$
$\displaystyle\leq
C\big{(}\mathbb{E}|X_{N}^{i}(t)|^{4}+\mathbb{E}|X^{i}(t)|^{4}\big{)}^{1/2}\bigg{(}\frac{1}{RN}\sum\limits_{i=1}^{N}\mathbb{E}\sup_{0\leq
s\leq
t}|X^{i}_{N}(s)|^{4}+\frac{1}{RN}\sum\limits_{i=1}^{N}\mathbb{E}\sup_{0\leq
s\leq t}|X^{i}(s)|^{4}\bigg{)}^{1/2}.$

We get the following estimate for $E_{1}(t)$ by applying Lemma 3.3 and Theorem
3.6:

$\displaystyle E_{1}(t)\leq\frac{C}{R},$ (4.40)

where $C$ is a positive constant independent of $N$ and $R$.

Now, we estimate $E_{2}(t)$. We have
$\mathbb{E}(|X_{N}^{i}(t)-X^{i}(t)|^{2}I_{\Omega_{2}(t)})\leq\mathbb{E}(|X_{N}^{i}(t\wedge\tau_{R})-X^{i}(t\wedge\tau_{R})|^{2})$.
Using Ito’s formula, we have

$\displaystyle|X_{N}^{i}$
$\displaystyle(t\wedge\tau_{R})-X^{i}(t\wedge\tau_{R})|^{2}=|X^{i}_{N}(0)-X^{i}(0)|^{2}$
$\displaystyle\;\;-2\mathbb{E}\int_{0}^{t\wedge\tau_{R}}\beta(s)(X_{N}^{i}(s)-X^{i}(s))\cdot(X_{N}^{i}(s)-\bar{X}_{N}(s)-X^{i}(s)+\bar{X}(s))ds$
$\displaystyle\;\;+2\int_{0}^{t\wedge\tau_{R}}\sigma^{2}(s)|\operatorname{Diag}(X_{N}^{i}(s)-\bar{X}_{N}(s)-X^{i}(s)+\bar{X}(s))|^{2}ds$
$\displaystyle\;\;+2\sqrt{2}\int_{0}^{t\wedge\tau_{R}}\sigma(s)\big{(}(X_{N}^{i}(s)-X^{i}(s))\cdot\operatorname{Diag}(X_{N}^{i}(s)-\bar{X}_{N}(s)-X^{i}(s)+\bar{X}(s))dW^{i}(s)\big{)}$
$\displaystyle\;\;+\int_{0}^{t\wedge\tau_{R}}\int_{\mathbb{R}^{d}}\Big{(}|X_{N}^{i}(s^{-})-X^{i}(s^{-})+\gamma(s)\operatorname{Diag}(X_{N}^{i}(s^{-})-\bar{X}_{N}(s^{-}))z$
$\displaystyle\;\;\;\;\;\;\;\;-\gamma(s)\operatorname{Diag}(X^{i}(s^{-})-\bar{X}(s^{-}))z|^{2}-|X^{i}_{N}(s^{-})-X^{i}(s^{-})|^{2}\Big{)}\mathcal{N}^{i}(ds,dz).$
(4.41)

The Cauchy-Bunyakowsky-Schwartz inequality and Young’s inequality provide the
following estimates:

$\displaystyle(X_{N}^{i}(s)-X^{i}(s))\cdot(X_{N}^{i}(s)-\bar{X}_{N}(s)-X^{i}(s)+\bar{X}(s))\leq
C(|X^{i}_{N}(s)-X^{i}(s)|^{2}+|\bar{X}_{N}(s)-\bar{X}(s)|^{2}),$ (4.42)
$\displaystyle|\operatorname{Diag}(X_{N}^{i}(s)-\bar{X}_{N}(s)-X^{i}(s)+\bar{X}(s))|^{2}\leq
C(|X^{i}_{N}(s)-X^{i}(s)|^{2}+|\bar{X}_{N}(s)-\bar{X}(s)|^{2}),$ (4.43)

and

$\displaystyle\Big{(}|X_{N}^{i}(s^{-})-X^{i}(s^{-})+\gamma(s)\operatorname{Diag}(X_{N}^{i}(s^{-})-\bar{X}_{N}(s^{-}))z-\gamma(s)\operatorname{Diag}(X^{i}(s^{-})$
$\displaystyle\;\;\;\;-\bar{X}(s^{-}))z|^{2}-|X^{i}_{N}(s^{-})-X^{i}(s^{-})|^{2}\Big{)}=\gamma^{2}(s)|\big{(}(X^{i}_{N}(s^{-})-\bar{X}_{N}(s^{-})-X^{i}(s^{-})+\bar{X}(s^{-}))\cdot
z\big{)}|^{2}$
$\displaystyle\;\;\;\;+2\gamma(s)\Big{(}\big{(}X_{N}^{i}(s^{-})-X^{i}(s^{-})\big{)}\cdot\big{(}\operatorname{Diag}(X_{N}^{i}(s^{-})-\bar{X}_{N}(s^{-})-X^{i}(s^{-})+\bar{X}(s^{-}))z\big{)}\Big{)}$
$\displaystyle\leq
C(|X^{i}_{N}(s^{-})-X^{i}(s^{-})|^{2}+|\bar{X}_{N}(s^{-})-\bar{X}(s^{-})|^{2})|z|^{2}$
$\displaystyle\;\;\;\;+2\gamma(s)\Big{(}\big{(}X_{N}^{i}(s^{-})-X^{i}(s^{-})\big{)}\cdot\big{(}\operatorname{Diag}(X_{N}^{i}(s^{-})-\bar{X}_{N}(s^{-})-X^{i}(s^{-})+\bar{X}(s^{-}))z\big{)}\Big{)}.$
(4.44)

Taking expectations on both sides of (4.41), using estimates (4.42)-(4.44) and
applying Doob’s optional stopping theorem [App04, Theorem 2.2.1], we get

