Convex order preservation for graphon mean-field systems

Daorong Cui Thanks: Email: dcuiab@connect.ust.hk, Department of Mathematics, The Hong Kong University of Science and Technology, Clearwater Bay, Kowloon, Hong Kong. Shuoqing Deng Thanks: Email: masdeng@ust.hk, Department of Mathematics, The Hong Kong University of Science and Technology, Clearwater Bay, Kowloon, Hong Kong. S. Deng is supported by the Hong Kong University of Science and Technology Start-up grant no. R9826 and Hong Kong RGC Early Career Scheme (ECS) under grant no. 26307125. Yang Xiang Thanks: Email: maxiang@ust.hk, Department of Mathematics, The Hong Kong University of Science and Technology, Clearwater Bay, Kowloon, Hong Kong.

Abstract

In this paper, we study the convex order preservation for the graphon mean-field systems on $\mathbb{R}^{d}$ . First, we establish the uniform-in-time Euler approximations for the graphon mean-field systems of interest, which extends previous results 7 on graphon particle systems and is necessary for our main theorem. Building on this together with appropriate conditions on the graphon functions, we then proceed to prove the marginal convex order preservation. Second, by employing the weighted $L^{2}$ -norm, we are able to further establish an infinite-horizon trajectory-level convergence result, which is key to the functional convex ordering as in 22. Finally, we apply our results to the study of value function comparison of graphon mean-field games(MFGs).

Mathematics Subject Classification (2020): 60G65, 60H35, 60J60, 60K35

Keywords: Convex order, Graphon mean-field systems, Euler schemes.

1 Introduction

Convex order between two distributions $\mu,\nu\in\mathcal{P}_{1}(\mathbb{R}^{d})$ , i.e. probability measures on $\mathbb{R}^{d}$ with finite first moment, is defined by

\mu\preceq_{cv}\nu\quad\text{if $\forall\varphi:\mathbb{R}^{d}\rightarrow\mathbb{R}$ convex, }\int_{\mathbb{R}^{d}}\varphi(x)\mu(\mathrm{d}x)\leq\int_{\mathbb{R}^{d}}\varphi(y)\nu(\mathrm{d}y).

By taking two specific linear functions $\varphi(x)=\pm x$ , this naturally implies that both measures have the same mean. Two $\mathbb{R}^{d}$ -valued random vectors $X$ and $Y$ are in convex order if their corresponding distributions are.

The study of convex ordering for stochastic processes dates back to the work of Hajek 16 on the case of increasing convex order, and has been explored by various authors since then, see, for instance 19; 25; 27; 9 and references therein. We also refer to 22; 26; 18; 17 for more recent articles. In 26, Pagès established functional convex order results for two Brownian martingale diffusion processes under the assumption that diffusion coefficient is convex in the spatial variable. Such a convexity condition was further relaxed by 17 in the one-marginal case while an assumption comparable to the spatial convexity of the diffusion coefficient is needed for the two-marginal case. Liu and Pagès 22 extended such functional convex order results to the McKean-Vlasov equations, and Jourdain and Pagès 18 compared the convex order of stochastic Volterra equations.

In this paper, we will investigate in depth the comparison of marginal and functional convex orders for scaled graphon mean-field systems, where “scaled” means that the drift coefficient is affine in the space variable. Such a system is defined on some filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ by

\mathrm{d}X^{u}_{t}=b(t,X^{u}_{t})\mathrm{d}t+\sigma(t,X^{u}_{t},[G\bm{\mu}_{t}]^{u})\mathrm{d}B^{u}_{t},\quad u\in I:=[0,1]\text{ and }t\geq 0,

where $b$ and $\sigma$ are suitable coefficient functions; $G(\cdot,\cdot)$ is the graphon function, defined as a symmetric measurable function from $I^{2}$ to $I$ . The graphon-weighted measure $[G\bm{\mu}_{t}]^{u}$ , defined as

[G\bm{\mu}_{t}]^{u}(\mathrm{d}x):=\int_{I}G(u,v)\mu^{v}_{t}(\mathrm{d}x)\mathrm{d}v,\quad\text{$\mu^{u}_{t}:=\mathcal{L}(X^{u}_{t})$ and $\bm{\mu}_{t}:=\{\mu^{u}_{t}\}_{u}$},

represents the population’s influence on player $u$ ’s state, and $\{B^{u}\}_{u}$ is a sequence of i.i.d. $q$ -dimensional standard Brownian motions. As in the McKean-Vlasov dynamics, the coefficients of this system depend on the distributions of the state variables. However, we also remark that each $X^{u}$ alone is not a standard McKean-Vlasov dynamic, as its own law $\mu^{u}$ plays a negligible role in the evolution, see also 3. Apart from the introduction of the graphon interractions, another novelty of our framework is the extension of the time horizon to infinity.

Building on the graphon theories developped by Lovász 23, the study on graphon mean-field systems has gained considerable attention recently. They play a crucial role in modeling heterogeneous network structures of many systems beyond the scope of classical mean-field models. It is well known that the graphon serves as a natural continuum limit of the graph sequence. Motivated by this, 7; 5; 3; 13; 4 quantitatively studied the convergence of finite particle systems towards their graphon counterparts by establishing results such as propagation of chaos (POC) 7; 13, law of large numbers (LLN) 7; 3; 13; 4, uniform-in-time convergence 7, and concentration bounds on the Wasserstein distance 5. Moreover, Caines and Huang 11 incorporated graphon into the MFG framework, proved a well-posedness result for the graphon mean-field games (GMFGs), and in particular, provided an $\epsilon$ -Nash result which relates the mean-field equilibrium to the equilibria of $N$ players on the graph sequence. We should also mention that an approximate Nash-equilibrium theory for GMFG was established in 20 under a label-state formulation, which defines different strengths of approximation under different regularities of the model. We refer to 2; 6; 28 for other works on GMFG, and 14; 15 for works on graphon mean-field type control (MFC) problems.

The main contributions of our work are two-fold. First, under some dissipativity condition, we are able to establish a convergence result for the Euler scheme of the graphon mean-field systems that are uniform in both time and label. This extends the results of (7, Section 6), where the interraction is on the drift coefficient and linear, and the convergence is for the graphon particle system. To consider the convex ordering, it is necessary that the graphon interraction occurs on the diffusion coefficients, and more importantly, the Euler scheme should be established for the graphon mean-field system. The reason is that the particle system does not propagate the convex order, see (22, Remark 5.1). Under a more general non-linear interraction formulation and suitable conditions on the coefficients, we are able to make the comparison of marginal and functional convex order for graphon mean-field systems. The results is obtained by combining the forward-backward inductions with convergence of the Euler schemes, and then applied to the study of a one-dimensional linear-quadratic (LQ) GMFG.

