2. Well-posedness of Lévy-type McKean–Vlasov SDEs

We start with some notations used in the sequel. Let $|\cdot|$ and $\langle \cdot, \cdot\rangle$ be the Euclidean vector norm and the scalar product in $\mathbb{R}^d$ , respectively. For a matrix A, we use the Frobenius norm defined as $\|A\|=\sqrt{tr[AA^{\text{T}}]}$ , where $A^{\text{T}}$ represents the transpose of the matrix A. Let $\mathcal{M}(\mathbb{R}^d)$ denote the space of all probability measures on $\mathbb{R}^d$ carrying the usual topology of weak convergence. Furthermore, for $p\geqslant 1$ , let $\mathcal{M}_p(\mathbb{R}^d)$ represent the subspace of $\mathcal{M}(\mathbb{R}^d)$ as follows:

$$\mathcal{M}_p(\mathbb{R}^d)\;:\!=\;\left\{\mu\in\mathcal{M}(\mathbb{R}^d): \mu(|\cdot|^p)\;:\!=\;\int_{\mathbb{R}^d}|x|^p\mu({\textrm{d}} x)\lt\infty\right\}.$$

For $\mu_1, \mu_2\in\mathcal{M}_p(\mathbb{R}^d)$ , the $L^p$ -Wasserstein metric between $\mu_1$ and $\mu_2$ is defined as

$$W_p(\mu_1,\mu_2)\;:\!=\;\inf_{\pi\in\mathscr{C}(\mu_1,\mu_2)}\left(\int_{\mathbb{R}^d\times \mathbb{R}^d}|x-y|^p\pi({\textrm{d}} x,{\textrm{d}} y)\right)^{\frac{1}{p}},$$

where $\mathscr{C}(\mu_1,\mu_2)$ means the collection of all the probability measures whose marginal distributions are $\mu_1$ , $\mu_2$ , respectively. Then $\mathcal{M}_p(\mathbb{R}^d)$ endowed with the above metric is a Polish space.

Let $\delta_x$ be the Dirac delta measure centered at the point $x\in\mathbb{R}^d$ . A direct calculation shows that $\delta_x$ belongs to $\mathcal{M}_p(\mathbb{R}^d)$ for any $x\in\mathbb{R}^d$ . Another remark is that if $\mu_1=\mathscr{L}_X$ and $\mu_2=\mathscr{L}_Y$ are the distributions of the random variables X and Y, respectively, then

$$(W_p(\mu_1,\mu_2))^p\leqslant \int_{\mathbb{R}^d\times \mathbb{R}^d}|x-y|^p\mathscr{L}_{(X,Y)}({\textrm{d}} x,{\textrm{d}} y)=\mathbb{E}|X-Y|^p,$$

where $\mathscr{L}_{(X,Y)}$ denotes the joint distribution of the random vector (X, Y).

Given $T\gt 0$ , let $D([0,T];\;\mathbb{R}^d)$ be the collection of all càdlàg (i.e. right continuous with left limits) functions from [0, T] to $\mathbb{R}^d$ . Note that, at the endpoints of the closed interval [0, T], we stipulate that an element in $D([0,T];\;\mathbb{R}^d)$ is right continuous at 0 and has a left limit at T, respectively. For $1\leqslant p \lt\infty$ , we use $L^p(\Omega;\;\mathbb{R}^d)$ to denote the family of all $\mathbb{R}^d$ -valued random variables Y such that $\mathbb{E}|Y|^p\lt\infty$ . Similarly, we denote by $L^p(\Omega;\;D([0,T];\;\mathbb{R}^d))$ the subspace of all $D([0,T];\;\mathbb{R}^d)$ -valued random variables X that satisfy $\mathbb{E}[\sup_{0\leqslant t\leqslant T}|X(t)|^p]\lt\infty.$ Then, we present the following proposition.

We recall several useful inequalities that will be employed frequently throughout this paper. The first is Young’s inequality, stated as

(2.1) \begin{equation}ab\leqslant\epsilon \frac{a^p}{p}+\epsilon^{-\frac{q}{p}}\frac{b^q}{q}, \quad \mbox{for all }\epsilon, a, b\gt 0, \hbox{ where } p\gt 1,\ \frac{1}{p}+\frac{1}{q}=1.\end{equation}

Next, we list two elementary inequalities:

(2.2) \begin{equation}\left(\sum_{i=1}^k|a_i|\right)^l\leqslant\left(k\max_{1\leqslant i\leqslant k}|a_i|\right)^l\leqslant k^l\sum_{i=1}^k|a_i|^l, \quad \mbox{for all }l\gt 0, \ a_i\in\mathbb{R},\ k\in\mathbb{N},\end{equation}

and

(2.3) \begin{equation} (a+b+c)^{l}\leqslant 3^{l-1}(|a|^l+|b|^l+|c|^l), \quad \mbox{for all }l\geqslant1,\ a,b,c\in\mathbb{R}.\end{equation}

In addition, noting that stochastic integrals w.r.t. compensated Poisson random measures are local martingales, we require the following preparatory results to proceed with the analysis.

Proof. We emphasize that this proposition can be viewed as a special instance of Novikov’s result, which is rigorously established in [Reference Novikov31, Theorem 1]; see also [Reference Kühn and Schilling21, Theorem 4.20] for applications of Novikov’s result and its relation to variants of the Burkholder–Davis–Gundy (BDG) inequality. For the case $p\geqslant 2$ , a proof of the conclusion (i) based on the BDG inequality for local martingales is presented in [Reference Mikulevicius and Pragarauskas29, Lemma 1], whereas an alternative approach utilizing Itô’s formula (applied to $x\mapsto x^p$ ) and Doob’s martingale inequality can be found in [Reference Applebaum1, Theorem 4.4.23]. For the case $1\leqslant p\leqslant 2$ , the conclusion (ii) is stated in [Reference Dareiotis, Kumar and Sabanis7, Lemma 2.1] without proof. To ensure clarity for readers and maintain mathematical rigor, we provide a detailed proof for this case here. For convenience, we define the processes

$$A_t\;:\!=\;\int_0^t\int_U|H(s,z)|^2\nu({\textrm{d}} z)\, {\textrm{d}} s$$

and

$$J_t\;:\!=\;\int_0^t\int_U(A_s+\varepsilon)^{-\frac{2-p}{4}}H(s,z)\tilde{N}({\textrm{d}} s,{\textrm{d}} z)$$

for $0\leqslant t\leqslant T$ , where $1\leqslant p\leqslant 2$ and $\varepsilon\gt0$ is a small parameter.

On the one hand, applying the integration by parts formula (also referred to Itô’s product formula; see [Reference Applebaum1, Theorem 4.4.13]), we obtain

$$J_t\cdot(A_t+\varepsilon)^{\frac{2-p}{4}}=\int_0^t J_s\,{\textrm{d}} \left[(A_s+\varepsilon)^{\frac{2-p}{4}}\right]+\int_0^t\int_UH(s,z)\tilde{N}({\textrm{d}} s,{\textrm{d}} z).$$

Noting that $(A_t+\varepsilon)^{\frac{2-p}{4}}$ is a nonnegative and nondecreasing process, we deduce the bound

\begin{align}|I_t|\leqslant |J_t|(A_t+\varepsilon)^{\frac{2-p}{4}}+\int_0^t|J_s|\,{\textrm{d}} \left[(A_s+\varepsilon)^{\frac{2-p}{4}}\right]\leqslant 2\left(\sup_{0\leqslant s\leqslant t} |J_s|\right)(A_t+\varepsilon)^{\frac{2-p}{4}}.\notag\end{align}

Since this estimate holds for all $t\geqslant 0$ and the right-hand side remains nondecreasing, it follows that

$$\sup_{0\leqslant s\leqslant t}|I_s|^p\leqslant 2^p\left(\sup_{0\leqslant s\leqslant t} |J_s|\right)^p(A_t+\varepsilon)^{\frac{(2-p)p}{4}}.$$

By Hölder inequality [Reference Mao26, p. 5], we further derive

(2.6) \begin{align}\mathbb{E}\left(\sup_{0\leqslant s\leqslant t}|I_s|^p\right)&\leqslant 2^p\mathbb{E}\left[\left(\sup_{0\leqslant s\leqslant t} |J_s|\right)^p(A_t+\varepsilon)^{\frac{(2-p)p}{4}}\right]\notag\\&\leqslant 2^p\left[\mathbb{E}\left(\sup_{0\leqslant s\leqslant t} |J_s|\right)^{p\cdot\frac{2}{p}}\right]^{\frac{p}{2}}\left[\mathbb{E}(A_t+\varepsilon)^{\frac{p(2-p)}{4}\cdot\frac{2}{2-p}}\right]^{\frac{2-p}{2}}\notag\\&= 2^p\left[\mathbb{E}\left(\sup_{0\leqslant s\leqslant t} |J_s|\right)^2\right]^{\frac{p}{2}}\left[\mathbb{E}(A_t+\varepsilon)^{\frac{p}{2}}\right]^{\frac{2-p}{2}}.\end{align}

On the other hand, applying the Itô isometry for integrals with respect to compensated Poisson random measures yields

\begin{equation}\mathbb{E}|J_t|^2=\mathbb{E}\int_0^t\int_U(A_s+\varepsilon)^{\frac{p-2}{2}}|H(s,z)|^2\nu({\textrm{d}} z)\,{\textrm{d}} s=\mathbb{E}\int_0^t(A_s+\varepsilon)^{\frac{p-2}{2}}\,{\textrm{d}} A_s\leqslant \frac{2}{p}\mathbb{E}(A_t+\varepsilon)^{\frac{p}{2}}.\notag\end{equation}

By Doob’s martingale inequality [Reference Applebaum1, Theorem 2.1.5], we then obtain

(2.7) \begin{equation} \mathbb{E}\left(\sup_{0\leqslant s\leqslant t} |J_s|^2\right)\leqslant4\mathbb{E}|J_t|^2\leqslant\frac{8}{p}\mathbb{E}(A_t+\varepsilon)^{\frac{p}{2}}.\end{equation}

Combining (2.6) and (2.7), and using the fact that $\mathbb{E}\left(\sup_{0\leqslant s\leqslant t} |J_s|\right)^2=\mathbb{E}\left(\sup_{0\leqslant s\leqslant t} |J_s|^2\right)$ , we arrive at

$$\mathbb{E}\left(\sup_{0\leqslant s\leqslant t}|I_s|^p\right) \leqslant\left(\frac{32}{p}\right)^{\frac{p}{2}}\mathbb{E}[\varepsilon+A_t]^{\frac{p}{2}}.$$

The result follows by letting $\varepsilon\to0$ . This completes the proof.

2.1. Formulation of the well-posedness results

This section is dedicated to establishing the existence and uniqueness theorem for the solutions of the d-dimensional Lévy-type McKean–Vlasov SDEs, i.e.

(2.8) \begin{align}{\textrm{d}} X(t)&=b\left(t,X(t-\!),\mathscr{L}_{X(t)}\right) \,{\textrm{d}} t +\sigma\left(t,X(t-\!),\mathscr{L}_{X(t)}\right)\,{\textrm{d}} W(t)\nonumber \\ &\quad +\int_{U}h\left(t,X(t-\!),\mathscr{L}_{X(t)},z\right)\tilde{N}({\textrm{d}} t,{\textrm{d}} z)\end{align}

for $t\in[0,T]$ with initial condition $X(0)=x_0$ . The functions b, $\sigma$ , and h are defined as follows:

\begin{align*} b:[0,T]\times\mathbb{R}^d\times M(\mathbb{R}^d)\to\mathbb{R}^d, \quad &\sigma:[0,T]\times\mathbb{R}^d\times M(\mathbb{R}^d)\to\mathbb{R}^{d\times m},\nonumber\\ & h:[0,T]\times\mathbb{R}^d\times M(\mathbb{R}^d)\times U\to\mathbb{R}^d,\end{align*}

where b, $\sigma$ , and h are Borel measurable functions. We now proceed by providing the precise definition of a solution to (2.8).

Definition 1. We say that (2.8) admits a unique strong solution if there exists an $\{\mathcal{F}_t\}_{0\leqslant t\leqslant T}$ -adapted $\mathbb{R}^d$ -valued càdlàg stochastic process $(X(t))_{t\in[0,T]}$ such that

1. (i) $X(t)=x_0+\int_0^t b\left(s,X(s-\!),\mathscr{L}_{X(s)}\right) \,{\textrm{d}} s+\sigma\left(s,X(s-\!),\mathscr{L}_{X(s)}\right)\,{\textrm{d}} W(s)+\int_{U}h\left(s,X(s-\!),\mathscr{L}_{X(s)},z\right)\tilde{N}({\textrm{d}} s,{\textrm{d}} z)$ , $t\in[0,T]$ , $\mathbb{P}$ -almost surely;

2. (ii) if $Y=(Y(t))_{t\in[0,T]}$ is another solution with $Y(0)=x_0$ , then $\mathbb{P}(X(t)=Y(t) \hbox{ for all } t\in[0,T])=1.$

Assume that there exists a constant $\kappa\geqslant2$ such that the following assumptions hold.

Assumption 1. (One-sided locally Lipschitz condition on the state variable.) For every $R\gt0$ , there exists a positive constant $L_R$ such that for any $t\in[0,T]$ , $x,y\in\mathbb{R}^d$ with $|x|\vee|y|\leqslant R$ , and $\mu\in\mathcal{M}_2(\mathbb{R}^d)$ ,

\begin{align} &\langle x-y, b(t,x,\mu)-b(t,y,\mu)\rangle\vee\|\sigma(t,x,\mu)-\sigma(t,y,\mu)\|^2 \notag\\&\quad\vee\int_U\left|h(t,x,\mu,z)-h(t,y,\mu,z)\right|^2\nu({\textrm{d}} z)\leqslant L_R|x-y|^2.\notag\end{align}

Here, the symbol ‘ $\vee$ ’ denotes the maximum of the multiple terms.

Assumption 2. (Globally Lipschitz condition on the measure.) There exists a positive constant L such that, for any $t\in[0,T]$ , $x\in\mathbb{R}^d$ , and $\mu_1,\mu_2\in\mathcal{M}_2(\mathbb{R}^d)$ ,

\begin{align} &|b(t,x,\mu_1)-b(t,x,\mu_2)|^2+\|\sigma(t,x,\mu_1)-\sigma(t,x,\mu_2)\|^2 \notag\\&\quad+\int_U\left|h(t,x,\mu,z)-h(t,y,\mu,z)\right|^2\nu({\textrm{d}} z)\leqslant LW_2^2(\mu_1,\mu_2).\notag\end{align}

Assumption 3. (Continuity.) For any $t\in[0,T]$ , $b(t,\cdot,\cdot), \sigma(t,\cdot,\cdot)$ , and $\int_Uh(t,\cdot,\cdot,z)\nu({\textrm{d}} z)$ are continuous on $\mathbb{R}^d\times\mathcal{M}_2(\mathbb{R}^d).$

Assumption 4. (One-sided linear and global linear growth condition.) There exists a positive constant K such that, for any $t\in[0,T]$ , $x\in\mathbb{R}^d$ , and $\mu\in\mathcal{M}_2(\mathbb{R}^d)$ ,

$$\langle x, b(t,x,\mu)\rangle\vee \|\sigma(t,x,\mu)\|^2\vee \int_U|h(t,x,\mu,z)|^2\nu({\textrm{d}} z)\leqslant K(1+|x|^2+W_2^2(\mu,\delta_0)).$$

Assumption 5. ( $\kappa$ -order growth condition on the drift coefficient.) There exists a positive constant $K_1$ such that, for any $t\in[0,T]$ , $x\in\mathbb{R}^d$ , and $\mu\in\mathcal{M}_{2}(\mathbb{R}^d)$ ,

$$|b(t,x,\mu)|^2\leqslant K_1(1+|x|^{\kappa}+W_2^{\kappa}(\mu,\delta_0)).$$

Assumption 6. (r-order moment condition for the initial data.) Consider $x_0\in L^r(\Omega;\;\mathbb{R}^d)$ for some $r\geqslant \max\{\kappa^2/2,4\}$ , i.e. $\mathbb{E}|x_0|^r\lt\infty.$

Assumption 7. (Additional growth conditions and Lipschitz type conditions on the jump coefficient h.) There exists a positive $K_2$ such that, for any $t\in[0,T]$ , $x\in\mathbb{R}^d$ , and $\mu\in\mathcal{M}_{2}(\mathbb{R}^d)$ ,

$$\int_U|h(t,x,\mu,z)|^r\nu({\textrm{d}} z)\leqslant K_2(1+|x|^r+W_2^{r}(\mu,\delta_0)).$$

In addition, if $\kappa \gt 2$ , there exist constants $K_3, L^{\prime}\gt 0$ such that, for any $t\in[0,T]$ , $x,y\in\mathbb{R}^d$ , and $\mu,\mu_1,\mu_2\in\mathcal{M}_{2}(\mathbb{R}^d)$ ,

$$\int_U|h(t,x,\mu,z)|^\kappa\nu({\textrm{d}} z)\leqslant K_3(1+|x|^\kappa+W_2^{\kappa}(\mu,\delta_0)),$$

$$\int_U\left|h(t,x,\mu_1,z)-h(t,x,\mu_2,z)\right|^{\kappa}\nu({\textrm{d}} z)\leqslant L^{\prime}W_2^{\kappa}(\mu_1,\mu_2),$$

and for every $R\gt0$ , there exists a constant $L_R^{\prime}\gt 0$ such that for any $t\in[0,T]$ , $x,y\in\mathbb{R}^d$ with $|x|\vee|y|\leqslant R$ , and $\mu\in\mathcal{M}_2(\mathbb{R}^d)$ ,

$$\int_U\left|h(t,x,\mu,z)-h(t,y,\mu,z)\right|^{\kappa}\nu({\textrm{d}} z)\leqslant L_R^{\prime}|x-y|^{\kappa}.$$

The main result of this section is stated as follows.

Theorem 1. (Well-posedness.) Let Assumptions 1–7 be satisfied. Then (2.8) admits a unique strong solution $(X(t))_{t\in[0,T]}$ $\in L^{\kappa}(\Omega;\;\mathbb{R}^d)$ with the initial value $X(0)=x_0$ , where $\kappa\geqslant2$ . Moreover, the following estimate holds:

(2.9) \begin{equation}\mathbb{E}\left[\sup_{0\leqslant t\leqslant T}|X(t)|^r\right]\leqslant C,\end{equation}

where $C\;:\!=\;C(T,r,\mathbb{E}|x_0|^r)$ is a positive constant. Here, $r\geqslant \max\{\kappa^2/2,4\}$ .

Remark 1. We emphasize that the conditions in Assumptions 1–7 are carefully chosen, and the results in Theorem 1 are broadly applicable.

