Proof. With the assumed integrability on $X_0$ , it is straightforward to adapt the Picard iteration and the Grönwall type estimates in proof of [Reference Wang35, Theorem 1.1] to allow for random initial conditions $X_0$ , as stated in Lemma 3.

Proof. With the assumed integrability on $X_0$ , it is straightforward to adapt the proof of [Reference Prömel and Scheffels30, Theorem 2.3] to the case that $X_0$ is a random variable.

Proof. The weak existence of some $L^{p}$ -solution to the SVE (14) follows from [Reference Prömel and Scheffels29, Theorem 3.3] with the straightforward adaptation to random initial conditions $X_0$ . For the pathwise uniqueness, we can adapt the proof from [Reference Abi Jaber and El Euch2, Proposition B.3] using the Lipschitz and Hölder continuity of $\mu,\sigma$ uniformly in t.

Proof of Theorem 1. We define the solution map $\Phi$ by

(16) \begin{equation} \Phi \colon C\big([0,T];\,\mathcal{P}_{\delta}(\mathbb{R}^d)\big) \to C\big([0,T];\,\mathcal{P}_{\delta}(\mathbb{R}^d)\big), \qquad \rho \mapsto \Phi(\rho) \,:\!=\, \big(\mathcal{L}(X^\rho_t)\big)_{t\in[0,T]}, \end{equation}

where $X^\rho$ is the unique $L^{p}$ -solution to the stochastic Volterra equation

(17) \begin{equation} X_t = X_0 + \int_0^tK_{\mu}(s,t)\mu(s,X_s,\rho_s)\,\mathrm{d}s + \int_0^tK_{\sigma}(s,t)\sigma(s,X_s,\rho_s)\,\mathrm{d}B_s, \quad t\in [0,T]. \end{equation}

Note that a unique fixed point of the solution map $\Phi$ implies the existence of a unique $L^{p}$ -solution $X=(X_t)_{t\in[0,T]}$ to the mean-field SVE (2) satisfying $\sup_{t\in[0,T]}\mathbb{E}[|X_t|^{q}] < \infty$ for every $q\geq 1$ ; cf. Step 1 below. Hence, it is sufficient to prove that the solution map $\Phi$ has a unique fixed point.

Step 1. We show the well-definedness of the solution map $\Phi$ .

For a fixed $\rho=(\rho_t)_{t\in[0,T]}\in C([0,T];\,\mathcal{P}_{\delta}(\mathbb{R}^d))$ , the integral equation (17) is an ordinary stochastic Volterra equation. Due to Assumption 2, the linear growth and Lipschitz condition of Lemma 3 are satisfied. Hence, there exists a unique strong $L^{p}$ -solution $X^\rho=(X^\rho_t)_{t\in[0,T]}$ to the SVE (17) and, by Remark 4, $\sup_{t\in[0,T]}\mathbb{E}[|X^\rho_t|^q]<\infty$ for $q\geq q$ , provided $X_0\in L^q$ , and the sample paths of $X^\rho$ are almost surely continuous. Moreover, note that $(\mathcal{L}(X^\rho_t))_{t\in[0,T]}\in C([0,T];\,\mathcal{P}_{\delta}(\mathbb{R}^d))$ , since, by the representation of the Wasserstein distance in terms of random variables (see [Reference Carmona and Delarue12, (5.14)]) and by Remark 4, we have

\begin{equation*} W_\delta\big( \mathcal{L}(X_t^\rho),\mathcal{L}(X_s^\rho) \big) \leq \mathbb{E}\big[|X_t^\rho-X_s^\rho|^\delta\big]^{{1}/{\delta}} \lesssim |t-s|^\beta, \quad s,t\in[0,T], \end{equation*}

for any $\beta\in (0,\gamma-1/p)$ with $\gamma\in\big(0,{1}/{2}\big]$ , where the parameters are given in Assumption 1.

Step 2: For $\rho,\tilde{\rho}\in C\big([0,T];\,\mathcal{P}_{\delta}(\mathbb{R}^d)\big)$ , we show that

(18) \begin{equation} \sup\limits_{s\in[0,t]}W_\delta(\Phi(\rho)_s,\Phi(\tilde{\rho})_s)^\delta \lesssim \int_0^t W_\delta(\rho_s,\tilde{\rho}_s)^\delta \,\mathrm{d}s, \quad t\in [0,T]. \end{equation}

We have

(19) \begin{align} & \mathbb{E}\big[|X_t^\rho-X_t^{\tilde{\rho}}|^{\delta}\big] \notag \\ & \quad \lesssim \mathbb{E}\bigg[\bigg|\int_0^tK_\mu(s,t)\big(\mu(s,X^\rho_s,\rho_s) - \mu(s,X^{\tilde{\rho}}_s,\tilde{\rho}_s) \big)\,\mathrm{d}s\bigg|^{\delta}\bigg] \notag \\ & \qquad + \mathbb{E}\bigg[\bigg|\int_0^tK_\sigma(s,t)\big(\sigma(s,X^\rho_s,\rho_s) - \sigma(s,X^{\tilde{\rho}}_s,\tilde{\rho}_s)\big)\,\mathrm{d}B_s\bigg|^{\delta}\bigg] \notag \\ & \quad \lesssim \bigg(\int_0^t|K_\mu(s,t)|^{({4+2\epsilon})/({4+\epsilon})}\,\mathrm{d}s\bigg)^{({4+\epsilon})/{\epsilon}} \int_0^t\mathbb{E}\big[|\mu(s,X^\rho_s,\rho_s)-\mu(s,X^{\tilde{\rho}}_s,\tilde{\rho}_s)|^{\delta}\big]\, \mathrm{d}s \notag \\ & \qquad + \mathbb{E}\bigg[\bigg(\int_0^t\big|K_\sigma(s,t)\big(\sigma(s,X^\rho_s,\rho_s) - \sigma(s,X^{\tilde{\rho}}_s,\tilde{\rho}_s)\big)\big|^2\,\mathrm{d}s\bigg)^{{\delta}/{2}}\bigg] \notag \\ & \quad \lesssim \int_0^t\mathbb{E}\big[\big|\mu(s,X^\rho_s,\rho_s) - \mu(s,X^{\tilde{\rho}}_s,\tilde{\rho}_s)\big|^{\delta}\big]\,\mathrm{d}s \notag \\ & \qquad + \bigg(\int_0^t|K_\sigma(s,t)|^{2+\epsilon}\,\mathrm{d}s\bigg)^{({4+2\epsilon})/({\epsilon(2+\epsilon)})} \int_0^t\mathbb{E}\big[\big|\sigma(s,X^\rho_s,\rho_s) - \sigma(s,X^{\tilde{\rho}}_s,\tilde{\rho}_s)\big|^\delta\big]\,\mathrm{d}s \notag \\ & \quad \lesssim \int_0^t\big(\mathbb{E}\big[\big|X^\rho_s-X^{\tilde{\rho}}_s\big|^{\delta}\big] + W_\delta(\rho_s,\tilde{\rho}_s)^\delta\big)\,\mathrm{d}s \end{align}

for $t\in[0,T]$ , where we used Hölder’s inequality in the drift integral with

\begin{align*}\frac{4+2\epsilon}{4+\epsilon}<1+\epsilon\end{align*}

(noting that $({4+2\epsilon})/({4+\epsilon})$ is the conjugate of $\delta/2$ ) such that, by the choice of $\delta$ in (3),

\begin{align*}\frac{4+\epsilon}{4+2\epsilon}+\frac{1}{\delta}=1,\end{align*}

and in the diffusion integral with $({2+\epsilon})/{2}$ such that (4) holds, Burkholder–Davis–Gundy’s inequality applied to the stochastic processes

\begin{align*}\bigg(\int_0^rK_\sigma(s,t)\big(\sigma(s,X^\rho_s,\rho_s) - \sigma(s,X^{\tilde{\rho}}_s,\tilde{\rho}_s)\big)\,\mathrm{d}B_s\bigg)_{r\in[0,t]},\end{align*}