Second, we are able to study the graphon mean-field system of infinite horizon. Under the space $L^{2}_{K}(0,\infty;\mathbb{R}^{d})$ of $\mathbb{R}^{d}$ -valued adapted stochastic processes, equipped with the weighted $L^{2}$ -norm, defined by

\mathbb{E}\Big[\|x\|_{K}^{2}\Big]:=\mathbb{E}\bigg[\int_{0}^{\infty}\mathrm{e}^{-Kt}|x_{t}|^{2}\mathrm{d}t\bigg],

we are able to establish the trajectory-level convergence results for the Euler scheme, which is key to the functional convex ordering results in infinite horizon. This weighted space was also employed in 8 to study the infinite-horizon McKean-Vlasov FBSDEs for the purpose of solving the infinite-horizon MFC problems and MFGs.

The rest of the paper is organized as follows. We conclude Section 1 with notations used throughout the paper. In Section 2, we introduce the graphon mean-field dynamics and the associated well-posedness and Euler scheme. In Section 3, we provide all main results on convex ordering, namely the convex ordering of the Euler scheme and the functional convex ordering. Proofs related to well-posedness and Euler scheme are collected in Section 4. Finally, in Section 5, we apply our results to the comparison of the value functions of the graphon mean-field games.

1.1 Notations

1. Let $I:=[0,1]$ . The set $I$ represents the labels of a continuum of agents. A graphon $G$ is a symmetric measurable function from $I\times I$ to $I$ , associated with the cut-norm defined as:

\|G\|_{\square}:=\sup_{S,T\in\mathcal{B}(I)}\bigg|\int_{S\times T}G(u,v)\mathrm{d}u\mathrm{d}v\bigg|.

2. We denote $\mathcal{P}(E)$ , $\mathcal{P}_{p}(E)$ , $\mathcal{M}(E)$ , and $\mathcal{M}_{p}(E)$ respectively the collection of probability measures, probability measures with finite $p$ -th moment, finite positive measures, and finite positive measures with finite $p$ -th moment on $(E,|\cdot|_{E})$ . Also, we define the space $\mathcal{H}(E)\subset\mathcal{P}_{2}(E)^{I}$ by

	$\displaystyle\mathcal{H}(E):=\bigg\{$	$\displaystyle\bm{\mu}\in\mathcal{P}_{2}(E)^{I}\text{ s.t. }\forall B\in\mathcal{B}(E),\;\bm{\mu}(B):u\mapsto\mu^{u}(B)\text{ is $\lambda_{I}$-measurable }$
		$\displaystyle\text{and s.t. }\sup_{u\in I}\int_{E}\|x\|_{E}^{2}\mu^{u}(\mathrm{d}x)<\infty\bigg\}.$

3. The $p$ -Wasserstein distance $\mathcal{W}_{p}$ on $\mathcal{P}_{p}(E)$ is defined for any $\mu,\nu\in\mathcal{P}_{p}(E)$ :

\mathcal{W}_{p}(\mu,\nu):=\inf_{\pi\in\Pi(\mu,\nu)}\left(\int_{E\times E}|x-y|_{E}^{p}\pi(\mathrm{d}x,\mathrm{d}y)\right)^{1/p},

where $\Pi(\mu,\nu)$ denotes the set of probability measures in $\mathcal{P}(E\times E)$ with the first marginal $\mu$ and the second marginal $\nu$ .

Fix an arbitrary reference point $x_{0}\in E$ , the 2-Wasserstein On Positive measures ( $\mathcal{WOP}_{2}$ ) metric, firstly introduced in 21, is defined for any $\mu,\nu\in\mathcal{M}_{2}(E)$ by

	$\displaystyle\mathcal{WOP}_{2}^{2}(\mu,\nu)$	$\displaystyle=(m_{\mu}-m_{\nu})^{2}+\mathcal{W}_{2}^{2}(T_{m_{\mu}}\#\overline{\mu},T_{m_{\nu}}\#\overline{\nu})$
		$\displaystyle=(m_{\mu}-m_{\nu})^{2}+(m_{\mu}-m_{\nu})(M_{x_{0}}(\mu)-M_{x_{0}}(\nu))+m_{\mu}m_{\nu}\mathcal{W}_{2}^{2}(\overline{\mu},\overline{\nu}),$

where $m_{\mu}$ denotes the total mass of measure $\mu$ ; $\overline{\mu}:=\mu/m_{\mu}$ if $m_{\mu}>0$ and $\delta_{x_{0}}$ otherwise, where $\delta_{x_{0}}$ is the Dirac measure at $x_{0}$ ; $M_{x_{0}}(\mu):=\int_{E}|x-x_{0}|_{E}^{2}\mu(\mathrm{d}x)$ ; for $a>0$ , we define $T_{a}(x):=a(x-x_{0})+x_{0}$ , and $T\#\mu$ represents the pushforward of a measure $\mu$ by a measurable mapping $T$ .

The space $\mathcal{H}(E)$ is a metric space itself when endowed with the distance defined for any $\bm{\mu},\bm{\nu}\in\mathcal{P}_{2}(E)^{I}$ by

\bm{d}_{\mathcal{H}}(\bm{\mu},\bm{\nu}):=\sup_{u\in I}\mathcal{W}_{2}(\mu^{u},\nu^{u})=\sup_{u\in I}\bigg(\inf_{\pi\in\Pi(\mu^{u},\nu^{u})}\int_{E\times E}|x-y|_{E}^{2}\pi(\mathrm{d}x,\mathrm{d}y)\bigg)^{1/2}.

(1.1)

4. Let $\mathbb{M}^{d\times q}(\mathbb{R})$ denote the set of matrices with $d$ rows and $q$ columns equipped with the operator norm $\|\cdot\|$ defined by $\|A\|:=\sup_{|z|_{q}\leq 1}|Az|_{d}$ , for any $A\in\mathbb{M}^{d\times q}(\mathbb{R})$ , where $|\cdot|_{d}$ denotes the canonical Euclidean norm on $\mathbb{R}^{d}$ generated by the canonical inner product $\langle\cdot|\cdot\rangle$ . When there is no ambiguity, we write $|\cdot|$ instead of $|\cdot|_{d}$ for the Euclidean norm on $\mathbb{R}^{d}$ to simplify notation.