$\displaystyle\mathbb{E}|X_{N}^{i}(t\wedge\tau_{R})-X^{i}(t\wedge\tau_{R})|^{2}\leq\mathbb{E}|X_{N}^{i}(0)-X^{i}(0)|^{2}$
$\displaystyle\;\;\;\;+C\mathbb{E}\int_{0}^{t\wedge\tau_{R}}\big{(}|X_{N}^{i}(s)-X^{i}(s)|^{2}+|\bar{X}_{N}(s)-\bar{X}(s)|^{2}\big{)}ds$
$\displaystyle\;\;\;\;+C\mathbb{E}\int_{0}^{t\wedge\tau_{R}}\int_{\mathbb{R}^{d}}(|X^{i}_{N}(s)-X^{i}(s)|^{2}+|\bar{X}_{N}(s)-\bar{X}(s)|^{2})|z|^{2}\rho_{z}(z)dzds$
$\displaystyle\leq\mathbb{E}|X_{N}^{i}(0)-X^{i}(0)|^{2}+C\mathbb{E}\int_{0}^{t\wedge\tau_{R}}|X_{N}^{i}(s)-X^{i}(s)|^{2}ds$
$\displaystyle\;\;\;\;+C\mathbb{E}\int_{0}^{t\wedge\tau_{R}}|\bar{X}_{N}(s)-\bar{X}^{\mathcal{E}_{s}}(s)|^{2}ds+C\mathbb{E}\int_{0}^{t\wedge\tau_{R}}|\bar{X}^{\mathcal{E}_{s}}(s)-\bar{X}(s)|^{2}ds.$
(4.45)

Substituting (4.33) and (4.36) in (4.45), we obtain

$\displaystyle\mathbb{E}$
$\displaystyle\big{(}|X_{N}^{i}(t\wedge\tau_{R})-X^{i}(t\wedge\tau_{R})|^{2}\big{)}\leq\mathbb{E}|X_{N}^{i}(0)-X^{i}(0)|^{2}$
$\displaystyle\;\;\;\;+CRe^{4\alpha
K_{u}\sqrt{R}}\int_{0}^{t}\frac{1}{N}\sum\limits_{i=1}^{N}\mathbb{E}\big{(}|X^{i}_{N}(s\wedge\tau_{R})-X^{i}(s\wedge\tau_{R})|^{2}\big{)}ds+C\frac{e^{2\alpha
K_{u}\sqrt{R}}}{N},$

where $C>0$ is independent of $N$ and $R$. Taking supremum over $i=1,\dots,N$,
we get

$\displaystyle\sup_{i=1,\dots,N}\mathbb{E}\big{(}|$ $\displaystyle
X_{N}^{i}(t\wedge\tau_{R})-X^{i}(t\wedge\tau_{R})|^{2}\big{)}\leq\sup_{i=1,\dots,N}\mathbb{E}|X_{N}^{i}(0)-X^{i}(0)|^{2}$
$\displaystyle+CRe^{4\alpha
K_{u}\sqrt{R}}\int_{0}^{t}\sup_{i=1,\dots,N}\mathbb{E}\big{(}|X^{i}_{N}(s\wedge\tau_{R})-X^{i}(s\wedge\tau_{R})|^{2}\big{)}ds+C\frac{e^{2\alpha
K_{u}\sqrt{R}}}{N}.$

Using Grönwall’s inequality, we have

$\displaystyle\sup_{i=1,\dots,N}\mathbb{E}\big{(}|X_{N}^{i}(t\wedge\tau_{R})$
$\displaystyle-X^{i}(t\wedge\tau_{R})|^{2}\big{)}\leq\frac{C}{N}e^{CRe^{4\alpha
K_{u}\sqrt{R}}}e^{2\alpha K_{u}R}\leq\frac{C}{N}e^{e^{C_{u}\sqrt{R}}},$ (4.46)

where $C>0$ and $C_{u}>0$ are constants independent of $N$ and $R$. In the
above calculations, we have used the facts that $R<e^{2\alpha K_{u}\sqrt{R}}$
and $2\alpha K_{u}\sqrt{R}<e^{2\alpha K_{u}\sqrt{R}}$ for sufficiently large
$R$.

We choose $R=\frac{1}{C_{u}^{2}}(\ln{(\ln({N^{1/2})})})^{2}$. Therefore,

$\displaystyle\sup_{i=1,\dots,N}\mathbb{E}(|X_{N}^{i}(t)-X^{i}(t)|^{2}I_{\Omega_{2}(t)})\leq\sup_{i=1,\dots,N}\mathbb{E}\big{(}|X_{N}^{i}(t\wedge\tau_{R})$
$\displaystyle-X^{i}(t\wedge\tau_{R})|^{2}\big{)}\leq\frac{C}{N^{1/2}},$

which implies

$\displaystyle\lim\limits_{N\rightarrow\infty}\sup_{i=1,\dots,N}\mathbb{E}(|X_{N}^{i}(t)-X^{i}(t)|^{2}I_{\Omega_{2}(t)})=\lim\limits_{N\rightarrow\infty}\sup_{i=1,\dots,N}\mathbb{E}\big{(}|X_{N}^{i}(t\wedge\tau_{R})$
$\displaystyle-X^{i}(t\wedge\tau_{R})|^{2}\big{)}=0.$ (4.47)

The term (4.40) and the choice of $R$ provide the following estimate:

$\displaystyle\mathbb{E}\big{(}|X_{N}^{i}(t)-X^{i}(t)|^{2}I_{\Omega_{1}}(t)\big{)}\leq\frac{C}{R}\leq\frac{C}{(\ln{(\ln({N^{1/2})}))^{2}}},$

where $C>0$ is independent of $N$ and $R$. This yields

$\displaystyle\lim_{N\rightarrow\infty}\sup_{i=1,\dots,N}\mathbb{E}\big{(}|X^{i}_{N}(t)-X^{i}(t)|^{2}I_{\Omega_{1}(t)}\big{)}=0.$
(4.48)

As a consequence of (4.47) and (4.48) , we get

$\displaystyle\lim_{N\rightarrow\infty}\sup_{i=1,\dots,N}\mathbb{E}|X_{N}^{i}(t)-X^{i}(t)|^{2}=0,$

for all $t\in[0,T]$. ∎

######  Remark 4.2.

It is not difficult to see from the above theorem that the empirical measure
of the particle system (2.2) converges to the law of the mean-field SDEs (2.2)
in $2-$Wasserstein metric, i.e. for all $t\in[0,T]$:

$\displaystyle\lim_{N\rightarrow\infty}\mathcal{W}_{2}^{2}(\mathcal{E}_{t}^{N},\mathcal{L}_{X(t)})=0,$
(4.49)

where $\mathcal{E}_{t}^{N}=\frac{1}{N}\sum_{i=1}^{N}\delta_{X^{i}_{N}(t)}$. ∎

######  Remark 4.3.

Theorem 4.7 implies weak convergence of the empirical measure,
$\mathcal{E}_{t}^{N}$ of interacting particle system towards
$\mathcal{L}_{X(t)}$ which is the law of the mean-field limit process $X(t)$
(see [Shi13, Szn91]). ∎