1. (i) The one-sided locally Lipschitz condition in Assumption 1 is weaker than the classical locally Lipschitz condition. In fact, it is clear that the locally Lipschitz condition implies the one-sided locally Lipschitz condition (via the mean value inequality). However, the converse is false. For example, consider $b(t,x,\mu)=x^3-x^{\frac{1}{3}}+t+\int_{\mathbb{R}} z\mu({\textrm{d}} z)$ in $\mathbb{R}$ . For $|x|\vee|y|\leqslant R$ , we have

   \begin{align} \langle x-y, b(t,x,\mu)-b(t,y,\mu)\rangle &=|x-y|^2\left(x^2+xy+y^2\right)-(x-y)\left(x^{\frac{1}{3}}-y^{\frac{1}{3}}\right) \notag\\&\leqslant 3R^2|x-y|^2, \notag \end{align}
   since $(x-y)(x^{\frac{1}{3}}-y^{\frac{1}{3}})\geqslant0$ for all x, y. Thus, b is one-sided locally Lipschitz but not locally Lipschitz.

2. (ii) In contrast to the one-sided (globally) Lipschitz condition in the recent paper [Reference Neelima, Kumar, Dos Reis and Reisinger30], which asserts that there exists a constant $C\gt 0$ such that for any $x,y\in\mathbb{R}^d$ and $\mu\in\mathcal{M}_2(\mathbb{R}^d),$

   \begin{align} &\langle x-y, b(t,x,\mu)-b(t,y,\mu)\rangle+\|\sigma(t,x,\mu)-\sigma(t,y,\mu)\|^2\notag\\ &\quad +\int_U\left|h(t,x,\mu,z)-h(t,y,\mu,z)\right|^2\nu({\textrm{d}} z)\leqslant C|x-y|^2,\notag \end{align}
   the one-sided locally Lipschitz condition in Assumption 1 is expressed using the operation ‘ $\vee$ ’ instead of ‘ $+$ ’. This makes the condition in Assumption 1 weaker in some cases. For instance, consider b as a one-sided locally Lipschitz function and $\sigma=h=x$ with $\nu(U)\lt\infty$ . In this case, Assumption 1 holds, but the one-sided (globally) Lipschitz condition in [Reference Neelima, Kumar, Dos Reis and Reisinger30] is not satisfied.

3. (iii) The result simplifies to the case of pure Brownian motion when $h\equiv 0$ . In contrast to the Brownian motion model considered in [Reference Li, Mao, Song, Wu and Yin22], where the drift coefficient is required to satisfy a linear growth condition, the present framework imposes only a one-sided linear growth condition on the drift coefficient b. Furthermore, b is permitted to exhibit polynomial growth w.r.t. the state variable, as specified in Assumptions 4 and 5.

4. (iv) Referring to [Reference Sato35, Theorem 25.3], when the jump coefficient h is a submultiplicative function with respect to z, the growth conditions in Assumption 7 can be interpreted as requiring that the jump measure $[\nu]_{U}$ has a bounded $|h|^r$ -moment or, equivalently, the associated Lévy motion has bounded $|h|^r$ -moments, for every $t\in[0,T]$ , $x\in\mathbb{R}^d$ , and $\mu\in\mathcal{M}_{2}(\mathbb{R}^d)$ , where $r\geqslant \max\{\kappa^2/2,4\}$ and $\kappa\geqslant 2$ . In particular, the associated Lévy motion can be said to have bounded r-order moments when $h(t,x,\mu,z)=z$ . We remark that while Brownian motion can be considered a 2-stable Lévy motion, our assumptions exclude applications involving jump measures associated with $\alpha$ -stable Lévy motions for $0\lt\alpha\lt 2$ . This exclusion arises because such $\alpha$ -stable Lévy motions process r-order moments only for $r\lt\alpha$ ; see [Reference Sato35, Example 25.10]. For a recent study addressing the strong well-posedness of McKean–Vlasov SDEs driven by Lévy noise with finite moments of order $\beta\in[1,2]$ , we refer to [Reference Cavallazzi6]. However, it should be pointed out that the assumptions in [Reference Cavallazzi6] regarding the coefficients with respect to both the space variable and the measure remain within the globally Lipschitz framework.

2.2. Euler-type approximation and auxiliary lemmas

A key aspect of our approach to proving Theorem 1 is the construction of an Eulerlike sequence for the McKean–Vlasov SDEs (2.8). Once we demonstrate that this sequence is Cauchy in an appropriate complete space (specifically, $L^{\kappa}(\Omega;\;D([0,T];\;\mathbb{R}^d))$ , as is shown later), we can conclude that there exists a limiting process, which is indeed the desired solution to (2.8).

To this end, let $T\gt 0$ be given, and consider the equidistant partition of the interval [0, T]. For any integer $n\geqslant1$ , define $h_n=\frac{T}{n}$ and $t_k^n=kh_n$ , $k=0,1,\ldots, n$ . For a fixed k ( $0\leqslant k \leqslant n-1$ ) and $t\in(t_k^n, t_{k+1}^n]$ , we analyze the following approximation:

(2.10) \begin{align}{\textrm{d}} X^{(n)}(t)&=b\left(t,X^{(n)}(t-\!),\mu^{(n)}_{t_k^n}\right)\,{\textrm{d}} t+\sigma\left(t,X^{(n)}(t-\!),\mu^{(n)}_{t_k^n}\right)\,{\textrm{d}} W(t)\notag\\ &\quad +\int_{U}h\left(t,X^{(n)}(t-\!),\mu^{(n)}_{t_k^n},z\right)\tilde{N}({\textrm{d}} t,{\textrm{d}} z),\end{align}

where $\mu^{(n)}_{t_k^n}=\mathscr{L}_{X^{(n)}(t_k^n)}$ denotes the law of $X^{(n)}(t_k^n)$ . Observe that for each fixed k, if the initial value $X^{(n)}(t_k^n)$ and the distribution $\mu^{(n)}_{t_k^n}$ (at the left endpoint $t_k^n$ ) are known, then (2.10) reduces to a standard SDE that is independent of the law of $X^{(n)}(t)$ . We now establish, by induction, the existence and uniqueness of the solution to (2.10).

In fact, for $k=0$ and $t\in[0,t_1^n]$ , the distribution is $ \mu^{(n)}_{0}=\mathscr{L}_{X^{(n)}(0)}=\mathscr{L}_{x_0}$ . Applying Assumptions 1 and 4, we observe that the coefficients in (2.10) (with $k=0$ ) satisfy

\begin{align}&\left\langle x-y, b\left(t,x,\mu^{(n)}_{0}\right)-b\left(t,y,\mu^{(n)}_{0}\right)\right\rangle+\left\|\sigma\left(t,x,\mu^{(n)}_{0}\right)-\sigma\left(t,y,\mu^{(n)}_{0}\right)\right\|^2\notag\\&\quad +\int_U\left|h\left(t,x,\mu^{(n)}_{0},z\right)-h\left(t,y,\mu^{(n)}_{0},z\right)\right|^2\nu({\textrm{d}} z)\leqslant 3L_R|x-y|^2\notag\end{align}

and

\begin{align} &\left\langle x, b\left(t,x,\mu^{(n)}_{0}\right)\right\rangle+\left\|\sigma\left(t,x,\mu^{(n)}_{0}\right)\right\|^2+\int_U\left|h\left(t,x,\mu^{(n)}_{0},z\right)\right|^2\nu({\textrm{d}} z)\notag\\ &\quad\leqslant 3K\left(1+|x|^2+W_2^2\left(\mu^{(n)}_{0},\delta_0\right)\right) \leqslant3K\left(1+|x|^2\right)\left(1+\mathbb{E}\left|X^{(n)}(0)\right|^2\right).\notag\end{align}

Referring to [Reference Majka25, Theorem 1.1], it admits a unique solution on $[0,t_1^n]$ . Furthermore, by Assumption 5, it follows that for $r\geqslant\max\{\frac{\kappa^2}{2},4\}$ , there exits a positive constant C such that

\begin{equation}\mathbb{E}\left[\sup_{0\leqslant t\leqslant t_1^n}\left|X^{(n)}(t)\right |^r \right]\leqslant C\left(1+\mathbb{E}\left|X^{(n)}(0)\right|^r\right)\!, \notag\end{equation}

whose proof is quite similar to Lemma 1, and we omit the details here. Therefore, we can define $X^{(n)}(t_1^n)$ (which satisfies $\mathbb{E}|X^{(n)}(t_1^n)|^r\lt\infty$ ) and $\mu^{(n)}_{t_1^n}=\mathscr{L}_{X^{(n)}(t_1^n)}$ .

For $k=1$ and $t\in(t_1^n,t_2^n]$ , we can use $(X^{(n)}(t_1^n),\mu^{(n)}_{t_1^n})$ in place of $(X^{(n)}(0),\mu^{(n)}_{0})$ and repeat the above procedure. Inductively, for any $k=0,1,\ldots, n-1$ and $t\in(t_k^n, t_{k+1}^n]$ , we obtain the existence and uniqueness of the solution to the SDE (2.10) as well as the corresponding estimate

(2.11) \begin{equation}\mathbb{E}\left[\sup_{t_k^n\leqslant t\leqslant t_{k+1}^n}\left|X^{(n)}(t)\right|^r\right]\leqslant C\left(1+\mathbb{E}\left|X^{(n)}(t_k^n)\right|^r\right)\!,\end{equation}

by similar arguments.

At this point, we define by $[t]_n=t_k^n$ for all $t\in (t_k^n,t_{k+1}^n]$ , where $k=0,1,\ldots, n-1$ . Then, for $t\in[0,T]$ , we introduce the following approximating SDE

(2.12) \begin{align}{\textrm{d}} X^{(n)}(t)&=b\left(t,X^{(n)}(t-\!),\mu^{(n)}_{[t]_{n}}\right)\,{\textrm{d}} t+\sigma\left(t,X^{(n)}(t-\!),\mu^{(n)}_{[t]_{n}}\right)\, {\textrm{d}} W(t)\notag\\ &\quad +\int_{U}h\left(t,X^{(n)}(t-\!),\mu^{(n)}_{[t]_{n}},z\right)\tilde{N}({\textrm{d}} t,{\textrm{d}} z),\end{align}

with the initial value $X^{(n)}(0)=x_0$ , where $\mu^{(n)}_{[t]_{n}}=\mathscr{L}_{X^{(n)}([t]_{n})}$ . According to the previously presented procedures and results for (2.10), we conclude that there exists a unique solution to (2.12). In fact, for each $n\geqslant 1$ and $t\in[0,T]$ , we can always find a certain $k_\ast$ ( $0\leqslant k_\ast \leqslant n-1$ ) such that $t\in (t_{k_\ast}^n,t_{{k_\ast}+1}^n]$ . Then, the solution to (2.12) can be written as

\begin{align}X^{n}(t)&=x_0+\sum_{k=0}^{k_\ast}\int_{t_k^n}^{t_{k+1}^n \wedge t}b\left(t,X^{(n)}(s-\!),\mu^{(n)}_{t_k^n}\right)\,{\textrm{d}} s+\sigma\left(s,X^{(n)}(s-\!),\mu^{(n)}_{t_k^n}\right)\, {\textrm{d}} W(s)\notag\\&\quad +\int_{U}h\left(s,X^{(n)}(s-\!),\mu^{(n)}_{t_k^n},z\right)\tilde{N}({\textrm{d}} s,{\textrm{d}} z) \notag\\&=X^{n}(t_1^n)+\sum_{k=1}^{k_\ast}\int_{t_k^n}^{t_{k+1}^n \wedge t}b\left(t,X^{(n)}(s-\!),\mu^{(n)}_{t_k^n}\right)\, {\textrm{d}} s+\sigma\left(s,X^{(n)}(s-\!),\mu^{(n)}_{t_k^n}\right)\, {\textrm{d}} W(s)\notag\\&\quad +\int_{U}h\left(s,X^{(n)}(s-\!),\mu^{(n)}_{t_k^n},z\right)\tilde{N}({\textrm{d}} s,{\textrm{d}} z) \notag\\&\quad \cdots \;\;\;\; \cdots\notag\\&=X^{n}(t_{k_\ast}^n)+\int_{t_{k_\ast}^n}^{t}b\left(t,X^{(n)}(s-\!),\mu^{(n)}_{t_{k_\ast}^n}\right)\, {\textrm{d}} s+\sigma\left(s,X^{(n)}(s-\!),\mu^{(n)}_{t_{k_\ast}^n}\right)\, {\textrm{d}} W(s)\notag\\&\quad +\int_{U}h\left(s,X^{(n)}(s-\!),\mu^{(n)}_{t_{k_\ast}^n},z\right)\tilde{N}({\textrm{d}} s,{\textrm{d}} z),\notag\end{align}

and it is well-defined based on the results for (2.10) with $k=0,1,\ldots,k_\ast$ . Moreover, we have the following estimate

(2.13) \begin{align}\mathbb{E}\left[\sup_{0\leqslant t\leqslant T}\left|X^{(n)}(t)\right|^r\right]&=\mathbb{E}\left[\max_{0\leqslant k\leqslant n-1}\sup_{t_k^n\leqslant t\leqslant t_{k+1}^{n}}\left|X^{(n)}(t)\right|^r\right]\leqslant \sum_{k=0}^{n-1}\mathbb{E}\left[\sup_{t_k^n\leqslant t\leqslant t_{k+1}^{n}}\left|X^{(n)}(t)\right|^r\right]\notag\\&\leqslant C(n)\lt\infty.\end{align}

Under Assumption 6, which requires that the initial data $x_0$ satisfies $\mathbb{E}|x_0|^r\lt\infty$ with $r\geqslant\max\{\frac{\kappa^2}{2},4\}\geqslant\kappa$ , we deduce that $X^{(n)}\in L^{r}(\Omega;\;D([0,T];\;\mathbb{R}^d))\subset L^{\kappa}(\Omega;\;D([0,T];\;\mathbb{R}^d))$ . Hence, the stochastic processes $\{X^{(n)}(t)\}_{n\geqslant1}$ given by (2.12) form a sequence in $L^{\kappa}(\Omega;\;D([0,T];\;\mathbb{R}^d))$ . To demonstrate that this sequence is Cauchy, we require the following two auxiliary lemmas.

Lemma 1. (Uniform boundedness property.) Under Assumptions 4, 6, and 7, for any $T\gt 0$ , there exists a positive constant $C_r$ (independent of n) such that

(2.14) \begin{equation}\mathbb{E}\left[\sup_{0\leqslant t\leqslant T}\left|X^{(n)}(t)\right|^r\right]\leqslant C_r.\end{equation}

Proof. For $r\geqslant\max\{\frac{\kappa^2}{2},4\}$ and $t\in[0,T]$ , applying Itô’s formula [Reference Applebaum1, Theorem 4.4.7] to $|x|^r$ , along with the identity $ \nu({\textrm{d}} z) dt= N({\textrm{d}} t, {\textrm{d}} z) - \tilde{N}({\textrm{d}} t, {\textrm{d}} z)$ , yields that

(2.15) \begin{align}\left|X^{(n)}(t)\right|^r&=|x_0|^r+r\int_0^{t}\left|X^{(n)}(s-\!)\right|^{r-2}\left\langle X^{(n)}(s-\!), b\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_n}\right)\right\rangle\, {\textrm{d}} s\notag\\&\quad +\frac{r}{2}\int_0^{t}\left|X^{(n)}(s-\!)\right|^{r-2} \left\|\sigma(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_n})\right\|^2\, {\textrm{d}} s\notag\\&\quad +\frac{r(r-2)}{2}\int_0^{t}\left|X^{(n)}(s-\!)\right|^{r-4} \left\|(X^{(n)}(s-\!))^{\text{T}}\sigma\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_n}\right)\right\|^2\, {\textrm{d}} s\notag\\&\quad +r\int_0^{t}\left|X^{(n)}(s-\!)\right|^{r-2}\left\langle X^{(n)}(s-\!), \sigma\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_n}\right)\, {\textrm{d}} W(s)\right\rangle\notag\\&\quad +r\int_0^{t}\int_U\left|X^{(n)}(s-\!)\right|^{r-2}\left\langle X^{(n)}(s-\!), h\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_n},z\right)\right\rangle \tilde{N}({\textrm{d}} s,{\textrm{d}} z)\notag\\&\quad +\int_0^{t}\int_U\Big[\left|X^{(n)}(s-\!)+h\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_n},z\right)\right|^{r}\notag\\&\quad -\left|X^{(n)}(s-\!)\right|^r-r\left|X^{(n)}(s-\!)\right|^{r-2}\!\cdot\left\langle X^{(n)}(s-\!), h\!\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_n},z\right)\right\rangle\!\Big] N({\textrm{d}} s,{\textrm{d}} z).\end{align}

By virtue of Assumption 4, Young’s inequality (2.1) (with $\epsilon=1$ , $p=\frac{r}{r-2}$ , and $q=\frac{r}{2}$ ), Hölder inequality and the elementary inequality (2.3) (with $l=\frac{r}{2}$ ), one can estimate the second term of (2.15) by

(2.16) \begin{align}&rK\int_0^{t}\left|X^{(n)}(s-\!)\right|^{r-2}\left(1+\left|X^{(n)}(s-\!)\right|^2+\mathbb{E}\left|X^{(n)}([s]_{n}-)\right|^2\right)\, {\textrm{d}} s\notag\\&\leqslant (r-2)K\int_0^t\left|X^{(n)}(s-\!)\right|^r \, {\textrm{d}} s+2K\int_0^t\left(1+\left|X^{(n)}(s-\!)\right|^2+\mathbb{E}\left|X^{(n)}([s]_{n}-)\right|^2\right)^{\frac{r}{2}} \, {\textrm{d}} s\notag\\&\leqslant (r-2)K\int_0^t\left|X^{(n)}(s-\!)\right|^r \, {\textrm{d}} s+2\cdot3^{\frac{r}{2}-1}K\int_0^t\left(1+\left|X^{(n)}(s-\!)\right|^r+\mathbb{E}\left|X^{(n)}([s]_{n}-)\right|^r\right)\, {\textrm{d}} s.\end{align}

Analogously, the third and fourth terms of (2.15) can be estimated by

(2.17) \begin{equation}\frac{r-2}{2}K\int_0^t\left|X^{(n)}(s-\!)\right|^r \, {\textrm{d}} s+3^{\frac{r}{2}-1}K\int_0^t\left(1+\left|X^{(n)}(s-\!)\right|^r+\mathbb{E}\left|X^{(n)}([s]_{n}-)\right|^r\right)\, {\textrm{d}} s,\end{equation}

and

(2.18) \begin{equation}\frac{(r-2)^2}{2}K\int_0^t\left|X^{(n)}(s-\!)\right|^r \, {\textrm{d}} s+3^{\frac{r}{2}-1}(r-2)K\int_0^t\left(1+\left|X^{(n)}(s-\!)\right|^r+\mathbb{E}\left|X^{(n)}([s]_{n}-)\right|^r\right)\, {\textrm{d}} s,\end{equation}

respectively. Furthermore, note that the map $y\to|y|^r$ is of class $C^2$ and the remainder formula for $|y|^r$ gives

