1. Introduction
Mean-field stochastic differential equations (mean-field SDEs), also known as McKean–Vlasov stochastic differential equations, provide mathematical descriptions of random systems of interacting particles whose time evolutions depend, in some manner, on the probability distribution of the entire systems. A crucial reason for the frequent use of mean-field SDEs in applied mathematics is the fact that they allow the modeling of the ‘propagation of chaos’ within large interacting particle systems. Recall that, on a microscopic scale, the trajectory of each individual particle can often be appropriately modeled by a stochastic process. However, when the number of particles becomes very large, the microscopic scale usually contains too much information, making the interactions of individual particles intractable. Fortunately, sending the number of particles to infinity, the propagation of chaos states that the behavior of an individual particle depends only on the probability distribution of the entire system, i.e. on the macroscopic scale the interaction of individual particles becomes negligible.
Mean-field SDEs, as well as the propagation of chaos, originated in statistical physics and were first studied by Kac [Reference Kac20], McKean [Reference McKean23] and Vlasov [Reference Vlasov34]. Since then, these concepts have found a wide range of applications in a variety of fields such as physics, finance, and data science. We refer, e.g., to [Reference Carmona and Delarue12–Reference Chaintron and Diez15, Reference Jabin and Wang19, Reference Sznitman33] for comprehensive introductions to mean-field SDEs and their numerous applications. Except for a very small number of publications, like the rough path-based approaches to mean-field SDEs [Reference Bailleul, Catellier and Delarue5, Reference Bailleul, Catellier and Delarue6, Reference Coghi, Deuschel, Friz and Maurelli16], the vast majority of literature on mean-field SDEs and the propagation of chaos is restricted to Markovian systems of interacting particles, i.e. the behavior of each particle has to be independent of all past states of the system. On the contrary, it is often observed that many real-world dynamical systems do have memory effects, and thus do indeed depend on past states of the underlying systems. Well-known examples of such systems are the growth of populations, the spread of epidemics, and turbulence flows.
Classical mathematical models for random dynamical systems with memory effects are given by stochastic Volterra equations (SVEs), as introduced in the seminal works of Berger and Mizel [Reference Berger and Mizel7, Reference Berger and Mizel8]; see also, e.g., [Reference Pardoux and Protter26, Reference Protter27]. While SVEs allow for generating non-Markovian stochastic processes, the solutions of SVEs, in contrast to mean-field SDEs, do not depend directly on the probability distributions of the generated random systems.
In the present paper we aim to unify the theories of mean-field stochastic differential equations and stochastic Volterra equations, enabling us to combine the desirable modeling advantages of both classes of equations. More precisely, we introduce mean-field stochastic Volterra equations (mean-field SVEs),
\begin{equation} X_t = X_0 + \int_0^t K_{\mu}(s,t)\mu(s,X_s,\mathcal{L}(X_s))\,\mathrm{d}s + \int_0^t K_{\sigma}(s,t)\sigma(s,X_s,\mathcal{L}(X_s))\,\mathrm{d}B_s,\quad t\in [0,T],\end{equation}
where
$X_0$
is a random variable, B is a Brownian motion, and the coefficients
$\mu, \sigma$
, as well as the kernels
$K_\mu, K_\sigma$
, are measurable functions. Here,
$\mathcal{L}(X_s)$
denotes the law of the random variable
$X_s$
. In words, mean-field SVEs are a class of stochastic integral equations that describe the dynamics of random systems with both nonlinear interactions and memory effects. They constitute a generalization of mean-field SDEs and of classical SVEs. Notice that a solution to the mean-field SVE (1) is, in general, neither a Markov process nor a semimartingale.
Our first contribution is to establish the (strong) well-posedness of the mean-field SVE (1), meaning that there exists a unique strong solution to (1), under two sets of assumptions. On the one hand, we show the existence of a unique solution to the mean-field SVE (1) in a multi-dimensional setting with standard assumptions on the kernels and coefficients, i.e. we assume some integrability of the kernels as well as Lipschitz continuity and a linear growth condition for the coefficients, cf., e.g., [Reference Carmona11, Reference Wang35]. The proof is based on a classical fixed-point argument in combination with techniques from the theories of mean-field SDEs and SVEs. On the other hand, we show the existence of a unique solution to the mean-field SVE (1) in a one-dimensional setting, assuming sufficiently smooth kernels and Hölder-continuous diffusion coefficients that are independent of the law of the solution. To that end, we rely on a Yamada–Watanabe approach [Reference Yamada and Watanabe36] to SVEs with sufficiently smooth kernels, as recently generalized in [Reference Abi Jaber, Larsson and Pulido3, Reference Prömel and Scheffels30]. As comparison, for well-posedness results in the case of mean-field SDEs we refer to [Reference Bahlali, Mezerdi and Mezerdi4, Reference Huang and Wang18, Reference Kalinin, Meyer-Brandis and Proske21], and in the case of SVEs to [Reference Abi Jaber, Larsson and Pulido3, Reference Prömel and Scheffels30, Reference Wang35]. Furthermore, we remark that a specific type of mean-field SVEs was studied in [Reference Shi, Wang and Yong31], where the coefficients may depend on the law of the solution but only through an expectation operator.
Our second contribution is to establish quantitative, pointwise propagation of chaos results of Volterra-type systems of interacting particles. In words, sending the number of Volterra-type interacting particles to infinity, we obtain a macroscopic description of the systems based on a mean-field stochastic Volterra equation. The approach developed is based on a synchronous coupling method; it was initiated in [Reference McKean24] and extended in [Reference Sznitman33]. In the case of mean-field SDEs, synchronous coupling methods are widely used for systems that are described by systems of McKean–Vlasov diffusions, and often lead to pathwise propagation of chaos; see, e.g., [Reference Chaintron and Diez14, Theorem 3.20], [Reference Carmona11, Theorem 1.10], and [Reference Huang and Wang18]. In the present case of mean-field SVEs, implementing a synchronous coupling method becomes more challenging as the underlying McKean–Vlasov processes are of Volterra type and, thus, in general, lack the semimartingale and Markov property. As for our well-posedness theory of mean-field SVEs, we distinguish between the aforementioned multi- and one-dimensional settings. The pointwise nature of the our propagation of chaos results for mean-field SVEs is caused by the non-availability of a Burkholder–Davis–Gundy inequality in the multi-dimensional setting and by the Hölder continuity of the diffusion coefficients in the one-dimensional setting. The latter setting requires us to combine the synchronous coupling method with a Yamada–Watanabe approach.