Fubini’s theorem, the integrability of the kernels from Assumption 1, and the Lipschitz continuity of $\mu$ and $\sigma$ from Assumption 2. Since $\sup_{s\in[0,T]}\mathbb{E}[|X_s^\rho-X_s^{\tilde{\rho}}|^\delta]<\infty$ , we can apply Grönwall’s inequality to conclude that

(20) \begin{equation} \mathbb{E}\big[|X_t^\rho-X_t^{\tilde{\rho}}|^{\delta}\big] \lesssim \int_0^t W_\delta(\rho_s,\tilde{\rho}_s)^\delta \,\mathrm{d}s. \end{equation}

Since, by assumption, $\rho,\tilde{\rho}\in C\big([0,T];\,\mathcal{P}_{\delta}(\mathbb{R}^d)\big)$ , we can bound the Wasserstein distance by

\begin{equation*} W_\delta(\Phi(\rho)_t,\Phi(\tilde{\rho})_t) = W_\delta(\mathcal{L}(X_t^\rho),\mathcal{L}(X_t^{\tilde{\rho}})) \leq \mathbb{E}[|X_t^\rho-X_t^{\tilde{\rho}}|^\delta]^{{1}/{\delta}}, \end{equation*}

cf. [Reference Carmona and Delarue12, (5.14)], and plugging this into (20) and taking the supremum, we obtain (18).

Step 3: We show that the solution map $\Phi$ has a unique fixed point.

First note that it is sufficient to show that $\Phi^k$ is a contraction (see [Reference Bryant10, Theorem]), since the Wasserstein space $C([0,T];\,\mathcal{P}_\delta(\mathbb{R}^d))$ is a complete metric space (see, e.g., [Reference Panaretos and Zemel25, Proposition 2.2.8]), where $\Phi^k$ denotes the kth composition of $\Phi$ with itself. Let $C>0$ denote the generic constant in (18). Then, iteratively for $k\in\mathbb{N}$ ,

\begin{align*} \sup\limits_{s\in[0,T]}W_\delta(\Phi^k(\rho)_s,\Phi^k(\tilde{\rho})_s)^\delta & \leq C^k\int_0^T\frac{(T-s)^{k-1}}{(k-1)!}W_\delta(\rho_s,\tilde{\rho}_s)^\delta\,\mathrm{d}s \\ & \leq \frac{C^kT^k}{k!}\sup\limits_{s\in[0,T]} W_\delta(\rho_s,\tilde{\rho}_s)^\delta. \end{align*}

Thus, choosing k large enough that ${C^kT^k}/{k!}<1$ , we see that the mapping $\Phi^k$ is a contraction and, hence, $\Phi$ admits a unique fixed point, which completes the proof.

Proof of Theorem 3. We again consider the solution map $\Phi$ , as defined in (16), but choose $\delta=1$ and $d=1$ , that is,

\begin{equation*} \Phi\colon C\big([0,T];\,\mathcal{P}_{1}(\mathbb{R})\big) \to C\big([0,T];\,\mathcal{P}_{1}(\mathbb{R})\big), \qquad \rho \mapsto \Phi(\rho) \,:\!=\, \big(\mathcal{L}(X^\rho_t)\big)_{t\in[0,T]}, \end{equation*}

where $X^\rho$ is the unique $L^{p}$ -solution to the stochastic Volterra equation

\begin{equation*} X_t = X_0 + \int_0^tK_{\mu}(s,t)\mu(s,X_s,\rho_s)\,\mathrm{d}s + \int_0^tK_{\sigma}(s,t)\sigma(s,X_s)\,\mathrm{d}B_s, \quad t\in [0,T]. \end{equation*}

In the following we show that the solution map $\Phi$ possesses a unique fixed point. We proceed as in the proof of Theorem 1. Step 1 works exactly the same, using Lemmas 4 and 5, respectively, instead of Lemma 3, and Step 3 works exactly the same. That means we only need to show Step 2 or, more precisely, estimate (20) with $\delta=1$ . To do that, we treat separately the cases that Assumption 3 or Assumption 4 holds.

First, suppose the kernels $K_\mu,K_\sigma$ and initial condition $X_0$ satisfy Assumption 3. To get an analogous estimate as in (20), we use the semimartingale property of a solution $(X_t^\rho)_{t\in[0,T]}$ to (2) with fixed $\rho\in C([0,T];\,\mathcal{P}_1(\mathbb{R}))$ (cf. [Reference Prömel and Scheffels30, Lemma 3.6] or [Reference Protter27, Theorem 3.3]),

\begin{align*} X_t^\rho - X_0 & = \int_0^tK_\sigma(s,s)\sigma(s,X_s^\rho)\,\mathrm{d}B_s + \int_0^tK_\mu(s,s)\mu(s,X_s^\rho,\rho_s)\,\mathrm{d}s \\& \quad + \int_0^t\bigg(\int_0^s\partial_2K_\mu(u,s)\mu(u,X_u^\rho,\rho_u)\,\mathrm{d}u + \int_0^s\partial_2K_\sigma(u,s)\sigma(u,X_u^\rho)\,\mathrm{d}B_u\bigg)\,\mathrm{d}s, \end{align*}

and the Yamada–Watanabe functions $\phi_n$ for $n\in\mathbb{N}$ (cf. [Reference Prömel and Scheffels30, Proof of Theorem 5.3] or the original work, [Reference Yamada and Watanabe36]) that approximate the absolute value function in the following way. Let $(a_n)_{n\in\mathbb{N}}$ be a strictly decreasing sequence with $a_0=1$ such that $a_n\to 0$ as $n\to \infty$ and

\begin{equation*} \int_{a_n}^{a_{n-1}}\frac{1}{|x|^{1+2\xi}}\,\mathrm{d} x=n, \end{equation*}

where $\frac{1}{2}+\xi$ is the Hölder regularity of $\sigma$ . Furthermore, we define a sequence of mollifiers: let $(\psi_n)_{n\in\mathbb{N}}\in C_0^{\infty}(\mathbb{R})$ be smooth functions with compact support such that $\mathrm{supp}(\psi_n)\subset (a_n,a_{n-1})$ , and with the properties

(21) \begin{equation} 0 \leq \psi_n(x) \leq \frac{2}{n|x|^{1+2\xi}} \ \text{for all }x\in\mathbb{R}, \quad\text{and}\quad \int_{a_n}^{a_{n-1}}\psi_n(x)\,\mathrm{d} x=1. \end{equation}

We set $\phi_n(x)\,:\!=\,\int_0^{|x|}\big(\int_0^y\psi_n(z)\,\mathrm{d}z\big)\,\mathrm{d}y$ , $x\in \mathbb{R}$ . By (21) and the compact support of $\psi_n$ , it follows that $\phi_n(\cdot)\to |\cdot|$ uniformly as $n\to \infty$ . Since every $\psi_n$ , and thus every $\phi_n$ , is zero in a neighborhood around zero, the functions $\phi_n$ are smooth with $\|\phi_n'\|_\infty\leq 1$ , $\phi_n'(x)=\mathrm{sgn}(x)\int_0^{|x|}\psi_n(y)\,\mathrm{d}y$ , and $\phi_n''(x)=\psi_n(|x|)$ for $x\in\mathbb{R}$ , where $\|\cdot\|_\infty$ denotes the $\sup$ -norm on $\mathbb{R}$ .