(2.19) \begin{align}|y|^r-|b|^r-r|b|^{r-2}\langle b, y-b\rangle&\leqslant C_1\int_0^1|y-b|^2|b+\theta(y-b)|^{r-2} \, {\textrm{d}} \theta \notag\\&\leqslant C_1(|b|^{r-2}|y-b|^2+|y-b|^r),\end{align}

for any $y, b\in\mathbb{R}^d$ . The last term on the right-hand side of (2.15) can thus be estimated as

(2.20) \begin{equation}C_1\!\int_0^{t}\!\!\int_U\left(\!\left|X^{(n)}(s-\!)\right|^{r-2}\left|h\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_n},z\right)\right|^2\!+\left|h\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_n},z\right)\right|^{r}\right)N({\textrm{d}} s,{\textrm{d}} z).\end{equation}

Denote the above upper-bound (2.20) by $N_t$ . Substituting (2.16)–(2.20) into (2.15), taking the supremum over [0, u] for $u\in[0,T]$ and then taking expectations gives that

(2.21) \begin{align}\mathbb{E}\sup_{0\leqslant t\leqslant u}|X^{(n)}(t)|^r&\leqslant\mathbb{E}|x_0|^r+\mathbb{E}\sup_{0\leqslant t\leqslant u}|M_t|+\mathbb{E}\sup_{0\leqslant t\leqslant u}|N_t|\notag\\&\quad +\frac{(r+1)(r-2)}{2}K\int_0^u\mathbb{E}\left|X^{(n)}(s-\!)\right|^r \, {\textrm{d}} s\notag\\&\quad +3^{\frac{r}{2}-1}(r+1)K\int_0^u\left(1+\mathbb{E}\left|X^{(n)}(s-\!)\right|^r+\mathbb{E}\left|X^{(n)}([s]_{n}-)\right|^r\right)\, {\textrm{d}} s,\end{align}

where

\begin{align} M_t:&=r\int_0^{t}\left|X^{(n)}(s-\!)\right|^{r-2}\left\langle X^{(n)}(s-\!), \sigma\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_n}\right)\, {\textrm{d}} W(s)\right\rangle\notag\\ &\quad +r\int_0^{t}\int_U\left|X^{(n)}(s-\!)\right|^{r-2}\left\langle X^{(n)}(s-\!), h\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_n},z\right)\right\rangle \tilde{N}({\textrm{d}} s,{\textrm{d}} z)\notag\end{align}

is indeed a local martingale. On the one hand, by the BDG inequality (for the Brownian case) [Reference Mao26, Theorem 7.3 in Chapter 1] and the inequality (2.5) (with $p=1$ ) in Proposition 2, there exists a constant $C_2\gt 0$ such that

\begin{align}\mathbb{E}\sup_{0\leqslant t\leqslant u}|M_t|&\leqslant C_2r\mathbb{E}\bigg[\int_0^{u}\left|X^{(n)}(s-\!)\right|^{2r-2}\left\|\sigma\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_n}\right)\right\|^2\, {\textrm{d}} s\bigg]^{\frac{1}{2}}\notag\\&\quad +C_2r\mathbb{E}\bigg[\int_0^{u}\int_U\left|X^{(n)}(s-\!)\right|^{2r-2}\left|h\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_n}\right)\right|^2\nu({\textrm{d}} z)\, {\textrm{d}} s\bigg]^{\frac{1}{2}}\notag\\&\leqslant C_2r\mathbb{E}\bigg\{\sup_{0\lt t\leqslant u}\left|X^{(n)}(t-\!)\right|^{r-1}\bigg[\Big(\int_0^{u}\left\|\sigma\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_n}\right)\right\|^2\, {\textrm{d}} s\Big)^{\frac{1}{2}}\notag\\&\quad +\Big(\int_0^{u}\int_U\left|h\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_n}\right)\right|^2\nu({\textrm{d}} z)\, {\textrm{d}} s\Big)^{\frac{1}{2}}\bigg]\bigg\}.\notag\end{align}

Applying Assumption 4 yields

\begin{align}&\mathbb{E}\sup_{0\leqslant t\leqslant u}|M_t|\notag\\&\leqslant C_2r\mathbb{E}\left[\!\sup_{0\lt t\leqslant u}\left|X^{(n)}(t-\!)\right|^{r-1}\!\cdot2K\left(\int_0^u\sup_{0\lt t\leqslant s}\!\left(\!1+\left|X^{(n)}(t-\!)\right|^2\!+\mathbb{E}\left|X^{(n)}([t]_n-)\right|^2\right) {\textrm{d}} s\right)^{\frac{1}{2}}\right].\notag\end{align}

Then, due to Young’s inequality (2.1) (with $\epsilon=\frac{1}{2C_2(r-1)}$ , $p=\frac{r}{r-1}$ , and $q=r$ ), Hölder inequality, the elementary inequality (2.3) (with $l=\frac{r}{2})$ and Lyapunov inequality, one can further conclude

(2.22) \begin{align}&\mathbb{E}\sup_{0\leqslant t\leqslant u}|M_t|\notag\\&\leqslant \frac{1}{2}\mathbb{E}\sup_{0\lt t\leqslant u}\left|X^{(n)}(t-\!)\right|^r+ C_2^r(2(r-1))^{r-1}(2K)^{r}\mathbb{E}\notag\\&\quad\times\left[\int_0^u\sup_{0\lt t\leqslant s}\left(1+\left|X^{(n)}(t-\!)\right|^2+\mathbb{E}\left|X^{(n)}([t]_n-)\right|^2\right)\, {\textrm{d}} s\right]^{\frac{r}{2}}\notag\\&\leqslant\frac{1}{2}\mathbb{E}\sup_{0\lt t\leqslant u}\left|X^{(n)}(t-\!)\right|^r+ C_2^r(2(r-1))^{r-1}(2K)^{r}u^{\frac{r}{2}-1}\mathbb{E}\notag\\&\quad\times\left[\int_0^u\sup_{0\lt t\leqslant s}\left(1+\left|X^{(n)}(t-\!)\right|^2+\mathbb{E}\left|X^{(n)}([t]_n-)\right|^2\right)^{\frac{r}{2}} \, {\textrm{d}} s\right]\notag\\&\leqslant\frac{1}{2}\mathbb{E}\sup_{0\lt t\leqslant u}\left|X^{(n)}(t-\!)\right|^r+ C_2^r(2(r-1))^{r-1}(2K)^{r}(3u)^{\frac{r}{2}-1}\mathbb{E}\notag\\&\quad\times\int_0^u \sup_{0\lt t\leqslant s}\left(1+\left|X^{(n)}(t-\!)\right|^r+\mathbb{E}\left|X^{(n)}([t]_n-)\right|^r\right)\, {\textrm{d}} s.\end{align}

On the other hand, utilizing Assumptions 4 and 7, along with Young’s inequality (2.1) (with $\epsilon=1$ , $p=\frac{r}{r-2}$ , and $q=\frac{r}{2}$ ), the elementary inequality (2.3) (with $l=\frac{r}{2})$ , and Lyapunov’s inequality, we obtain the following estimate for the supremum of $|N_t|$ :

(2.23) \begin{align}\mathbb{E}\sup_{0\leqslant t\leqslant u}|N_t|&\leqslant C_1\mathbb{E}\int_0^{u}\int_U\bigg(\left|X^{(n)}(s-\!)\right|^{r-2}\left|h\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_n},z\right)\right|^2\notag\\&\quad+\left|h\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_n},z\right)\right|^{r}\bigg)\nu({\textrm{d}} z)\, {\textrm{d}} s\notag\\&\leqslant C_1\mathbb{E}\int_0^{u}\bigg[K\left|X^{(n)}(s-\!)\right|^{r-2}\left(1+\left|X^{(n)}(s-\!)\right|^2+\mathbb{E}\left|X^{(n)}([s]_{n}-)\right|^2\right)\notag\\&\quad +K_2\left(1+\left|X^{(n)}(s-\!)\right|^{r}+\left(\mathbb{E}\left|X^{(n)}([s]_{n}-)\right|^2\right)^{\frac{r}{2}}\right)\bigg]\, {\textrm{d}} s\notag\\&\leqslant C_1K\frac{r-2}{r}\int_0^u\mathbb{E}\sup_{0\lt t\leqslant s}\left|X^{(n)}(t-\!)\right|^r \, {\textrm{d}} s+C_1\left(\frac{2K}{r}3^{\frac{r}{2}-1}+K_2\right)\notag\\&\quad\times \int_0^u\left(1+\mathbb{E}\sup_{0\lt t\leqslant s}\left|X^{(n)}(t-\!)\right|^r+\mathbb{E}\sup_{0\lt t\leqslant s}\left|X^{(n)}([t]_{n}-)\right|^r\right)\, {\textrm{d}} s.\end{align}

Note that, by (2.13), we have

(2.24) \begin{align} \mathbb{E}\sup_{0\lt t\leqslant u}|X(t-\!)|^r\leqslant\mathbb{E}\sup_{0\leqslant t\leqslant u}|X(t)|^r\lt\infty.\end{align}

By combining all the estimates from (2.22) to (2.24) and applying Grönwall’s inequality, we deduce from (2.21) that

\begin{equation}\mathbb{E}\sup_{0\leqslant t\leqslant T}\left|X^{(n)}(t)\right|^r\leqslant2(1+\mathbb{E}|x_0|^r){\textrm{e}}^{\widehat{C}_rT}\leqslant C_r,\notag\end{equation}

where $\widehat{C}_r=K(r-2)\frac{r^2+r+2C_1}{r}+4\cdot3^{\frac{r}{2}-1}(r+1)+C_2^{r}2^{r+1}(r-1)^{r-1}(2K)^{r}(3T)^{\frac{r}{2}-1}+4C_1\cdot\left(\frac{2K}{r}3^{\frac{r}{2}-1}+K_2\right)$ . It is evident that the positive constant $C_r$ depends on r, T, K, $K_2$ , and the initial condition $x_0$ , but is independent of n. Thus, the proof is complete.

Lemma 2. (Time Hölder continuity.) Let Assumptions A4-A7 hold. For any initial condition $x_0\in L^r(\Omega;\;\mathbb{R}^d)$ with $r\geqslant\kappa^2/2$ , there exists a positive constant $C_{\kappa}$ such that, for any $0\leqslant s\leqslant t\leqslant T$ with $|t-s|\leqslant1$ ,

(2.25) \begin{equation}\sup_{n\geqslant1}\mathbb{E}\left[\left|X^{(n)}(t)-X^{(n)}(s)\right|^{\kappa}\right]\leqslant C_{\kappa}|t-s|.\end{equation}

Proof. It follows from (2.12) that

\begin{align} X^{(n)}(t)-X^{(n)}(s) &=\int_s^tb\left(u,X^{(n)}(u-),\mu^{(n)}_{[u]_{n}}\right)\, {\textrm{d}} u+\int_s^t\sigma\left(u,X^{(n)}(u-),\mu^{(n)}_{[u]_{n}}\right)\, {\textrm{d}} W(u)\notag\\ &\quad +\int_s^t\int_{U}h\left(u,X^{(n)}(u-),\mu^{(n)}_{[u]_{n}},z\right)\tilde{N}({\textrm{d}} u,{\textrm{d}} z).\notag\end{align}

By taking expectations on both sides, one obtains

(2.26) \begin{align}\mathbb{E}\left|X^{(n)}(t)-X^{(n)}(s)\right|^{\kappa}&\leqslant3^{\kappa-1}\mathbb{E}\left[\left|\int_s^tb\left(u,X^{(n)}(u-),\mu^{(n)}_{[u]_{n}}\right)\, {\textrm{d}} u\right|^{\kappa}\right]\notag\\&\quad +3^{\kappa-1}\mathbb{E}\left[\left|\int_s^t\sigma\left(u,X^{(n)}(u-),\mu^{(n)}_{[u]_{n}}\right)\, {\textrm{d}} W(u)\right|^{\kappa}\right]\notag\\&\quad +3^{\kappa-1}\mathbb{E}\left[\left|\int_s^t\int_{U}h\left(u,X^{(n)}(u-),\mu^{(n)}_{[u]_{n}},z\right)\tilde{N}({\textrm{d}} u,{\textrm{d}} z)\right|^{\kappa}\right]\notag\\&\;=\!:\;B_1+B_2+B_3.\end{align}

We proceed by estimating $B_1$ , $B_2$ , and $B_3$ individually. To maintain clarity, we present only the core estimation steps for each term. By applying Hölder inequality, Assumption 5, Lyapunov inequality, and estimate (2.14), we derive the following bound for $B_1$ :

\begin{align}B_1&\leqslant 3^{\kappa-1}(t-s)^{\kappa-1}\int_s^t\mathbb{E}\left[\left|b\left(u,X^{(n)}(u-),\mu^{(n)}_{[u]_{n}}\right)\right|^{\kappa}\right]\, {\textrm{d}} u\notag\\&\leqslant3^{\frac{3\kappa}{2}-2}K_1^{\frac{\kappa}{2}}(t-s)^{\kappa-1}\int_s^t\left(1+\mathbb{E}\left|X^{(n)}(u-)\right|^{\frac{\kappa^2}{2}}+\left(\mathbb{E}\left|X^{(n)}([u]_n-)\right|^2\right)^{\frac{\kappa^2}{4}}\right)\, {\textrm{d}} u\notag\\&\leqslant2\cdot3^{\frac{3\kappa}{2}-2}K_1^{\frac{\kappa}{2}}(t-s)^{\kappa-1}\int_s^t\left(1+\mathbb{E}\sup_{0\leqslant u\leqslant T}\left|X^{(n)}(u)\right|^{\frac{\kappa^2}{2}}\right)\, {\textrm{d}} u\leqslant C_{\kappa}(t-s)^{\kappa}.\notag\end{align}

For $B_2$ , using the BDG inequality, Hölder inequality, Assumption 4, and estimate (2.14), we have

\begin{align*}B_2&\leqslant 3^{\kappa-1}M_{\kappa}\mathbb{E}\left[\left|\int_s^t\left\|\sigma\left(u,X^{(n)}(u-),\mu^{(n)}_{[u]_{n}}\right)\right\|^2\, {\textrm{d}} u\right|^{\frac{\kappa}{2}}\right]\notag\\&\leqslant 3^{\kappa-1}M_{\kappa}(t-s)^{\frac{\kappa-2}{2}}\int_s^t\mathbb{E}\left[\left\|\sigma\left(u,X^{(n)}(u-),\mu^{(n)}_{[u]_{n}}\right)\right\|^{\kappa}\right]\, {\textrm{d}} u\notag\\&\leqslant2\cdot3^{\frac{3\kappa}{2}-2}M_{\kappa}K^{\frac{\kappa}{2}}(t-s)^{\frac{\kappa-2}{2}}\int_s^t\left(1+\mathbb{E}\sup_{0\leqslant u\leqslant T}\left|X^{(n)}(u)\right|^{\kappa}\right)\, {\textrm{d}} u\leqslant C_{\kappa}(t-s)^{\frac{\kappa}{2}},\notag\end{align*}

where $M_{\kappa}=[\kappa^{\kappa+1}/2(\kappa-1)^{\kappa-1}]^{\frac{\kappa}{2}}$ . For $B_3$ , using Kunita’s first inequality (i.e. the inequality (2.4) in Proposition 2), Hölder inequality, Assumptions 4 and 7, and estimate (2.14), we obtain

\begin{align*}B_3&\leqslant 3^{\kappa-1}D\mathbb{E}\left[\left|\int_s^t\int_U\left|h\left(u,X^{(n)}(u-),\mu^{(n)}_{[u]_{n}},z\right)\right|^2\nu({\textrm{d}} z)\, {\textrm{d}} u\right|^{\frac{\kappa}{2}}\right]\notag\\&\quad + 3^{\kappa-1}D\int_s^t\mathbb{E}\left[\int_U\left|h\left(u,X^{(n)}(u-),\mu^{(n)}_{[u]_{n}},z\right)\right|^{\kappa}\nu({\textrm{d}} z)\right]\, {\textrm{d}} u\notag\\&\leqslant 3^{\kappa-1}D(t-s)^{\frac{\kappa-2}{2}}\int_s^t\mathbb{E}\left[\left(\int_U\left|h\left(u,X^{(n)}(u-),\mu^{(n)}_{[u]_{n}},z\right)\right|^2\nu({\textrm{d}} z)\right)^{\frac{\kappa}{2}}\right]\, {\textrm{d}} u\notag\\&\quad + 3^{\kappa-1}D\int_s^t\mathbb{E}\left[\int_U\left|h\left(u,X^{(n)}(u-),\mu^{(n)}_{[u]_{n}},z\right)\right|^{\kappa}\nu({\textrm{d}} z)\right]\, {\textrm{d}} u\notag\\&\leqslant 3^{\kappa-1}DK^{\frac{\kappa}{2}}(t-s)^{\frac{\kappa-2}{2}}\int_s^t\mathbb{E}\left[\left(1+\left|X^{(n)}(u-)\right|^2+\mathbb{E}X^{(n)}([u]_n-)^2\right)^{\frac{\kappa}{2}}\right]\, {\textrm{d}} u\notag\\&\quad + 3^{\kappa-1}DK_3\int_s^t\mathbb{E}\left[\left(1+\left|X^{(n)}(u-)\right|^{\kappa}+\left(\mathbb{E}X^{(n)}([u]_n-)^2\right)^{\frac{\kappa}{2}}\right)\right]\, {\textrm{d}} u\notag\\&\leqslant2\cdot3^{\frac{3\kappa}{2}-2}DK^{\frac{\kappa}{2}}(t-s)^{\frac{\kappa-2}{2}}\int_s^t\left(1+\mathbb{E}\sup_{0\leqslant u\leqslant T}\left|X^{(n)}(u)\right|^{\kappa}\right)\, {\textrm{d}} u\notag\\&\quad +2\cdot3^{\kappa-1}DK_3\int_s^t\left(1+\mathbb{E}\sup_{0\leqslant u\leqslant T}\left|X^{(n)}(u)\right|^{\kappa}\right)\, {\textrm{d}} u\notag\\&\leqslant C_{\kappa}[(t-s)+(t-s)^{\frac{\kappa}{2}}],\notag\end{align*}

where D is a positive constant dependent on $\kappa$ . Consequently, the desired assertion follows by substituting the above estimates on $B_1$ , $B_2$ , and $B_3$ into (2.26) and then taking the supremum over n.

With Lemmas 1 and 2 established, we proceed to demonstrate the following result.