As (1.5) in 22, we define a partial order between two matrices in $\mathbb{M}^{d\times q}(\mathbb{R})$ as follows:

\forall A,B\in\mathbb{M}^{d\times q}(\mathbb{R}),\quad A\preceq B\quad\text{if $BB^{\top}-AA^{\top}$ is a positive semi-definite (PSD) matrix},

(1.2)

where $A^{\top}$ stands for the transpose of $A$ .
5. Given a graphon $G$ and a $\bm{\mu}\in\mathcal{H}(E)$ , we define the map $I\ni u\mapsto[G\bm{\mu}]^{u}\in\mathcal{M}_{2}(E)$ by

[G\bm{\mu}]^{u}(\mathrm{d}x):=\int_{I}G(u,v)\mu^{v}(\mathrm{d}x)\mathrm{d}v.

Similar to (3.2) in 13, we have for any $\bm{\mu},\bm{\nu}\in\mathcal{H}(E)$ that

\sup_{u\in I}\mathcal{WOP}^{2}_{2}([G\bm{\mu}]^{u},[G\bm{\nu}]^{u})\leq\int_{I}\mathcal{W}_{2}^{2}(\mu^{u},\nu^{u})\mathrm{d}u\leq\sup_{u\in I}\mathcal{W}^{2}_{2}(\mu^{u},\nu^{u}).

(1.3)

6. Denote by $\mathcal{C}_{\infty}(E):=\mathbb{C}([0,\infty),E)$ the space of continuous functions from $[0,\infty)$ to $E$ , equipped with the exponentially weighted $L^{2}$ -norm $\|\cdot\|_{K}$ , which is defined as

\|x\|^{2}_{K}:=\int_{0}^{\infty}\mathrm{e}^{-Kt}|x_{t}|_{E}^{2}\mathrm{d}t,\quad\text{for any $x:=\{x_{t}\}_{t\geq 0}\in C_{\infty}(E)$ and some $K\in\mathbb{R}^{*}_{+}$}.

(1.4)

In the case of $E=\mathbb{R}^{d}$ , we simply write $\mathcal{C}^{d}_{\infty}$ .
7. Let $\mathbb{C}_{cv}(E_{1},E_{2}):=\{\varphi:E_{1}\rightarrow E_{2}\text{ convex function}\}$ .
8. Denote by $\mathcal{C}_{\infty}(\mathcal{H}(E)):=\mathbb{C}([0,\infty),\mathcal{H}(E))$ the space of continuous functions from $[0,\infty)$ to $\mathcal{H}(E)$ , equipped with the distance $\bm{d}_{\mathcal{C}}$ defined as follows:

\bm{d}_{\mathcal{C}}(\bm{\mu},\bm{\nu}):=\sup_{t\geq 0}\bm{d}_{\mathcal{H}}(\bm{\mu}_{t},\bm{\nu}_{t}),\quad\text{for any $\bm{\mu},\;\bm{\nu}\in\mathcal{C}_{\infty}(\mathcal{H}(E)).$ }

(1.5)

9. For any $m_{1},m_{2}\in\mathbb{N}$ with $m_{1}\leq m_{2}$ , we denote by $x_{m_{1}:m_{2}}:=(x_{m_{1}},\cdots,x_{m_{2}})$ , and similarly, $\bm{\mu}_{m_{1}:m_{2}}:=(\bm{\mu}_{m_{1}},\cdots,\bm{\mu}_{m_{2}})$ . Moreover, we denote by $x_{0:\infty}:=(x_{0},x_{1},x_{2},\cdots)$ , and $\bm{\mu}_{0:\infty}:=(\bm{\mu}_{0},\bm{\mu}_{1},\bm{\mu}_{2},\cdots)$ .

2 Well-posedness and Euler scheme for the graphon mean-field systems

2.1 Assumptions and well-posedness

Given two graphons $G$ and $K$ , on some filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ , consider for any $u\in I$ the following two graphon mean-field systems:

		$\displaystyle X^{u}_{t}=X^{u}_{0}+\int_{0}^{t}b(s,X^{u}_{s})\mathrm{d}s+\int_{0}^{t}\sigma(s,X^{u}_{s},[G\bm{\mu}_{s}]^{u})\mathrm{d}B^{u}_{s},\quad t\geq 0,$		(2.1)
		$\displaystyle Y^{u}_{t}=Y^{u}_{0}+\int_{0}^{t}b(s,Y^{u}_{s})\mathrm{d}s+\int_{0}^{t}\theta(s,Y^{u}_{s},[K\bm{\nu}_{s}]^{u})\mathrm{d}B^{u}_{s},\quad t\geq 0,$		(2.2)

where we denote $\mu^{u}_{t}:=\mathcal{L}(X^{u}_{t})\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\bm{\mu}_{t}:=\{\mu^{u}_{t}\}_{u\in I}\in\mathcal{P}_{2}(\mathbb{R}^{d})^{I}$ , and similarly, $\nu^{u}_{t}:=\mathcal{L}(Y^{u}_{t})\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\bm{\nu}_{t}:=\{\nu^{u}_{t}\}_{u\in I}\in\mathcal{P}_{2}(\mathbb{R}^{d})^{I}$ ; $b\in\mathbb{R}^{d}$ and $\sigma,\theta\in\mathbb{M}^{d\times q}(\mathbb{R})$ are suitable functions; $\{B^{u}:u\in I\}$ are i.i.d. $q$ -dimensional Brownian motions, and $\{X^{u}_{0},Y^{u}_{0},B^{u}:u\in I\}$ are mutually independent.

Assumption 2.1.

(i) The functions $b,\;\sigma$ and $\theta$ are $\rho$ -Hölder continuous in $t$ , $\rho\in(0,1]$ , and Lipschitz continuous in $x$ and in $\mu$ in the following sense: for every $s\leq t$ , there exists a positive constant $\tilde{L}$ such that $\forall x\in\mathbb{R}^{d},\;\forall\mu\in\mathcal{M}_{2}(\mathbb{R}^{d})$

		$\displaystyle\|b(t,x)-b(s,x)\|\vee\\|\sigma(t,x,\mu)-\sigma(s,x,\mu)\\|\vee\\|\theta(t,x,\mu)-\theta(s,x,\mu)\\|$
		$\displaystyle\qquad\leq\tilde{L}\big(1+\|x\|+\mathcal{WOP}_{2}(\mu,\overline{\mu}\cdot\delta_{0})\big)(t-s)^{\rho};$

for every fixed $t\in\mathbb{R}_{+}$ , there exists $L>0$ such that $\forall x,y\in\mathbb{R}^{d},\;\forall\mu,\nu\in\mathcal{M}_{2}(\mathbb{R}^{d})$

\displaystyle|b(t,x)-b(t,y)|\vee\|\sigma(t,x,\mu)-\sigma(t,y,\nu)\|\vee\|\theta(t,x,\mu)-\theta(t,y,\nu)\|\leq L\big(|x-y|+\mathcal{WOP}_{2}(\mu,\nu)\big).