1.1. Organization of the paper
In Section 2 we present the main results regarding the well-posedness and propagation of chaos for mean-field stochastic Volterra equations. Section 3 provides some necessary well-posedness results for ordinary stochastic Volterra equations. The proofs of the main results are contained in Sections 4, 5, and 6.
2. Main results: Well-posedness and propagation of chaos
Let
$T\in (0,\infty)$
,
$d,m\in\mathbb{N}$
, and let
$(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in [0,T]},\mathbb{P})$
be a filtered probability space that satisfies the usual conditions. Suppose
$B=(B_t)_{t\in [0,T]}$
is an m-dimensional Brownian motion with respect to
$(\mathcal{F}_t)_{t\in [0,T]}$
. The law of a random variable X is denoted by
$\mathcal{L}(X)$
and, for
$p\geq 1$
, the space of probability measures on
$\mathbb{R}^d$
with finite pth moments by
$\mathcal{P}_{p}(\mathbb{R}^d)$
. Let
$C([0,T];\,\mathbb{R}^d)$
be the space of continuous functions from [0,T] to
$\mathbb{R}^d$
, equipped with the supremum norm
$\|\cdot\|_\infty$
, and
$\mathcal{P}_p(C([0,T];\,\mathbb{R}^d))$
be the space of probability measures on
$C([0,T];\mathbb{R}^d)$
with finite pth moment. For
$\rho,\tilde{\rho}\in\mathcal{P}_p(\mathbb{R}^d)$
, we write
$W_p(\rho,\tilde{\rho})$
for the p-Wasserstein distance between
$\rho$
and
$\tilde{\rho}$
(see [Reference Carmona and Delarue12, Chapter 5] for its definition) and, with a slight abuse of notation, for
$\rho,\tilde{\rho} \in \mathcal{P}_p(C([0,T];\,\mathbb{R}^d))$
we define the p-Wasserstein distance by
\begin{equation*} W_p(\rho,\tilde{\rho}) = \inf_{\pi \in \Pi(\rho,\tilde{\rho})}\bigg[\int_{C([0,T];\,\mathbb{R}^d)^2}\|x-y\|_\infty^p\,\mathrm{d}\pi(x,y)\bigg]^{1/p},\end{equation*}
where
$\Pi(\rho,\tilde{\rho})$
denotes the set of all probability measures on
$C([0,T];\,\mathbb{R}^d)^2$
with marginal distributions given by
$\rho$
and
$\tilde{\rho}$
, respectively. The space
$\mathbb{R}^d$
is always equipped with the Euclidean norm
$| \cdot |$
, and on the space
$\mathbb{R}^{d\times m}$
we use the Frobenius norm, also denoted by
$| \cdot |$
. Moreover, we set
$\Delta_T\,:\!=\,\lbrace (s,t)\in [0,T]\times [0,T]\colon 0\leq s\leq t\leq T \rbrace$
and use the notation
$A_{\eta}\lesssim B_{\eta}$
for a generic parameter
$\eta$
, meaning that
$A_{\eta}\le CB_{\eta}$
for some constant
$C>0$
independent of
$\eta$
.
We consider the d-dimensional mean-field stochastic Volterra equation
\begin{equation} X_t = X_0 + \int_0^t K_{\mu}(s,t)\mu(s,X_s,\mathcal{L}(X_s))\,\mathrm{d}s + \int_0^t K_{\sigma}(s,t)\sigma(s,X_s,\mathcal{L}(X_s))\,\mathrm{d}B_s,\quad t\in [0,T],\end{equation}
where
$X_0$
is a d-dimensional,
$\mathcal{F}_0$
-measurable random variable that is independent of B, and the coefficients
$\mu\colon[0,T]\times\mathbb{R}^d\times\mathcal{P}_{p}(\mathbb{R}^d)\to\mathbb{R}^d$
,
$\sigma\colon[0,T]\times\mathbb{R}^d\times\mathcal{P}_{p}(\mathbb{R}^{d})\to\mathbb{R}^{d\times m}$
and the kernels
$K_\mu, K_\sigma\colon \Delta_T\to \mathbb{R}$
are measurable functions. The integral
$\int_0^t K_{\sigma}(s,t)\sigma(s,X_s,\mathcal{L}(X_s))\,\mathrm{d}B_s$
is defined as a stochastic Itô integral.
Let us briefly recall the concepts of well-posedness, strong solutions, and pathwise uniqueness. We use, for measure spaces
$\mathcal{X},\mathcal{Y}$
and
$p\geq 1$
, the notation
$L^p=L^p(\mathcal{X};\,\mathcal{Y})$
for the space of all
$\mathcal{Y}$
-valued, measurable, p-integrable functions on
$\mathcal{X}$
, and, for two Banach spaces
$\mathcal{X},\mathcal{Y}$
,
$C(\mathcal{X};\,\mathcal{Y})$
for the space of all
$\mathcal{Y}$
-valued, continuous functions on
$\mathcal{X}$
. An
$(\mathcal{F}_t)_{t\in[0,T]}$
-progressively measurable stochastic process
$(X_t)_{t\in [0,T]}$
in
$L^p(\Omega\times [0,T];\,\mathbb{R}^d)$
, on the given probability space
$(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in[0,T]},\mathbb{P})$
, is called a (strong)
$L^{p}$
-solution of the mean-field SVE (2) if
\begin{equation*} \int_0^t(|K_\mu(s,t)\mu(s,X_s,\mathcal{L}(X_s))| + |K_\sigma(s,t)\sigma(s,X_s,\mathcal{L}(X_s))|^2 )\,\mathrm{d}s < \infty \quad \text{for all }t\in[0,T],\end{equation*}
and the integral equation (2) holds
$\mathbb{P}$
-almost surely. We say that pathwise uniqueness in
$L^{p}$
holds for the mean-field SVE (2) if
$\mathbb{P}(X_t=\tilde{X}_t$
for all
$t\in [0,T])=1$
for any two
$L^{p}$
-solutions
$(X_t)_{t\in[0,T]}$
and
$(\tilde{X}_t)_{t\in[0,T]}$
of (2) defined on the same probability space
$(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in[0,T]},\mathbb{P})$
. We say that the mean-field SVE (2) is well-posed in
$L^{p}$
(or that there exists a unique
$L^{p}$
-solution) for
$p\geq 1$
if there exists a strong
$L^{p}$
-solution to (2) and pathwise uniqueness in
$L^{p}$
holds.