###  4.3 Convergence of the numerical scheme

To implement the particle system (2.2), we have proposed to utilize the Euler
scheme introduced in Section 2.2.3. The jump-diffusion SDEs (2.2), governing
interacting particle system, have locally Lipschitz and linearly growing
coefficients. Due to non-global Lipschitzness of the coefficients, it is not
straightforward to deduce convergence of the Euler scheme to (2.2). In this
section, we go one step further and prove this convergence result uniform in
$N$. To this end, we introduce the function $\kappa_{h}(t)=t_{k}$, $t_{k}\leq
t<t_{k+1}$, where $0=t_{0}<\dots<t_{n}=T$ is a uniform partition of $[0,T]$,
i.e. $t_{k+1}-t_{k}=h$ for all $k=0,\dots,n-1$. We write the continuous
version of the numerical scheme (2.2.3) as follows:

$\displaystyle dY^{i}_{N}(t)$
$\displaystyle=-\beta(t)(Y^{i}_{N}(\kappa_{h}(t))-\bar{Y}_{N}(\kappa_{h}(t)))dt+\sqrt{2}\sigma(t)\operatorname{Diag}(Y^{i}_{N}(\kappa_{h}(t))-\bar{Y}_{N}(\kappa_{h}(t)))dW^{i}(t)$
$\displaystyle\;\;\;\;+\int_{\mathbb{R}^{d}}\operatorname{Diag}(Y^{i}_{N}(\kappa_{h}(t))-\bar{Y}_{N}(\kappa_{h}(t)))z\mathcal{N}^{i}(dt,dz).$
(4.50)

In this section, our aim is to show mean-square convergence of $Y^{i}_{N}(t)$
to $X^{i}_{N}(t)$ uniformly in $N$, i.e.

$\displaystyle\lim_{h\rightarrow
0}\sup_{i=1,\dots,N}\mathbb{E}|Y^{i}_{N}(t)-X^{i}_{N}(t)|^{2}=0,$ (4.51)

where $h\rightarrow 0$ means that keeping $T$ fixed the time-step of uniform
partition of $[0,T]$ goes to zero.

Let Assumptions 3.1-3.2 hold. Let $\mathbb{E}|Y^{i}_{N}(0)|^{2}<\infty$ and
$\mathbb{E}|Z|^{2}<\infty$, then the particle system (4.3) is well-posed (cf.
Theorem 3.2). Moreover, if $\mathbb{E}|Y^{i}_{N}(0)|^{2p}<\infty$ and
$\mathbb{E}|Z|^{2p}<\infty$ for some $p\geq 1$, then, due to Lemma 3.3, the
following holds:

$\displaystyle\mathbb{E}\sup_{0\leq t\leq T}|Y^{i}_{N}(t)|^{2p}\leq K,$ (4.52)

where we cannot say that $K$ is independent of $h$. However, to prove the
convergence of numerical scheme we need the uniform in $h$ and $N$ moment
bound, which we prove in the next lemma.

######  Lemma 4.8.

Let Assumptions 3.1, 3.3-3.5 hold. Let $p\geq 1$,
$\mathbb{E}|Y^{i}_{N}(0)|^{2p}<\infty$ and $\mathbb{E}|Z|^{2p}<\infty$. Then,
the following holds:

$\displaystyle\sup_{i=1,\dots,N}\mathbb{E}\sup_{0\leq t\leq
T}|Y^{i}_{N}(t)|^{2p}\leq K_{d},$ (4.53)

where $K_{d}$ is a positive constant independent of $h$ and $N$.

###### Proof.

Let $p$ be a positive integer. Using Ito’s formula, the Cauchy-Bunyakowsky-
Schwartz inequality and Young’s inequality, we have

$\displaystyle|Y^{i}_{N}$
$\displaystyle(t)|^{2p}=|Y^{i}_{N}(0)|^{2p}-2p\int_{0}^{t}\beta(s)|Y^{i}_{N}(s)|^{2p-2}\big{(}Y^{i}_{N}(s)\cdot(Y^{i}_{N}(\kappa_{h}(s))-\bar{Y}_{N}(\kappa_{h}(s)))\big{)}ds$
$\displaystyle\;\;\;\;+2\sqrt{2}p\int_{0}^{t}\sigma(s)|Y^{i}_{N}(s)|^{2p-2}\big{(}Y^{i}_{N}(s)\cdot\operatorname{Diag}(Y^{i}_{N}(\kappa_{h}(s))-\bar{Y}_{N}(\kappa_{h}(s)))dW^{i}(s)\big{)}$
$\displaystyle\;\;\;\;+4p(p-1)\int_{0}^{t}\sigma^{2}(s)|Y^{i}_{N}(s)|^{2p-4}|\operatorname{Diag}(Y^{i}_{N}(\kappa_{h}(s))-\bar{Y}_{N}(\kappa_{h}(s)))Y^{i}_{N}(s)|^{2}ds$
$\displaystyle\;\;\;\;+2p\int_{0}^{t}\sigma^{2}(s)|Y^{i}_{N}(s)|^{2p-2}|\operatorname{Diag}(Y^{i}_{N}(\kappa_{h}(s))-\bar{Y}_{N}(s)|^{2}ds$
$\displaystyle\;\;\;\;+\int_{0}^{t}\int_{\mathbb{R}^{d}}\Big{(}|Y^{i}_{N}(s^{-})+\gamma(s)\operatorname{Diag}(Y^{i}_{N}(\kappa_{h}(s))-\bar{Y}_{N}(\kappa_{h}(s)))z|^{2p}-|Y^{i}_{N}(s^{-})|^{2p}\Big{)}\mathcal{N}^{i}(ds,dz)$
$\displaystyle\leq|Y^{i}_{N}(0)|^{2p}+C\int_{0}^{t}(|Y^{i}_{N}(s)|^{2p}+|Y^{i}_{N}(\kappa_{h}(s))|^{2p}+|\bar{Y}_{N}(\kappa_{h}(s))|^{2p})ds$
$\displaystyle\;\;\;\;+2\sqrt{2}p\int_{0}^{t}\sigma(s)|Y^{i}_{N}(s)|^{2p-2}(Y^{i}_{N}(s)\cdot\operatorname{Diag}(Y^{i}_{N}(\kappa_{h}(s))-\bar{Y}_{N}(\kappa_{h}(s)))dW^{i}(s))$
$\displaystyle\;\;\;\;+C\int_{0}^{t}\int_{\mathbb{R}^{d}}\Big{(}|Y^{i}_{N}(s^{-})|^{2p}+(|Y^{i}_{N}(\kappa_{h}(s))|^{2p}+|\bar{Y}_{N}(\kappa_{h}(s))|^{2p})(1+|z|^{2p})\Big{)}\mathcal{N}^{i}(ds,dz).$