(ii) The map $I\ni u\mapsto\mu^{u}_{0}\in\mathcal{P}(\mathbb{R}^{d})$ is measurable, and $\sup_{u\in I}\mathbb{E}[|X^{u}_{0}|^{2}]<\infty$ $(\textit{idem for }\{\nu^{u}_{0}\}_{u\in I})$ .

(iii) Dissipativity: There exists some $c_{0}\in(0,\infty)$ such that

(x_{1}-x_{2})\cdot(b(t,x_{1})-b(t,x_{2}))\leq-c_{0}|x_{1}-x_{2}|^{2},\quad\forall t\in\mathbb{R}_{+},\;\forall x_{1},x_{2}\in\mathbb{R}^{d},

(2.3)

and

\kappa_{1}:=c_{0}-8L^{2}q>0.

(2.4)

(iv) In addition, suppose that

\kappa_{2}:=\sup_{t\geq 0}\big|b(t,0)\big|\vee\sup_{t\geq 0}\sup_{u\in I}\big\|\sigma(t,0,[G\bm{\delta}_{0}]^{u})\big\|\vee\sup_{t\geq 0}\sup_{u\in I}\big\|\theta(t,0,[K\bm{\delta}_{0}]^{u})\big\|<\infty,

(2.5)

where $\bm{\delta}_{0}:=\{\delta^{u}_{0}\}_{u\in I}$ and $\delta^{u}_{0}=\delta_{0}$ , $\forall u\in I$ .

Assumption 2.2.

(i) The function $b$ is affine in $x$ .

(ii) For every fixed $t\in\mathbb{R}_{+}$ and $\mu\in\mathcal{M}_{2}(\mathbb{R}^{d})$ , the function $x\mapsto\sigma(t,x,\mu)$ is convex in the sense that

\forall x,y\in\mathbb{R}^{d},\;\forall\lambda\in[0,1],\quad\sigma(t,\lambda x+(1-\lambda)y,\mu)\preceq\lambda\sigma(t,x,\mu)+(1-\lambda)\sigma(t,y,\mu).

(iii) For every fixed $(t,u,x)\in\mathbb{R}_{+}\times I\times\mathbb{R}^{d}$ , the function $\bm{\mu}\mapsto\sigma(t,x,[G\bm{\mu}]^{u})$ is nondecreasing with respect to the convex order, namely,

\forall\bm{\mu},\bm{\nu}\in\mathcal{H}(\mathbb{R}^{d})\text{ satisfying }\mu^{u}\preceq_{cv}\nu^{u},\;\forall u\in I\;\Longrightarrow\;\sigma(t,x,[G\bm{\mu}]^{u})\preceq\sigma(t,x,[G\bm{\nu}]^{u}).

(iv) For every $(t,u,x,\bm{\mu})\in\mathbb{R}_{+}\times I\times\mathbb{R}^{d}\times\mathcal{H}(\mathbb{R}^{d})$ , we have

\sigma(t,x,[G\bm{\mu}]^{u})\preceq\theta(t,x,[K\bm{\mu}]^{u}).

(v) $X^{u}_{0}\preceq_{cv}Y^{u}_{0},\;\forall u\in I$ .

Example 2.1.

Graphon mean-field systems (2.1)-(2.2) are nonlinear with respect to the measures, which is a more general setting than the scalar framework adopted in 7; 3; 6. Under the linear interaction assumption with $d=q=1$ , i.e.

		$\displaystyle\mathrm{d}X^{u}_{t}=b(t,X^{u}_{t})\mathrm{d}t+\int_{I}\int_{\mathbb{R}}G(u,v)\sigma(t,X^{u}_{t},z)\mu^{v}_{t}(\mathrm{d}z)\mathrm{d}v\mathrm{d}B^{u}_{t},\quad\text{with }\mu^{u}_{t}=\mathcal{L}(X^{u}_{t}),$
		$\displaystyle\mathrm{d}Y^{u}_{t}=b(t,Y^{u}_{t})\mathrm{d}t+\int_{I}\int_{\mathbb{R}}K(u,v)\theta(t,Y^{u}_{t},z)\nu^{v}_{t}(\mathrm{d}z)\mathrm{d}v\mathrm{d}B^{u}_{t},\quad\text{with }\nu^{u}_{t}=\mathcal{L}(Y^{u}_{t}),$

Assumption 2.2 (ii)-(iv) can be implied by the following conditions:

$\bullet$ for every $t\in\mathbb{R}_{+}$ , the function $\mathbb{R}\times\mathbb{R}\ni(x,z)\mapsto\sigma(t,x,z)\in\mathbb{R}_{+}$ is convex.

$\bullet$ for every $(u,t,x)\in I\times\mathbb{R}_{+}\times\mathbb{R}$ , we have for any $(v_{1},v_{2},z_{1},z_{2})\in I^{2}\times\mathbb{R}^{2}$ ,

G(u,v_{1})G(u,v_{2})\sigma(t,x,z_{1})\sigma(t,x,z_{2})\leq K(u,v_{1})K(u,v_{2})\theta(t,x,z_{1})\theta(t,x,z_{2}).

To see this, Assumption 2.2 (ii) follows directly from the first point, and (iii) is satisfied due to the non-negativity as well as the convexity of $\sigma$ , i.e.

0\leq\int_{I}G(u,v)\bigg(\int_{\mathbb{R}}\sigma(t,X^{u}_{t},z)\mu^{v}_{t}(\mathrm{d}z)\bigg)\mathrm{d}v\leq\int_{I}G(u,v)\bigg(\int_{\mathbb{R}}\sigma(t,X^{u}_{t},z)\nu^{v}_{t}(\mathrm{d}z)\bigg)\mathrm{d}v,

which verifies (iii) by noticing that

\bigg[\int_{I}G(u,v)\bigg(\int_{\mathbb{R}}\sigma(t,X^{u}_{t},z)\mu^{v}_{t}(\mathrm{d}z)\bigg)\mathrm{d}v\bigg]^{2}\leq\bigg[\int_{I}G(u,v)\bigg(\int_{\mathbb{R}}\sigma(t,X^{u}_{t},z)\nu^{v}_{t}(\mathrm{d}z)\bigg)\mathrm{d}v\bigg]^{2}.

Based on the second point, a straightforward calculation shows that Assumption 2.2 (iv) holds.

The following lemma gives well-posedness of systems (2.1) and (2.2), which extends the existing results, see Proposition 2.1 in 3 or Proposition 3.1 in 13, from finite-horizon to infinite-horizon.