Using $\phi_n$ , we apply Itô’s formula to $\tilde{X}_t\,:\!=\,X_t^\rho-X_t^{\tilde{\rho}}$ , with the notation

\begin{equation*} \tilde{Z}_t\,:\!=\,\int_0^t\big(\mu(s,X^\rho_s,\rho_s)-\mu(s,X^{\tilde{\rho}}_s,\tilde{\rho}_s)\big)\,\mathrm{d}s, \qquad Y_t^\rho\,:\!=\,\int_0^t \sigma(s,X_s^\rho)\,\mathrm{d}B_s, \end{equation*}

$H_t^\rho\,:\!=\,\int_0^t\partial_2 K_\sigma(s,t)\,\mathrm{d}Y_s^\rho$ , and $Y_t^{\tilde{\rho}}$ and $H_t^{\tilde{\rho}}$ analogously, as well as $\tilde{Y}_t\,:\!=\,Y_t^\rho-Y_t^{\tilde{\rho}}$ , and $\tilde{H}_t\,:\!=\,H_t^\rho-H_t^{\tilde{\rho}}$ , for $t\in [0,T]$ , to obtain

\begin{align*} \phi_n(\tilde{X}_t) & = \int_0^t\phi_n'(\tilde{X}_s)\,\mathrm{d}\tilde{X}_s + \frac{1}{2}\int_0^t\phi_n''(\tilde{X}_s)\,\mathrm{d}\langle\tilde{X}\rangle_s\\ & = \int_0^t\phi_n'(\tilde{X}_s)K_\mu(s,s)(\mu(s,X_s^\rho,\rho_s)-\mu(s,X_s^{\tilde{\rho}},\tilde{\rho}_s))\, \mathrm{d}s \\ & \quad + \int_0^t\phi_n'(\tilde{X}_s)\bigg(\int_0^s\partial_2K_\mu(u,s)\,\mathrm{d}\tilde{Z}_u\bigg)\, \mathrm{d}s \\ \end{align*}

(22) \begin{align}& \quad + \int_0^t\phi_n'(\tilde{X}_s)\tilde{H}_s\,\mathrm{d}s + \int_0^t\phi_n'(\tilde{X}_s)K_\sigma(s,s)\,\mathrm{d}\tilde{Y}_s \notag \\ & \quad + \frac{1}{2}\int_0^t\phi_n''(\tilde{X}_s)K_\sigma(s,s)^2 \big(\sigma(s,X_s^\rho)-\sigma(s,X_s^{\tilde{\rho}})\big)^2\,\mathrm{d}s \notag \\ & \,=\!:\, I_{1,t}^n + I_{2,t}^n + I_{3,t}^n + I_{4,t}^n + I_{5,t}^n. \end{align}

Note that $H_t^\rho$ and $H_t^{\tilde{\rho}}$ are well-defined stochastic Itô integrals due to Assumption 3.

For $I_{1,t}^n$ , the bound $\|\phi'_n\|_\infty \leq 1$ , the boundedness of $K_\mu$ , the Lipschitz continuity of $\mu$ , and Jensen’s inequality yield

(23) \begin{equation} \mathbb{E}[I_{1,t}^n] \lesssim \int_0^t \big(\mathbb{E}[|\tilde{X}_{s}|]+W_1(\rho_s,\tilde{\rho}_s)\big)\,\mathrm{d} s. \end{equation}

For $I_{2,t}^n$ , we additionally use the boundedness of $\partial_2 K_\mu(u,s)$ on $\Delta_T$ to obtain

(24) \begin{equation} \mathbb{E}[I_{2,t}^n] \lesssim \int_0^t \big(\mathbb{E}[|\tilde{X}_{s}|]+W_1(\rho_s,\tilde{\rho}_s)\big)\,\mathrm{d} s. \end{equation}

For $I_{3,t}^n$ , we use $\|\phi'_n\|_\infty \leq 1$ and the integration by parts formula to estimate

(25) \begin{align} \mathbb{E}[I_{3,t}^n] & \leq \int_0^t\mathbb{E}[|\tilde{H}_{s}|]\,\mathrm{d}s \notag \\ & \leq \int_0^t|\partial_2 K_\sigma(s,s)|\,\mathbb{E}[|\tilde{Y}_{s}|]\,\mathrm{d}s + \int_0^t\int_0^s|\partial_{21}K_\sigma(u,s)|\,\mathbb{E}[|\tilde{Y}_{u}|]\,\mathrm{d}u\,\mathrm{d}s \notag \\ & \leq \int_0^t\mathbb{E}[|\tilde{Y}_{s}|] \bigg(\partial_2K_\sigma(s,s) + \int_s^t|\partial_{21}K_\sigma(s,u)|\,\mathrm{d}u\bigg)\,\mathrm{d}s \lesssim \int_0^t\mathbb{E}[|\tilde{Y}_s|]\,\mathrm{d}s, \end{align}

with the boundedness of $\partial_2 K_\sigma(s,s)$ and $\int_s^t\partial_{21}K_\sigma(s,u)\,\mathrm{d}u$ from Assumption 3. For $I_{4,t}^n$ , since $I_{4,t}^n$ is a martingale by [Reference Protter28, p. 73, Corollary 3] due to the boundedness of $K_\sigma$ , the growth bound on $\sigma$ and the finiteness of the moments of $X^\rho$ and $X^{\tilde{\rho}}$ (cf. [Reference Prömel and Scheffels30, Theorem 2.3]), we get

(26) \begin{equation} \mathbb{E}[I^n_{4,t}] = \mathbb{E}\bigg[\int_0^{t}\phi_n'(\tilde{X}_s)K_\sigma(s,s)(\sigma(s,X_s^\rho) - \sigma(s,X_s^{\tilde{\rho}}))\,\mathrm{d}B_s\bigg] = 0. \end{equation}

For $I_{5,t}^n$ , by using the boundedness of $K_\sigma$ , the Hölder continuity of $\sigma$ , and the inequality $\phi_n''(x)\leq {2}/{n|x|^{1+2\xi}}$ , we get

(27) \begin{equation} \mathbb{E}[I_{5,t}^n] \lesssim \mathbb{E}\bigg[\int_0^t \phi_n''(\tilde{X}_{s})|\tilde{X}_{s}|^{1+2\xi}\,\mathrm{d}s\bigg] \leq \mathbb{E}\bigg[\int_0^t\frac{2}{n|\tilde{X}_{s}|^{1+2\xi}}|\tilde{X}_{s}|^{1+2\xi}\,\mathrm{d}s\bigg] \lesssim \frac{1}{n}. \end{equation}

Sending $n\to\infty$ and combining the five previous estimates (23), (24), (25), (26), and (27) with (22) yields

(28) \begin{equation} \mathbb{E}[|\tilde{X}_{t}|] \lesssim \int_0^t\big(\mathbb{E}[|\tilde{X}_{s}|] + \mathbb{E}[|\tilde{Y}_{s}|] + W_1(\rho_s,\tilde{\rho}_s)\big)\,\mathrm{d}s. \end{equation}

To apply Grönwall’s lemma, we set $M(t)\,:\!=\,\mathbb{E}[|\tilde{X}_{t}|] + \mathbb{E}[|\tilde{Y}_{t}|]$ for $t\in[0,T]$ . To find a bound for $\mathbb{E}[|\tilde{Y}_{t}|]$ , we apply the integration by parts formula to obtain

(29) \begin{align} \tilde{X}_t & = \int_0^tK_\mu(s,t)(\mu(s,X_s^\rho,\rho_s)-\mu(s,X_s^{\tilde{\rho}},\tilde{\rho}_s))\,\mathrm{d}s + \int_0^tK_\sigma(s,t)\,\mathrm{d}\tilde{Y}_s \notag \\ & = \int_0^tK_\mu(s,t)(\mu(s,X_s^\rho,\rho_s)-\mu(s,X_s^{\tilde{\rho}},\tilde{\rho}_s))\,\mathrm{d}s + K_\sigma(t,t)\tilde{Y}_t - \int_0^t\partial_1K_\sigma(s,t)\tilde{Y}_s\,\mathrm{d}s, \end{align}

keeping in mind that $K_\sigma(\cdot,t)$ is absolutely continuous for every $t\in [0,T]$ . Due to $|K_\sigma(t,t)|> C$ for some constant $C>0$ , we can rearrange (29) and use (28) to get