Lemma 3. (Cauchy sequences.) The sequence $\{X^{(n)}(t)\}_{n\geqslant1}$ given by (2.12) is a Cauchy sequence in $L^{\kappa}(\Omega;\;D([0,T];\;\mathbb{R}^d))$ . Specifically, for any $n,m\geqslant1$ , the following holds:

(2.27) \begin{equation} \left\|X^{(n)}-X^{(m)}\right\|_{L^{\kappa}}=\left(\mathbb{E}\left[\sup_{0\leqslant t\leqslant T}\left|X^{(n)}(t)-X^{(m)}(t)\right|^{\kappa}\right]\right)^{\frac{1}{\kappa}}\to0, \quad \mbox{as }n, m\to\infty.\end{equation}

Proof. Note that, for $t\in[0,T]$ , the difference between $X^{(n)}$ and $X^{(m)}$ satisfies the following equation:

\begin{align*}X^{(n)}(t)-X^{(m)}(t)&=\int_0^t\left[b\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_{n}}\right)-b\left(s,X^{(m)}(s-\!),\mu^{(m)}_{[s]_{m}}\right)\right]\, {\textrm{d}} s\notag\\&\quad +\int_0^t\left[\sigma\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_{n}}\right)-\sigma\left(s,X^{(m)}(s-\!),\mu^{(m)}_{[s]_{m}}\right)\right]\, {\textrm{d}} W(s)\notag\\&\quad +\int_0^t\int_{U}\left[h\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_{n}},z\right)-h\left(s,X^{(m)}(s-\!),\mu^{(m)}_{[s]_{m}},z\right)\right]\tilde{N}({\textrm{d}} s,{\textrm{d}} z).\notag\end{align*}

To facilitate the analysis, we define the stopping time:

$$\tau_R\;:\!=\;\inf\left\{t\in[0,T]: \left|X^{(n)}(t)\right|\vee\left|X^{(m)}(t)\right|\gt R\right\},$$

for each $R\gt 0$ . The stopping time technique is employed here to ensure boundedness of the processes $X^{(n)}$ and $X^{(m)}$ up to $\tau_R$ , leveraging the fact that (2.12) describes a classical (nondistribution-dependent) SDE. It is clear that, for $0\leqslant t\leqslant \tau_R\wedge T$ , we have $|X^{(n)}(t-\!)|\leqslant R$ and $|X^{(m)}(t-\!)|\leqslant R$ . Then, by De Morgan’s law, we arrive at

(2.28) \begin{align}\mathbb{E}\left[\sup_{0\leqslant t\leqslant T}\left|X^{(n)}(t)-X^{(m)}(t)\right|^{\kappa}\right]&=\mathbb{E}\Big[\sup_{0\leqslant t\leqslant T}\left|X^{(n)}(t)-X^{(m)}(t)\right|^{\kappa}\mathbb{I}_{\{\tau_R\gt T\}}\Big]\notag\\&\quad +\mathbb{E}\Big[\sup_{0\leqslant t\leqslant T}\left|X^{(n)}(t)-X^{(m)}(t)\right|^{\kappa}\mathbb{I}_{\{\tau_R\leqslant T\}}\Big]\notag\\&\;=\!:\;J_1+J_2,\end{align}

where $\mathbb{I}_A$ is the indicator function of the set A.

In the subsequent analysis, we estimate each term $J_1$ and $J_2$ on the right-hand side of (2.28).

(1) Estimation of the term $J_1$ . We note that

(2.29) \begin{equation}J_1\leqslant\mathbb{E}\left[\sup_{0\leqslant t\leqslant T}\left|X^{(n)}(t\wedge\tau_R)-X^{(m)}(t\wedge\tau_R)\right|^{\kappa}\right]\lt\infty,\end{equation}

where the finiteness of the term is guaranteed by Lemma 1. By applying Itô’s formula, we obtain the following representation:

\begin{equation*}\left|X^{(n)}(t\wedge\tau_R)-X^{(m)}(t\wedge\tau_R)\right|^{\kappa}=J_{1,R}(t)+J_{2,R}(t)+J_{3,R}(t)+J_{4,R}(t)+J_{5,R}(t)+J_{6,R}(t),\end{equation*}

where the individual terms $J_{i,R}$ , for $i=1,\ldots,6$ , are given by

\begin{align*}J_{1,R}(t)&=\kappa\int_0^{t\wedge\tau_R}\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)\right|^{\kappa-2}\notag\\&\quad\cdot\Big\langle X^{(n)}(s-\!)-X^{(m)}(s-\!), b\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_{n}}\right)-b\left(s,X^{(m)}(s-\!),\mu^{(m)}_{[s]_{m}}\right)\Big\rangle\, {\textrm{d}} s,\notag\\J_{2,R}(t)&=\frac{\kappa}{2}\int_0^{t\wedge\tau_R}\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)\right|^{\kappa-2}\left\| \sigma\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_{n}}\right)\right .\notag\\&\left .\quad -\sigma\left(s,X^{(m)}(s-\!),\mu^{(m)}_{[s]_{m}}\right) \right\|^2\, {\textrm{d}} s,\notag\\J_{3,R}(t)&=\frac{\kappa(\kappa-2)}{2}\int_0^{t\wedge\tau_R}\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)\right|^{\kappa-4}\notag\\&\quad\cdot\left\| \left(X^{(n)}(s-\!)-X^{(m)}(s-\!)\!\right)^{\text{T}}\left(\!\sigma\!\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_{n}}\right)-\sigma\!\left(s,X^{(m)}(s-\!),\mu^{(m)}_{[s]_{m}}\right)\!\right) \right\|^2 {\textrm{d}} s,\notag\\J_{4,R}(t)&=\int_0^{t}\int_U\bigg[\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)+h\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_n},z\right)\right .\notag\\&\left .\quad -h\left(s,X^{(m)}(s-\!),\mu^{(m)}_{[s]_{m}},z\right)\right|^{\kappa}\notag\\&\quad -\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)\right|^{\kappa}-\kappa\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)\right|^{\kappa-2}\notag\\&\quad\cdot\left\langle X^{(n)}(s-\!)-X^{(m)}(s-\!), h\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_n},z\right)\right .\notag\\&\left .\quad -h\left(s,X^{(m)}(s-\!),\mu^{(m)}_{[s]_{m}},z\right)\right\rangle \bigg] N({\textrm{d}} s,{\textrm{d}} z),\notag\\J_{5,R}(t)&=\kappa\int_0^{t\wedge\tau_R}\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)\right|^{\kappa-2}\notag\\&\quad\cdot\left\langle X^{(n)}(s-\!)-X^{(m)}(s-\!),\sigma\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_{n}}\right)-\sigma\left(s,X^{(m)}(s-\!),\mu^{(m)}_{[s]_{m}}\right)\, {\textrm{d}} W(s)\right\rangle,\notag\\J_{6,R}(t)&=\kappa\int_0^{t\wedge\tau_R}\int_U\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)\right|^{\kappa-2}\notag\\&\quad\cdot\left\langle X^{(n)}(s-\!)-X^{(m)}(s-\!),h\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_{n}},z\right)\right .\notag\\&\left .\quad -h\left(s,X^{(m)}(s-\!),\mu^{(m)}_{[s]_{m}},z\right)\right\rangle\tilde{N}({\textrm{d}} s,{\textrm{d}} z).\notag\end{align*}

In order to take the supremum over time and the expectation, we need to estimate $\mathbb{E}\big[\sup_{0\leqslant t\leqslant u}J_{i,R}(t)\big]$ for $i=1, \ldots, 6$ .

Note that the terms $J_{i,R}$ for $i=1,2,3$ are standard Lebesgue integrals, and can be estimated in a similar manner. For any $u\in[0,T]$ , applying Assumptions 1 and 2, we derive

\begin{align*}\mathbb{E}\left[\sup_{0\leqslant t\leqslant u}J_{1,R}(t)\right]&\leqslant\kappa\mathbb{E}\bigg[\sup_{0\leqslant t\leqslant u}\int_0^{t\wedge\tau_R}\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)\right|^{\kappa-2}\notag\\&\quad\cdot\Big\langle X^{(n)}(s-\!)-X^{(m)}(s-\!), b\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_{n}}\right)\notag\\&\quad -b\left(s,X^{(m)}(s-\!),\mu^{(n)}_{[s]_{n}}\right)\Big\rangle\,{\textrm{d}} s\bigg]\notag\\&\quad +\kappa\mathbb{E}\bigg[\int_0^{u\wedge\tau_R}\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)\right|^{\kappa-1}\notag\\&\quad\cdot\left|b\left(s,X^{(m)}(s-\!),\mu^{(n)}_{[s]_{n}}\right)-b\left(s,X^{(m)}(s-\!),\mu^{(m)}_{[s]_{m}}\right)\right|\, {\textrm{d}} s\bigg]\notag\\&\leqslant\kappa L_R\mathbb{E}\left[\int_0^{u\wedge\tau_R}\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)\right|^{\kappa} \, {\textrm{d}} s\right]\notag\\&\quad +\kappa\sqrt{L}\mathbb{E}\left[\int_0^{u\wedge\tau_R}\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)\right|^{\kappa-1} W_2\left(\mu^{(n)}_{[s]_{n}},\mu^{(m)}_{[s]_{m}}\right)\, {\textrm{d}} s\right].\notag\end{align*}

By further applying Young’s inequality (2.1) (with $\epsilon=1$ , $p=\frac{\kappa}{\kappa-1}$ and $q=\kappa$ ), we obtain

(2.30) \begin{align}&\mathbb{E}\left[\sup_{0\leqslant t\leqslant u}J_{1,R}(t)\right]\notag\\&\leqslant\left(\kappa L_R+(\kappa-1)\sqrt{L}\right)\mathbb{E}\left[\int_0^{u\wedge\tau_R}\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)\right|^{\kappa} \, {\textrm{d}} s\right]\notag\\&\quad +3^{\kappa-1}\sqrt{L}\int_0^{u\wedge\tau_R}\left[W_2^{\kappa}(\mu^{(n)}_{[s]_{n}},\mu^{(n)}_{s})+W_2^{\kappa}\left(\mu^{(n)}_{s},\mu^{(m)}_{s}\right)+W_2^{\kappa}\left(\mu^{(m)}_{s},\mu^{(m)}_{[s]_{m}}\right)\right]\, {\textrm{d}} s\notag\\&\leqslant\left[\kappa L_R+\sqrt{L}\left(\kappa-1+3^{\kappa-1}\right)\right]\int_0^{u\wedge\tau_R}\mathbb{E}\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)\right|^{\kappa} \, {\textrm{d}} s\notag\\&\quad +3^{\kappa-1}\sqrt{L}\int_0^{u\wedge\tau_R}\left(\mathbb{E}\left|X^{(n)}(s-\!)-X^{(n)}([s]_{n}-)\right|^{\kappa}+\mathbb{E}\left|X^{(m)}(s-\!)-X^{(m)}([s]_{m}-)\right|^{\kappa}\right)\, {\textrm{d}} s.\end{align}

Analogously, by Assumptions 1 and 2 and Young’s inequality (2.1) (with $\epsilon=1$ , $p=\frac{\kappa}{\kappa-2}$ , $q=\frac{\kappa}{2}$ ), we derive the following bounds for $J_{2,R}$ and $J_{3,R}$ :

(2.31) \begin{align}&\mathbb{E}\left[\sup_{0\leqslant t\leqslant u}J_{2,R}(t)\right]\notag\\[-1pt]&\leqslant\left[\kappa L_R+L\left(\kappa-2+2\cdot3^{\kappa-1}\right)\right]\int_0^{u\wedge\tau_R}\mathbb{E}\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)\right|^{\kappa} \, {\textrm{d}} s\notag\\[-1pt]&\quad +2L\cdot3^{\kappa-1}\int_0^{u\wedge\tau_R}\left(\mathbb{E}\left|X^{(n)}(s-\!)-X^{(n)}([s]_{n}-)\right|^{\kappa}+\mathbb{E}\left|X^{(m)}(s-\!)-X^{(m)}([s]_{m}-)\right|^{\kappa}\right)\, {\textrm{d}} s,\end{align}

and

(2.32) \begin{align}&\mathbb{E}\left[\sup_{0\leqslant t\leqslant u}J_{3,R}(t)\right]\notag\\[-1pt]&\leqslant(\kappa-2)\left[\kappa L_R+\left(\kappa-2+2\cdot3^{\kappa-1}\right)L\right]\int_0^{u\wedge\tau_R}\mathbb{E}\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)\right|^{\kappa} \, {\textrm{d}} s\notag\\[-1pt]&\quad +2L\cdot3^{\kappa-1}(\kappa-2)\int_0^{u\wedge\tau_R}\left(\mathbb{E}\left|X^{(n)}(s-\!)-X^{(n)}([s]_{n}-)\right|^{\kappa}\right .\notag\\[-1pt]&\quad \left . +\mathbb{E}\left|X^{(m)}(s-\!)-X^{(m)}([s]_{m}-)\right|^{\kappa}\right)\, {\textrm{d}} s.\end{align}

As for the last three terms, we first use the remainder formula in (2.19) and Assumption 7 to obtain

(2.33) \begin{align}&\mathbb{E}\left[\sup_{0\leqslant t\leqslant u}J_{4,R}(t)\right]\notag\\[-1pt]&\leqslant C_1\mathbb{E}\int_0^{u\wedge\tau_R}\int_U\bigg(\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)\right|^{\kappa-2}\left|h\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_n},z\right)\right .\notag\\&\quad \left . -\,h\left(s,X^{(m)}(s-\!),\mu^{(m)}_{[s]_{m}},z\right)\right|^2\notag\\[-1pt]&\quad +\left|h\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_n},z\right)-h\left(s,X^{(m)}(s-\!),\mu^{(m)}_{[s]_{m}},z\right)\right|^{\kappa}\bigg)\nu({\textrm{d}} z)\, {\textrm{d}} s.\notag\\[-1pt]&\leqslant C_1\left(L_R+\frac{(\kappa-2+2\cdot3^{\kappa-1})L}{\kappa}\right)\int_0^{u\wedge\tau_R}\mathbb{E}\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)\right|^{\kappa} \, {\textrm{d}} s\notag\\[-1pt]&\quad +2C_1L\cdot\frac{3^{\kappa-1}}{\kappa}\!\int_0^{u\wedge\tau_R}\!\left(\mathbb{E}\left|X^{(n)}(s-\!)-X^{(n)}([s]_{n}-)\right|^{\kappa}+\mathbb{E}\left|X^{(m)}(s-\!)-X^{(m)}([s]_{m}-)\right|^{\kappa}\right) {\textrm{d}} s\notag\\[-1pt]&\quad +C_12^{\kappa-1}\mathbb{E}\int_0^{u\wedge\tau_R}\int_U\left|h\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_n},z\right)-h\left(s,X^{(m)}(s-\!),\mu^{(n)}_{[s]_{n}},z\right)\right|^{\kappa}\nu({\textrm{d}} z)\, {\textrm{d}} s \notag\\[-1pt]&\quad +C_12^{\kappa-1}\mathbb{E}\int_0^{u\wedge\tau_R}\int_U\left|h\left(s,X^{(m)}(s-\!),\mu^{(n)}_{[s]_n},z\right)-h\left(s,X^{(m)}(s-\!),\mu^{(m)}_{[s]_{m}},z\right)\right|^{\kappa}\nu({\textrm{d}} z)\, {\textrm{d}} s\notag\\[-1pt]&\leqslant C_1\left(L_R+\frac{(\kappa-2+2\cdot3^{\kappa-1})L}{\kappa}+2^{\kappa-1}L_R^{\prime}+L^{\prime}6^{\kappa-1}\right)\int_0^{u\wedge\tau_R}\mathbb{E}\sup_{0\leqslant t\leqslant s}\left|X^{(n)}(t)-X^{(m)}(t)\right|^{\kappa} \, {\textrm{d}} s\notag\\[-1pt]&\quad +3^{\kappa-1}C_1\left(\frac{2L}{\kappa}+2^{\kappa-1}L^{\prime}\right)\int_0^{u\wedge\tau_R}\left(\mathbb{E}\left|X^{(n)}(s-\!)-X^{(n)}([s]_{n}-)\right|^{\kappa}\right .\notag\\&\quad \left . +\mathbb{E}\left|X^{(m)}(s-\!)-X^{(m)}([s]_{m}-)\right|^{\kappa}\right)\, {\textrm{d}} s.\end{align}

We exploit the BDG inequality and Assumptions 2 and 3 to obtain

\begin{align*}&\mathbb{E}\left[\sup_{0\leqslant t\leqslant u}J_{5,R}(t)\right]\notag\\&\leqslant\kappa\cdot\sqrt{32}\mathbb{E}\bigg(\int_0^{u\wedge\tau_R}\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)\right|^{2\kappa-2}\left\|\sigma\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_{n}}\right)\right .\notag\\&\quad \left . -\,\sigma\left(s,X^{(m)}(s-\!),\mu^{(m)}_{[s]_{m}}\right)\right\|^2\, {\textrm{d}} s\bigg)^{\frac{1}{2}}\notag\\&\leqslant6\kappa\mathbb{E}\Bigg[\sup_{0\leqslant t\leqslant u}\left|X^{(n)}(t\wedge\tau_R)-X^{(m)}(t\wedge\tau_R)\right|^{\kappa-1}\bigg(\int_0^{u\wedge\tau_R}2\left\|\sigma\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_{n}}\right)\right .\notag\\&\quad \left . -\,\sigma\left(s,X^{(m)}(s-\!),\mu^{(n)}_{[s]_{n}}\right)\right\|^2\notag\\&\quad +2\left\|\sigma\left(s,X^{(m)}(s-\!),\mu^{(n)}_{[s]_{n}}\right)-\sigma\left(s,X^{(m)}(s-\!),\mu^{(m)}_{[s]_{m}}\right)\right\|^2\, {\textrm{d}} s\bigg)^{\frac{1}{2}}\Bigg]\notag\\&\leqslant6\kappa\mathbb{E}\Bigg[\sup_{0\leqslant t\leqslant u}\left|X^{(n)}(t\wedge\tau_R)-X^{(m)}(t\wedge\tau_R)\right|^{\kappa-1}\bigg(\int_0^{u\wedge\tau_R}2L_R\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)\right|^2\notag\\&\quad +2LW_2^2\left(\mu^{(n)}_{[s]_{n}},\mu^{(m)}_{[s]_{m}}\right)\, {\textrm{d}} s\bigg)^{\frac{1}{2}}\Bigg].\notag\end{align*}

Then owing to Young’s inequality (2.1) (with $\epsilon=\frac{1}{24(\kappa-1)}$ , $p=\frac{\kappa}{\kappa-1}$ , and $q=\kappa$ ), Hölder inequality, the elementary inequality (2.2) (with $k=2$ and $l=\frac{\kappa}{2}$ ), the elementary inequality (2.3) (with $l=\kappa$ ), and Lyapunov inequality, we further have