Lemma 2.3.

Suppose that Assumption 2.1 holds. Then, there exists a unique pathwise solution $\{X^{u}\}_{u\in I}$ to the graphon mean-field system (2.1). In addition, $\sup_{u\in I}\mathbb{E}\big[\|X^{u}\|^{2}_{K}\big]<\infty$ holds, and the map $I\ni u\mapsto\mu^{u}:=\mathcal{L}(X^{u})\in\mathcal{P}_{2}(\mathcal{C}^{d}_{\infty})$ is measurable $(\textit{idem for $\{Y^{u}\}_{u\in I}$})$ .

2.2 Euler scheme

In this section, we analyze the Euler scheme of systems (2.1)-(2.2) with step size $h>0$ . For $m\in\mathbb{N}$ , we define $t^{h}_{m}:=m\cdot h$ . Without ambiguity, we will write $t_{m}$ instead of $t_{m}^{h}$ . Let $Z^{u}_{m+1}:=\frac{1}{\sqrt{h}}(B^{u}_{t_{m+1}}-B^{u}_{t_{m}})$ be i.i.d. random variables with probability distribution $\mathcal{N}(0,\mathbf{I}_{q})$ . The Euler schemes of equations (2.1) and (2.2) are defined by

		$\displaystyle\bar{X}^{u,h}_{t_{m+1}}=\bar{X}^{u,h}_{t_{m}}+h\cdot b(t_{m},\bar{X}^{u,h}_{t_{m}})+\sqrt{h}\cdot\sigma(t_{m},\bar{X}^{u,h}_{t_{m}},[G\bar{\bm{\mu}}^{h}_{t_{m}}]^{u})Z^{u}_{m+1},\quad\bar{X}^{u,h}_{0}=X^{u}_{0},$		(2.6)
		$\displaystyle\bar{Y}^{u,h}_{t_{m+1}}=\bar{Y}^{u,h}_{t_{m}}+h\cdot b(t_{m},\bar{Y}^{u,h}_{t_{m}})+\sqrt{h}\cdot\theta(t_{m},\bar{Y}^{u,h}_{t_{m}},[K\bar{\bm{\nu}}^{h}_{t_{m}}]^{u})Z^{u}_{m+1},\quad\bar{Y}^{u,h}_{0}=Y^{u}_{0},$		(2.7)

where $\bar{\mu}^{u,h}_{t_{m}}:=\mathcal{L}(\bar{X}^{u,h}_{t_{m}})$ and $\bar{\nu}^{u,h}_{t_{m}}:=\mathcal{L}(\bar{Y}^{u,h}_{t_{m}})$ ; $\bar{\bm{\mu}}^{h}_{t_{m}}$ and $\bar{\bm{\nu}}^{h}_{t_{m}}$ are respectively the measure ensembles of $\{\bar{\mu}^{u,h}_{t_{m}}\}_{u\in I}$ and $\{\bar{\nu}^{u,h}_{t_{m}}\}_{u\in I}$ . Moreover, we can naturally extend the Euler scheme to continuous time as below (denoting $\bar{X}^{u,h}$ , $\bar{Y}^{u,h}$ the corresponding processes): for every $t\in[t_{m},t_{m+1})$ ,

		$\displaystyle\bar{X}^{u,h}_{t}=\bar{X}^{u,h}_{t_{m}}+b(t_{m},\bar{X}^{u,h}_{t_{m}})(t-t_{m})+\sigma(t_{m},\bar{X}^{u,h}_{t_{m}},[G\bar{\bm{\mu}}^{h}_{t_{m}}]^{u})(B^{u}_{t}-B^{u}_{t_{m}}),$		(2.8)
		$\displaystyle\bar{Y}^{u,h}_{t}=\bar{Y}^{u,h}_{t_{m}}+b(t_{m},\bar{Y}^{u,h}_{t_{m}})(t-t_{m})+\theta(t_{m},\bar{Y}^{u,h}_{t_{m}},[K\bar{\bm{\nu}}^{h}_{t_{m}}]^{u})(B^{u}_{t}-B^{u}_{t_{m}}).$		(2.9)

The following Proposition 2.4 is the convergence results of the Euler scheme. It is valid for both processes $X$ and $Y$ , and its proof is postponed to Section 4.

Proposition 2.4.

Assume that Assumption 2.1 is in force.

(i)

There exist $h_{0},C>0$ such that

\sup_{u\in I}\mathbb{E}\Big[\big\|X^{u}\big\|_{K}^{2}\Big]\vee\sup_{h\in(0,h_{0})}\sup_{u\in I}\mathbb{E}\Big[\big\|\bar{X}^{u,h}\big\|_{K}^{2}\Big]<C.

Moreover, there exists a constant $\kappa$ such that for any $h\in(0,h_{0})$ ,

\sup_{t\geq 0}\sup_{u\in I}\mathbb{E}\Big[\big|\bar{X}^{u,h}_{t}-\bar{X}^{u,h}_{[t]^{h}}\big|^{2}\Big]\leq\kappa(t-[t]^{h})\leq\kappa h,

where we used the notation $[t]^{h}:=\lfloor\frac{t}{h}\rfloor\cdot h$ .

(ii)

Recall $h_{0}>0$ defined in (i). Then, there exists $\gamma>0$ such that for any $h\in(0,h_{0})$ ,

$\sup_{u\in I}\mathbb{E}\Big[\big\|X^{u}-\bar{X}^{u,h}\big\|_{K}^{2}\Big]\leq\gamma h^{1\wedge 2\rho}.$

Notice from Lemma 4.2 (i) that $\{\bar{\mu}^{u,h}_{t}\}_{t\geq 0}:=\{\mathcal{L}(\bar{X}^{u,h}_{t})\}_{t\geq 0}\in\mathbb{C}([0,\infty),\mathcal{P}_{2}(\mathbb{R}^{d})).$ In addition, we have the following property, which justifies that the graphon-weighted measure, i.e. $[G\bar{\bm{\mu}}^{h}_{t_{m}}]^{u}$ or $[K\bar{\bm{\nu}}^{h}_{t_{m}}]^{u}$ , is well-defined.

Lemma 2.5.

The map $I\ni u\mapsto\{\bar{\mu}^{u,h}_{t}\}_{t\geq 0}:=\{\mathcal{L}(\bar{X}^{u,h}_{t})\}_{t\geq 0}\in\mathbb{C}([0,\infty),\mathcal{P}_{2}(\mathbb{R}^{d}))$ is measurable $($ idem for $\{\bar{\nu}^{u,h}_{t}\}_{t\geq 0}$ $)$ .