(30) \begin{align} \mathbb{E}[|\tilde{Y}_{t}|] & \leq C\bigg(\int_0^t\mathbb{E}\big[|\mu(s,X_s^\rho,\rho_s)-\mu(s,X_s^{\tilde{\rho}},\tilde{\rho}_s)|\big]\, \mathrm{d}s \notag \\ & \qquad\quad + \mathbb{E}[|\tilde{X}_{t}|] + \int_0^t|\partial_1K_\sigma(s,t)|\mathbb{E}[|\tilde{Y}_{s}|]\,\mathrm{d}s\bigg) \notag \\ & \lesssim \int_0^t\big(\mathbb{E}[|\tilde{X}_{s}|] + \mathbb{E}[|\tilde{Y}_s|] + W_1(\rho_s,\tilde{\rho}_s)\big)\,\mathrm{d}s. \end{align}

Now, Grönwall’s lemma applied to (28) and (30) yields $M(t) \lesssim \int_0^tW_1(\rho_s,\tilde{\rho_s})\,\mathrm{d}s$ and hence $\mathbb{E}[|X_t^\rho-X_t^{\tilde{\rho}}|] \lesssim \int_0^tW_1(\rho_s,\tilde{\rho}_s)\,\mathrm{d}s$ , which is the analogous estimate of (20).

For the second case, suppose the kernels $K_\mu,K_\sigma$ and initial condition $X_0$ satisfy Assumption 4. We need to find an analogue to estimate (20). By using the notation $\tilde{X}_t\,:\!=\,X_t^\rho-X_t^{\tilde{\rho}}$ and $Y_t^\rho\,:\!=\,\int_0^t\mu(s,X_s^\rho,\rho_s)\,\mathrm{d}s+\int_0^t\sigma(s,X_s^\rho)\,\mathrm{d}B_s$ , $Y^{\tilde{\rho}}_t$ analoguously, $\tilde{Y}_t\,:\!=\,Y^{\rho}_t-Y^{\tilde{\rho}}_t$ , and the semimartingale property

\begin{equation*} X_t^\rho - X_0 = \int_0^t\tilde{K}(0)\,\mathrm{d}Y_s^\rho + \int_0^t\int_0^s\tilde{K}^\prime(s-u)\,\mathrm{d}Y_u^\rho\,\mathrm{d}s, \end{equation*}

we can implement the Yamada–Watanabe approach with

(31) \begin{align} \phi_n(\tilde{X}_t) & = \int_0^t\phi_n'(\tilde{X}_s)\tilde{K}(0)\,\mathrm{d}\tilde{Y}_s + \int_0^t\phi_n'(\tilde{X}_s)\int_0^s\tilde{K}'(s-u)\,\mathrm{d}\tilde{Y}_u \,\mathrm{d}s \notag \\ & \quad + \frac{1}{2}\int_0^t \phi_n''(\tilde{X}_s)\tilde{K}(0)^2 \big(\sigma(s,X_s^\rho)-\sigma(s,X_s^{\tilde{\rho}})\big)^2\,\mathrm{d}s \notag \\ & \,=\!:\, I_{1,t}^n + I_{2,t}^n + I_{3,t}^n. \end{align}

Now, the Lipschitz assumption on $\mu$ applied to $I_{1,t}^n$ and $I_{2,t}^n$ , the Hölder assumption on $\sigma$ applied to $I_{3,t}^n$ , the boundedness of $\tilde{K}$ and $\tilde{K}^\prime$ , the inequalities $\|\phi_n\|_\infty\leq 1$ and $\phi_n^{\prime\prime}(x) \leq {2}/{n|x|^{1+2\xi}}$ , and sending $n\to\infty$ yields, as in the first case, with Grönwall’s lemma the inequality $\mathbb{E}[|X_t^\rho-X_t^{\tilde{\rho}}|] \lesssim \int_0^tW_1(\rho_s,\tilde{\rho}_s)\,\mathrm{d}s$ , which implies the estimate (20) and, hence, yields the claimed well-posedness of the mean-field SVE (10).

Proof. Let $\lbrace \sigma(1),\ldots,\sigma(N)\rbrace$ be an arbitrary permutation of $\lbrace 1,\ldots,N\rbrace$ . By (32), we have

(33) \begin{align} Y^{\sigma(1)} & = G_1\big((\zeta^{\sigma(1)},\zeta^{\sigma(2)},\ldots,\zeta^{\sigma(N-1)},\zeta^{\sigma(N)})\big), \notag \\ Y^{\sigma(2)} & = G_1\big((\zeta^{\sigma(2)},\zeta^{\sigma(3)},\ldots,\zeta^{\sigma(N)},\zeta^{\sigma(1)})\big), \notag \\ &\quad\!\!\!\! \vdots \notag \\ Y^{\sigma(N)} & = G_1\big((\zeta^{\sigma(N)},\zeta^{\sigma(1)},\ldots,\zeta^{\sigma(N-2)},\zeta^{\sigma(N-1)})\big), \end{align}

where $G_1$ denotes the first component of the N-dimensional mapping G. Define $W^i\,:\!=\,(X^i,Y^i)$ for $i\in\lbrace 1,\ldots,N\rbrace$ . Then, by the definition of $X^i$ and (33),

(34) \begin{align} W^{\sigma(1)} & = \big(F(\zeta^{\sigma(1)}), G_1\big((\zeta^{\sigma(1)},\zeta^{\sigma(2)},\ldots,\zeta^{\sigma(N-1)},\zeta^{\sigma(N)})\big)\big), \notag \\ W^{\sigma(2)} & = \big(F(\zeta^{\sigma(2)}), G_1\big((\zeta^{\sigma(2)},\zeta^{\sigma(3)},\ldots,\zeta^{\sigma(N)},\zeta^{\sigma(1)})\big)\big), \notag \\ &\quad\!\!\!\! \vdots \notag \\ W^{\sigma(N)} & = \big(F(\zeta^{\sigma(N)}), G_1\big((\zeta^{\sigma(N)},\zeta^{\sigma(1)},\ldots,\zeta^{\sigma(N-2)},\zeta^{\sigma(N-1)})\big)\big). \end{align}

Analogously, we have

(35) \begin{align} W^{1} & = \big(F(\zeta^{1}),G_1\big((\zeta^{1},\zeta^{2},\ldots,\zeta^{N-1},\zeta^{N})\big) \big), \notag \\ W^{2} & = \big(F(\zeta^{2}),G_1\big((\zeta^{2},\zeta^{3},\ldots,\zeta^{N},\zeta^{1})\big) \big), \notag \\ &\quad\!\!\!\! \vdots \notag \\ W^{N} & = \big(F(\zeta^{N}),G_1\big((\zeta^{N},\zeta^{1},\ldots,\zeta^{N-2},\zeta^{N-1})\big)\big). \end{align}

Now, since, by assumption, $(\zeta^1,\ldots,\zeta^N)$ and $(\zeta^{\sigma(1)},\ldots,\zeta^{\sigma(N)})$ have the same joint distribution, (34) and (35) yield that $(W^1,\ldots,W^N)$ and $(W^{\sigma(1)},\ldots,W^{\sigma(N)})$ also have the same joint distribution, which proves the claimed exchangeability.

Proof of Theorem 2. Let us briefly outline the main steps of the proof:

1. Step 1. We show the existence of the system of processes $(X^{N,i})_{i=1,\ldots,N}$ uniquely solving (6), for every $N\in\mathbb{N}$ .

2. Step 2. We prove the inequality

   (36) \begin{equation} \mathbb{E}[|X_t^{N,i}-\underline{\textrm{X}}\,_t^i|^\delta] \lesssim \int_0^t\mathbb{E}\Bigg[W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \mathcal{L}({\underline{\textrm{X}}_s^i})\Bigg)^\delta\Bigg]\,\mathrm{d}s, \quad t \in [0,T], \end{equation}
   for any $ 1\leq i\leq N$ . Recall that $X^{N,i}$ is defined in (6) and $\underline{X}^i$ is defined as the solution of the mean-field SVE (2) with initial condition $X_0^i$ and driving Brownian motion $B^i$ .