(2.34) \begin{align}&\mathbb{E}\left[\sup_{0\leqslant t\leqslant u}J_{5,R}(t)\right]-\frac{1}{4}\mathbb{E}\left[\sup_{0\leqslant t\leqslant u}\left|X^{(n)}(t\wedge\tau_R)-X^{(m)}(t\wedge\tau_R)\right|^{\kappa}\right]\notag\\& \leqslant6^{\kappa}\cdot2^{\frac{\kappa}{2}}(4(\kappa-1))^{\kappa-1}\mathbb{E}\left(\int_0^{u\wedge\tau_R}L_R\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)\right|^2+LW_2^2\left(\mu^{(n)}_{[s]_{n}},\mu^{(m)}_{[s]_{m}}\right)\, {\textrm{d}} s\right)^{\frac{\kappa}{2}}\notag\\& \leqslant6^{\kappa}\cdot2^{\frac{\kappa}{2}}(4(\kappa-1))^{\kappa-1}u^{\frac{\kappa}{2}-1}\mathbb{E}\left[\int_0^{u\wedge\tau_R}\left(L_R\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)\right|^2\right .\right .\notag\\&\quad \left .\left . +\,LW_2^2\left(\mu^{(n)}_{[s]_{n}},\mu^{(m)}_{[s]_{m}}\right)\right)^{\frac{\kappa}{2}} \, {\textrm{d}} s\right]\notag\\& \leqslant 12^{\kappa}\cdot(4(\kappa-1))^{\kappa-1}u^{\frac{\kappa}{2}-1}\mathbb{E}\left[\int_0^{u\wedge\tau_R}\left(L_R^{\frac{\kappa}{2}}\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)\right|^{\kappa}\right .\right .\notag\\&\quad \left .\left . +\,L^{\frac{\kappa}{2}}W_2^{\kappa}\left(\mu^{(n)}_{[s]_{n}},\mu^{(m)}_{[s]_{m}}\right)\right)\, {\textrm{d}} s\right]\notag\\& \leqslant12^{\kappa}\cdot(4(\kappa-1))^{\kappa-1}u^{\frac{\kappa}{2}-1}\left(L_R^{\frac{\kappa}{2}}+L^{\frac{\kappa}{2}}3^{\kappa-1}\right)\int_0^{u\wedge\tau_R}\mathbb{E}\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)\right|^{\kappa} \, {\textrm{d}} s\notag\\&\quad +12^{\kappa}\cdot(L)^{\frac{\kappa}{2}}(12(\kappa-1))^{\kappa-1}u^{\frac{\kappa}{2}-1}\int_0^{u\wedge\tau_R}\bigg(\mathbb{E}\left|X^{(n)}(s-\!)-X^{(n)}([s]_{n}-)\right|^{\kappa}\notag\\&\quad +\mathbb{E}\left|X^{(m)}(s-\!)-X^{(m)}([s]_{m}-)\right|^{\kappa}\bigg)\, {\textrm{d}} s.\end{align}

Finally, we apply the inequality (2.5) (with $p=1$ ) in Proposition 2 and Young’s inequality (2.1) (with $\epsilon=\frac{1}{4D(\kappa-1)}$ , $p=\frac{\kappa}{\kappa-1}$ , and $q=\kappa$ ) to obtain

(2.35) \begin{align}\mathbb{E}&\left[\sup_{0\leqslant t\leqslant u}J_{6,R}(t)\right]\notag\\&\leqslant\kappa D\mathbb{E}\bigg(\int_0^{u\wedge\tau_R}\int_U\left|X^{(n)}(s-\!)-X^{(m)}(s-\!)\right|^{2\kappa-2}\left|h\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_{n} -},z\right)\right .\notag\\&\quad \left . -\,h\left(s,X^{(m)}(s-\!),\mu^{(m)}_{[s]_{m}},z\right)\right|^2\nu({\textrm{d}} z)\, {\textrm{d}} s\bigg)^{\frac{1}{2}}\notag\\\leqslant &\frac{1}{4}\mathbb{E}\left[\sup_{0\leqslant t\leqslant u}\left|X^{(n)}(t\wedge\tau_R)-X^{(m)}(t\wedge\tau_R)\right|^{\kappa}\right]\notag\\&\quad +(2D)^{\kappa}\cdot(4(\kappa-1))^{\kappa-1}u^{\frac{\kappa}{2}-1}\left(L_R^{\frac{\kappa}{2}}+L^{\frac{\kappa}{2}}3^{\kappa-1}\right)\int_0^{u\wedge\tau_R}\mathbb{E}\sup_{0\leqslant t\leqslant s}\left|X^{(n)}(t)-X^{(m)}(t)\right|^{\kappa} \, {\textrm{d}} s\notag\\&\quad +(2D)^{\kappa}\cdot L^{\frac{\kappa}{2}}(12(\kappa-1))^{\kappa-1}u^{\frac{\kappa}{2}-1}\int_0^{u\wedge\tau_R}\Big(\mathbb{E}\left|X^{(n)}(s-\!)-X^{(n)}([s]_{n}-)\right|^{\kappa}\notag\\&\quad +\mathbb{E}\left|X^{(m)}(s-\!)-X^{(m)}([s]_{m}-)\right|^{\kappa}\Big)\, {\textrm{d}} s.\end{align}

Substituting the estimates derived from (2.30)–(2.35) into (2.29) yields the inequality

\begin{align*}&\mathbb{E}\left[\sup_{0\leqslant t\leqslant u}\left|X^{(n)}(t\wedge\tau_R)-X^{(m)}(t\wedge\tau_R)\right|^\kappa\right]\notag\\&\leqslant2\widehat{M}_1(u)\int_0^{u}\mathbb{E}\sup_{0\leqslant t\leqslant s}\left|X^{(n)}(t\wedge\tau_R)-X^{(m)}(t\wedge\tau_R)\right|^{\kappa} \, {\textrm{d}} s\notag\\&\quad +2\cdot3^{\kappa-1}\widehat{M}_2(u)\!\int_0^{u\wedge\tau_R}\!\left(\mathbb{E}\left|X^{(n)}(s-\!)-X^{(n)}([s]_{n}-\!)\right|^{\kappa}\!+\mathbb{E}\left|X^{(m)}(s-\!)-X^{(m)}([s]_{m}-\!)\right|^{\kappa}\right)\, {\textrm{d}} s.\notag\end{align*}

Here, $\widehat{M}_1(u)=\kappa^2 L_R+(\kappa-1+3^{\kappa-1})\sqrt{L}+(\kappa-1)(\kappa-2+2\cdot3^{\kappa-1})L+C_1\Big(L_R+\frac{(\kappa-2+2\cdot3^{\kappa-1})L}{\kappa}+2^{\kappa-1}L_R^{\prime}+6^{\kappa-1}L^{\prime}\Big)+(12^{\kappa}+(2D)^{\kappa})(4(\kappa-1))^{\kappa-1}\left(L_R^{\frac{\kappa}{2}}+L^{\frac{\kappa}{2}}3^{\kappa-1}\right)u^{\frac{\kappa}{2}-1}$ and $\widehat{M}_2(u)=\sqrt{L}+2L(\kappa-1)+C_1\left(\frac{2L}{\kappa}+2^{\kappa-1}L^{\prime}\right)+(12^{\kappa}+(2D)^{\kappa})L^{\frac{\kappa}{2}}(4(\kappa-1))^{\kappa-1}u^{\frac{\kappa}{2}-1}$ . In addition, for any $t\in[0,T]$ , the result in Lemma 2 implies that

$$\mathbb{E}\left|X^{(n)}(t)-X^{(n)}([t]_{n})\right|^{\kappa}\leqslant Ch_n \quad\mbox{and}\quad \mathbb{E}\left|X^{(m)}(t)-X^{(m)}([t]_{n})\right|^{\kappa}\leqslant Ch_m.$$

By these estimates, together with Grönwall’s inequality, we conclude that

(2.36) \begin{align}J_1&\leqslant\mathbb{E}\left[\sup_{0\leqslant t\leqslant T}\left|X^{(n)}(t\wedge\tau_R)-X^{(m)}(t\wedge\tau_R)\right|^{\kappa}\right]\leqslant 2CT(h_n+h_m)\cdot3^{\kappa-1}\widehat{M}_2(T){\textrm{e}}^{2\widehat{M}_1(T)\cdot T}\notag\\&\;=\!:\;T(h_n+h_m)C(\kappa,T,L,L^{\prime}){\textrm{e}}^{T\cdot C(\kappa,T,L,L_R,L^{\prime},L_R^{\prime})}.\end{align}

(2) Estimation of the term $J_2$ . With the aid of the Cauchy–Schwarz inequality, we have

(2.37) \begin{align}J_2&=\mathbb{E}\left[\sup_{0\leqslant t\leqslant T}\left|X^{(n)}(t)-X^{(m)}(t)\right|^{\kappa}\mathbb{I}_{\{\tau_R\leqslant T\}}\right]\notag\\&\leqslant \sqrt{\mathbb{E}\left(\sup_{0\leqslant t\leqslant T}\left|X^{(n)}(t)-X^{(m)}(t)\right|^{{\kappa}}\right)^2}\sqrt{\mathbb{E}\left(\mathbb{I}_{\{\tau_R\leqslant T\}}\right)^2}\notag\\&\leqslant \sqrt{\mathbb{E}\left(\sup_{0\leqslant t\leqslant T}\left|X^{(n)}(t)-X^{(m)}(t)\right|^{2\kappa}\right)}\sqrt{\mathbb{E}\left(\mathbb{I}_{\{\tau_R\leqslant T\}}\right)}\notag\\&\leqslant2^{\kappa-\frac{1}{2}}\sqrt{\mathbb{E}\left[\sup_{0\leqslant t\leqslant T}\left|X^{(n)}(t)\right|^{2\kappa}+\sup_{0\leqslant t\leqslant T}\left|X^{(m)}(t)\right|^{2\kappa}\right]}\sqrt{\mathbb{P}\Big(\tau_R\leqslant T\Big)}\notag\\&\leqslant C_1\sqrt{\mathbb{P}\Big(\tau_R\leqslant T\Big)}.\end{align}

Here, the result of Lemma 1 has been utilized. Further, by employing the subadditivity of probability and invoking Lemma 1 once more, we can estimate

\begin{align}\mathbb{P}\Big(\tau_R\leqslant T\Big)&\leqslant\mathbb{E}\left(\mathbb{I}_{\{\tau_R\leqslant T\}}\frac{|X^{(n)}(\tau_R)|^{4}+|X^{(m)}(\tau_R)|^{4}}{R^4}\right)\notag\\&\leqslant\frac{1}{R^4}\left(\mathbb{E}\sup_{0\leqslant t\leqslant T}\left|X^{(n)}(t)\right|^4+\mathbb{E}\sup_{0\leqslant t\leqslant T}\left|X^{(m)}(t)\right|^4\right)\notag\\&\leqslant \frac{C}{R^4}.\notag\end{align}

By substituting this into (2.37), we further obtain

(2.38) \begin{equation}J_2\leqslant \frac{C}{R^2}.\end{equation}

At this point, we can estimate (2.28) by combining (2.36) and (2.38) as follows:

(2.39) \begin{equation}\mathbb{E}\sup_{0\leqslant t\leqslant T}\left|X^{(n)}(t)-X^{(m)}(t)\right|^\kappa\leqslant T(h_n+h_m)C(\kappa,T,L,L^{\prime}){\textrm{e}}^{T\cdot C(\kappa,T,L,L_R,L^{\prime},L_R^{\prime})}+\frac{C}{R^2}.\end{equation}

Note that R is independent of n and m, and $\frac{C}{R^2}$ converges to 0 as $R\to\infty$ . For any given $\varepsilon\gt 0$ , there exists a sufficiently large number $R(\varepsilon)\gt 0$ , such that,

$$\frac{C}{R_{*}^2}\lt\frac{\varepsilon}{2},$$

when $R_{*}\geqslant R(\varepsilon)$ . Since both $h_n$ and $h_m$ converge to 0 as $n,m\to\infty$ , for the $\varepsilon\gt 0$ chosen previously, we have

$$T(h_n+h_m)C(\kappa,T,L,L^{\prime}){\textrm{e}}^{T\cdot C(\kappa,T,L,L_{R_*},L^{\prime},L_{R^{\prime}_*})}\lt\frac{\varepsilon}{2},$$

by letting $n,m\to\infty$ . Consequently, we conclude that (2.27) holds.

2.3. Proof of Theorem 1

In this subsection, we turn to proving the main theorem in this section. The proof consists of three steps.

Step 1: Existence. Let $\{X^{(n)}(t)\}_{n\geqslant1}$ be the Cauchy sequence in $L^{\kappa}(\Omega;\;D([0,T];\;\mathbb{R}^d))$ given by (2.12). Keep in mind that the space $L^{\kappa}(\Omega;\;D([0,T];\;\mathbb{R}^d))$ , equipped with the norm $||X||_{L^{\kappa}}\;:\!=\;\left(\mathbb{E}\left[\sup_{0\leqslant t\leqslant T}|X(t)|^{\kappa}\right]\right)^{\frac{1}{\kappa}}$ , is a Banach space (see Proposition 1). Thus, there exists an $\{\mathcal{F}_t\}_{0\leqslant t\leqslant T}$ -adapted $\mathbb{R}^d$ -valued càdlàg stochastic process $\{X(t)\}_{t\in[0,T]}$ with $X(0)=x_0$ and $\mu_t=\mathcal{L}_{X(t)}$ such that

(2.40) \begin{equation}\lim_{n\to \infty}\left[\mathbb{E}\left|X^{(n)}(t)-X(t)\right|^{\kappa}\right]^{\frac{1}{\kappa}}\leqslant\lim_{n\to \infty}\left[\mathbb{E}\sup_{0\leqslant t\leqslant T}\left|X^{(n)}(t)-X(t)\right|^{\kappa}\right]^{\frac{1}{\kappa}}=0.\end{equation}

We next prove that $\{X(t)\}_{t\in[0,T]}$ is a solution to (2.8). Indeed, the main idea is to show that the right-hand side of (2.12) converges in probability to

$$x_0+\!\int_0^t \!b(s,X(s-\!),\mu_s)\, {\textrm{d}} s+\!\int_0^t\!\sigma(s,X(s-\!),\mu_s)\, {\textrm{d}} W(s)+\!\int_0^t\!\int_Uh(s,X(s-\!),\mu_s,z)\tilde{N}({\textrm{d}} z,{\textrm{d}} s),$$

by taking the limit on both sides of (2.12). Here $\mu_s=\mathcal{L}(X(s))$ for any $s\in[0,T]$ .

First, it follows from (2.40) that there exists a subsequence (for notational simplicity, still indexed by n) such that, for all $s\in[0,T]$ ,

$$X^{(n)}(s,\omega) \to X(s,\omega),\quad \mathbb{P}\text{-almost surely.}$$

By applying Lemma 2, the Wasserstein distance between $\mu^{(n)}_{[s]_n}$ and $\mu_s$ satisfies

(2.41) \begin{align}&\lim_{n\to \infty}\sup_{0\leqslant s\leqslant t}W_{2}^{\kappa}\left(\mu^{(n)}_{[s]_n},\mu_s\right)\notag\\&\leqslant2^{\kappa-1}\lim_{n\to \infty}\sup_{0\leqslant s\leqslant t}\mathbb{E}\left|X^{(n)}(s)-X^{(n)}([s]_n)\right|^{\kappa}+2^{\kappa-1}\lim_{n\to \infty}\mathbb{E}\left[\sup_{0\leqslant s\leqslant t}\left|X^{(n)}(s)-X(s)\right|^{\kappa}\right]\notag\\&\leqslant2^{\kappa-1}C\lim_{n\to\infty}h_n=0.\end{align}

Taking Assumption 3 into account, it follows immediately that, for all $s\in[0,T]$ and almost all $\omega\in\Omega$ ,

\begin{equation}b\left(s, X^{(n)}(s), \mu^{(n)}_{[s]_n}\right)\to b\left(s, X(s),\mu_s\right)\!,\;\sigma\left(s, X^{(n)}(s), \mu^{(n)}_{[s]_n}\right)\to \sigma\left(s, X(s),\mu_s\right)\!,\notag\end{equation}

(2.42) \begin{equation}\int_Uh\left(s, X^{(n)}(s-\!), \mu^{(n)}_{[s]_n},z\right)\nu({\textrm{d}} z)\to\int_Uh\left(s,X(s-\!),\mu_s,z\right)\nu({\textrm{d}} z),\end{equation}

as $n\to\infty$ .

Next, we claim that the sequences $\{b(s, X^{(n)}(s), \mu^{(n)}_{[s]_n})\}_{n\geqslant1}$ and $\{\sigma(s, X^{(n)}(s), \mu^{(n)}_{[s]_n})\}_{n\geqslant1}$ are uniformly integrable. In fact, from Assumptions 4 and 5 and Lemma 1, we obtain the following uniform boundedness,

\begin{align} &\sup_{n\geqslant1}\mathbb{E}|b(s, X^{(n)}(s), \mu^{(n)}_{[s]_n})| \leqslant\sqrt{3K_1}\sup_{n\geqslant1}\mathbb{E}[1+|X^{(n)}(s)|^{\frac{\kappa}{2}}+W_2^{\frac{\kappa}{2}}( \mu^{(n)}_{[s]_n},\delta_0)]\leqslant \sqrt{3K_1}(1+2C), \notag\\ &\sup_{n\geqslant1}\mathbb{E}\|\sigma(t, X^{(n)}(s), \mu^{(n)}_{[s]_n})\|^2 \leqslant K\sup_{n\geqslant1}\mathbb{E}[1+|X^{(n)}(s)|^{2}+W_2^{2}( \mu^{(n)}_{[s]_n},\delta_0)]\leqslant K(1+2C),\notag \end{align}

and the following uniform absolute continuity,

\begin{align} \sup_{n\geqslant1}\mathbb{E}\left[\left|b\left(s, X^{(n)}(s), \mu^{(n)}_{[s]_n}\right)\right| \mathbb{I}_A\right] &\leqslant K_1\sup_{n\geqslant1}\left[\mathbb{E}\left(1+\left|X^{(n)}(s)\right|^{\kappa}+W_2^{\kappa}\left( \mu^{(n)}_{[s]_n},\delta_0\right)\right)\right]^{\frac{1}{2}}(\mathbb{P}(A))^{\frac{1}{2}}\notag\\ &\leqslant K_1\sqrt{1+2C}(\mathbb{P}(A))^{\frac{1}{2}}\to 0,\notag\\ \sup_{n\geqslant1}\mathbb{E}\left[\left\|\sigma\left(s, X^{(n)}(s), \mu^{(n)}_{[s]_n}\right)\right\|^2\mathbb{I}_A\right] &\leqslant K\sup_{n\geqslant1}\left[\mathbb{E}\left(1+\left|X^{(n)}(s)\right|^{2}+W_2^{2}\left( \mu^{(n)}_{[s]_n},\delta_0\right)\!\right)^2\right]^{\frac{1}{2}}\!(\mathbb{P}(A))^{\frac{1}{2}}\notag\\ &\leqslant K\sqrt{3(1+2C)}(\mathbb{P}(A))^{\frac{1}{2}}\to 0,\notag\end{align}

when $\mathbb{P}(A)\to 0$ . The uniform integrability of $\{b(s, X^{(n)}(s), \mu^{(n)}_{[s]_n})\}_{n\geqslant1}$ and $\{\sigma(s, X^{(n)}(s), \mu^{(n)}_{[s]_n})\}_{n\geqslant1}$ follows from [Reference Shiryaev38, Lemma 3 in p. 190].