The proof of Lemma 2.5 is also postponed to Section 4.

3 Main results

3.1 Convex order results for Euler schemes

In this section, we will establish the convex ordering results for the random variables $\bar{X}^{u,h}_{t_{m}}$ and $\bar{Y}^{u,h}_{t_{m}}$ , $m\in\mathbb{N}$ , defined by the Euler schemes (2.6) and (2.7). In order to simplify the notation, we rewrite (2.6) and (2.7) by setting

\bar{X}^{u}_{m}:=\bar{X}^{u,h}_{t_{m}},\qquad\bar{Y}^{u}_{m}:=\bar{Y}^{u,h}_{t_{m}},\qquad\bar{\mu}^{u}_{m}:=\bar{\mu}^{u,h}_{t_{m}},\qquad\bar{\nu}^{u}_{m}:=\bar{\nu}^{u,h}_{t_{m}}.

Then, we have

		$\displaystyle\bar{X}^{u}_{m+1}=b_{m}(\bar{X}^{u}_{m})+\sigma^{u}_{m}(\bar{X}^{u}_{m},\bar{\bm{\mu}}_{m})Z^{u}_{m+1},\quad\bar{X}^{u}_{0}=X^{u}_{0},\quad\text{where }\bar{\bm{\mu}}_{m}:=\{\bar{\mu}^{u}_{m}\}_{u\in I},$		(3.1)
		$\displaystyle\bar{Y}^{u}_{m+1}=b_{m}(\bar{Y}^{u}_{m})+\theta^{u}_{m}(\bar{Y}^{u}_{m},\bar{\bm{\nu}}_{m})Z^{u}_{m+1},\quad\bar{Y}^{u}_{0}=Y^{u}_{0},\quad\text{where }\bar{\bm{\nu}}_{m}:=\{\bar{\nu}^{u}_{m}\}_{u\in I},$		(3.2)

where for every $m\in\mathbb{N}$ , we define:

		$\displaystyle b_{m}(x):=x+h\cdot b(t_{m},x),\qquad\sigma^{u}_{m}(x,\bm{{\mu}}):=\sqrt{h}\cdot\sigma(t_{m},x,[G\bm{\mu}]^{u}),$
		$\displaystyle\theta^{u}_{m}(x,\bm{{\mu}}):=\sqrt{h}\cdot\theta(t_{m},x,[K\bm{\mu}]^{u}).$

First, we notice that it follows from Lemma 4.2 (i) and Lemma 2.5 that $\bar{\bm{\mu}}_{m}\in\mathcal{H}(\mathbb{R}^{d})$ for any $m=0,1,2,\cdots$ . Then, by Assumption 2.2, $\bar{X}^{u}_{0},\;\bar{Y}^{u}_{0},\;b_{m},\;\sigma^{u}_{m}\;\text{and }\theta^{u}_{m}$ , for any $m\in\mathbb{N}$ and $u\in I$ , satisfy the discrete-time counterpart.

Assumption 3.1.

(i) The function $b_{m}$ is affine in $x$ .

(ii) The function $\sigma^{u}_{m}$ is convex in $x$ in the sense that

\forall x,y\in\mathbb{R}^{d},\;\forall\lambda\in[0,1],\quad\sigma^{u}_{m}(\lambda x+(1-\lambda)y,\bm{\mu})\preceq\lambda\sigma^{u}_{m}(x,\bm{\mu})+(1-\lambda)\sigma^{u}_{m}(y,\bm{\mu}).

(iii) The function $\sigma^{u}_{m}$ is nondecreasing in $\bm{\mu}$ with respect to the convex order:

\forall\bm{\mu},\bm{\nu}\in\mathcal{H}(\mathbb{R}^{d})\text{ satisfying }\mu^{u}\preceq_{cv}\nu^{u},\;\forall u\in I\;\Longrightarrow\;\sigma^{u}_{m}(x,\bm{\mu})\preceq\sigma^{u}_{m}(x,\bm{\nu}).

(iv) We have the following order between $\sigma^{u}_{m}$ and $\theta^{u}_{m}$ :

\forall(x,\bm{\mu})\in\mathbb{R}^{d}\times\mathcal{H}(\mathbb{R}^{d}),\quad\sigma^{u}_{m}(x,\bm{\mu})\preceq\theta^{u}_{m}(x,\bm{\mu}).

(v) $\bar{X}^{u}_{0}\preceq_{cv}\bar{Y}^{u}_{0},\;\forall u\in I$ .

Let us recall an elementary characterization of convex order between two integrable $\mathbb{R}^{d}$ -valued random variables or their distributions, see for instance, (22, Lemma 4.1), or similarly (1, Lemma A.1). This characterization allows us to restrict test functions from all convex functions to those with linear growth.

Lemma 3.2.

Let $\mu,\nu\in\mathcal{P}_{1}(\mathbb{R}^{d})$ . Then, we have $\mu\preceq_{cv}\nu$ if and only if

\forall\varphi:\mathbb{R}^{d}\rightarrow\mathbb{R}\text{ convex and such that }\sup_{x\in\mathbb{R}^{d}}\frac{|\varphi(x)|}{1+|x|}<\infty,\;\int_{\mathbb{R}^{d}}\varphi(x)\mu(\mathrm{d}x)\leq\int_{\mathbb{R}^{d}}\varphi(y)\nu(\mathrm{d}y).

3.1.1 Single-marginal case

For every $m\in\mathbb{N}$ and $u\in I$ , we define an operator $Q^{u}_{m+1}:\mathbb{C}_{cv}(\mathbb{R}^{d},\mathbb{R})\rightarrow\mathbb{C}(\mathbb{R}^{d}\times\mathbb{M}^{d\times q}(\mathbb{R}),\mathbb{R})$ associated with $Z^{u}_{m+1}$ by

\displaystyle\mathbb{R}^{d}\times\mathbb{M}^{d\times q}(\mathbb{R})\ni(x,e)\longmapsto

\displaystyle\big(Q^{u}_{m+1}\varphi\big)(x,e):=\mathbb{E}\Big[\varphi\big(b_{m}(x)+eZ^{u}_{m+1}\big)\Big]\in\mathbb{R}.

(3.3)

For every $m\in\mathbb{N}^{*}$ , let $\mathcal{F}_{m}$ denote the $\sigma$ -algebra generated by $\{X^{u}_{0},Y^{u}_{0},Z^{u}_{1},\cdots,Z^{u}_{m}\}_{u\in I}$ , and in particular, let $\mathcal{F}_{0}$ denote the $\sigma$ -algebra generated by $\{X^{u}_{0},Y^{u}_{0}\}_{u\in I}$ , and let $\mathcal{F}_{\infty}$ denote the $\sigma$ -algebra generated by $\{X^{u}_{0},Y^{u}_{0},B^{u}\}_{u\in I}$ . We can then obtain the following single-marginal result.