3. Step 3. We prove that the right-hand side of (36) tends to zero.

4. Step 4. We show that Steps 2 and 3 imply the statement.

Step 1: By the Lipschitz continuity of $\mu$ and $\sigma$ , and the observation that $W_\delta(\bar{\rho}_x^N,\bar{\rho}_y^N)^\delta\leq({1}/{N})\sum_{j=1}^N |x_j-y_j|^\delta$ for $x,y\in\mathbb{R}^{N\times d}$ with the notation $\bar{\rho}_x^N=({1}/{N})\sum_{j=1}^N\delta_{x_j}\in \mathcal{P}_\delta(\mathbb{R}^d)$ , we obtain, for every $i\in\lbrace 1,\ldots,N\rbrace$ , the Lipschitz condition

\begin{align*} \big|\mu(t,x_i,\bar{\rho}_x^N)-\mu(t,y_i,\bar{\rho}_y^N)\big|^\delta + \big|\sigma(t,x_i,\bar{\rho}_x^N)-\sigma(t,y_i,\bar{\rho}_y^N)\big|^\delta & \lesssim |x_i-y_i|^\delta + \frac{1}{N}\sum\limits_{j=1}^N |x_j-y_j|^\delta \\ & \lesssim \|x-y\|_{N\times d}^\delta, \end{align*}

where $\|\cdot\|_{N\times d}$ denotes the row sum norm on $\mathbb{R}^{N\times d}$ . With the notation $\tilde{\mu}_i(t,x)\,:\!=\,\mu(t,x_i,\bar{\rho}_x^N)$ and $\tilde{\sigma}_i(t,x)$ analogously for any $1\leq i\leq N$ , we directly conclude that the growth condition is fulfilled by

\begin{align*} \big|\tilde{\mu}_i(t,x)\big| + \big|\tilde{\sigma}_i(t,x)\big| & \leq \big|\tilde{\mu}_i(t,x)-\tilde{\mu}_i(t,0)\big| + \big|\tilde{\sigma}_i(t,x)-\tilde{\sigma}_i(t,0)\big| + \big|\tilde{\mu}_i(t,0)\big| + \big|\tilde{\sigma}_i(t,0)\big| \\ & \lesssim \|x\|_{N\times d} + \big|\mu(t,0,\delta_0)\big| + \big|\sigma(t,0,\delta_0)\big| \\ & \lesssim \|x\|_{N\times d} + C_{\delta_0} \lesssim 1+\|x\|_{N\times d} \end{align*}

for all $t\in[0,T]$ , $x\in\mathbb{R}^{N\times d}$ . Thus, due to the equivalence of all norms on the finite-dimensional vector space $\mathbb{R}^{N\times d}$ , we can apply the standard Volterra well-posedness result for Lipschitz coefficients from Lemma 3 to obtain the system of processes $(X^{N,i})_{i=1,\ldots,N,}$ that uniquely solves (6) for every $N\in\mathbb{N}$ .

Step 2: We consider the first summand on the left-hand side of (7), i.e. $\mathbb{E}[|X_t^{N,i}-\underline{\textrm{X}}\,_t^i|^\delta]$ . Using Hölder’s inequality as in (19), Fubini’s theorem, and the Burkholder–Davis–Gundy inequality such as the Lipschitz continuity of $\mu$ and $\sigma$ , we can bound, for $1\leq i\leq N$ ,

\begin{align*} & \mathbb{E}[|X_t^{N,i}-\underline{\textrm{X}}\,_t^i|^\delta] \notag \\ & \quad = \mathbb{E}\bigg[\bigg|\int_0^tK_\mu(s,t) \big(\mu(s,X_s^{N,i},\bar{\rho}_s^N)-\mu(s,{\underline{\textrm{X}}_s^i},\mathcal{L}({\underline{\textrm{X}}_s^i}))\big)\, \mathrm{d}s \notag \\ & \qquad\qquad + \int_0^tK_\sigma(s,t)\big(\sigma(s,X_s^{N,i},\bar{\rho}_s^N) - \sigma(s,{\underline{\textrm{X}}_s^i},\mathcal{L}({\underline{\textrm{X}}_s^i}))\big)\,\mathrm{d}B_s^i\bigg|^\delta\bigg] \notag \\ & \quad \lesssim \bigg(\int_0^t|K_\mu(s,t)|^{({4+2\epsilon})/({4+\epsilon})}\,\mathrm{d}s\bigg)^{({4+\epsilon})/{\epsilon}} \int_0^t\mathbb{E}\big[\big|\mu(s,X_s^{N,i},\bar{\rho}_s^N)-\mu(s,{\underline{\textrm{X}}_s^i}, \mathcal{L}({\underline{\textrm{X}}_s^i}))\big|^\delta\big]\,\mathrm{d}s \notag \\ & \qquad + \mathbb{E}\bigg[\bigg( \int_0^t\big|K_\sigma(s,t)\big(\sigma(s,X_s^{N,i},\bar{\rho}_s^N)-\sigma(s,{\underline{\textrm{X}}_s^i}, \mathcal{L}({\underline{\textrm{X}}_s^i}))\big|^2\,\mathrm{d}s\big)^{{\delta}/{2}}\bigg]\end{align*}

(37) \begin{align} & \quad \lesssim \int_0^t\mathbb{E}\big[\big|\mu(s,X_s^{N,i},\bar{\rho}_s^N)-\mu(s,{\underline{\textrm{X}}_s^i}, \mathcal{L}({\underline{\textrm{X}}_s^i}))\big|^\delta\big]\,\mathrm{d}s \notag \\& \qquad + \bigg(\int_0^t|K_\sigma(s,t)|^{2+\epsilon}\,\mathrm{d}s\bigg)^{{4}/{\epsilon}} \int_0^t\mathbb{E}\big[\big|\sigma(s,X_s^{N,i},\bar{\rho}_s^N)-\sigma(s,{\underline{\textrm{X}}_s^i}, \mathcal{L}({\underline{\textrm{X}}_s^i}))\big|^\delta\big]\,\mathrm{d}s \notag \\ & \quad \lesssim \int_0^t\mathbb{E}\big[|X_s^{N,i}-{\underline{\textrm{X}}_s^i}|^\delta + W_\delta(\bar{\rho}_s^N, \mathcal{L}({\underline{\textrm{X}}_s^i}))^\delta\big]\,\mathrm{d}s \end{align}

for any $t\in[0,T]$ . By Remark 6, we obtain that $\mathcal{L}({\underline{\textrm{X}}_s^i})=\mathcal{L}(\underline{\textrm{X}}_s^1)$ . Hence, we get

(38) \begin{align} W_\delta\big(\bar{\rho}_s^N,\mathcal{L}({\underline{\textrm{X}}_s^i})\big)^\delta & = W_\delta\Bigg(\bar{\rho}_s^N,\mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg)^\delta \notag \\ & \leq 2^\delta W_\delta\Bigg(\bar{\rho}_s^N,\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}\Bigg)^\delta + 2^\delta W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg)^\delta \notag \\ & \lesssim \frac{1}{N}\sum\limits_{j=1}^N\big|X_s^{N,j}-\underline{X}_s^j\big|^\delta + W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j},\mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg)^\delta. \end{align}

Moreover, by Remark 6, we can find a measurable map $G\colon \mathbb{R}^d\times C([0,T];\,\mathbb{R}^m)\to C([0,T];\,\mathbb{R}^d)$ such that, for any $1\leq i\leq N$ , $\underline{X}^i=G(X_0^i,B^i)$ . In the same way, there is a measurable map $G_N\colon (\mathbb{R}^{d}\times C([0,T];\,\mathbb{R}^m))^N\to C([0,T];\,\mathbb{R}^d)^N$ , such that

\begin{equation*} (X^{N,1},\ldots ,X^{N,N})=G_N\big((X_0^1,\ldots ,X_0^N),(B^1,\ldots ,B^N)\big). \end{equation*}