Hence, by applying the dominated convergence theorem [Reference Shiryaev38, Theorem 4 on p. 188], together with (2.42), we obtain, for any $s\in[0,T]$ ,

(2.43) \begin{align}&\,\,\,\lim_{n\to \infty}\mathbb{E}\left|b\left(s, X^{(n)}(s), \mu^{(n)}_{[s]_n}\right)-b\left(s, X(s),\mu_s\right)\right|=0,\end{align}

(2.44) \begin{align}&\lim_{n\to \infty}\mathbb{E}\left\|\sigma\left(s, X^{(n)}(s), \mu^{(n)}_{[s]_n}\right)-\sigma\left(s, X(s),\mu_s\right)\right\|^2=0.\end{align}

In addition, note that, following from (2.40),

\begin{equation}\mathbb{E}\left[\sup_{0\leqslant t\leqslant T}\left|X(t)\right|^{\kappa}\right]\leqslant C.\notag\end{equation}

We further have the following estimates based on Assumptions 4 and 5 and Lemma 1:

(2.45) \begin{align}&\sup_{n\geqslant1}\sup_{s\in[0,t]}\mathbb{E}\left|b\left(s, X^{(n)}(s), \mu^{(n)}_{[s]_n}\right)-b(s, X(s),\mu_s)\right|\notag\\&\leqslant \sqrt{3K_1}\sup_{n\geqslant1}\sup_{s\in[0,t]}\mathbb{E}\left[2+\left|X^{(n)}(s)\right|^{\frac{\kappa}{2}}+|X(s)|^{\frac{\kappa}{2}}+W_2^{\frac{\kappa}{2}}\left( \mu^{(n)}_{[s]_n},\delta_0\right)+W_2^{\frac{\kappa}{2}}(\mu_s,\delta_0)\right]\notag\\&\leqslant 2\sqrt{3K_1}(1+2C),\end{align}

(2.46) \begin{align}&\sup_{n\geqslant1}\sup_{s\in[0,t]}\mathbb{E}\left\|\sigma\left(s, X^{(n)}(s), \mu^{(n)}_{[s]_n}\right)-\sigma(s, X(s),\mu_s)\right\|^2\notag\\&\leqslant 2K\sup_{n\geqslant1}\sup_{s\in[0,t]}\mathbb{E}\left[2+\left|X^{(n)}(s)\right|^{2}+|X(s)|^{2}+W_2^{\kappa}\left( \mu^{(n)}_{[s]_n},\delta_0\right)+W_2^{2}(\mu_s,\delta_0)\right]\leqslant 4K(1+2C).\end{align}

For any $t\in[0,T]$ , by applying the dominated convergence theorem in conjunction with (2.43) and (2.45), we eventually obtain

(2.47) \begin{align}&\lim_{n\to \infty}\mathbb{E}\left|\int_0^t\left(b\left(s, X^{(n)}(s), \mu^{(n)}_{[s]_n}\right)-b(s, X(s),\mu_s)\right)\, {\textrm{d}} s\right|\notag\\&\leqslant\lim_{n\to \infty}\int_0^t\mathbb{E}\left|b\left(s, X^{(n)}(s), \mu^{(n)}_{[s]_n}\right)-b(s, X(s),\mu_s)\right|\, {\textrm{d}} s=0.\end{align}

Similarly, in view of (2.44) and (2.46), we arrive at

(2.48) \begin{align}&\lim_{n\to \infty}\mathbb{E}\left|\int_0^t\left(\sigma\left(s, X^{(n)}(s), \mu^{(n)}_{[s]_n}\right)-\sigma(s, X(s),\mu_s)\right)\, {\textrm{d}} W(s)\right|^2\notag\\&=\lim_{n\to \infty}\int_0^t\mathbb{E}\left\|\sigma\left(s, X^{(n)}(s), \mu^{(n)}_{[s]_n}\right)-\sigma(s, X(s),\mu_s)\right\|^2\, {\textrm{d}} s=0.\end{align}

Finally, we examine the estimates for the integral w.r.t. the Poisson random measure. For any $u\in[0,T]$ , it follows from the inequality (2.5) (with $p=1$ ) in Proposition 2 and Assumptions 1 and 2 that

By (2.40), (2.41), and Lyapunov inequality, we deduce that

(2.49) \begin{align}\lim_{n\to\infty}\mathbb{E}\sup_{0\leqslant u\leqslant t}\left|\int_0^u\int_U\left(h\left(s,X^{(n)}(s-\!),\mu^{(n)}_{[s]_n},z\right)-h(s,X(s-\!),\mu_s,z)\right)\tilde{N}({\textrm{d}} s,{\textrm{d}} z)\right|=0.\end{align}

As a consequence, by (2.47), (2.48), and (2.49), we conclude that the process $\{X(t)\}_{t\in[0,T]}$ is a strong solution to (2.8). This completes the proof of existence.

Step 2: Boundedness. For $t\in[0,T]$ , let $X(t)\in L^{\kappa}(\Omega;\;D([0,T];\;\mathbb{R}^d))$ be a solution to (2.8). In the following, we estimate the rth moment of the solution $(X(t))_{t\in[0,T]}$ , where $r\geqslant \max\{\frac{\kappa^2}{2},4\}$ and the initial value $X(0) = x_0$ satisfies $\mathbb{E}|x_0|^r \lt \infty$ , as specified in Assumption 6.

For every $R\gt 0$ , we define the stopping time

$$\pi_R\;:\!=\;\inf\big\{t\in[0,T]:|X(t)|\gt R\big\}\wedge T.$$

It is clear that $|X(t-\!)|\leqslant R$ for $0\leqslant t\leqslant \pi_R$ , and $\mathbb{E}\sup_{0\leqslant t\leqslant u\wedge\pi_R}|X(t-\!)|^r\lt\infty$ for any $u\in[0,T]$ . To derive an upper-bound for $\mathbb{E}\sup_{0\leqslant t\leqslant u\wedge\pi_R}|X(t)|^r$ , we employ the procedure similar to that in the proof of Lemma 1, where the case for $X^{(n)}(t)$ with $t\in[0,T]$ was considered. Specifically, for $r\geqslant\max\{\frac{\kappa^2}{2},4\}$ and $t\in[0,u\wedge\pi_R]$ , and utilizing tools such as Itô’s formula, as demonstrated in the proof of Lemma 1, we estimate that

(2.50) \begin{align}\mathbb{E}\sup_{0\leqslant t\leqslant u\wedge\pi_R}|X(t)|^r&\leqslant\mathbb{E}|x_0|^r+\frac{1}{2}\mathbb{E}\sup_{0\lt t\leqslant u\wedge\pi_R}|X(t-\!)|^r+\widehat{D}\!\int_0^u\!\left(\!1+\mathbb{E}\sup_{0\lt t\leqslant s\wedge\pi_R}|X(t-\!)|^r\right) {\textrm{d}} s\notag\\&\quad +K(r-2)\frac{r^2+r+2C_1}{2r}\int_0^u\left(1+\mathbb{E}\sup_{0\lt t\leqslant s\wedge\pi_R}|X(t-\!)|^r\right)\, {\textrm{d}} s\lt\infty,\end{align}

with $\widehat{D}=2\cdot3^{\frac{r}{2}-1}(r+1)K+C_2^r2^{r}(r-1)^{r-1}(2K)^{r}(3T)^{\frac{r}{2}-1}+2C_1\cdot\left(\frac{2K}{r}3^{\frac{r}{2}-1}+K_2\right).$ Note that

$$\mathbb{E}\sup_{0\lt t\leqslant u\wedge\pi_R}|X(t-\!)|^r\leqslant\mathbb{E}\sup_{0\leqslant t\leqslant u\wedge\pi_R}|X(t)|^r.$$

Since $\pi_R\to T,$ $\mathbb{P}$ -almost surely, we conclude the proof of the estimate (2.9) by applying Grönwall’s inequality and the Fatou’s lemma. Specifically, we have

$$ \mathbb{E}\sup_{0\leqslant t\leqslant T}|X(t)|^r\leqslant \liminf_{R\to\infty}\mathbb{E}\sup_{0\leqslant t\leqslant T\wedge\pi_R}|X(t)|^r\leqslant C_r\lt\infty. $$

Step 3: Uniqueness. Let X(t), Y(t) be two solutions of (2.8) on the same probability space with $X(0)=Y(0)$ . By (2.14), for a fixed $r\geqslant\max\{\kappa^2/2, 4\}$ , there exists a positive constant $C_r$ such that

$$\mathbb{E}\left[\sup_{0\leqslant t\leqslant T}|X(t)|^r\right]\leqslant C_r,\quad \mathbb{E}\left[\sup_{0\leqslant t\leqslant T}|Y(t)|^r\right]\leqslant C_r.$$

For a sufficiently large $R\gt 0$ , we define the stopping time

$$\bar{\tau}_R\;:\!=\;\inf\big\{t\in[0,T]: |X(t)|\vee|Y(t)|\gt R\big\}.$$

To proceed, we compare $|X(t)-Y(t)|$ and $\bar{\tau}_R$ in this context with $|X^{(n)}(t)-X^{(m)}(t)|$ and $\tau_R$ , as introduced in the proof of Lemma 3. Clearly, the same method as used in the proof of Lemma 3 can be applied here, yielding the following estimate:

\begin{align}\mathbb{E}\left[\sup_{0\leqslant t\leqslant T}|X(t)-Y(t)|^{2}\right]&=\mathbb{E}\left[\sup_{0\leqslant t\leqslant T}|X(t)-Y(t)|^{2}\mathbb{I}_{\{\bar{\tau}_R\gt T\}}\right]\notag\\&\quad +\mathbb{E}\left[\sup_{0\leqslant t\leqslant T}|X(t)-Y(t)|^{2}\mathbb{I}_{\{\bar{\tau}_R\leqslant T\}}\right]\notag\\&\leqslant\mathbb{E}\left[\sup_{0\leqslant t\leqslant T}\left|X(t\wedge\bar{\tau}_R)-Y(t\wedge\bar{\tau}_R)\right|^{2}\right]+C_1\sqrt{\mathbb{P}\Big(\bar{\tau}_R\leqslant T\Big)}\leqslant\frac{C}{R^2}.\notag\end{align}

Letting $R\to\infty$ gives the uniqueness of the solution to (2.8).

This completes the proof of Theorem 1.

3. Stochastic averaging principle

In this section, we establish a stochastic averaging principle for the following stochastic integral equation

(3.1) \begin{align} X_{\varepsilon}(t)&=x_0+\int_0^t b\left(\frac{s}{\varepsilon },X_{\varepsilon}(s-\!),\mathscr{L}_{X_{\varepsilon}(s)}\right)\, {\textrm{d}} s+\int_0^t\sigma\left(\frac{s}{\varepsilon },X_{\varepsilon}(s-\!),\mathscr{L}_{X_{\varepsilon}(s)}\right)\, {\textrm{d}} W(s)\notag\\ &\quad +\int_0^t\int_{U}h\left(\frac{s}{\varepsilon },X_{\varepsilon}(s-\!),\mathscr{L}_{X_{\varepsilon}(s)},z\right)\tilde{N}({\textrm{d}} s,{\textrm{d}} z),\quad t\in[0,T],\end{align}

where $\varepsilon$ is a small positive parameter ( $0\lt\varepsilon\ll1$ ). Assuming that (3.1) satisfies the conditions specified in Assumptions 1–7, the existence and uniqueness of its solution follow directly as a consequence of Theorem 1.

As mentioned in Section 1, our main goal is to demonstrate that the solution $(X_{\varepsilon}(t))_{ t\in[0,T]}$ of (3.1) can be approximated by a simpler (or averaged) process in an appropriate sense. To proceed, we associate (3.1) with the following averaged McKean–Vlasov SDE:

(3.2) \begin{align} \bar{X}(t)&=x_0+\int_0^t \bar{b}\left(\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)}\right)\, {\textrm{d}} s+\int_0^t\bar{\sigma}\left(\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)}\right)\, {\textrm{d}} W(s)\notag\\ &\quad +\int_0^t\int_{U}\bar{h}\left(\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)},z\right)\tilde{N}({\textrm{d}} s,{\textrm{d}} z),\quad t\in[0,T],\end{align}

where $\bar{b}: \mathbb{R}^d\times M_2(\mathbb{R}^d)\to \mathbb{R}^d$ , $\bar{\sigma}: \mathbb{R}^d\times M_2(\mathbb{R}^{d})\to\mathbb{R}^{d\times m}$ , and $\bar{h}: \mathbb{R}^d\times M_2(\mathbb{R}^d)\times U\to \mathbb{R}^d$ are Borel measurable functions. To ensure that (3.2) also admits a unique solution and to facilitate the application of stochastic averaging techniques, we impose specific averaging conditions. It is worth noting that these conditions differ slightly from the classical ones (see, e.g., [Reference Shen, Song and Wu36, Reference Xu, Duan and Xu42]) due to the distinct characteristics of the nonlinear terms involved in the equation.

Assumption 8. (Averaging conditions.) There exist positive bounded functions (sometimes referred to as rate functions of convergence) $\varphi_i$ , defined on [0,T], with $\lim_{t\to\infty}\varphi_i(t)=0$ for $i=1, 2, 3$ , such that

$$\frac{1}{t}\int_{0}^{t}|b(s,x,\mu)-\bar{b}(x,\mu)|^2\, {\textrm{d}} s\leqslant\varphi_1(t)C_R^b(1+|x|^2),$$

$$\frac{1}{t}\int_{0}^{t}\|\sigma(s,x,\mu)-\bar{\sigma}(x,\mu)\|^2\, {\textrm{d}} s\leqslant\varphi_2(t)C_R^{\sigma}(1+|x|^2),$$

$$\frac{1}{t}\int_{0}^{t}\int_{U}|h(s,x,\mu,z)-\bar{h}(x,\mu,z)|^2\nu({\textrm{d}} z)\, {\textrm{d}} s\leqslant\varphi_3(t)C_R^{h}(1+|x|^2),$$

respectively, for all $t\in[0,T]$ , $x,y\in\mathbb{R}^d$ with $|x|\vee|y|\leqslant R$ , and $\mu\in\mathcal{M}_2(\mathbb{R}^d)$ . Here, $C_R^b$ , $C_R^\sigma$ , and $C_R^h$ are positive constants.

Furthermore, if $\kappa\gt 2$ , an additional condition is required.

Assumption 9. (Additional averaging conditions on the jump coefficients.) There exists a positive bounded function $\varphi$ , defined on [0, T], with $\lim_{t\to\infty}\varphi(t)=0$ , such that

$$\frac{1}{t}\int_{0}^{t}\int_{U}|h(s,x,\mu,z)-\bar{h}(x,\mu,z)|^r\nu({\textrm{d}} z)\, {\textrm{d}} s\leqslant\varphi(t)C_R^{h}(1+|x|^r),$$

$$\frac{1}{t}\int_{0}^{t}\int_{U}|h(s,x,\mu,z)-\bar{h}(x,\mu,z)|^{\kappa}\nu({\textrm{d}} z)\, {\textrm{d}} s\leqslant\varphi(t)C_R^{h}(1+|x|^{\kappa}),$$

respectively, for all $t\in[0,T]$ , $x,y\in\mathbb{R}^d$ with $|x|\vee|y|\leqslant R$ , and $\mu\in\mathcal{M}_2(\mathbb{R}^d)$ .

The main theorem on the averaging principle for (3.1) is thus formulated as follows.

Theorem 2. (Averaging principle.) Suppose that Assumptions 1–9 hold. Then, the following averaging principle holds:

(3.3) \begin{equation}\lim_{\varepsilon\to0}\mathbb{E}\sup_{0\leqslant t\leqslant T}|X_{\varepsilon}(t)-\bar{X}(t)|^{2}=0.\end{equation}

As a direct consequence of Theorem 2 and by applying the Chebyshev–Markov inequality, we have the following corollary.

Corollary 1. The solution $X_{\varepsilon}(t)$ converges in probability to the averaged solution $\bar{X}(t)$ . Specifically, for any $\delta\gt 0$ ,

\begin{equation}\mathbb{P}(\sup_{0\leqslant t\leqslant T}\left|X_{\varepsilon}(t)-\bar{X}(t)\right|\gt\delta)\to 0,\notag \quad \text{as} \ \varepsilon \to 0.\end{equation}

Prior to establishing Theorem 2, it is necessary to address the well-posedness of the averaged equation (3.2). The following lemma ensures this property.

We now complete the proof of Theorem 2 as follows.

Proof of Theorem 2. For any $t\in[0,T]$ , it follows from (3.1) and (3.2) that

\begin{align}X_{\varepsilon}(t)-\bar{X}(t)&=\int_0^t\left[b\left(\frac{s}{\varepsilon},X_{\varepsilon}(s-\!),\mathscr{L}_{X_{\varepsilon}(s)}\right)-\bar{b}\left(\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)}\right)\right]\, {\textrm{d}} s\notag\\&\quad +\int_0^t\left[\sigma\left(\frac{s}{\varepsilon},X_{\varepsilon}(s-\!),\mathscr{L}_{X_{\varepsilon}(s)}\right)-\bar{\sigma}\left(\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)}\right)\right]\, {\textrm{d}} W(s)\notag\\&\quad +\int_0^t\int_{U}\left[h\left(\frac{s}{\varepsilon},X_{\varepsilon}(s-\!),\mathscr{L}_{X_{\varepsilon}(s)},z\right)-\bar{h}\left(\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)},z\right)\right]\tilde{N}({\textrm{d}} s,{\textrm{d}} z).\notag\end{align}

To handle the one-sided locally Lipschitz case, we introduce a stopping time $\eta_R$ for each $R\gt 0$ defined as

$$\eta_R\;:\!=\;\inf\big\{t\in[0,T]: |X_{\varepsilon}(t)|\vee|\bar{X}(t)|\gt R\big\}.$$

Using De Morgan’s law, the following decomposition holds:

(3.3) \begin{align}\mathbb{E}\left[\sup_{0\leqslant t\leqslant T}|X_{\varepsilon}(t)-\bar{X}(t)|^{2}\right]&=\mathbb{E}\left[\sup_{0\leqslant t\leqslant T}|X_{\varepsilon}(t)-\bar{X}(t)|^{2}\mathbb{I}_{\{\eta_R\gt T\}}\right]\notag\\&\quad +\mathbb{E}\left[\sup_{0\leqslant t\leqslant T}|X_{\varepsilon}(t)-\bar{X}(t)|^{2}\mathbb{I}_{\{\eta_R\leqslant T\}}\right]\notag\\&\;=\!:\; I_1+I_2.\notag\end{align}

We now proceed to estimate each term on the right-hand side of the equation above.