Proposition 3.3.

Let $\{\bar{X}^{u}_{m}\}$ and $\{\bar{Y}^{u}_{m}\}$ be random variables defined by (3.1) and (3.2). Under Assumption 3.1, we have

\bar{X}^{u}_{m}\preceq_{cv}\bar{Y}^{u}_{m},\quad\forall m\in\mathbb{N},\;\forall u\in I.

Proposition 3.3 relies on the following lemma whose proof is similar to (22, Proofs of Lemmas 4.2 and 4.3), hence is omitted.

Lemma 3.4.

Let $\varphi\in\mathbb{C}_{cv}(\mathbb{R}^{d},\mathbb{R})$ with linear growth. Then, for every $m\in\mathbb{N}^{*}$ and $u\in I$ :

(i) the function $(x,e)\mapsto(Q^{u}_{m}\varphi)(x,e)$ is (finite and) convex;

(ii) for any fixed $x\in\mathbb{R}^{d}$ , the function $e\mapsto(Q^{u}_{m}\varphi)(x,e)$ attains its minimum at $\mathbf{0}^{d\times q}$ , where $\mathbf{0}^{d\times q}$ is the zero-matrix of size $d\times q$ ;

(iii) for any fixed $x\in\mathbb{R}^{d}$ , the function $e\mapsto(Q^{u}_{m}\varphi)(x,e)$ is nondecreasing with respect to the partial order of $d\times q$ matrix, defined in (1.2);

(iv) for any $\bm{\mu}\in\mathcal{H}(\mathbb{R}^{d})$ , the function $x\mapsto\mathbb{E}\big[\varphi\big(b_{m}(x)+\sigma^{u}_{m}(x,\bm{\mu})Z^{u}_{m+1}\big)\big]$ is convex with linear growth.

Proof of Proposition 3.3.

Assumption 3.1 directly implies $\bar{X}^{u}_{0}\preceq_{cv}\bar{Y}^{u}_{0}$ , $\forall u\in I$ . Assume $\bar{X}^{u}_{m}\preceq_{cv}\bar{Y}^{u}_{m}$ , $\forall u\in I$ , or equivalently, $\bar{\mu}^{u}_{m}\preceq_{cv}\bar{\nu}^{u}_{m}$ , $\forall u\in I$ . Let $\varphi\in\mathbb{C}_{cv}(\mathbb{R}^{d},\mathbb{R})$ with linear growth. Then, for any $u\in I$ :

	$\displaystyle\mathbb{E}\Big[\varphi\big(\bar{X}^{u}_{m+1}\big)\Big]$	$\displaystyle=\mathbb{E}\Big[\varphi\big(b_{m}(\bar{X}^{u}_{m})+\sigma^{u}_{m}(\bar{X}^{u}_{m},\bar{\bm{\mu}}_{m})Z^{u}_{m+1}\big)\Big]$
		$\displaystyle=\mathbb{E}\Big[\mathbb{E}\Big[\varphi\big(b_{m}(\bar{X}^{u}_{m})+\sigma^{u}_{m}(\bar{X}^{u}_{m},\bar{\bm{\mu}}_{m})Z^{u}_{m+1}\big)\Big\|\mathcal{F}_{m}\Big]\Big]$
		$\displaystyle=\int_{\mathbb{R}^{d}}\bar{\mu}^{u}_{m}(\mathrm{d}x)\mathbb{E}\Big[\varphi\big(b_{m}(x)+\sigma^{u}_{m}(x,\bar{\bm{\mu}}_{m})Z^{u}_{m+1}\big)\Big]$
		(the integrability is due to Lemma 4.2 (i) and Lemma 3.4 (iv))
		$\displaystyle\leq\int_{\mathbb{R}^{d}}\bar{\mu}^{u}_{m}(\mathrm{d}x)\mathbb{E}\Big[\varphi\big(b_{m}(x)+\sigma^{u}_{m}(x,\bar{\bm{\nu}}_{m})Z^{u}_{m+1}\big)\Big]$
		(by Assumption 3.1 (iii) and Lemma 3.4 (iii), since $\bar{\mu}^{u}_{m}\preceq_{cv}\bar{\nu}^{u}_{m}$ , $\forall u\in I$ )
		$\displaystyle\leq\int_{\mathbb{R}^{d}}\bar{\nu}^{u}_{m}(\mathrm{d}x)\mathbb{E}\Big[\varphi\big(b_{m}(x)+\sigma^{u}_{m}(x,\bar{\bm{\nu}}_{m})Z^{u}_{m+1}\big)\Big]$
		(by Lemma 3.4 (iv), since $\bar{\mu}^{u}_{m}\preceq_{cv}\bar{\nu}^{u}_{m}$ )
		$\displaystyle\leq\int_{\mathbb{R}^{d}}\bar{\nu}^{u}_{m}(\mathrm{d}x)\mathbb{E}\Big[\varphi\big(b_{m}(x)+\theta^{u}_{m}(x,\bar{\bm{\nu}}_{m})Z^{u}_{m+1}\big)\Big]$
		(by Assumption 3.1 (iv) and Lemma 3.4 (iii))
		$\displaystyle=\mathbb{E}\Big[\varphi\big(\bar{Y}^{u}_{m+1}\big)\Big].$

Thus, $\bar{X}^{u}_{m+1}\preceq_{cv}\bar{Y}^{u}_{m+1}$ by applying Lemma 3.2. One concludes the proof by a forward induction. ∎

3.1.2 Multi-marginal case

Fix $T>0$ and define $M:=\frac{T}{h}$ if $\frac{T}{h}\in\mathbb{N}$ , otherwise $M:=\lfloor\frac{T}{h}\rfloor+1$ . For every $u\in I$ , we recursively define a sequence of functions

\Phi^{u}_{m}:(\mathbb{R}^{d})^{m+1}\times(\mathcal{P}_{2}(\mathbb{R}^{d})^{I})^{\infty}\longrightarrow\mathbb{R},\quad m=\mathbb{N}\cup\{\infty\},

in a backward way as follows:
$\bullet$ Set for $m\geq M$ or $m=\infty$ :

\Phi^{u}_{m}\big(x_{0:M-1},\underbrace{x_{M},\cdots,x_{M}}_{\text{\#: $m-M+1$}};\bm{\mu}_{0:\infty}\big)=F\big(x_{0:M-1},\underbrace{x_{M},x_{M},\cdots}_{\text{\#: $\infty$}}\big),