More generally, by the symmetry of the system (6), for any permutation $\varsigma$ of $\lbrace 1,\ldots ,N \rbrace$ ,

\begin{equation*} (X^{N,\varsigma(1)},\ldots,X^{N,\varsigma(N)}) = G_N\big((X_0^{\varsigma(1)},\ldots,X_0^{\varsigma(N)}),(B^{\varsigma(1)},\ldots,B^{\varsigma(N)})\big). \end{equation*}

Hence, since the random variables $((X_0^i,B^i))_{1\leq i\leq N}$ are i.i.d. and, in particular, exchangeable, we can apply Lemma 6 to obtain that the coupled processes $((X^{N,i},\underline{X}^i))_{1\leq i\leq N}$ are exchangeable and hence, in particular, are identically distributed. For $i=1$ we can insert (38) into (37) and conclude by Jensen’s inequality that

\begin{align*} & \mathbb{E}[|X_t^{N,1}-\underline{\textrm{X}}\,_t^1|^\delta] \\ & \qquad \lesssim \int_0^t\mathbb{E}\Bigg[|X_s^{N,1}-\underline{\textrm{X}}_s^1|^\delta + \frac{1}{N}\sum\limits_{j=1}^N|X_s^{N,j}-\underline{X}_s^j|^\delta + W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg)^\delta\Bigg]\,\mathrm{d}s \\ & \qquad = \int_0^t\mathbb{E}\Bigg[2|X_s^{N,1}-\underline{\textrm{X}}_s^1|^\delta + W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg)^\delta\Bigg]\,\mathrm{d}s. \end{align*}

Using Grönwall’s lemma, we deduce that

\begin{equation*} \mathbb{E}[|X_t^{N,1}-\underline{\textrm{X}}\,_t^1|^\delta] \lesssim \int_0^t\mathbb{E}\Bigg[W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N \delta_{\underline{\textrm{X}}_s^j},\mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg)^\delta\Bigg]\,\mathrm{d}s, \end{equation*}

and since the processes $((X^{N,i},\underline{X}^i))_{1\leq i\leq N}$ are identically distributed, this completes Step 2.

Step 3: First, we show that

(39) \begin{equation} \lim\limits_{N\to\infty}\mathbb{E}\Bigg[W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg)^\delta\Bigg] = 0 \end{equation}

for any $s\in[0,T]$ by showing convergence in probability and uniform integrability. By the Glivenko–Cantelli theorem (see [Reference Shorack and Wellner32, Chapter 26, Theorem 1] for a general version) and since the $\underline{X}^j$ are i.i.d., we get the convergence $({1}/{N})\sum_{j=1}^N\delta_{\underline{\textrm{X}}_s^j} \to \mathcal{L}(\underline{\textrm{X}}_s^1)$ as $N\to \infty$ almost surely, and hence in probability. Furthermore, again using the notation $\bar{\rho}^N_s = ({1}/{N})\sum_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}$ , we can bound using Hölder’s inequality and the boundedness of all moments of ${\underline{\textrm{X}}_s^i}$ , $1\leq i\leq N$ , in (5), to get

(40) \begin{align} & \sup\limits_{N\in\mathbb{N}}\mathbb{E}\big[W_\delta(\bar{\rho}^N_s,\mathcal{L}(\underline{\textrm{X}}_s^1))^\delta \mathbf{1}_{\lbrace W_\delta(\bar{\rho}^N_s,\mathcal{L}(\underline{\textrm{X}}_s^1))>K\rbrace}\big] \notag \\ & \qquad \leq K^{-1}\sup\limits_{N\in\mathbb{N}}\mathbb{E}\big[ W_\delta(\bar{\rho}^N_s,\mathcal{L}(\underline{\textrm{X}}_s^1))^{\delta+1}\big] \notag \\ & \qquad \leq K^{-1}\sup\limits_{N\in\mathbb{N}}\mathbb{E}\big[ W_{\delta+1}(\bar{\rho}^N_s,\mathcal{L}(\underline{\textrm{X}}_s^1))^{\delta+1}\big] \notag \\ & \qquad \leq K^{-1}\sup\limits_{N\in\mathbb{N}}\mathbb{E}\big[ W_{\delta+1}(\bar{\rho}^N_s,\delta_0)^{\delta+1} + W_{\delta+1}(\delta_0,\mathcal{L}(\underline{\textrm{X}}_s^1))^{\delta+1}\big] \notag \\ & \qquad = K^{-1}\sup\limits_{N\in\mathbb{N}}\mathbb{E}\Bigg[ \frac{1}{N}\Bigg(\sum\limits_{i=1}^N |{\underline{\textrm{X}}_s^i}|^{\delta+1}\Bigg) + |\underline{\textrm{X}}_s^1|^{\delta+1} \Bigg] \notag \\ & \qquad = 2K^{-1}\mathbb{E}[|\underline{\textrm{X}}_s^1|^{\delta+1}] \to 0 \end{align}

as $K\to\infty$ , which shows uniform $\delta$ -integrability of the family of random variables

\begin{equation*} \Bigg(W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg)\Bigg)_{N\in\mathbb{N}}. \end{equation*}

Hence, Vitali’s convergence theorem (see [Reference Bogachev9, Theorem 4.5.4]) reveals the $L^\delta$ -convergence as claimed in (39).

To conclude Step 3, it remains to show that the convergence (39) is uniform in s. Therefore, we first notice that, for any $p\geq \delta$ ,

(41) \begin{align} \mathbb{E}\Bigg[W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg)^{p}\Bigg] & \leq \mathbb{E}\Bigg[W_p\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg)^{p}\Bigg] \notag \\ & \lesssim \mathbb{E}\Bigg[W_p\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \delta_0\Bigg)^{p}\Bigg] + W_p(\delta_0,\mathcal{L}(\underline{\textrm{X}}_s^1))^{p} \notag \\ & = \frac{1}{N}\mathbb{E}\Bigg[\sum\limits_{j=1}^N|\underline{\textrm{X}}_s^j|^{p}\Bigg] + \mathbb{E}[|\underline{\textrm{X}}_s^1|^{p}] = 2\mathbb{E}[|\underline{\textrm{X}}_s^1|^{p}] < \infty, \end{align}

by (5). With Jensen’s inequality, (41) also follows for $1\leq p< \delta$ .

Let $k \,:\!=\,\lceil \delta\rceil\geq\delta$ denote the smallest integer greater than or equal to $\delta$ . Notice that with the same argument as in (40), by substituting the exponent $\delta$ by k and then bounding from above using the $(k+1)$ -Wasserstein distance and again by Vitali’s convergence theorem, the $L^k$ -convergence of the $\delta$ -Wasserstein distance in (39) also follows. Once we show that this $L^k$ -convergence is uniform in s, then it will follow that

(42) \begin{multline} \lim\limits_{N\to\infty}\sup\limits_{s\in[0,T]}\mathbb{E}\Bigg[ W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg)^\delta\Bigg] \\ \leq \lim\limits_{N\to\infty}\sup\limits_{s\in[0,T]}\mathbb{E}\Bigg[ W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg)^k\Bigg]^{{\delta}/{k}} = 0. \end{multline}

Therefore, using the factorization $a^{k}-b^{k} =(a-b) \sum_{r=0}^{k-1} a^{k-1-r}b^r$ and Hölder’s inequality with $\delta$ and $q=({4+2\epsilon})/({4+\epsilon})$ such that ${1}/{\delta}+{1}/{q}=1$ , we get