In the following we distinguish between multi-dimensional and one-dimensional settings, since these settings allow us to establish the well-posedness of the mean-field SVE (2) with different regularity assumptions on the kernels and coefficients. The main existence and uniqueness results regarding mean-field SVEs as well as propagation of chaos are stated in Subsections 2.1 and 2.2. In the multi-dimensional setting (Subsection 2.1) we make standard Lipschitz assumptions on the coefficients
$\mu,\sigma$
, whereas in the one-dimensional setting (Subsection 2.2) we assume that
$\mu$
is Lipschitz continuous but allow
$\sigma$
to be only Hölder continuous. We prove the corresponding results in Sections 4, 5, and 6.
2.1. Mean-field SVEs with Lipschitz-continuous coefficients
In this subsection we consider the multi-dimensional stochastic Volterra equation (2) with dimensions
$d,m\in\mathbb{N}$
and coefficients
$\mu,\sigma$
that are Lipschitz continuous in the space and distributional component, uniformly in the time component, allowing for potentially singular kernels. We start by stating the assumptions on the kernels.
Assumption 1. Assume there are constants
$\gamma\in\big(0,\frac{1}{2}\big]$
,
$\epsilon>0$
, and
$L>0$
such that
$K_\mu, K_\sigma\colon \Delta_T\to \mathbb{R}$
are measurable functions fulfilling
\begin{align*} \int_0^t|K_{\mu}(s,t')-K_{\mu}(s,t)|^{1+\epsilon}\,\mathrm{d}s + \int_t^{t'}|K_{\mu}(s,t')|^{1+\epsilon}\,\mathrm{d}s & \leq L|t'-t|^{\gamma(1+\epsilon)}, \\ \int_0^t|K_{\sigma}(s,t')-K_{\sigma}(s,t)|^{2+\epsilon}\,\mathrm{d}s + \int_t^{t'}|K_{\sigma}(s,t')|^{2+\epsilon}\,\mathrm{d}s & \leq L|t'-t|^{\gamma(2+\epsilon)} \end{align*}
for all
$(t,t^\prime)\in \Delta_T$
.
Note that Assumption 1 allows for singular kernels, like the fractional convolutional kernel
$K(s,t)=(t-s)^{-\alpha}$
for
$\alpha\in (0,1/2)$
and the examples provided in [Reference Abi Jaber, Cuchiero, Larsson and Pulido1, Example 1.3]. Moreover, for
$\epsilon>0$
given by Assumption 1, let the fixed parameter
$\delta>2$
be defined by
\begin{equation} \delta\,:\!=\,\frac{4+2\epsilon}{\epsilon},\end{equation}
such that
\begin{equation} \frac{2}{2+\epsilon}+\frac{2}{\delta}=1.\end{equation}
In the following we use the
$\delta$
-Wasserstein distance on the space
$\mathcal{P}_\delta(\mathbb{R}^d)$
of probability measures on
$\mathbb{R}^d$
with finite
$\delta$
th moments. Relying on the
$\delta$
-Wasserstein distance, we specify the assumptions on the regularity of the coefficients
$\mu$
and
$\sigma$
, which are a classical linear growth condition and a Lipschitz assumption.
Assumption 2. Let
$\mu\colon[0,T]\times\mathbb{R}^d\times\mathcal{P}_{\delta}(\mathbb{R}^d)\to\mathbb{R}^d$
and
$\sigma\colon [0,T]\times\mathbb{R}^d\times\mathcal{P}_{\delta}(\mathbb{R}^d)\to\mathbb{R}^{d\times m}$
be measurable functions such that:
-
(i) for any bounded set
$\mathcal{K}\subset\mathcal{P}_{\delta}(\mathbb{R}^d)$
, there is a constant
$C_{\mathcal{K}}>0$
such that the linear growth condition
$|\mu(t,x,\rho)|+|\sigma(t,x,\rho)|\leq C_{\mathcal{K}} (1+|x|)$
holds for all
$\rho \in \mathcal{K}$
,
$t\in[0,T]$
, and
$x\in\mathbb{R}^d$
; -
(ii)
$\mu$
and
$\sigma$
are Lipschitz continuous in x and in
$\rho$
with respect to the
$\delta$
-Wasserstein distance uniformly in t, i.e. there is a constant
$C_{\mu,\sigma}>0$
such that
holds for all
\begin{equation*} |\mu(t,x,\rho)-\mu(t,\tilde{x},\tilde{\rho})|+|\sigma(t,x,\rho)-\sigma(t,\tilde{x},\tilde{\rho})| \leq C_{\mu,\sigma}\big(|x-\tilde{x}|+W_{\delta}(\rho,\tilde{\rho})\big) \end{equation*}
$t\in [0,T]$
,
$x,\tilde{x}\in \mathbb{R}^d$
, and
$\rho,\tilde{\rho}\in \mathcal{P}_{\delta}(\mathbb{R}^d)$
.
Our first result is the well-posedness of the mean-field stochastic Volterra equation (2).