1. (1) Estimation of the term $I_1$ . We begin by bounding the term $I_1$ as follows

   (3.4) \begin{equation}I_1=\mathbb{E}\left[\sup_{0\leqslant t\leqslant T}|X_{\varepsilon}(t)-\bar{X}(t)|^{2}\mathbb{I}_{\{\eta_R\gt T\}}\right]\leqslant\mathbb{E}\left[\sup_{0\leqslant t\leqslant T}|X_{\varepsilon}(t\wedge\eta_R)-\bar{X}(t\wedge\eta_R)|^{2}\right]\lt\infty.\end{equation}
   Here, we are effectively considering the process up to the stopping time $\eta_R$ , which ensures that $|X_{\varepsilon}(t-\!)|$ and $|\bar{X}(t-\!)|$ are bounded by R for all $t\leqslant T$ . By applying the Itô’s formula, we obtain
   $$|X_{\varepsilon}(t\wedge\eta_R)-\bar{X}(t\wedge\tau_R)|^{2}=\sum_{i=1}^5\Lambda_{i}(t),$$
   where the terms $\Lambda_i$ for $i=1, \ldots,5$ are defined as follows:
   \begin{align*} \Lambda_{1}(t)&=2\int_0^{t\wedge\eta_R}\left\langle X_{\varepsilon}(s-\!)-\bar{X}(s-\!), b\left(\frac{s}{\varepsilon},X_{\varepsilon}(s-\!),\mathscr{L}_{X_{\varepsilon}(s)}\right)-\bar{b}\left(\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)}\right)\right\rangle\, {\textrm{d}} s,\notag\\ \Lambda_{2}(t)&=\int_0^{t\wedge\eta_R}\left\| \sigma\left(\frac{s}{\varepsilon},X_{\varepsilon}(s-\!),\mathscr{L}_{X_{\varepsilon}(s)}\right)-\bar{\sigma}\left(\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)}\right)\right\|^2\, {\textrm{d}} s,\notag\\ \Lambda_{3}(t)&=\int_0^{t\wedge\eta_R}\int_U\left|h\left(\frac{s}{\varepsilon},X_{\varepsilon}(s-\!),\mathscr{L}_{X_{\varepsilon}(s)},z\right)-\bar{h}\left(\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)},z\right)\right|^2 N({\textrm{d}} s,{\textrm{d}} z),\notag\\ \Lambda_{4}(t)&=2\int_0^{t\wedge\eta_R}\left\langle X_{\varepsilon}(s-\!)-\bar{X}(s-\!), \sigma\left(\frac{s}{\varepsilon},X_{\varepsilon}(s-\!),\mathscr{L}_{X_{\varepsilon}(s)}\right)\right .\notag\\&\quad \left .-\,\bar{\sigma}\left(\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)}\right)\, {\textrm{d}} W(s)\right\rangle,\notag\\ \Lambda_{5}(t)&=2\int_0^{t\wedge\eta_R}\left\langle X_{\varepsilon}(s-\!)-\bar{X}(s-\!),h\left(\frac{s}{\varepsilon},X_{\varepsilon}(s-\!),\mathscr{L}_{X_{\varepsilon}(s)},z\right)\right .\notag\\&\quad \left .-\,\bar{h}\left(\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)},z\right)\right\rangle\tilde{N}({\textrm{d}} s,{\textrm{d}} z).\notag\end{align*}
   By taking the supremum over [0, u] for $u\in[0,T]$ and then taking expectations, we can now estimate $\mathbb{E}\big[\sup_{0\leqslant t\leqslant u}\Lambda_{i}(t)\big]$ for $i=1, \ldots, 5$ , respectively. In view of Assumptions 1, 2, and 8, we obtain
   (3.5) \begin{align}\mathbb{E}&\left[\sup_{0\leqslant t\leqslant u}\Lambda_{1}(t)\right]\notag\\&\leqslant2\mathbb{E}\left[\sup_{0\leqslant t\leqslant u}\int_0^{t\wedge\eta_R}\left\langle X_{\varepsilon}(s-\!)-\bar{X}(s-\!), b\left(\frac{s}{\varepsilon},X_{\varepsilon}(s-\!),\mathscr{L}_{X_{\varepsilon}(s)}\right)\right .\right .\notag\\&\quad \left .\left .-\,b\left(\frac{s}{\varepsilon},\bar{X}(s-\!),\mathscr{L}_{X_{\varepsilon}(s)}\right)\right\rangle\, {\textrm{d}} s\right]\notag\\&\quad +2\mathbb{E}\left[\sup_{0\leqslant t\leqslant u}\int_0^{t\wedge\eta_R}|X_{\varepsilon}(s-\!)-\bar{X}(s-\!)|\cdot\left|b\left(\frac{s}{\varepsilon},\bar{X}(s-\!),\mathscr{L}_{X_{\varepsilon}(s)}\right)\right .\right .\notag\\&\quad \left .\left .-\,b\left(\frac{s}{\varepsilon},\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)}\right)\right|\, {\textrm{d}} s\right]\notag\\&\quad +2\mathbb{E}\left[\sup_{0\leqslant t\leqslant u}\int_0^{t\wedge\eta_R}|X_{\varepsilon}(s-\!)-\bar{X}(s-\!)|\cdot\left|b\left(\frac{s}{\varepsilon},\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)}\right)\right .\right .\notag\\&\quad \left .\left .-\,\bar{b}\left(\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)}\right)\right|\, {\textrm{d}} s\right]\notag\\&\leqslant2L_R\mathbb{E}\left[\int_0^{u\wedge\eta_R}|X_{\varepsilon}(s-\!)-\bar{X}(s-\!)|^2\, {\textrm{d}} s\right]\notag\\&\quad +2\sqrt{L}\mathbb{E}\left[\int_0^{u\wedge\eta_R}|X_{\varepsilon}(s-\!)-\bar{X}(s-\!)|\cdot W_2\left(\mathscr{L}_{X_{\varepsilon}(s)},\mathscr{L}_{\bar{X}(s)}\right)\, {\textrm{d}} s\right]\notag\\&\quad +2\mathbb{E}\left[\int_0^{u\wedge\eta_R}|X_{\varepsilon}(s-\!)-\bar{X}(s-\!)|\cdot\left|b\left(\frac{s}{\varepsilon},\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)}\right)\right .\right .\notag\\&\quad \left .\left .-\,\bar{b}\left(\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)}\right)\right|\, {\textrm{d}} s\right]\notag\\&\leqslant(2L_R+2\sqrt{L}+1)\mathbb{E}\left[\int_0^{u\wedge\eta_R}|X_{\varepsilon}(s-\!)-\bar{X}(s-\!)|^2\, {\textrm{d}} s\right]\notag\\&\quad +u\mathbb{E}\left[\frac{\varepsilon}{u\wedge\eta_R}\int_0^{\frac{u\wedge\eta_R}{\varepsilon}}\left|b\left(s,\bar{X}(s\varepsilon-),\mathscr{L}_{\bar{X}(s\varepsilon)}\right)-\bar{b}\left(\bar{X}(s\varepsilon-),\mathscr{L}_{\bar{X}(s\varepsilon)}\right)\right|^2\, {\textrm{d}} s\right]\notag\\&\leqslant(2L_R+2\sqrt{L}+1)\int_0^u\mathbb{E}\sup_{0\leqslant t\leqslant s}|X_{\varepsilon}(t\wedge\eta_R)-\bar{X}(t\wedge\eta_R)|^2\, {\textrm{d}} s+uC_R^b\varphi_1\left(\frac{u\wedge\eta_R}{\varepsilon}\right)\notag\\&\quad\times\left(1+\mathbb{E}\sup_{0\leqslant t\leqslant u}|\bar{X}(t)|^2\right)\notag\\&\leqslant(2L_R+2\sqrt{L}+1)\int_0^u\mathbb{E}\sup_{0\leqslant t\leqslant s}|X_{\varepsilon}(t\wedge\eta_R)-\bar{X}(t\wedge\eta_R)|^2\, {\textrm{d}} s+uC_R^b\cdot C\varphi_1\left(\frac{u\wedge\eta_R}{\varepsilon}\right).\end{align}
   Here, we have used the fact that for each $u\in[0,T]$ ,
   $$\left(\mathbb{E}\sup_{0\leqslant t\leqslant u}|\bar{X}(t)|^2\right)^{\frac{1}{2}}\leqslant(\mathbb{E}\sup_{0\leqslant t\leqslant u}|\bar{X}(t)|^r)^{\frac{1}{r}}\lt\infty,\quad \text{if}\ \mathbb{E}|X_{\varepsilon}(0)|^r\lt\infty.$$
   By applying Assumptions 1, 2, 8, and employing techniques analogous to those used in deriving the estimate for $\Lambda_1$ , we establish the following bound for $\Lambda_2$ :
   (3.6) \begin{align}&\mathbb{E}\left[\sup_{0\leqslant t\leqslant u}\Lambda_{2}(t)\right]\notag\\&\leqslant3\mathbb{E}\left[\sup_{0\leqslant t\leqslant u}\int_0^{t\wedge\eta_R}\left\|\sigma\left(\frac{s}{\varepsilon},X_{\varepsilon}(s-\!),\mathscr{L}_{X_{\varepsilon}(s)}\right)-\sigma\left(\frac{s}{\varepsilon},\bar{X}(s-\!),\mathscr{L}_{X^{\varepsilon}(s)}\right)\right\|^2\, {\textrm{d}} s\right]\notag\\&\quad +3\mathbb{E}\left[\sup_{0\leqslant t\leqslant u}\int_0^{t\wedge\eta_R}\left\|\sigma\left(\frac{s}{\varepsilon},\bar{X}(s-\!),\mathscr{L}_{X_{\varepsilon}(s)}\right)-\sigma\left(\frac{s}{\varepsilon},\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)}\right)\right\|^2\, {\textrm{d}} s\right]\notag\\&\quad +3\mathbb{E}\left[\sup_{0\leqslant t\leqslant u}\int_0^{t\wedge\eta_R}\left\|\sigma\left(\frac{s}{\varepsilon},\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)}\right)-\bar{\sigma}\left(\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)}\right)\right\|^2\, {\textrm{d}} s\right]\notag\\&\leqslant3L_R\mathbb{E}\left[\int_0^{u\wedge\eta_R}|X_{\varepsilon}(s-\!)-\bar{X}(s-\!)|^2\, {\textrm{d}} s\right]+3L\mathbb{E}\left[\int_0^{u\wedge\eta_R}W_2^2\left(\mathscr{L}_{X_{\varepsilon}(s)},\mathscr{L}_{\bar{X}(s)}\right)\, {\textrm{d}} s\right]\notag\\&\quad +3\mathbb{E}\left[\int_0^{u\wedge\eta_R}\left\|\sigma\left(\frac{s}{\varepsilon},\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)}\right)-\bar{\sigma}\left(\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)}\right)\right\|^2\, {\textrm{d}} s\right]\notag\\&\leqslant3(L_R+L)\int_0^u\mathbb{E}\sup_{0\leqslant t\leqslant s}|X_{\varepsilon}(t\wedge\eta_R)-\bar{X}(t\wedge\eta_R)|^2\, {\textrm{d}} s+3uC_R^{\sigma}\cdot C\varphi_2\left(\frac{u\wedge\eta_R}{\varepsilon}\right).\end{align}
   Similarly, using Assumptions 1, 2, and 8, we obtain the following estimate for $\Lambda_3$ :
   (3.7) \begin{align}&\mathbb{E}\left[\sup_{0\leqslant t\leqslant u}\Lambda_{3}(t)\right]\notag\\&\leqslant\mathbb{E}\left[\int_0^{t\wedge\eta_R}\int_U\left|h\left(\frac{s}{\varepsilon},X_{\varepsilon}(s-\!),\mathscr{L}_{X_{\varepsilon}(s)},z\right)-\bar{h}\left(X(s-\!),\mathscr{L}_{\bar{X}(s)},z\right)\right|^2\nu({\textrm{d}} z)\, {\textrm{d}} s\right] \notag\\&\leqslant3(L_R+L)\int_0^u\mathbb{E}\sup_{0\leqslant t\leqslant s}|X_{\varepsilon}(t\wedge\eta_R)-\bar{X}(t\wedge\eta_R)|^2\, {\textrm{d}} s+3uC_R^{h}\cdot C\varphi_3\left(\frac{u\wedge\eta_R}{\varepsilon}\right).\end{align}
   Next, we apply the BDG inequality, along with Young’s inequality (2.1) (with $\epsilon=\frac{1}{24}$ and $p=q=2$ ) and the estimate (3.6), to derive the following bound for $\Lambda_4$ :
   (3.8) \begin{align}\mathbb{E}&\left[\sup_{0\leqslant t\leqslant u}\Lambda_{4}(t)\right]\notag\\&\leqslant2\sqrt{32}\mathbb{E}\left(\int_0^{u\wedge\eta_R}|X_{\varepsilon}(s-\!)-\bar{X}(s-\!)|^2\cdot\left\|\sigma\left(\frac{s}{\varepsilon},X_{\varepsilon}(s-\!),\mathscr{L}_{X_{\varepsilon}(s)}\right)\right .\right .\notag\\&\quad \left .\left .-\bar{\sigma}\left(\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)}\right)\right\|^2\, {\textrm{d}} s\right)^{\frac{1}{2}}\notag\\&\leqslant12\mathbb{E}\Bigg[\sup_{0\leqslant t\leqslant u}|X_{\varepsilon}(t\wedge\eta_R)-\bar{X}(t\wedge\eta_R)|\left(\int_0^{u\wedge\eta_R}\left\|\sigma\left(\frac{s}{\varepsilon},X_{\varepsilon}(s-\!),\mathscr{L}_{X_{\varepsilon}(s)}\right)\right .\right .\notag\\&\quad \left .\left .-\bar{\sigma}\left(\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)}\right)\right\|^2\, {\textrm{d}} s\right)^{\frac{1}{2}}\Bigg]\notag\\&\leqslant\frac{1}{4}\mathbb{E}\left[\sup_{0\leqslant t\leqslant u}|X_{\varepsilon}(t\wedge\eta_R)-\bar{X}(t\wedge\eta_R)|^2\right]+144\mathbb{E}\left[\int_0^{u\wedge\eta_R}\left\|\sigma\left(\frac{s}{\varepsilon},X_{\varepsilon}(s-\!),\mathscr{L}_{X_{\varepsilon}(s)}\right)\right .\right .\notag\\&\quad \left .\left .-\bar{\sigma}\left(\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)}\right)\right\|^2\, {\textrm{d}} s\right]\notag\\&\leqslant\frac{1}{4}\mathbb{E}\left[\sup_{0\leqslant t\leqslant u}|X_{\varepsilon}(t\wedge\eta_R)-\bar{X}(t\wedge\eta_R)|^2\right]+432(L_R+L)\int_0^u\mathbb{E}\sup_{0\leqslant t\leqslant s}|X_{\varepsilon}(t\wedge\eta_R)\notag\\&\quad -\bar{X}(t\wedge\eta_R)|^2\, {\textrm{d}} s+432uC_R^{\sigma}\cdot C\varphi_2\left(\frac{u\wedge\eta_R}{\varepsilon}\right).\end{align}
   By applying the inequality (2.5) (with $p=1$ ) from Proposition 2, Young’s inequality (2.1) (with $\epsilon=\frac{1}{4D},p=q=2$ ) and the estimate (3.7), we arrive at the following estimate for $\Lambda_5$ :
   (3.9) \begin{align}\mathbb{E}&\left[\sup_{0\leqslant t\leqslant u}\Lambda_{5}(t)\right]\notag\\&\leqslant2D\mathbb{E}\left(\int_0^{u\wedge\eta_R}\int_U|X_{\varepsilon}(s-\!)-\bar{X}(s-\!)|^2\left|h\left(\frac{s}{\varepsilon},X_{\varepsilon}(s-\!),\mathscr{L}_{X^{\varepsilon}(s)},z\right)\right .\right .\notag\\&\quad \left .\left .-\,\bar{h}\left(\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)},z\right)\right|^2\nu({\textrm{d}} z)\, {\textrm{d}} s\right)^{\frac{1}{2}}\notag\\&\leqslant\frac{1}{4}\mathbb{E}\left[\sup_{0\leqslant t\leqslant u}|X_{\varepsilon}(t\wedge\eta_R)-\bar{X}(t\wedge\eta_R)|^2\right]\notag\\&\quad +4D^2\mathbb{E}\left[\int_0^{u\wedge\eta_R}\int_U\left|h\left(\frac{s}{\varepsilon},X_{\varepsilon}(s-\!),\mathscr{L}_{X_{\varepsilon}(s)},z\right)-\bar{h}\left(\bar{X}(s-\!),\mathscr{L}_{\bar{X}(s)},z\right)\right|^2\nu({\textrm{d}} z)\, {\textrm{d}} s\right]\notag\\&\leqslant\frac{1}{4}\mathbb{E}\left[\sup_{0\leqslant t\leqslant u}|X_{\varepsilon}(t\wedge\eta_R)-\bar{X}(t\wedge\eta_R)|^2\right]\notag\\&\quad +12D^2(L_R+L)\int_0^u\mathbb{E}\sup_{0\leqslant t\leqslant s}|X_{\varepsilon}(t\wedge\eta_R)-\bar{X}(t\wedge\eta_R)|^2\, {\textrm{d}} s\notag\\&\quad +12D^2uC_R^{h}\cdot C\varphi_3\left(\frac{u\wedge\eta_R}{\varepsilon}\right).\end{align}
   Finally, substituting the estimates (3.5)–(3.9) into the expression (3.4) for $I_1$ , and further utilizing Grönwall’s inequality, we obtain
   (3.10) \begin{align}I_1&\leqslant\mathbb{E}\left[\sup_{0\leqslant t\leqslant T}|X_{\varepsilon}(t\wedge\eta_R)-\bar{X}(t\wedge\eta_R)|^2\right]\leqslant \widehat{N}_2{\textrm{e}}^{\widehat{N}_1T}\end{align}
   with $\widehat{N}_1=4(L_R+\sqrt{L})+2+12\cdot(73+2D^2)(L_R+L)$ and $\widehat{N}_2=2TC_R^b\cdot C\varphi_1\left(\frac{T\wedge\eta_R}{\varepsilon}\right)+870TC_R^{\sigma}\cdot C\varphi_2\left(\frac{T\wedge\eta_R}{\varepsilon}\right)+6(1+4D^2)TC_R^{\sigma}C\varphi_3\left(\frac{T\wedge\eta_R}{\varepsilon}\right)$ .