(43) \begin{align} &\Bigg|\mathbb{E}\Bigg[W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}\,_t^j}, \mathcal{L}(\underline{\textrm{X}}\,_t^1)\Bigg)^{k}\Bigg] - \mathbb{E}\Bigg[W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg)^{k}\Bigg]\Bigg| \notag \\ & \qquad = \Bigg|\mathbb{E}\Bigg[\Bigg(W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}\,_t^j}, \mathcal{L}(\underline{\textrm{X}}\,_t^1)\Bigg) - W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg)\Bigg) \notag \\ & \qquad\qquad\quad \times \sum\limits_{r=0}^{k-1}W_\delta\Bigg( \frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}\,_t^j},\mathcal{L}(\underline{\textrm{X}}\,_t^1)\Bigg)^{k-1-r} W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg)^{r}\Bigg]\Bigg| \notag \\& \qquad \leq \Bigg|\mathbb{E}\Bigg[\Bigg(W_\delta\Bigg( \frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}\,_t^j},\mathcal{L}(\underline{\textrm{X}}\,_t^1)\Bigg) - W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg)\Bigg)^\delta\Bigg]^{{1}/{\delta}} \notag \\& \qquad\quad \times \mathbb{E}\Bigg[\Bigg(\sum\limits_{r=0}^{k-1} W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}\,_t^j}, \mathcal{L}(\underline{\textrm{X}}\,_t^1)\Bigg)^{k-1-r} W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg)^{r}\Bigg)^q\Bigg]^{{1}/{q}}\Bigg|. \end{align}

Again using Hölder’s inequality such as (41), we can bound the second expectation by

(44) \begin{align} & \mathbb{E}\Bigg[\Bigg(\sum\limits_{r=0}^{k-1}W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}\,_t^j}, \mathcal{L}(\underline{\textrm{X}}\,_t^1)\Bigg)^{k-1-r}W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg)^{r}\Bigg)^q\Bigg]^{{1}/{q}} \notag \\ & \lesssim \Bigg(\sum\limits_{r=0}^{k-1}\mathbb{E}\Bigg[ W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}\,_t^j},\mathcal{L}(\underline{\textrm{X}}\,_t^1)\Bigg)^{q(k-1-r)} W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg)^{qr}\Bigg]\Bigg)^{{1}/{q}} \notag \\ & \lesssim \Bigg(\sum\limits_{r=0}^{k-1}\mathbb{E}\Bigg[ W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}\,_t^j}, \mathcal{L}(\underline{\textrm{X}}\,_t^1)\Bigg)^{2q(k-1-r)}\Bigg]^{{1}/{2}} \mathbb{E}\Bigg[W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg)^{2qr}\Bigg]^{{1}/{2}}\Bigg)^{{1}/{q}} \notag \\ & < \infty. \end{align}

Inserting (44) into (43) and using the triangle inequality

\begin{align*} & W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}\,_t^j},\mathcal{L}(\underline{\textrm{X}}\,_t^1)\Bigg) \\ & \qquad \leq W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}\,_t^j}, \frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}\Bigg) + W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j},\mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg) + W_\delta\big(\mathcal{L}(\underline{\textrm{X}}_s^1),\mathcal{L}(\underline{\textrm{X}}\,_t^1)\big), \end{align*}

which also holds if we switch s and t, we arrive at

\begin{align*} & \Bigg|\mathbb{E}\Bigg[W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{X}\,_t^j}, \mathcal{L}(\underline{\textrm{X}}\,_t^1)\Bigg)^{k}\Bigg] - \mathbb{E}\Bigg[W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg)^{k}\Bigg]\Bigg| \\ & \qquad \lesssim \mathbb{E}\Bigg[\Bigg|\Bigg(W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}\,_t^j}, \mathcal{L}(\underline{\textrm{X}}\,_t^1)\Bigg) - W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg)\Bigg)\Bigg|^\delta\Bigg]^{{1}/{\delta}} \\ & \qquad \lesssim \mathbb{E}\Bigg[\Bigg(W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}\,_t^j}, \frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}\Bigg) + W_\delta\big(\mathcal{L}(\underline{\textrm{X}}\,_t^1), \mathcal{L}(\underline{\textrm{X}}_s^1)\big)\Bigg)^\delta\Bigg]^{{1}/{\delta}} \\ & \qquad \lesssim \mathbb{E}\Bigg[W_\delta\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}\,_t^j}, \frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}\Bigg)^\delta\Bigg]^{{1}/{\delta}} + \mathbb{E}\big[W_\delta\big(\mathcal{L}(\underline{\textrm{X}}\,_t^1), \mathcal{L}(\underline{\textrm{X}}_s^1)\big)^\delta\big]^{{1}/{\delta}} \\ &\qquad \lesssim \mathbb{E}[|\underline{\textrm{X}}\,_t^1-\underline{\textrm{X}}_s^1|^\delta]^{{1}/{\delta}} \\ & \qquad \lesssim |t-s|^\beta, \end{align*}

where the last line holds by Remark 4 for any $\beta\in (0,\gamma-1/p)$ with $\gamma\in\big(0,\frac{1}{2}\big]$ from Assumption 1. Hence, we obtain that (42) holds, which together with (36) shows that

(45) \begin{equation} \lim\limits_{N\to\infty}\sup\limits_{t\in [0,T]}\mathbb{E}[|X_t^{N,1}-\underline{\textrm{X}}\,_t^1|^\delta]=0, \end{equation}

and knowing that $((X^{N,i},\underline{X}^i))_{1\leq i\leq N}$ are identically distributed, this completes Step 3.

Step 4: We already know from Step 3 that the first summand in (7) converges to zero. For the second summand, we use the triangle inequality and Jensen’s inequality to obtain

(46) \begin{align} & \sup\limits_{t\in [0,T]}\mathbb{E}\Bigg[W_\delta\Bigg(\frac{1}{N}\sum\limits_{i=1}^N\delta_{X_t^{N,i}}, \mathcal{L}(\underline{\textrm{X}}\,_t^1)\Bigg)^\delta\Bigg] \notag \\ & \qquad \lesssim \sup\limits_{t\in[0,T]}\mathbb{E}\Bigg[ W_\delta\Bigg(\frac{1}{N}\sum\limits_{i=1}^N\delta_{X_t^{N,i}}, \frac{1}{N}\sum\limits_{i=1}^N\delta_{\underline{\textrm{X}}\,_t^i}\Bigg)^\delta\Bigg] + \sup\limits_{t\in[0,T]}\mathbb{E}\Bigg[W_\delta\Bigg(\frac{1}{N}\sum\limits_{i=1}^N\delta_{\underline{\textrm{X}}\,_t^i}, \mathcal{L}(\underline{\textrm{X}}\,_t^1)\Bigg)^\delta\Bigg] \notag \\ & \qquad \lesssim \sup\limits_{t\in[0,T]}\mathbb{E}\Bigg[ \frac{1}{N}\sum\limits_{i=1}^N|X_t^{N,i}-\underline{\textrm{X}}\,_t^i|^\delta\Bigg] + \sup\limits_{t\in[0,T]}\mathbb{E}\Bigg[W_\delta\Bigg(\frac{1}{N}\sum\limits_{i=1}^N\delta_{\underline{\textrm{X}}\,_t^i}, \mathcal{L}(\underline{\textrm{X}}\,_t^1)\Bigg)^\delta\Bigg], \end{align}

which also tends to 0 as $N\to\infty$ by (42) and (45).

Proof of Theorem 4. We prove the statement by using the same Steps 1–4 as in the proof of Theorem 2, but with $\delta=1$ . For Step 2, though, we need to consider separately the cases where Assumption 3, the first case in the proof of Theorem 1, and Assumption 4, the second case, hold.

Step 1: By Remark 3, we obtain as in the proof of Theorem 2 the unique system of stochastic processes $(X^{N,i})_{i=1,\ldots,N}$ that solves (6).