Theorem 1. Suppose that the initial value
$X_0$
is in
$L^p(\Omega;\,\mathbb{R}^d)$
, the kernels
$K_\mu, K_\sigma$
fulfill Assumption 1, the coefficients
$\mu,\sigma$
fulfill Assumption 2, and
$p>\max\{{1}/{\gamma},1+{2}/{\epsilon}\}$
, where
$\gamma\in\big(0,\frac{1}{2}\big]$
and
$\epsilon>0$
are given by Assumption 1. Then, the mean-field stochastic Volterra equation (2) is well posed in
$L^{p}$
. Moreover, for any
$q\geq p$
, if
$X_0\in L^q(\Omega;\,\mathbb{R}^d)$
, the unique
$L^{p}$
-solution X of (2) satisfies
\begin{equation} \sup\limits_{t\in[0,T]}\mathbb{E}[|X_t|^{q}]<\infty. \end{equation}
Our second result is the propagation of chaos for mean-field stochastic Volterra equations, i.e. we show that the unique
$L^{p}$
-solution to the mean-field stochastic Volterra equation (2) is the limit
$N\to\infty$
of the solutions to the following system of N mean-field stochastic Volterra equations:
\begin{equation} X_t^{N,i} = X_0^i + \int_0^tK_\mu(s,t)\mu(s,X_s^{N,i},\bar{\rho}_s^N)\,\mathrm{d}s + \int_0^tK_\sigma(s,t)\sigma(s,X_s^{N,i},\bar{\rho}_s^N)\,\mathrm{d}B_s^i, \quad t\in[0,T],\end{equation}
for
$i\in\lbrace 1,\ldots,N\rbrace$
, where
$\bar{\rho}_t^N\,:\!=\,({1}/{N})\sum_{i=1}^N\delta_{X_t^{N,i}}$
is the empirical distribution of
$(X_t^{N,i})_{i=1,\ldots,N}$
,
$(X_0^i)_{i\in\mathbb{N}} \, \subset \, L^q(\Omega;\,\mathbb{R}^d)$
is a sequence of
$\mathcal{F}_0$
-measurable, independent and identically distributed (i.i.d.) random variables for some
$q>4$
, and
$(B^i)_{i\in\mathbb{N}}$
is a sequence of independent m-dimensional Brownian motions, which are all defined on the given probability space
$(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in [0,T]},\mathbb{P})$
. Strong
$L^{p}$
-solutions, pathwise uniqueness in
$L^{p}$
, and well-posedness in
$L^{p}$
for the system (6) of mean-field SVEs are defined analogously to (2), and
$\delta_x$
denotes the Dirac measure at x for
$x\in \mathbb{R}^d$
. Moreover, for
$i\in\mathbb{N}$
, let
$\underline{\textbf{X}}^i$
be the solution of the mean-field SVE (2) with the initial condition
$X_0^i$
and driving Brownian motion
$B^i$
. In the present multi-dimensional setting, we obtain the following convergence result.
Theorem 2. (Volterra propagation of chaos). Suppose Assumptions 1 and 2, and that the sequence of initial conditions
$(X_0^i)_{i\in\mathbb{N}}\subset L^q(\Omega;\,\mathbb{R}^d)$
for some
$q>\max\{p,2\delta\}$
and
$p>\max\{{1}/{\gamma},1+{2}/{\epsilon}\}$
, where
$\delta$
is defined in (3). Then, the system (6) of mean-field SVEs is well posed in
$L^{p}$
for every
$N\geq 1$
, where the unique
$L^{p}$
-solution is denoted by
$(X_t^{N,i})_{i=1,\ldots,N}$
. Moreover, we have
\begin{equation} \lim\limits_{N\to\infty}\Bigg(\max\limits_{1\leq i\leq N}\bigg(\sup\limits_{t\in[0,T]} \mathbb{E}\big[|X_t^{N,i}-\underline{X}\,_t^i|^\delta\big]\bigg) + \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{X}\,_t^1)\Bigg)^\delta\Bigg]\Bigg) = 0. \end{equation}
The rate of convergence in (7) is explicitly stated in the next lemma.
Lemma 1. With the assumptions and notation of Theorem 2, we have
\begin{equation} \max\limits_{1\leq i\leq N}\bigg(\sup\limits_{t\in[0,T]}\mathbb{E}[|X_t^{N,i}-\underline{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_{X_t^{N,i}}, \mathcal{L}(\underline{X}\,_t^1)\Bigg)^\delta\Bigg] \lesssim \varepsilon_N, \end{equation}
where
$(\varepsilon_N)_{N\in\mathbb{N}}$
is given by
\begin{equation} \varepsilon_N = \begin{cases} N^{-1/2} & \text{if } d<2\delta, \\ N^{-1/2}\log_2(1+N) & \text{if } d=2\delta, \\ N^{-\delta/d} & \text{if } d>2\delta. \end{cases} \end{equation}
Remark 1. The rates of convergence obtained in (9) are analogous to the classical rates for ordinary mean-field SDEs with Lipschitz coefficients (see [Reference Chaintron and Diez14, Theorem 3.20]), using
$W_\delta(\cdots)^\delta$
instead of
$W_2(\cdots)^2$
and, consequently, replacing the exponent
$2/d$
by
$\delta/d$
in (9). Note that in the case of ordinary mean-field SDEs we obtain a pathwise propagation of chaos result (meaning that the
$\sup$
in (8) is inside the expectation operators), which is a stronger type of convergence than the pointwise convergence presented in Theorem 2. This weaker type of convergence is caused by the missing availability of the standard Burkholder–Davis–Gundy inequality for the solutions of stochastic Volterra equations since they are, in general, not semimartingales. However, the rates of convergence provided in Lemma 1 seem to be optimal for synchronous coupling methods, since it is shown in [Reference Fournier and Guillin17, Theorem 1ff] that for terms of the form
$\mathbb{E}[W_\delta(\bar{\rho}_N,\rho)^\delta]$
the rates in (9) are sharp. Consequently, optimality could only be lost in the inequalities (47) or (48), which, at least in general, appears not to be the case.
2.2. Mean-field SVEs with Hölder-continuous diffusion coefficients
In this subsection we consider mean-field SVEs in a one-dimensional setting, i.e. we assume
$d=m=1$
. This allows us to relax the Lipschitz assumption on the diffusion coefficient
$\sigma$
to Hölder continuity in the space variable, provided that
$\sigma$
is independent of the distribution of the solution and that the kernels are sufficiently regular. More precisely, we consider the one-dimensional mean-field stochastic Volterra equation
\begin{equation} X_t = X_0 + \int_0^tK_{\mu}(s,t)\mu(s,X_s,\mathcal{L}(X_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}
where
$(B_t)_{t\in[0,T]}$
is a one-dimensional Brownian motion,
$X_0$
is an
$\mathcal{F}_0$
-measurable random variable, the coefficients
$\mu\colon[0,T]\times\mathbb{R}\times\mathcal{P}_{p}(\mathbb{R})\to\mathbb{R}$
,
$\sigma\colon[0,T]\times\mathbb{R}\to\mathbb{R}$
and the kernels
$K_\mu, K_\sigma\colon \Delta_T\to \mathbb{R}$
are measurable functions. We consider two different sets of assumptions on the kernels and on the initial condition.