2. (2) Estimation of the term $I_2$ . Using the Cauchy–Schwarz inequality and Theorem 1, we deduce that

   (3.11) \begin{align}I_2&=\mathbb{E}\left[\sup_{0\leqslant t\leqslant T}|X_{\varepsilon}(t)-\bar{X}(t)|^2\mathbb{I}_{\{\eta_R\leqslant T\}}\right]\leqslant\sqrt{\mathbb{E}\left(\sup_{0\leqslant t\leqslant T}|X_{\varepsilon}(t)-\bar{X}(t)|^2\right)^2}\sqrt{\mathbb{E}\left(\mathbb{I}_{\{\eta_R\leqslant T\}}\right)^2}\notag\\&\leqslant2\sqrt{2}\sqrt{\mathbb{E}\left(\sup_{0\leqslant t\leqslant T}|X_{\varepsilon}(t)|^4+\sup_{0\leqslant t\leqslant T}|\bar{X}(t)|^4\right)}\sqrt{\mathbb{E}\left(\mathbb{I}_{\{\eta_R\leqslant T\}}\frac{|X_{\varepsilon}(\eta_R)|^4+|\bar{X}(\eta_R)|^4}{R^4}\right)}\notag\\&\leqslant\frac{2\sqrt{2}}{R^2}\left(\mathbb{E}\sup_{0\leqslant t\leqslant T}|X_{\varepsilon}(t)|^4+\mathbb{E}\sup_{0\leqslant t\leqslant T}|\bar{X}(t)|^4\right)\leqslant \frac{C}{R^2}.\end{align}

By combining the estimates on $I_1$ and $I_2$ , i.e. (3.10) and (3.11), we conclude that

\begin{align}\mathbb{E}\sup_{0\leqslant t\leqslant T}|X_{\varepsilon}(t)-\bar{X}(t)|^2\leqslant \widehat{N}_2{\textrm{e}}^{\widehat{N}_1T}+\frac{C}{R^2}.\notag\end{align}

Now, for any $\delta\gt 0$ , we can choose $R\gt 0$ large enough such that $\frac{C}{R^2}\lt\frac{\delta}{2}$ . In addition, by taking $\varepsilon$ sufficiently small and using the averaging condition 8, we obtain that

$$\widehat{N}_2{\textrm{e}}^{\widehat{N}_1T}\lt\frac{\delta}{2}.$$

Thus, the arbitrariness of $\delta$ implies that $\mathbb{E}\sup_{0\leqslant t\leqslant T}|X_{\varepsilon}(t)-\bar{X}(t)|^2$ converges to 0, as $\varepsilon$ goes to 0. This completes the proof.

4. Example

In this section, we provide an illustrative example to demonstrate the theoretical results established in this paper. We highlight that the model (4.1) is carefully designed to satisfy all the conditions of our assumptions and to facilitate the explicit derivation of the corresponding averaged equation.

Example 1. Consider the following one-dimensional McKean–Vlasov SDE

(4.1) \begin{align}{\textrm{d}} X_{\varepsilon}(t)&=\left[\left(X_{\varepsilon}(t-\!)-X_{\varepsilon}^3(t-\!)\right)\frac{\frac{t}{\varepsilon}}{1+\frac{t}{\varepsilon}}+\mathbb{E}X_{\varepsilon}(t-\!)\right]\, {\textrm{d}} t \notag\\&\quad +\left[X_{\varepsilon}(t-\!)\sin\left(\log^2\left(1+X_{\varepsilon}^2(t-\!)\right)\right)\frac{\frac{t}{\varepsilon}}{2+\frac{t}{\varepsilon}}+\mathbb{E}X_{\varepsilon}(t-\!)\right]\, {\textrm{d}} W(t) \notag\\&\quad +\int_{U}\left[X_{\varepsilon}(t-\!)\sin\left(\log^{\frac{3}{2}}\left(1+X_{\varepsilon}^2(t-\!)\right)\right)\left(1-{\textrm{e}}^{-\frac{t}{\varepsilon}}\right)+\mathbb{E}X_{\varepsilon}(t-\!)\right]\tilde{N}({\textrm{d}} t, {\textrm{d}} z),\end{align}

with $t\in[0,T]$ and the initial condition $X_{\varepsilon}(0)=x_0$ . Here, W(t) is a scalar Wiener process, $U=\mathbb{R}\backslash \{ 0\}$ , and $\nu$ is a finite measure with $\nu(U)=1$ . Define the following functions:

$$b(t,x,\mu)=(x-x^3)\frac{t}{1+t}+\int_{\mathbb{R}}y\mu({\textrm{d}} y),\quad\sigma(t,x,\mu)=\psi(x)\frac{t}{2+t}+\int_{\mathbb{R}}y\mu({\textrm{d}} y),$$

$$h(t,x,\mu,z)=\phi(x)(1-{\textrm{e}}^{-t})+\int_{\mathbb{R}}y\mu({\textrm{d}} y),$$

where $\psi(x)=x\sin(\log^2(1+x^2))$ and $\phi(x)=x\sin(\log^{\frac{3}{2}}(1+x^2))$ are continuously differentiable functions. For any $x\in \mathbb{R}$ , we can show that

(4.2) \begin{align}&|\psi(x)|\leqslant|x|, \quad |\phi(x)|\leqslant |x|,\;\;|(\partial_x\psi)(x)|\leqslant 1+4\log(1+x^2) \notag\\&\quad \text{and}\quad |(\partial_x\phi)(x)|\leqslant 1+3\sqrt{\log(1+x^2)}.\end{align}

1. (1) Well-posedness. To show that (4.1) has a unique solution $(X_{\varepsilon}(t))_{t\in[0,T]}$ , we need to verify that the conditions in Theorem 1 are satisfied. For any $R\gt 0$ , $x,y\in\mathbb{R}$ with $|x|\vee|y|\leqslant R$ and $\mu\in\mathcal{M}_2(\mathbb{R})$ , we provide the following estimates:

   \begin{align}(x-y)(b(t,x,\mu)-b(t,y,\mu))&=(x-y)(x-x^3-y+y^3)\frac{t}{1+t}\notag\\&\leqslant |x-y|^2-|x-y|^2(x^2+xy+y^2)\notag\\&\leqslant |x-y|^2(1-xy)\leqslant (1+R^2)|x-y|^2\;=\!:\; L_R^1|x-y|^2,\notag\end{align}
   \begin{align}|\sigma(t,x,\mu)-\sigma(t,y,\mu)|^2&=\left|\psi(x)-\psi(y)\right|^2\left(\frac{t}{2+t}\right)^2\notag\\&\leqslant \left|\int_0^1(\partial_x\psi)(y+\theta(x-y))(x-y)\, {\textrm{d}} \theta\right|^2\notag\\&\leqslant \Big[\sup_{|z|\leqslant R}\left|(\partial_x\psi)(z)\right|\Big]^2|x-y|^2\notag\\&\leqslant \Big[\sup_{|z|\leqslant R}\left(1+4\log(1+z^2)\right)\Big]^2|x-y|^2\notag\\&\leqslant \left(1+4\log(1+R^2)\right)^2|x-y|^2 \;=\!:\; L_R^2|x-y|^2,\notag\end{align}
   \begin{align}\int_U|h(t,x,\mu,z)-h(t,y,\mu,z)|^2\nu({\textrm{d}} z)&=\left|\phi(x)-\phi(y)\right|^2(1-{\textrm{e}}^{-t})^2\notag\\&\leqslant \left|\int_0^1(\partial_x\phi)(y+\theta(x-y))(x-y)\, {\textrm{d}} \theta\right|^2\notag\\&\leqslant \left[\sup_{|z|\leqslant R}|(\partial_x\phi)(z)|\right]^2\cdot|x-y|^2\notag\\&\leqslant \Big[\sup_{|z|\leqslant R}\left(1+3\sqrt{\log(1+z^2)}\right)\Big]^2|x-y|^2\notag\\&\leqslant \left(1+3\sqrt{\log(1+R^2)}\right)^2|x-y|^2 \;=\!:\; L_R^3|x-y|^2.\notag\end{align}
   These estimates imply that Assumption 1 is satisfied by denoting $L_R=\max\{L_R^1,L_R^2,L_R^3\}$ . Next, we estimate the following for any $x\in\mathbb{R}$ and $\mu_1,\mu_2\in\mathcal{M}_2(\mathbb{R})$
   
   which shows that b, $\sigma$ , and h satisfy Assumptions 2 and 3. Furthermore, using the bounds in (4.2) and the fact that $\frac{t}{1+t}$ , $\frac{t}{2+t}$ , and $1-{\textrm{e}}^{-t}$ are bounded, we deduce that for any $x\in\mathbb{R}^d$ and $\mu\in\mathcal{M}_2(\mathbb{R}),$
   \begin{align}x\cdot b(t,x,\mu)&\leqslant x(x-x^3)\left(\frac{t}{1+t}\right)+x\int_{\mathbb{R}}y\mu({\textrm{d}} y)\notag\\&\leqslant x^2+\frac{1}{2}x^2+\frac{1}{2}\left(\int_{\mathbb{R}}y\mu({\textrm{d}} y)\right)^2\leqslant 2\left(1+x^2+W_2^{2}(\mu,\delta_0)\right)\!, \notag\end{align}
   \begin{align}|\sigma(t,x,\mu)|^2&=\left|\psi(x)\frac{t}{2+t}+\int_{\mathbb{R}}y\mu({\textrm{d}} y)\right|^2\notag\\&\leqslant 2|\psi(x)|^2+2\left(\int_{\mathbb{R}}y\mu({\textrm{d}} y)\right)^2\leqslant 2\left(1+|x|^2+W_2^2(\mu,\delta_0)\right)\!,\notag\end{align}
   \begin{align}\int_U|h(t,x,\mu,z)|^2\nu({\textrm{d}} z)&=\left|\phi(x)(1-{\textrm{e}}^{-t})+\int_{\mathbb{R}}y\mu({\textrm{d}} y)\right|^2 \leqslant 2|\phi(x)|^2+2\left(\int_{\mathbb{R}}y\mu({\textrm{d}} y)\right)^2\notag\\&\leqslant 2\left(1+|x|^2+W_2^2(\mu,\delta_0)\right).\notag\end{align}
   Thus, Assumption 4 holds. In addition, for any $x\in\mathbb{R}^d$ and $\mu\in\mathcal{M}_2(\mathbb{R})$ , we have
   \begin{align}|b(t,x,\mu)|^2&\leqslant 2(x-x^3)^2\left(\frac{t}{1+t}\right)^2\!+2\left(\int_{\mathbb{R}}y\mu({\textrm{d}} y)\right)^2\!\leqslant 2x^2-4x^4+2x^6+2W_2^2(\mu,\delta_0)\notag\\&\leqslant 4\left(1+x^6+W_2^{6}(\mu,\delta_0)\right). \notag\end{align}
   Thus, Assumption 5 holds with $\kappa=6$ . Finally, since $X_{\varepsilon}(0)$ is a constant, Assumption 6 (with $r\geqslant 18$ ) naturally holds. Due to the expression of h and the finiteness of $\nu$ , Assumption 7 can be easily verified using the same technique as Assumptions 1 and 2 were checked.

2. (2) Averaging principle. Define

   \begin{align*}&\bar{b}(x,\mu)=x-x^3+\int_{\mathbb{R}}y\mu({\textrm{d}} y), \quad \bar{\sigma}(x,\mu)=\psi(x)+\int_{\mathbb{R}}y\mu({\textrm{d}} y), \notag\\&\quad\bar{h}(x,\mu,z)=\phi(x)+\int_{\mathbb{R}}y\mu({\textrm{d}} y).\end{align*}
   We can now verify that the averaging conditions in Assumptions 8 and 9 (with $\kappa=6$ and $r\geqslant 18$ ) are satisfied:
   \begin{align}\frac{1}{t}\int_{0}^{t}|b(s,x,\mu)-\bar{b}(x,\mu)|^2\, {\textrm{d}} s&=\frac{1}{t}\int_{0}^{t}|x-x^3|^2\left[1-\frac{s}{1+s}\right]^2\, {\textrm{d}} s=x^2(1-x^2)^2\frac{1}{1+t}\notag\\&\leqslant \varphi_1(t)C_R^b\left(1+|x|^2\right)\notag\end{align}
   \begin{align}\frac{1}{t}\int_{0}^{t}|\sigma(s,x,\mu)-\bar{\sigma}(x,\mu)|^2\, {\textrm{d}} s&=\frac{1}{t}\int_{0}^{t}\psi^2(x)\left[1-\frac{s}{2+s}\right]^2\, {\textrm{d}} s=\psi^2(x)\frac{2}{2+t}\notag\\&\leqslant \varphi_2(t)C_R^{\sigma}\left(1+|x|^2\right)\notag\end{align}
   \begin{align}\frac{1}{t}\int_{0}^{t}\int_U|h(s,x,\mu,z)-\bar{h}(x,\mu,z)|^2\nu({\textrm{d}} z)\, {\textrm{d}} s&=\frac{1}{t}\int_{0}^{t}\phi^2(x)[1-(1-{\textrm{e}}^{-s})]^2\, {\textrm{d}} s\notag\\&=\phi^2(x)\frac{1-{\textrm{e}}^{-2t}}{2t}\leqslant \varphi_3(t)C_R^{h}\left(1+|x|^2\right)\!,\notag\end{align}
   and
   
   for $x\in\mathbb{R}$ with $|x|\leqslant R$ , where the functions $\varphi_1(t)=\frac{1}{1+t}$ , $\varphi_2(t)=\frac{1}{2+t}$ , $\varphi_3(t)=\frac{1-{\textrm{e}}^{-2t}}{2t}$ , and $\varphi(t)=\frac{1-{\textrm{e}}^{-lt}}{lt}$ are continuous, positive, and bounded, with the property that $\lim_{t\to\infty}\varphi_i(t)=\lim_{t\to\infty}\varphi(t)=0$ , for $i=1,2,3$ .

Based on the discussion and the result of Theorem 2, the solution of (4.1) can be approximated by the following equation (for $t\in[0,T]$ and $\bar{X}(0)=x_0$ )

(4.3) \begin{align}d\bar{X}(t)&=\left[\left(\bar{X}(t-\!)-\bar{X}^3(t-\!)\right)+\mathbb{E}\bar{X}(t-\!)\right]\, {\textrm{d}} t \notag\\&\quad +\left[\bar{X}(t-\!)\sin\left(\log^2\left(1+\bar{X}^2(t-\!)\right)\right)+\mathbb{E}\bar{X}(t-\!)\right]\, {\textrm{d}} W(t) \notag\\&\quad +\int_{U}\left[\bar{X}(t-\!)\sin\left(\log^{\frac{3}{2}}\left(1+\bar{X}^2(t-\!)\right)\right)+\mathbb{E}\bar{X}(t-\!)\right]\tilde{N}({\textrm{d}} t,{\textrm{d}} z),\end{align}

in the sense of mean square.

We now carry out numerical simulations to compute the solutions of (4.1) and (4.3) with $x_0=1$ , $T=10$ , $\varepsilon=0.01$ and $x_0=1$ , $T=10$ , $\varepsilon=0.001$ , respectively. Figure 1(a) and (b) illustrate the comparison between the solution $X_{\varepsilon}(t)$ of (4.1) and the averaged solution $\bar{X}(t)$ of (4.3). As shown, the solutions of the original equation and the averaged equation exhibit strong agreement. In addition, one can find that, for fixed sample points, the error $\sup_{0\leqslant t \leqslant 10}|X_{\varepsilon}(t)-\bar{X}(t)|$ decreases when $\varepsilon$ changes from $0.01$ to $0.001$ . This observed behavior aligns with the predictions of the averaging principle stated in Theorem 2.

Figure 1.

Comparison of the solution $X_{\varepsilon}(t)$ for (4.1) with the averaged solution $\bar{X}(t)$ for (4.3): (a) $X_{\varepsilon}(0)=\bar{X}(0)=1$ , $\varepsilon=0.01$ ; (b) $X_{\varepsilon}(0)=\bar{X}(0)=1$ , $\varepsilon=0.001$ .

We remark that in our numerical simulations to approximate the McKean–Vlasov SDE (4.1) and (4.3), we use N-dimensional systems of interacting particles, which can be regarded as standard SDEs. This approach is based on the so-called propagation of chaos result (see Appendix B). Based on Proposition 3, we briefly introduce an Euler–Maruyama (EM) numerical scheme to approximate the solution of (B.2), which, in turn, serves as an approximation for the solution of the McKean–Vlasov SDE (2.8). To this end, we partition the time interval [0, T] into n subintervals of equal length and define $t_k^n=kh_n$ for $k=0,1,\ldots,n$ , where $n\in\mathbb{N}$ and the step size is given by $h_n=\frac{T}{n}$ . The EM scheme for the interacting particle system (B.2) is specified by the initial condition $X^{i,N,n}(0)=X^{i,N}(0)$ and the recurrence relation

(4.4) \begin{align}X^{i,N,n}(t_{k+1}^n)&=X^{i,N,n}(t_k^n)+b(t_k^n,X^{i,N,n}(t_k^n),\mu_{t_k^n}^{X,N,n})h_n+\sigma(t_k^n,X^{i,N,n}(t_k^n),\mu_{t_k^n}^{X,N,n})\Delta W^{i,n}(k)\notag\\&\quad +\int_{t_k^n}^{t_{k+1}^n}\int_{U}h(t_k^n,X^{i,N,n}(t_k^n),\mu_{t_k^n}^{X,N,n},z)\tilde{N}^i({\textrm{d}} t,{\textrm{d}} z), \quad i=1,2,\ldots,N,\end{align}

where $X^{i,N,n}(t_k^n)$ denotes the approximation of $X^{i,N}(t_k^n)$ , $\mu_{t_k^n}^{X,N,n}=\frac{1}{N}\sum_{j=1}^N\delta_{X^{j,N,n}(t_k^n)}$ is the empirical measure, and $\Delta W^{i,n}(t)=W^i(t_{k+1}^n)-W^i(t_k^n)$ is the Brownian increment. To simulate the integrals w.r.t. the compensated Poisson random measure $\tilde N({\textrm{d}} t,{\textrm{d}} z) = N({\textrm{d}} t,{\textrm{d}} z) - \nu ({\textrm{d}} z)\, {\textrm{d}} t$ , we also employ the technique of introducing a compound Poisson process $\int_U z\tilde N(t,{\textrm{d}} z)$ , as detailed in [Reference Applebaum1, Section 4.3.2].

For this example, we simulate $N=100$ particles with a time step 0.01, $T=10$ . Figure 2(a) and (b) depict the realizations of the interacting particle systems associated with the McKean–Vlasov SDEs (4.1) and (4.3), respectively, under the initial conditions $X_{\varepsilon}(0)=\bar{X}(0)=1$ and $\varepsilon=0.01$ . Numerically, the Wasserstein distance between the distributions of the solutions to (4.1) (i.e. $\mathscr{L}_{X_{\varepsilon}(t)}$ ) and (4.3) (i.e. $\mathscr{L}_{\bar{X}(t)}$ ) is approximated via the empirical distributions of the interacting particle systems, as illustrated in Figure 2(c).

Figure 2.

Comparison of the interacting N-particle systems associated with McKean–Vlasov SDEs (4.1) and (4.3), for $N=100$ and $\varepsilon=0.01$ .