First taking supremum over $0\leq t\leq T$ and then expectation, we obtain

$\displaystyle\mathbb{E}$ $\displaystyle\sup_{0\leq t\leq
T}|Y^{i}_{N}(t)|^{2p}\leq\mathbb{E}|Y^{i}_{N}(0)|^{2p}+C\mathbb{E}\int_{0}^{T}\Big{(}|Y^{i}_{N}(s)|^{2p}+|Y^{i}_{N}(\kappa_{h}(s))|^{2p}+|\bar{Y}_{N}(\kappa_{h}(s))|^{2p}\Big{)}ds$
$\displaystyle+2\sqrt{2}p\mathbb{E}\sup_{0\leq t\leq
T}\bigg{|}\int_{0}^{t}\sigma(s)|Y^{i}_{N}(s)|^{2p-2}(Y^{i}_{N}(s)\cdot\operatorname{Diag}(Y^{i}_{N}(\kappa_{h}(s))-\bar{Y}_{N}(\kappa_{h}(s)))dW^{i}(s))\bigg{|}$
$\displaystyle+C\mathbb{E}\int_{0}^{T}\int_{\mathbb{R}^{d}}\Big{(}|Y^{i}_{N}(s^{-})|^{2p}+(|Y^{i}_{N}(\kappa_{h}(s))|^{2p}+|\bar{Y}_{N}(\kappa_{h}(s))|^{2p})(1+|z|^{2p})\Big{)}\mathcal{N}^{i}(ds,dz),$

where $C$ is independent of $h$ and $N$. Using the Burkholder-Davis-Gundy
inequality (note that we can apply this inequality due to (4.52)) and the fact
that $\mathbb{E}|Z|^{2p}<\infty$, we get

$\displaystyle\mathbb{E}$ $\displaystyle\sup_{0\leq t\leq
T}|Y^{i}_{N}(t)|^{2p}\leq\mathbb{E}|Y^{i}_{N}(0)|^{2p}+C\mathbb{E}\int_{0}^{T}\Big{(}|Y^{i}_{N}(s)|^{2p}+|Y^{i}_{N}(\kappa_{h}(s))|^{2p}+|\bar{Y}_{N}(\kappa_{h}(s))|^{2p}\Big{)}ds$
$\displaystyle\;\;\;\;+C\mathbb{E}\bigg{(}\int_{0}^{T}|Y^{i}_{N}(s)|^{4p-4}\big{(}Y^{i}_{N}(s)\cdot(Y^{i}_{N}(\kappa_{h}(s))-\bar{Y}_{N}(\kappa_{h}(s)))\big{)}^{2}ds\bigg{)}^{1/2}$
$\displaystyle\;\;\;\;+C\mathbb{E}\int_{0}^{T}\int_{\mathbb{R}^{d}}\Big{(}|Y^{i}_{N}(s)|^{2p}+(|Y^{i}_{N}(\kappa_{h}(s))|^{2p}+|\bar{Y}_{N}(\kappa_{h}(s))|^{2p})(1+|z|^{2p})\Big{)}\rho_{z}(z)dzds$
$\displaystyle\leq\mathbb{E}|Y^{i}_{N}(0)|^{2p}+C\mathbb{E}\int_{0}^{T}\Big{(}|Y^{i}_{N}(s)|^{2p}+|Y^{i}_{N}(\kappa_{h}(s))|^{2p}+|\bar{Y}_{N}(\kappa_{h}(s))|^{2p}\Big{)}ds$
$\displaystyle\;\;\;\;+\mathbb{E}\sup_{0\leq t\leq
T}|Y^{i}_{N}(t)|^{2p-1}\bigg{(}\int_{0}^{T}|Y^{i}_{N}(\kappa_{h}(s))-\bar{Y}_{N}(\kappa_{h}(s))|^{2}ds\bigg{)}^{1/2}.$

Applying Young’s inequality and Holder’s inequality, we ascertain

$\displaystyle\mathbb{E}\sup\limits_{0\leq t\leq
T}|Y^{i}_{N}(t)|^{2p}\leq\mathbb{E}|Y_{N}^{i}(0)|^{2p}+C\int_{0}^{T}(|Y^{i}_{N}(s)|^{2p}+|Y^{i}_{N}(\kappa_{h}(s))|^{2p}+|\bar{Y}_{N}(\kappa_{h}(s))|^{2p})ds$
$\displaystyle\;\;\;\;+\frac{1}{2}\mathbb{E}\sup\limits_{0\leq t\leq
T}|Y^{i}_{N}(t)|^{2p}+C\mathbb{E}\Big{(}\int_{0}^{T}|Y^{i}_{N}(\kappa_{h}(s))-\bar{Y}_{N}(\kappa_{h}(s))|^{2}ds\Big{)}^{p}$
$\displaystyle\leq\mathbb{E}|Y_{N}^{i}(0)|^{2p}+C\int_{0}^{t}(|Y^{i}_{N}(s)|^{2p}+|Y^{i}_{N}(\kappa_{h}(s))|^{2p}+|\bar{Y}_{N}(\kappa_{h}(s))|^{2p})ds$
$\displaystyle\;\;\;\;+\frac{1}{2}\mathbb{E}\sup\limits_{0\leq t\leq
T}|Y^{i}_{N}(t)|^{2p}+C\mathbb{E}\int_{0}^{T}|Y^{i}_{N}(\kappa_{h}(s))-\bar{Y}_{N}(\kappa_{h}(s))|^{2p}ds.$
(4.54)

Using Jensen’s inequality and (3.4), we have

$\displaystyle|\bar{Y}_{N}(\kappa_{h}(s))|^{2}$
$\displaystyle\leq\sum\limits_{i=1}^{N}|Y^{i}_{N}(\kappa_{h}(s))|^{2}\frac{e^{-\alpha
f(Y^{i}_{N}(\kappa_{h}(s)))}}{\sum_{j=1}^{N}e^{-\alpha
f(Y^{j}_{N}(\kappa_{h}(s)))}}\leq
L_{1}+\frac{L_{2}}{N}\sum\limits_{i=1}^{N}|Y^{i}_{N}(\kappa_{h}(s))|^{2}.$
(4.55)