Assumption 3. Let
$\gamma\in\big(0,\frac{1}{2}\big]$
and
$\epsilon >0$
. Let
$X_0$
be an
$\mathcal{F}_0$
-measurable random variable and
$K_\mu, K_\sigma\colon \Delta_T\to \mathbb{R}$
be continuous functions such that:
-
(i)
$K_\mu(s,\cdot)$
is absolutely continuous for every
$s\in [0,T]$
, and
$\partial_2 K_\mu $
is bounded on
$\Delta_T$
; -
(ii)
$K_\sigma(\cdot,t)$
is absolutely continuous for every
$t\in [0,T]$
,
$K_\sigma(s,\cdot)$
is absolutely continuous for every
$s\in [0,T]$
with
$\partial_2 K_\sigma\in L^{2}(\Delta_T)$
, and
$\partial_2 K_\sigma(\cdot,t)$
is absolutely continuous for every
$t\in[0,T]$
. Furthermore, there is a constant
$C_1>0$
such that
$|K_\sigma(t,t)|\geq C_1$
for any
$t\in [0,T]$
, and there exists
$C_2>0$
such that
and
\begin{equation*} \int_0^s|K_{\sigma}(u,t)-K_{\sigma}(u,s)|^{2+\epsilon}\,\mathrm{d}u \leq C_2|t-s|^{\gamma (2+\epsilon)} \end{equation*}
$|\partial_1K_\sigma(s,t)| + |\partial_2K_\sigma(s,s)| + \int_s^t|\partial_{21}K_\sigma(s,u)|\,\mathrm{d}u \leq C_2$
hold for any
$(s,t)\in\Delta_T$
;
-
(iii)
$X_0 \in L^p(\Omega;\,\mathbb{R})$
for
$p>\max\{{1}/{\gamma},1+{2}/{\epsilon}\}$
.
Instead of Assumption 3, we can alternatively require
$K_\mu$
,
$K_\sigma$
, and
$X_0$
to fulfill the following assumption, where the kernels are supposed to be convolutional.
Assumption 4. Let
$X_0$
be an
$\mathcal{F}_0$
-measurable random variable and
$K_\mu, K_\sigma\colon \Delta_T\to \mathbb{R}$
be continuous functions such that:
-
(i)
$K_\mu(s,t)=K_\sigma(s,t)=\tilde{K}(t-s)$
for some
$\tilde{K}\in C^1([0,T];\,\mathbb{R})$
; -
(ii)
$X_0 \in L^p(\Omega;\,\mathbb{R})$
for
$p>2$
.
Next, we formulate the assumptions on the coefficients.
Assumption 5. Let
$\mu\colon [0,T]\times\mathbb{R}\times\mathcal{P}_{1}(\mathbb{R})\to\mathbb{R} $
and
$\sigma\colon [0,T]\times\mathbb{R}\to\mathbb{R}$
be measurable functions such that:
-
(i) for any bounded set
$\mathcal{K}\subset\mathcal{P}_{1}(\mathbb{R})$
, there is a constant
$C_{\mathcal{K}}>0$
such that the linear growth condition
$|\mu(t,x,\rho)|+|\sigma(t,x)|\leq C_\mathcal{K} \rho(1+|x|)$
holds for all
$\rho\in\mathcal{K}$
,
$t\in[0,T]$
, and
$x\in\mathbb{R}$
; -
(ii)
$\mu$
is Lipschitz continuous in x and
$\rho$
with respect to the 1-Wasserstein distance, uniformly in t, i.e. there is a constant
$C_{\mu}>0$
such that
holds for all
\begin{equation*} |\mu(t,x,\rho)-\mu(t,\tilde{x},\tilde{\rho})|\leq C_{\mu}\big(|x-\tilde{x}|+W_{1}(\rho,\tilde{\rho})\big) \end{equation*}
$t\in [0,T]$
,
$x,\tilde{x}\in \mathbb{R}$
, and
$\rho,\tilde{\rho}\in \mathcal{P}_{1}(\mathbb{R})$
, and
$\sigma$
is Hölder continuous of order
$\frac{1}{2}+\xi$
for some
$\xi\in \big[0,\frac{1}{2}\big]$
in x uniformly in t, i.e. there is a constant
$C_{\sigma}>0$
such that
$|\sigma(t,x)-\sigma(t,\tilde{x})|\leq C_{\sigma}|x-\tilde{x}|^{({1}/{2})+\xi}$
holds for all
$t\in [0,T]$
and
$x,\tilde{x}\in \mathbb{R}$
.
First, we establish the well-posedness of the mean-field stochastic Volterra equation (10) with Hölder-continuous diffusion coefficients. Its proof is based on a Yamada–Watanabe-type approach [Reference Yamada and Watanabe36], which requires essentially a one-dimensional setting and leads to the stronger assumptions on the kernels. Moreover, note that the Hölder-continuous diffusion coefficients are required to be independent of the law of the solution, which is essentially a standard assumption for ordinary mean-field stochastic differential equations as it appears to be a necessary assumption to implement a Yamada–Watanabe type approach, cf. [Reference Kalinin, Meyer-Brandis and Proske21].
Theorem 3. Suppose Assumption 5, and that the kernels
$K_{\mu},K_{\sigma}$
and the initial condition
$X_0$
satisfy Assumption 3 or 4 with p given therein. Then the mean-field stochastic Volterra equation (10) is well posed in
$L^{p}$
. Moreover, for any
$q\geq p$
, if
$X_0\in L^q(\Omega;\,\mathbb{R}^d)$
, the unique solution X of (10) satisfies
$\sup_{t\in[0,T]}\mathbb{E}[|X_t|^{q}]<\infty$
.
Second, we establish the propagation of chaos for one-dimensional stochastic mean-field SVEs with Hölder-continuous diffusion coefficients. To that end, we consider the symmetric system of N mean-field stochastic Volterra equations
\begin{equation} X_t^{N,i}=X_0^i + \int_0^tK_\mu(s,t)\mu(s,X_s^{N,i},\bar{\rho}_s^N)\,\mathrm{d}s + \int_0^tK_\sigma(s,t)\sigma(s,X_s^{N,i})\,\mathrm{d}B_s^i, \quad t\in[0,T],\end{equation}
for
$i\in\lbrace 1,\ldots,N\rbrace$
, where
$(X_0^i)_{i\in\mathbb{N}}\subset L^p(\Omega;\,\mathbb{R})$
is an i.i.d. sequence of initial conditions, and
$(B^i)_{i\in\mathbb{N}}$
is a sequence of independent one-dimensional Brownian motions. Moreover, for
$i\in\mathbb{N}$
,
$\underline{\textrm{X}}^i$
denotes the solution of the mean-field SVE (10) with initial condition
$X_0^i$
and driving Brownian motion
$B^i$
. In the present one-dimensional setting, we obtain the following convergence result.