Step 2: In the first case, suppose the kernels $K_\mu,K_\sigma$ and initial condition $X_0$ satisfy Assumption 3. To mimic the inequality (36), we use the semimartingale property

\begin{align*} X_t^{N,i} - \underline{\textrm{X}}\,_t^i & = \int_0^tK_\sigma(s,s)\big(\sigma(s,X_s^{N,i}) - \sigma(s,{\underline{\textrm{X}}_s^i})\big)\,\mathrm{d}B_s \\ & \quad + \int_0^tK_\mu(s,s)\big(\mu(s,X_s^{N,i},\bar{\rho}_s^N) - \mu(s,{\underline{\textrm{X}}_s^i},\mathcal{L}({\underline{\textrm{X}}_s^i}))\big)\,\mathrm{d}s \\ & \quad + \int_0^t\bigg(\int_0^s\partial_2K_\mu(u,s) \big(\mu(u,X_u^{N,i},\bar{\rho}_u^N)-\mu(u,{\underline{\textrm{X}}_u^i},\mathcal{L}({\underline{\textrm{X}}_u^i}))\big)\,\mathrm{d}u \\ & \qquad\qquad\quad + \int_0^s\partial_2K_\sigma(u,s)\big(\sigma(u,X_u^{N,i})-\sigma(u,{\underline{\textrm{X}}_u^i})\big)\, \mathrm{d}B_u\bigg)\,\mathrm{d}s \end{align*}

to perform a Yamada–Watanabe approach exactly as we did around equality (22), and obtain, for fixed $i\in\lbrace1,\ldots,N\rbrace$ with the notation $M^{N,i}(t)\,:\!=\,\mathbb{E}[|X_t^{N,i}-\underline{\textrm{X}}\,_t^i|]+\mathbb{E}[|\tilde{Y}_t|]$ , where $\tilde{Y}_t\,:\!=\,\int_0^t\sigma(s,X_s^{N,i})\,\mathrm{d}B_s^i-\int_0^t\sigma(s,{\underline{\textrm{X}}_s^i})\,\mathrm{d}B_s^i$ , that $M^{N,i}(t) \lesssim \int_0^t\big(M^{N,i}(s) + \mathbb{E}[W_1(\bar{\rho}_s^N,\mathcal{L}(\underline{\textrm{X}}^i_s))]\big)\,\mathrm{d}s$ , such that, proceeding as in the proof of Theorem 2, including applying Grönwall’s inequality, we obtain

(47) \begin{equation} \mathbb{E} \left[|X_t^{N,i}-\underline{\textrm{X}}\,_t^i|\right] \lesssim \int_0^t\mathbb{E}\Bigg[W_1\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \mathcal{L}({\underline{\textrm{X}}_s^i})\Bigg)\Bigg]\,\mathrm{d}s. \end{equation}

In the second case, suppose the kernels $K_\mu,K_\sigma$ and initial condition $X_0$ satisfy Assumption 4. As in the previous case, to mimic inequality (36) we use the semimartingale property

\begin{align*} X_t^{N,i}-\underline{\textrm{X}}\,_t^i & = \int_0^t\tilde{K}(0)\big(\sigma(s,X_s^{N,i}) - \sigma(s,{\underline{\textrm{X}}_s^i})\big)\,\mathrm{d}B_s \\ & \quad + \int_0^t\tilde{K}(0)\big(\mu(s,X_s^{N,i},\bar{\rho}_s^N) - \mu(s,{\underline{\textrm{X}}_s^i},\mathcal{L}({\underline{\textrm{X}}_s^i}))\big)\,\mathrm{d}s \\ & \quad + \int_0^t\bigg(\int_0^s\tilde{K}^\prime(s-u)\big(\mu(u,X_u^{N,i},\bar{\rho}_u^N) - \mu(u,{\underline{\textrm{X}}_u^i},\mathcal{L}({\underline{\textrm{X}}_u^i}))\big)\,\mathrm{d}u \\ & \qquad\qquad\quad + \int_0^s\tilde{K}^\prime(s-u)\big(\sigma(u,X_u^{N,i})-\sigma(u,{\underline{\textrm{X}}_u^i})\big)\, \mathrm{d}B_u\bigg)\,\mathrm{d}s \end{align*}

to perform a Yamada–Watanabe approach and apply Grönwall’s inequality as in (31), which yields

\begin{equation*} \mathbb{E}[|X_t^{N,i}-\underline{\textrm{X}}\,_t^i|] \lesssim \int_0^t\mathbb{E}\Bigg[W_1\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \mathcal{L}(\underline{\textrm{X}}^i_s)\Bigg)\Bigg]\,\mathrm{d}s. \end{equation*}

Step 3: Obtaining the convergence to zero uniformly in s of the right-hand side of (47) now follows easily by using

\begin{equation*} \mathbb{E}\Bigg[W_1\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg)\Bigg] \leq \mathbb{E}\Bigg[W_2\Bigg(\frac{1}{N}\sum\limits_{j=1}^N\delta_{\underline{\textrm{X}}_s^j}, \mathcal{L}(\underline{\textrm{X}}_s^1)\Bigg)^2\Bigg]^{{1}/{2}}, \end{equation*}

and then using [Reference Carmona and Delarue12, (5.19)], and proceeding as in [Reference Carmona and Delarue13, Proof of Theorem 2.12].

Step 4: As in (46), we obtain

(48) \begin{align} & \sup\limits_{0\leq t\leq T}\mathbb{E}\Bigg[W_1\Bigg(\frac{1}{N}\sum\limits_{i=1}^N\delta_{X_t^{N,i}}, \mathcal{L}(\underline{\textrm{X}}\,_t^1)\Bigg)\Bigg] \notag \\ & \qquad \lesssim \sup\limits_{0\leq t\leq T}\mathbb{E}\Bigg[\frac{1}{N}\sum\limits_{i=1}^N |X_t^{N,i}-\underline{\textrm{X}}\,_t^i|\Bigg] + \sup\limits_{0\leq t\leq T}\mathbb{E}\Bigg[W_1\Bigg(\frac{1}{N}\sum\limits_{i=1}^N\delta_{\underline{\textrm{X}}\,_t^i}, \mathcal{L}(\underline{\textrm{X}}\,_t^1)\Bigg)\Bigg], \end{align}

which tends to zero by the uniform convergence to zero of the right-hand side of (47), and finishes the proof.

Proof of Lemma 1. By Lemma 7, for any $t\in[0,T]$ ,

(49) \begin{equation} \mathbb{E}\Bigg[W_\delta\Bigg(\frac{1}{N}\sum\limits_{i=1}^N\delta_{\underline{\textrm{X}}\,_t^i}, \mathcal{L}(\underline{\textrm{X}}\,_t^1)\Bigg)^\delta\Bigg] \lesssim \varepsilon_N, \end{equation}

where $(\varepsilon_N)_{N\in\mathbb{N}}$ is given by (9) and the right-hand side does not depend on t. Plugging (49) into (36) and taking the supremum over [0, T] and maximum over $1,\ldots,N$ shows the desired convergence rate of the first term in (8). Then, using this and plugging (49) into (46) gives the desired rate for the second term.

Proof of Lemma 2. First, suppose the kernels $K_\mu,K_\sigma$ and initial condition $X_0$ satisfy Assumption 3. By Lemma 7 we obtain

(50) \begin{equation} \mathbb{E}\Bigg[W_1\Bigg(\frac{1}{N}\sum\limits_{i=1}^N\delta_{\underline{\textrm{X}}\,_t^i}, \mathcal{L}(\underline{\textrm{X}}\,_t^1)\Bigg)\Bigg] \lesssim N^{-1/2} \end{equation}

independent of $t\in[0,T]$ . Plugging (50) into (47) and (48) yields the statement.

Otherwise, suppose the kernels $K_\mu,K_\sigma$ and initial condition $X_0$ satisfy Assumption 4. Plugging (50) into the analogues of (47) and (48) yields the statement.