Therefore, substituting (4.55) in (4.54) yields

$\displaystyle\mathbb{E}\sup\limits_{0\leq t\leq T}|Y^{i}_{N}(t)|^{2p}\leq
2\mathbb{E}|Y_{N}^{i}(0)|^{2p}+C+C\mathbb{E}\int_{0}^{T}\Big{(}|Y^{i}_{N}(s)|^{2p}+|Y^{i}_{N}(\kappa_{h}(s))|^{p}+\frac{1}{N}\sum\limits_{i=1}^{N}|Y_{N}^{i}(\kappa_{h}(s))|^{2p}\Big{)}ds$
$\displaystyle\leq
2\mathbb{E}|Y_{N}^{i}(0)|^{2p}+C+C\int_{0}^{T}\Big{(}\mathbb{E}\sup_{0\leq
u\leq
s}|Y^{i}_{N}(u)|^{2p}+\frac{1}{N}\sum\limits_{i=1}^{N}\mathbb{E}\sup_{0\leq
u\leq s}|Y_{N}^{i}(u)|^{2p}\Big{)}ds,$

where $C>0$ is independent of $h$ and $N$. Taking supremum over $i=1,\dots,N$,
we get

$\displaystyle\sup\limits_{i=1,\dots,N}\mathbb{E}\sup\limits_{0\leq t\leq
T}|Y^{i}_{N}(t)|^{2p}\leq
2\mathbb{E}|Y^{i}_{N}(0)|^{2p}+C+C\int_{0}^{T}\sup_{i=1,\dots,N}\mathbb{E}\sup_{0\leq
u\leq s}|Y^{i}_{N}(u)|^{2p}ds,$

where $C>0$ is independent of $h$ and $N$. Using Grönwall’s lemma, we have the
desired result. ∎

######  Lemma 4.9.

Let Assumptions 3.1, 3.3-3.5 hold. Let
$\sup_{i=1,\dots,N}\mathbb{E}|X^{i}_{N}(0)|^{4}<\infty$,
$\sup_{i=1,\dots,N}\mathbb{E}|Y^{i}_{N}(0)|^{4}<\infty$,
$\mathbb{E}|Z|^{4}<\infty$. Then

$\displaystyle\sup_{i=1,\dots,N}\mathbb{E}|Y^{i}_{N}(t)-Y^{i}_{N}(\kappa_{h}(t))|^{2}\leq
Ch,$

where $C$ is a positive constant independent of $N$ and $h$.

###### Proof.

We have

$\displaystyle|Y^{i}_{N}(t)$ $\displaystyle-Y^{i}_{N}(\kappa_{h}(t))|^{2}\leq
C\bigg{(}\bigg{|}\int_{\kappa_{h}(t)}^{t}(Y^{i}_{N}(\kappa_{h}(s))-\bar{Y}_{N}(\kappa_{h}(s)))ds\bigg{|}^{2}$
$\displaystyle\;\;\;\;+\bigg{|}\int_{\kappa_{h}(t)}^{t}\operatorname{Diag}(Y^{i}_{N}(\kappa_{h}(s))-\bar{Y}_{N}(\kappa_{h}(s)))dW^{i}(s)\bigg{|}^{2}$
$\displaystyle\;\;\;\;+\bigg{|}\int_{\kappa_{h}(t)}^{t}\int_{\mathbb{R}^{d}}\operatorname{Diag}(Y^{i}_{N}(\kappa_{h}(s))-\bar{Y}_{N}(\kappa_{h}(s)))z\mathcal{N}^{i}(ds,dz)\bigg{|}^{2}\bigg{)},$

where $C$ is independent of $h$ and $N$. Taking expectation and using Ito’s
isometry (note that we can apply Ito’s isometry due to Lemma 4.8), we get

$\displaystyle\mathbb{E}|Y^{i}_{N}(t)$
$\displaystyle-Y^{i}_{N}(\kappa_{h}(t))|^{2}\leq
C(1+\mathbb{E}|Z|^{2})\bigg{(}\int_{\kappa_{h}(t)}^{t}\mathbb{E}|Y^{i}_{N}(\kappa_{h}(s))-\bar{Y}_{N}(\kappa_{h}(s))|^{2}ds\bigg{)}.$

Therefore, use of (4.55) gives

$\displaystyle\sup_{i=1,\dots,N}\mathbb{E}$
$\displaystyle|Y^{i}_{N}(t)-Y^{i}_{N}(\kappa_{h}(t))|^{2}\leq
C(1+\mathbb{E}|Z|^{2})\bigg{(}\int_{\kappa_{h}(t)}^{t}\sup_{i=1,\dots,N}\mathbb{E}|Y^{i}_{N}(\kappa_{h}(s))|^{2}$
$\displaystyle\;\;\;\;+2L_{1}+\frac{L_{2}}{N}\sum\limits_{i=1}^{N}\sup_{i=1,\dots,N}\big{(}\mathbb{E}|Y^{i}_{N}(\kappa_{h}(s))|^{2})ds\bigg{)}.$

Using Lemma 3.3 and Lemma 4.8, we get

$\displaystyle\sup_{i=1,\dots,N}\mathbb{E}|Y^{i}_{N}(t)-Y^{i}_{N}(\kappa_{h}(t))|^{2}\leq
C(t-\kappa_{h}(t))\leq Ch,$

where $C$ is independent of $N$ and $h$. ∎

######  Theorem 4.10.

Let Assumptions 3.1, 3.3-3.5 hold. Let $\mathbb{E}|Z|^{4}<\infty$,
$\sup_{i=1,\dots,N}\mathbb{E}|X^{i}_{N}(0)|^{4}<\infty$,
$\sup_{i=1,\dots,N}\mathbb{E}|Y^{i}_{N}(0)|^{4}<\infty$ and
$Y^{i}_{N}(0)=X^{i}_{N}(0)$, $i=1,\dots,N$. Then