Theorem 4. (Volterra propagation of chaos) Suppose Assumption 5, and that the kernels
$K_{\mu},K_{\sigma}$
and the initial conditions
$X_0^i$
for
$i\in\mathbb{N}$
satisfy Assumption 3 or 4 with p given therein. Then, the system (11) of mean-field SVEs is well posed in
$L^{p}$
, where the unique
$L^{p}$
-solution is denoted by
$(X_t^{N,i})_{i=1,\ldots,N}$
for every
$N\geq 1$
. Moreover,
\begin{equation} \lim\limits_{N\to\infty}\Bigg(\max\limits_{1\leq i\leq N}\bigg(\sup\limits_{t\in[0,T]} \mathbb{E}[|X_t^{N,i}-\underline{X}\,_t^i|]\bigg) + \sup\limits_{t\in[0,T]}\mathbb{E}\Bigg[W_1\Bigg(\frac{1}{N}\sum\limits_{i=1}^N\delta_{X_t^{N,i}}, \mathcal{L}(\underline{X}\,_t^1)\Bigg)\Bigg]\Bigg) = 0. \end{equation}
The rate of convergence in (12) is explicitly stated in the next lemma.
Lemma 2. Supposing the assumptions and notation of Theorem 4, we have
\begin{equation} \max\limits_{1\leq i\leq N}\bigg(\sup\limits_{t\in[0,T]}\mathbb{E}[|X_t^{N,i}-\underline{X}\,_t^i|]\bigg) + \sup\limits_{t\in[0,T]}\mathbb{E}\Bigg[W_1\Bigg(\frac{1}{N}\sum\limits_{i=1}^N\delta_{X_t^{N,i}}, \mathcal{L}(\underline{X}\,_t^1)\Bigg)\Bigg] \lesssim N^{-1/2}. \end{equation}
Remark 2. The rate of convergence in (13) is expected to be optimal for synchronous coupling methods, cf. Remark 1, since it is shown in [Reference Fournier and Guillin17, Theorem 1ff] that for terms of the form
$\mathbb{E}[W_1(\bar{\rho}_N,\rho)]$
the rate is sharp. Consequently, optimality could only be lost in the inequalities (36) or (46).
3. On the well-posedness of ordinary stochastic Volterra equations
In this section we provide various well-posedness results for ordinary stochastic Volterra equations with random initial conditions that are needed to prove the well-posedness results for mean-field stochastic Volterra equations presented in Section 2. We start with SVEs with Lipschitz-continuous coefficients, which is a slight modification of [Reference Wang35, Theorem 1.1].
Lemma 3. Let the kernels
$K_\mu, K_\sigma$
fulfill Assumption 1,
$p>\max\{{1}/{\gamma},1+{2}/{\epsilon}\}$
with
$\gamma\in\big(0,\frac{1}{2}\big]$
and
$\epsilon>0$
from Assumption 1, the initial value
$X_0\in L^p(\Omega;\,\mathbb{R}^d)$
be adapted, and the measurable coefficients
$\mu\colon [0,T] \times\mathbb{R}^d\to \mathbb{R}^d$
and
$\sigma\colon [0,T]\times\mathbb{R}^d\to \mathbb{R}^{d\times m}$
for some
$d,m\in\mathbb{N}$
fulfill the linear growth condition
$|\mu(t,x)| + |\sigma(t,x)|\leq C_{\mu,\sigma}(1+|x|)$
for some
$C_{\mu,\sigma}>0$
and all
$t\in[0,T]$
,
$x\in\mathbb{R}^d$
, and the Lipschitz condition
\begin{equation*} |\mu(t,x)-\mu(t,y)| + |\sigma(t,x)-\sigma(t,y)| \leq C_{\mu,\sigma}|x-y| \end{equation*}
for some
$C_{\mu,\sigma}>0$
and all
$t\in[0,T]$
,
$x,y\in\mathbb{R}^d$
. Then, the d-dimensional stochastic Volterra equation
\begin{equation*} X_t = X_0 + \int_0^tK_\mu(s,t)\mu(s,X_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*}
is well posed in
$L^{p}$
, where
$(B_t)_{t\in[0,T]}$
is an m-dimensional Brownian motion.
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.
For one-dimensional ordinary stochastic Volterra equations the Lipschitz assumption on the diffusion coefficients can be relaxed to Hölder continuity, provided the kernels are sufficiently regular or have a convolutional structure. The next results are a slight modification of [Reference Prömel and Scheffels30, Theorem 2.3], allowing for SVEs with random initial conditions.
Lemma 4. Let the kernels
$K_\mu, K_\sigma$
fulfill Assumption 3,
$p>\max\{{1}/{\gamma},1+{2}/{\epsilon}\}$
with
$\gamma\in \big(0,\frac{1}{2}\big]$
and
$\epsilon>0$
from Assumption 3, the initial value
$X_0\in L^p(\Omega;\,\mathbb{R})$
, and the measurable coefficients
$\mu\colon [0,T]\times\mathbb{R}\to \mathbb{R}$
and
$\sigma\colon [0,T]\times\mathbb{R}\to \mathbb{R}$
fulfill the linear growth condition
\begin{equation*} |\mu(t,x)| + |\sigma(t,x)|\leq C_{\mu,\sigma}(1+|x|) \end{equation*}
for some
$C_{\mu,\sigma}>0$
and all
$t\in[0,T]$
,
$x\in\mathbb{R}$
,
$\mu$
be the Lipschitz condition
\begin{equation*} |\mu(t,x)-\mu(t,y)| \leq C_{\mu}|x-y| \end{equation*}
for some
$C_{\mu}>0$
and all
$t\in[0,T]$
,
$x,y\in\mathbb{R}$
, and
$\sigma$
be the Hölder condition
\begin{equation*} |\sigma(t,x)-\sigma(t,y)| \leq C_{\sigma}|x-y|^{({1}/{2})+\xi} \end{equation*}
for
$\xi\in\big[0,\frac{1}{2}\big]$
, some
$C_{\sigma}>0$
, and all
$t\in[0,T]$
,
$x,y\in\mathbb{R}$
. Then the stochastic Volterra equation
\begin{equation*} X_t = X_0 + \int_0^tK_\mu(s,t)\mu(s,X_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*}
is well posed in
$L^{p}$
, where
$(B_t)_{t\in[0,T]}$
is a one-dimensional Brownian motion.
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.
The next lemma is a slight generalization of [Reference Abi Jaber and El Euch2, Proposition B.3], providing the well-posedness of one-dimensional SVEs with convolutional kernels and random initial conditions.