$\displaystyle\lim\limits_{h\rightarrow
0}\lim\limits_{N\rightarrow\infty}\sup_{i=1,\dots,N}\mathbb{E}|Y^{i}_{N}(t)-X^{i}_{N}(t)|^{2}=\lim\limits_{N\rightarrow\infty}\lim\limits_{h\rightarrow
0}\sup_{i=1,\dots,N}\mathbb{E}|Y^{i}_{N}(t)-X^{i}_{N}(t)|^{2}=0,$ (4.56)

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

###### Proof.

Let

$\displaystyle\tau_{1.R}=\inf\Big{\\{}t\geq
0\;;\;\frac{1}{N}\sum\limits_{i=1}^{N}|X^{i}_{N}(t)|^{4}\geq R\Big{\\}}$
$\displaystyle,\;\;\;\;\tau_{3,R}=\inf\Big{\\{}t\geq
0\;;\;\frac{1}{N}\sum\limits_{i=1}^{N}|Y^{i}_{N}(t)|^{4}\geq R\Big{\\}},$
$\displaystyle\tau^{h}_{R}$ $\displaystyle=\tau_{1,R}\wedge\tau_{3,R},$

and

$\displaystyle\Omega_{3}(t)$ $\displaystyle=\\{\tau_{1,R}\leq
t\\}\cup\\{\tau_{3,R}\leq
t\\},\;\;\;\Omega_{4}(t)=\Omega\backslash\Omega_{3}(t)=\\{\tau_{1,R}\geq
t\\}\cap\\{\tau_{3,R}\geq t\\}.$

We have

$\displaystyle\mathbb{E}|Y^{i}_{N}(t)-X^{i}_{N}(t)|^{2}$
$\displaystyle=\mathbb{E}\big{(}|Y^{i}_{N}(t)-X^{i}_{N}(t)|^{2}I_{\Omega_{3}(t)}\big{)}+\mathbb{E}\big{(}|Y^{i}_{N}(t)-X^{i}_{N}(t)|^{2}I_{\Omega_{4}(t)}\big{)}$
$\displaystyle=:E_{3}(t)+E_{4}(t).$

Let us first estimate the term $E_{3}(t)$. Using Cauchy-Bunyakowsky-Schwartz
inequality, Chebyshev’s inequality, Lemma 3.3 and Lemma 4.8, we get

$\displaystyle\mathbb{E}\big{(}|Y^{i}_{N}(t)-X^{i}_{N}(t)|^{2}I_{\Omega_{3}(t)}\big{)}$
$\displaystyle\leq\big{(}\mathbb{E}|Y^{i}_{N}(t)-X^{i}_{N}(t)|^{4}\big{)}^{1/2}\big{(}\mathbb{E}I_{\Omega_{3}(t)}\big{)}^{1/2}$
$\displaystyle\leq
C\bigg{(}\frac{1}{RN}\sum\limits_{i=1}^{N}\mathbb{E}\sup_{0\leq s\leq
t}|Y^{i}_{N}(s)|^{4}+\frac{1}{RN}\sum\limits_{i=1}^{N}\mathbb{E}\sup_{0\leq
s\leq t}|X^{i}_{N}(s)|^{4}\bigg{)}\leq\frac{C}{R},$ (4.57)

where $C$ is independent of $h$, $N$ and $R$.

Note that
$\mathbb{E}\big{(}|Y^{i}_{N}(t)-X^{i}_{N}(t)|^{2}I_{\Omega_{4}(t)}\big{)}\leq\mathbb{E}|Y^{i}_{N}(t\wedge\tau^{h}_{R})-X^{i}_{N}(t\wedge\tau^{h}_{R})|^{2}$.
Using Ito’s formula, we obtain

$\displaystyle|Y^{i}_{N}(t\wedge\tau^{h}_{R})-X^{i}_{N}(t\wedge\tau^{h}_{R})|^{2}=|Y^{i}_{N}(0)-X^{i}_{N}(0)|^{2}$
$\displaystyle-2\int_{0}^{t\wedge\tau^{h}_{R}}\beta(s)\big{(}(Y^{i}_{N}(s)-X^{i}_{N}(s))\cdot(Y^{i}_{N}(\kappa_{h}(s))-\bar{Y}_{N}(\kappa_{h}(s))-X^{i}_{N}(s)+\bar{X}_{N}(s))\big{)}ds$
$\displaystyle+2\sqrt{2}\int_{0}^{t\wedge\tau^{h}_{R}}\sigma(s)\big{(}(Y^{i}_{N}(s)-X^{i}_{N}(s))\cdot\operatorname{Diag}(Y^{i}_{N}(\kappa_{h}(s))-\bar{Y}_{N}(\kappa_{h}(s))-X^{i}_{N}(s)+\bar{X}_{N}(s))dW^{i}(s)\big{)}$
$\displaystyle+2\int_{0}^{t\wedge\tau^{h}_{R}}\sigma^{2}(s)|Y^{i}_{N}(\kappa_{h}(s))-\bar{Y}_{N}(\kappa_{h}(s))-X^{i}_{N}(s)+\bar{X}_{N}(s)|^{2}ds$
$\displaystyle+\int_{0}^{t\wedge\tau^{h}_{R}}\int_{\mathbb{R}^{d}}\big{(}|Y^{i}_{N}(s^{-})-X^{i}_{N}(s^{-})+\operatorname{Diag}(Y^{i}_{N}(\kappa_{h}(s))-\bar{Y}_{N}(\kappa_{h}(s)))z-\operatorname{Diag}(X^{i}_{N}(s)-\bar{X}_{N}(s))z|^{2}$
$\displaystyle\;\;\;\;\;\;-|Y^{i}_{N}(s^{-})-X^{i}_{N}(s^{-})|^{2}\big{)}\mathcal{N}^{i}(ds,dz).$

Taking expectation on both sides, and using the Cauchy-Bunyakowsky-Schwartz
inequality, Young’s inequality, Ito’s isometry (note that we can apply Ito’s
isometry due to Lemma 4.8) and Doob’s optional stopping theorem [App04,
Theorem 2.2.1], we get

$\displaystyle\mathbb{E}\big{(}|Y^{i}_{N}(t\wedge\tau^{h}_{R})-X^{i}_{N}(t\wedge\tau^{h}_{R})|^{2}\big{)}$
$\displaystyle\leq
Ch+C(1+|z|^{2})\mathbb{E}\int_{0}^{t\wedge\tau^{h}_{R}}\Big{(}|Y^{i}_{N}(\kappa_{h}(s))-X^{i}_{N}(s)|^{2}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;+|\bar{Y}_{N}(\kappa_{h}(s))-\bar{X}_{N}(s)|^{2}\Big{)}ds$
$\displaystyle\leq
C\mathbb{E}\int_{0}^{t\wedge\tau^{h}_{R}}\Big{(}|Y^{i}_{N}(\kappa_{h}(s))-Y^{i}_{N}(s)|^{2}+|Y^{i}_{N}(s)-X^{i}_{N}(s)|^{2}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;+|\bar{Y}_{N}(\kappa_{h}(s))-\bar{Y}_{N}(s)|^{2}+|\bar{Y}_{N}(s)-\bar{X}_{N}(s)|^{2}\Big{)}ds.$
(4.58)

Due to Lemma 4.9, we have

$\displaystyle\sup_{i=1,\dots,N}\mathbb{E}|Y^{i}_{N}(\kappa_{h}(s))-Y^{i}_{N}(s)|^{2}\leq
Ch,$ (4.59)

where $C$ is independent of $h$ and $N$.