Lemma 5. Suppose that
$X_0\in L^p(\Omega;\,\mathbb{R})$
for some
$p>2$
, the kernels are of the form
$K_\mu(s,t)=K_\sigma(s,t)=\tilde{K}(t-s)$
for some
$\tilde{K}\in C^1([0,T];\,\mathbb{R})$
, and the measurable coefficients
$\mu\colon [0,T] \times\mathbb{R}\to \mathbb{R}$
and
$\sigma\colon [0,T]\times\mathbb{R}\to \mathbb{R}$
fulfill the linear growth condition
\begin{equation*} |\mu(t,x)| + |\sigma(t,x)|\leq C_{\mu,\sigma}(1+|x|) \end{equation*}
for some
$C_{\mu,\sigma}>0$
and all
$t\in[0,T]$
,
$x\in\mathbb{R}$
,
$\mu$
satisfies the Lipschitz condition
\begin{equation*} |\mu(t,x)-\mu(t,y)| \leq C_{\mu}|x-y| \end{equation*}
for some
$C_{\mu}>0$
and all
$t\in[0,T]$
,
$x,y\in\mathbb{R}$
, and
$\sigma$
satisfies the Hölder condition
\begin{equation*} |\sigma(t,x)-\sigma(t,y)| \leq C_{\sigma}|x-y|^{({1}/{2})+\xi} \end{equation*}
for
$\xi\in\big[0,\frac{1}{2}\big]$
, some
$C_{\sigma}>0$
, and all
$t\in[0,T]$
,
$x,y\in\mathbb{R}$
. Then the stochastic Volterra equation
\begin{equation} X_t = X_0 + \int_0^t\tilde{K}(t-s)\mu(s,X_s)\,\mathrm{d}s + \int_0^t\tilde{K}(t-s)\sigma(s,X_s)\,\mathrm{d}B_s, \quad t\in[0,T], \end{equation}
is well posed in
$L^{p}$
, where
$(B_t)_{t\in[0,T]}$
is a one-dimensional Brownian motion.
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.
Moreover, for the well-posedness results of mean-field SVEs we need a multi-dimensional well-posedness result for stochastic Volterra equations where the Hölder-continuous coefficient
$\sigma$
is a diagonal matrix, where each entry only depends on the component of the solution of the respective dimension, as provided in the next remark.
Remark 3. For
$N\in\mathbb{N}$
let us consider the N-dimensional stochastic Volterra equation
\begin{equation} X_t = X_0 + \int_0^tK_\mu(s,t)\mu(s,X_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}
where
$(B_t)_{t\in[0,T]}$
is an N-dimensional Brownian motion, and
\begin{equation*} X_t=\left(\begin{array}{c} X_t^1 \\ \vdots \\ X_t^N \end{array}\right),\qquad X_0=\left(\begin{array}{c} X_0^1 \\ \vdots \\ X_0^N \end{array}\right), \end{equation*}
\begin{equation*} \mu(s,X_s)=\left(\begin{array}{c} \mu_1(s,X_s) \\ \vdots \\ \mu_N(s,X_s) \end{array}\right), \qquad \sigma(s,X_s)=\left( \begin{array}{c@{\quad}c@{\quad}c} \sigma_1(s,X_s^1) &\cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots& \sigma_N(s,X_s^N) \\ \end{array}\right). \end{equation*}
Suppose that the kernels
$K_\mu,K_\sigma$
and the initial value
$X_0$
fulfill Assumption 3 or 4 with p as defined there, that
$\mu\colon [0,T]\times \mathbb{R}^N\to \mathbb{R}^N$
is Lipschitz continuous in the space variable, uniformly in the time variable, and each
$\sigma_i\colon [0,T]\times\mathbb{R}\to \mathbb{R}$
for
$i\in\lbrace1,\ldots, N\rbrace$
is (
$1/2+\xi$
)-Hölder continuous in the space variable, uniformly in the time variable for some
$\xi\in\big[0,\frac12\big]$
. By considering each dimension separately, e.g. as done for SDEs in [Reference Yamada and Watanabe36, Theorem 1], it is straightforward to conclude the well-posedness in
$L^{p}$
of the SVE (15) from the corresponding one-dimensional results in Lemmas 4 and 5.
We conclude this section with a remark on the path regularity of solutions and one on the notion of
$L^{p}$
-well-posedness.
Remark 4. (Path regularity). Let X be the unique (d-, 1-, or N-dimensional) solution to the stochastic Volterra equation in any of the settings in Lemmas 3, 4, 5 or Remark 3 with
$p>\max\{{1}/{\gamma},1+{2}/{\epsilon}\}$
. In the case of Assumption 4, we can set
$\gamma=\frac{1}{2}$
and
$p>2$
as given there. Assuming
$X_0\in L^q$
for
$q\geq p$
, by adapting [Reference Prömel and Scheffels30, Lemmas 3.1 and 3.4] to the multi-dimensional setting, it follows that
$\sup_{t\in[0,T]}\mathbb{E}[|X_t|^q]<\infty$
and
$\mathbb{E}[|X_t-X_s|^q]\lesssim |t-s|^{\beta q}$
for any
$q\geq 1$
,
$\beta\in (0,\gamma-{1}/{p})$
,
$s,t\in[0,T]$
, and, hence, that the solution X has a modification with
$\beta$
-Hölder-continuous sample paths.
Remark 5. The notion of
$L^{p}$
-well-posedness, as used in Lemmas 3, 4, 5 and Remark 3, appears to be necessary to prove the existence of a strong solution and pathwise uniqueness. First, we need to assume that a solution X is in
$L^p(\Omega\times[0,T];\,\mathbb{R}^d)$
to conclude continuity of its sample paths with standard estimates, as in [Reference Prömel and Scheffels30, Lemma 3.1]. Second, in order to be able to apply Grönwall’s lemma to an inequality of the form
$\mathbb{E}[|X_t-Y_t|^p]\lesssim \int_0^t \mathbb{E}[|X_s-Y_s|^p]\,\mathrm{d}s$
, we need to assume that both solutions X,Y are in
$L^p(\Omega\times[0,T];\,\mathbb{R}^d)$
to guarantee finiteness of the expectations
$\sup_{s\in[0,t]}\mathbb{E}[|X_s|^p]$
and
$\sup_{s\in[0,t]}\mathbb{E}[|Y_s|^p]$
by standard estimates, as in [Reference Prömel and Scheffels30, Lemma 3.4].
4. Well-posedness: Proofs of Theorems 1 and 3
Proof of Theorem
1. We define the solution map
$\Phi$
by
\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
\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
\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
\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
\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.