Now, we will estimate the term $|\bar{Y}_{N}(s)-\bar{Y}_{N}(\kappa_{h}(s))|$.
Recall that we used discrete Jensen’s inequality, Assumptions 3.3-3.4 and
Cauchy-Bunyakowsky-Schwartz inequality to obtain (4.35). We apply the same set
of arguments as before to get

$\displaystyle|\bar{Y}_{N}(s)$
$\displaystyle-\bar{Y}_{N}(\kappa_{h}(s))|=\bigg{|}\sum\limits_{i=1}^{N}Y^{i}_{N}(s)\frac{e^{-\alpha
f(Y^{i}_{N}(s))}}{\sum_{j=1}^{N}e^{-\alpha
f(Y^{j}_{N}(s))}}-\sum\limits_{i=1}^{N}Y^{i}_{N}(\kappa_{h}(s))\frac{e^{-\alpha
f(Y^{i}_{N}(\kappa_{h}(s)))}}{\sum_{j=1}^{N}e^{-\alpha
f(Y^{j}_{N}(\kappa_{h}(s)))}}\bigg{|}$
$\displaystyle\leq\frac{1}{\frac{1}{N}\sum_{j=1}^{N}e^{-\alpha
f(Y^{j}_{N}(s))}}\bigg{|}\frac{1}{N}\sum\limits_{i=1}^{N}\big{(}Y^{i}_{N}(s)-Y^{i}_{N}(\kappa_{h}(s))\big{)}e^{-\alpha
f(Y^{i}_{N}(s))}\bigg{|}$
$\displaystyle\;\;\;\;+\frac{1}{\frac{1}{N}\sum_{j=1}^{N}e^{-\alpha
f(Y^{j}_{N}(s))}}\bigg{|}\frac{1}{N}\sum\limits_{i=1}^{N}Y^{i}_{N}(\kappa_{h}(s))\Big{(}e^{-\alpha
f(Y^{i}_{N}(s))}-e^{-\alpha f(Y^{i}_{N}(\kappa_{h}(s)))}\Big{)}\bigg{|}$
$\displaystyle\;\;\;\;+\bigg{|}\frac{1}{N}\sum\limits_{i=1}^{N}Y^{i}_{N}(\kappa_{h}(s))e^{-\alpha
f(Y^{i}_{N}(\kappa_{h}(s)))}\bigg{(}\frac{1}{\frac{1}{N}\sum_{j=1}^{N}e^{-\alpha
f(Y^{j}_{N}(s))}}-\frac{1}{\frac{1}{N}\sum_{j=1}^{N}e^{-\alpha
f(Y^{j}_{N}(\kappa_{h}(s)))}}\bigg{)}\bigg{|}$ $\displaystyle\leq
C\Bigg{(}e^{\frac{\alpha
K_{u}}{N}\sum_{j=1}^{N}|Y^{j}_{N}(s)|^{2}}\frac{1}{N}\sum\limits_{i=1}^{N}|Y^{i}_{N}(s)-Y^{i}_{N}(\kappa_{h}(s))|$
$\displaystyle\;\;\;\;+e^{\frac{\alpha
K_{u}}{N}\sum_{j=1}^{N}(|Y^{j}_{N}(s)|^{2}+|Y^{j}_{N}(\kappa_{h}(s))|^{2})}\times\bigg{(}\frac{1}{N}\sum\limits_{i=1}^{N}(1+|Y^{i}_{N}(s)|^{2}+|Y^{i}_{N}(\kappa_{h}(s))|^{2})^{2}\bigg{)}^{1/2}$
$\displaystyle\;\;\;\;\;\times\bigg{(}\frac{1}{N}\sum\limits_{i=1}^{N}|Y^{i}_{N}(s)-Y^{i}_{N}(\kappa_{h}(s))|^{2}\bigg{)}^{1/2}\Bigg{)},$

where $C>0$ is independent of $h$ and $N$. Squaring both sides, we ascertain

$\displaystyle|\bar{Y}_{N}(s)-\bar{Y}_{N}(\kappa_{h}(s))|^{2}\leq
C\Bigg{(}e^{\frac{2\alpha
K_{u}}{N}\sum_{j=1}^{N}|Y^{j}_{N}(s)|^{2}}\frac{1}{N}\sum\limits_{i=1}^{N}|Y^{i}_{N}(s)-Y^{i}_{N}(\kappa_{h}(s))|^{2}$
$\displaystyle\;\;\;\;+e^{\frac{2\alpha
K_{u}}{N}\sum_{j=1}^{N}(|Y^{j}_{N}(s)|^{2}+|Y^{j}_{N}(\kappa_{h}(s))|^{2})}\times\bigg{(}\frac{1}{N}\sum\limits_{i=1}^{N}(1+|Y^{i}_{N}(s)|^{2}+|Y^{i}_{N}(\kappa_{h}(s))|^{2})^{2}\bigg{)}$
$\displaystyle\;\;\;\;\;\times\bigg{(}\frac{1}{N}\sum\limits_{i=1}^{N}|Y^{i}_{N}(s)-Y^{i}_{N}(\kappa_{h}(s))|^{2}\bigg{)}\Bigg{)}.$
(4.60)

In the similar manner, we can obtain the following bound:

$\displaystyle|\bar{X}_{N}(s)-\bar{Y}_{N}(s)|^{2}\leq
C\Bigg{(}e^{\frac{2\alpha
K_{u}}{N}\sum_{j=1}^{N}|X^{j}_{N}(s)|^{2}}\frac{1}{N}\sum\limits_{i=1}^{N}|X^{i}_{N}(s)-Y^{i}_{N}(s)|^{2}$
$\displaystyle\;\;\;\;+e^{\frac{2\alpha
K_{u}}{N}\sum_{j=1}^{N}(|X^{j}_{N}(s)|^{2}+|Y^{j}_{N}(s)|^{2})}\times\bigg{(}\frac{1}{N}\sum\limits_{i=1}^{N}(1+|X^{i}_{N}(s)|^{2}+|Y^{i}_{N}(s)|^{2})^{2}\bigg{)}$
$\displaystyle\;\;\;\;\;\times\bigg{(}\frac{1}{N}\sum\limits_{i=1}^{N}|X^{i}_{N}(s)-Y^{i}_{N}(s)|^{2}\bigg{)}\Bigg{)},$
(4.61)

where $C>0$ is independent of $h$ and $N$. We substitute (4.59), (4.60) and
(4.61) in (4.58) to get

$\displaystyle\mathbb{E}\big{(}|Y^{i}_{N}(t\wedge\tau^{h}_{R})-X^{i}_{N}(t\wedge\tau^{h}_{R})|^{2}\big{)}\leq
C\mathbb{E}\int_{0}^{t\wedge\tau_{R}^{h}}\big{(}|X^{i}_{N}(s)-Y^{i}_{N}(s)|^{2}\big{)}ds+Ch$
$\displaystyle\;\;+CRe^{4\alpha
K_{u}\sqrt{R}}\bigg{(}\mathbb{E}\int_{0}^{t\wedge\tau_{R}^{h}}\frac{1}{N}\sum\limits_{i=1}^{N}\big{(}|Y^{i}_{N}(s)-Y^{i}_{N}(\kappa_{h}(s))|^{2}\big{)}ds+\mathbb{E}\int_{0}^{t\wedge\tau_{R}^{h}}\frac{1}{N}\sum\limits_{i=1}^{N}\big{(}|X^{i}_{N}(s)-Y^{i}_{N}(s)|^{2}\big{)}ds\bigg{)}$
$\displaystyle\leq
C\int_{0}^{t}\mathbb{E}\big{(}|X^{i}_{N}(s\wedge\tau_{R}^{h})-Y^{i}_{N}(s\wedge\tau_{R}^{h})|^{2}\big{)}ds+Ch+CRe^{4\alpha
K_{u}\sqrt{R}}\int_{0}^{t}\frac{1}{N}\sum\limits_{i=1}^{N}\mathbb{E}\big{(}|Y^{i}_{N}(s)-Y^{i}_{N}(\kappa_{h}(s))|^{2}\big{)}ds$
$\displaystyle\;\;+CRe^{4\alpha
K_{u}\sqrt{R}}\int_{0}^{t}\frac{1}{N}\sum\limits_{i=1}^{N}\mathbb{E}\big{(}|X^{i}_{N}(s\wedge\tau_{R}^{h})-Y^{i}_{N}(s\wedge\tau_{R}^{h})|^{2}\big{)}ds,$

where $C>0$ is independent of $h$, $N$ and $R$. Taking supremum over
$i=1,\dots,N$ and using Lemma 4.9, we obtain

$\displaystyle\sup_{i=1,\dots,N}\mathbb{E}\big{(}|Y^{i}_{N}(t\wedge\tau^{h}_{R})$
$\displaystyle-X^{i}_{N}(t\wedge\tau^{h}_{R})|^{2}\big{)}\leq CRe^{4\alpha
K_{u}\sqrt{R}}h$ $\displaystyle+CRe^{4\alpha
K_{u}\sqrt{R}}\int_{0}^{t}\sup_{i=1,\dots,N}\mathbb{E}\big{(}|Y^{i}_{N}(s\wedge\tau^{h}_{R})-X^{i}_{N}(s\wedge\tau^{h}_{R})|^{2}\big{)}ds\bigg{)},$

where $C$ is independent of $h$, $N$ and $R$. Using Grönwall’s lemma, we get

$\displaystyle\sup_{i=1,\dots,N}\mathbb{E}\big{(}|Y^{i}_{N}(t\wedge\tau^{h}_{R})$
$\displaystyle-X^{i}_{N}(t\wedge\tau^{h}_{R})|^{2}\big{)}\leq CRe^{4\alpha
K_{u}\sqrt{R}}e^{CRe^{4\alpha K_{u}\sqrt{R}}}h\leq Ce^{e^{C_{u}\sqrt{R}}}h,$

where $C>0$ and $C_{u}>0$ are constants independent of $h$, $N$ and $R$.

We choose $R=\frac{1}{C_{u}^{2}}(\ln{(\ln{(h^{-1/2})})})^{2}$. Consequently,
we have

$\displaystyle\sup_{i=1,\dots,N}\mathbb{E}\big{(}|Y^{i}_{N}(t)-X^{i}_{N}(t)|^{2}I_{\Omega_{4}(t)}\big{)}\leq\sup_{i=1,\dots,N}\mathbb{E}\big{(}|Y^{i}_{N}(t\wedge\tau^{h}_{R})$
$\displaystyle-X^{i}_{N}(t\wedge\tau^{h}_{R})|^{2}\big{)}\leq Ch^{1/2},$

where $C>0$ is independent of $h$ and $N$. This implies

$\displaystyle\lim\limits_{h\rightarrow
0}\lim\limits_{N\rightarrow\infty}\sup_{i=1,\dots,N}\mathbb{E}\big{(}|Y^{i}_{N}(t)-X^{i}_{N}(t)|^{2}I_{\Omega_{4}(t)}\big{)}=\lim\limits_{N\rightarrow\infty}\lim\limits_{h\rightarrow
0}\sup_{i=1,\dots,N}\mathbb{E}\big{(}|Y^{i}_{N}(t)-X^{i}_{N}(t)|^{2}I_{\Omega_{4}(t)}\big{)}=0.$
(4.62)

The term (4.57) and the choice of $R$ provide the following estimate:

$\displaystyle\sup_{i=1,\dots,N}\mathbb{E}\big{(}|Y^{i}_{N}(t)-X^{i}_{N}(t)|^{2}I_{\Omega_{3}(t)}\big{)}\leq\frac{C}{(\ln{(\ln{(h^{-1/2})})})^{2}},$

where $C$ is independent of $h$ and $N$. This gives

$\displaystyle\lim\limits_{h\rightarrow