Next, we provide the proof of Theorem 3. We keep its presentation fairly short since it is in parts similar to the proof of Theorem 1.
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
\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*}
\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
\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
\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
\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
\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
\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
\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
\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
\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
\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).
Remark 6. The well-posedness from Theorems 1 and 3, together with a general version of the classical Yamada–Watanabe result (see, e.g., [Reference Kurtz22, Theorem 1.5]; see also [Reference Kurtz22, Example 2.14]), implies that there is some measurable map
$G\colon \mathbb{R}^d\times C([0,T];\,\mathbb{R}^m)\to C([0,T];\,\mathbb{R}^d)$
such that any solution X of (2) and (10), respectively, given some initial value
$X_0$
and Brownian motion B, can be represented as
$X=G(X_0,B)$
. Hence, if
$X,\tilde{X}$
are solutions of (2) and (10), respectively, for initial values
$X_0,\tilde{X}_0$
with the same law and Brownian motions
$B,\tilde{B}$
, it is straightforward that
$\mathcal{L}(X_t)=\mathcal{L}(\tilde{X}_t)$
almost surely for all
$t\in[0,T]$
.
5. Propagation of chaos: Proofs of Theorems 2 and 4
An important argument in the proofs of the propagation of chaos results will be to show that the coupled processes
$((X^{N,i},\underline{X}^i))_{1\leq i\leq N}$
are identically distributed. The following lemma plays a crucial role. Recall that a sequence of random variables
$(\zeta^1,\zeta^2,\ldots)$
is called exchangeable if, for any
$N\in\mathbb{N}$
, the vectors
$(\zeta^1,\ldots,\zeta^N)$
and
$(\zeta^{\sigma(1)},\ldots,\zeta^{\sigma(N)})$
have the same joint distribution, where
$\lbrace\sigma(1),\ldots,\sigma(N)\rbrace$
is an arbitrary permutation of
$\lbrace 1,\ldots,N\rbrace$
.
Lemma 6. Let
$(A,\mathcal{F}_A)$
and
$(B,\mathcal{F}_B)$
be measurable spaces and for some fixed
$N\in\mathbb{N}$
, let
$(\zeta^1,\ldots,\zeta^N)$
be an exchangeable family of A-valued random variables. Let
$F\colon A\to B$
be a measurable function and define the family of random variables
$(X^1,\ldots,X^N)$
by
$X^i\,:\!=\,F(\zeta^i)$
for
$i\in \lbrace 1,\ldots,N\rbrace$
. Further, let
$G\colon A^N\to B^N$
be a measurable function that fulfills the exchangeability property
\begin{equation} (y_1,\ldots,y_N)=G((x_1,\ldots,x_N))\Rightarrow (y_{\sigma(1)},\ldots,y_{\sigma(N)})=G((x_{\sigma(1)},\ldots,x_{\sigma(N)})) \end{equation}
for arbitrary
$x_1,\ldots,x_N\in A$
and any permutation
$\lbrace\sigma(1),\ldots,\sigma(N)\rbrace$
of
$\lbrace 1,\ldots,N\rbrace$
. Define the family of random variables
$(Y^1,\ldots,Y^N)$
by
$(Y^1,\ldots,Y^N)\,:\!=\,G((\zeta^1,\ldots,\zeta^N))$
. Then, the coupled family of random variables
$((X^i,Y^i))_{1\leq i\leq N}$
is exchangeable.
Proof. Let
$\lbrace \sigma(1),\ldots,\sigma(N)\rbrace$
be an arbitrary permutation of
$\lbrace 1,\ldots,N\rbrace$
. By (32), we have
\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),
\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
\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.
We start with the proof of Theorem 2.
Proof of Theorem 2. Let us briefly outline the main steps of the proof:
-
Step 1. We show the existence of the system of processes
$(X^{N,i})_{i=1,\ldots,N}$
uniquely solving (6), for every
$N\in\mathbb{N}$
. -
Step 2. We prove the inequality
(36)for any
\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}
$ 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
$B^i$
.
-
Step 3. We prove that the right-hand side of (36) tends to zero.
-
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*}
\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
\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
\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
\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
$p\geq \delta$
,
\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
\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
\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
\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
\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
\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}
We continue with the proof of Theorem 4. Since the proof is similar to that of Theorem 4, we focus, for the sake of brevity, on the main differences.
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
\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
\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.
6. Rate of convergence: Proofs of Lemmas 1 and 2
The proofs of Lemmas 1 and 2 rely on a quantitative Glivenko–Cantelli theorem due to Fournier and Guillin [Reference Fournier and Guillin17], which provides a sharp estimate of the
$\delta$
-Wasserstein distance. For the sake of completeness, we recall [Reference Fournier and Guillin17, Theorem 1] in the following lemma.
Lemma 7. Let
$\delta>0$
and
$\bar{\rho}^N\,:\!=\,({1}/{N})\sum_{i=1}^N \delta_{X^i}$
be the empirical distribution of i.i.d. random variables
$(X^i)_{i=1,\ldots,N}$
with common distribution
$\rho$
such that
$\rho\in\mathcal{P}_p(\mathbb{R}^d)$
for every
$p\geq 1$
. Then
$\mathbb{E}[W_\delta(\bar{\rho}^N,\rho)^\delta]\lesssim \varepsilon_N$
, where
$(\varepsilon_N)_{N\in\mathbb{N}}$
is given by (9), i.e.
\begin{equation*} \varepsilon_N= \begin{cases} N^{-1/2} & \text{if } d<2\delta, \\ N^{-1/2}\log_2(1+N) & \text{if } d=2\delta, \\ N^{-\delta/d} & \text{if } d>2\delta, \end{cases} \end{equation*}
and
$\mathbb{E}[W_1(\bar{\rho}^N,\rho)]\lesssim N^{-1/2}$
.
With this lemma at hand, we can prove Lemmas 1 and 2.
Proof of Lemma
1. By Lemma 7, for any
$t\in[0,T]$
,
\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
\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.
Acknowledgements
D. J. Prömel and D. Scheffels would like to thank P. Nikolaev for fruitful discussions which helped to improve the present work.
Funding information
D. Scheffels gratefully acknowledges financial support by the Research Training Group ‘Statistical Modeling of Complex Systems’ (RTG 1953) funded by the German Science Foundation (DFG).
Competing interests
There were no competing interests to declare which arose during the preparation or publication process of this article.
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