(3.4)

where $F:(\mathbb{R}^{d})^{\infty}\rightarrow\mathbb{R}$ is a convex function with quadratic growth in the sense that

\exists C>0,\;\forall x:=x_{0:\infty}\in(\mathbb{R}^{d})^{\infty}\text{ such that }|F(x)|\leq C\Big(1+\|x\|^{2}_{K}\Big),

(3.5)

where $\|x\|^{2}_{K}$ is defined as follows:

\|x\|^{2}_{K}:=\sum_{m=0}^{\infty}\mathrm{e}^{-Kt_{m}}\frac{|x_{m}|^{2}+|x_{m+1}|^{2}}{2}\Delta t_{m}=\sum_{m=0}^{\infty}\mathrm{e}^{-Kt_{m}}\frac{|x_{m}|^{2}+|x_{m+1}|^{2}}{2}h

with $t_{m}:=m\cdot h$ and $\Delta t_{m}:=t_{m+1}-t_{m}$ .
$\bullet$ Set for $m<M$ :

	$\displaystyle\Phi^{u}_{m}\big(x_{0:m};\bm{\mu}_{0:\infty}\big)$	$\displaystyle=\Big(Q^{u}_{m+1}\Phi^{u}_{m+1}\big(x_{0:m},\cdot;\bm{\mu}_{0:\infty}\big)\Big)\big(x_{m},\sigma^{u}_{m}(x_{m},\bm{\mu}_{m})\big)$
		$\displaystyle=\mathbb{E}\Big[\Phi^{u}_{m+1}\big(x_{0:m},b_{m}(x_{m})+\sigma^{u}_{m}(x_{m},\bm{\mu}_{m})Z^{u}_{m+1};\bm{\mu}_{0:\infty}\big)\Big].$		(3.6)

The functions $\{\Phi^{u}_{m}\}$ share the following properties.

Lemma 3.5 (Lemma 4.4 in 22).

For every $m\in\mathbb{N}\cup\{\infty\}$ and $u\in I$ :

(i) For a fixed $\bm{\mu}_{0:\infty}\in(\mathcal{P}_{2}(\mathbb{R}^{d})^{I})^{\infty}$ , the function $\Phi^{u}_{m}(\cdot;\bm{\mu}_{0:\infty})$ is convex and has a quadratic growth in $x_{0:m}$ so that $\Phi^{u}_{m}$ is well-defined.

(ii) For a fixed $x_{0:m}\in(\mathbb{R}^{d})^{m+1}$ , the function $\Phi^{u}_{m}(x_{0:m};\cdot)$ is nondecreasing in $\bm{\mu}_{0:\infty}$ with respect to the convex order in the sense that for any $\bm{\mu}_{0:\infty},\bm{\nu}_{0:\infty}\in(\mathcal{P}_{2}(\mathbb{R}^{d})^{I})^{\infty}$ with $\mu^{u}_{j}\preceq_{cv}\nu^{u}_{j},\;\forall j\in\mathbb{N},\;\forall u\in I$ ,

\Phi^{u}_{m}\big(x_{0:m};\bm{\mu}_{0:\infty}\big)\leq\Phi^{u}_{m}\big(x_{0:m};\bm{\nu}_{0:\infty}\big).

Proof.

(i) For every $m\geq M$ or $m=\infty$ , the function $\Phi^{u}_{m}$ is convex due to the convexity of $F$ . Suppose that the map $x_{0:m+1}\mapsto\Phi^{u}_{m+1}(x_{0:m+1};\bm{\mu}_{0:\infty})$ is convex for some $m\leq M-1$ . For any $x_{0:m},y_{0:m}\in(\mathbb{R}^{d})^{m+1}$ and $\lambda\in[0,1]$ , we have

		$\displaystyle\Phi^{u}_{m}\big(\lambda x_{0:m}+(1-\lambda)y_{0:m};\bm{\mu}_{0:\infty}\big)$
		$\displaystyle\quad=\mathbb{E}\Big[\Phi^{u}_{m+1}\big(\lambda x_{0:m}+(1-\lambda)y_{0:m},b_{m}\big(\lambda x_{m}+(1-\lambda)y_{m}\big)$
		$\displaystyle\qquad+\sigma^{u}_{m}\big(\lambda x_{m}+(1-\lambda)y_{m},\bm{\mu}_{m}\big)Z^{u}_{m+1};\bm{\mu}_{0:\infty}\big)\Big]$
		$\displaystyle\quad\leq\mathbb{E}\Big[\Phi^{u}_{m+1}\big(\lambda x_{0:m}+(1-\lambda)y_{0:m},\lambda b_{m}(x_{m})+(1-\lambda)b_{m}(y_{m})$
		$\displaystyle\qquad+\big(\lambda\sigma^{u}_{m}\big(x_{m},\bm{\mu}_{m}\big)+(1-\lambda)\sigma^{u}_{m}\big(y_{m},\bm{\mu}_{m}\big)\big)Z^{u}_{m+1};\bm{\mu}_{0:\infty}\big)\Big]$
		(by Assumption 3.1 (ii) and Lemma 3.4 (iii), since $\Phi^{u}_{m+1}(x_{0:m},\cdot;\bm{\mu}_{0:\infty})$ is convex)
		$\displaystyle\quad\leq\lambda\mathbb{E}\Big[\Phi^{u}_{m+1}\big(x_{0:m},b_{m}(x_{m})+\sigma^{u}_{m}\big(x_{m},\bm{\mu}_{m}\big)Z^{u}_{m+1};\bm{\mu}_{0:\infty}\big)\Big]$
		$\displaystyle\qquad+(1-\lambda)\mathbb{E}\Big[\Phi^{u}_{m+1}\big(y_{0:m},b_{m}(y_{m})+\sigma^{u}_{m}\big(y_{m},\bm{\mu}_{m}\big)Z^{u}_{m+1};\bm{\mu}_{0:\infty}\big)\Big]$
		$\displaystyle\quad=\lambda\Phi^{u}_{m}\big(x_{0:m};\bm{\mu}_{0:\infty}\big)+(1-\lambda)\Phi^{u}_{m}\big(y_{0:m};\bm{\mu}_{0:\infty}\big).$

Thus, the function $\Phi^{u}_{m}(\cdot;\bm{\mu}_{0:\infty})$ is convex, and one completes the proof by a backward induction. If $t_{M}>T$ , then for the case of $m=M-1$ , one should replace $b_{m}$ , $\sigma^{u}_{m}$ , $\theta^{u}_{m}$ and $Z^{u}_{m+1}$ in the above derivation by

		$\displaystyle b_{m}(x):=x+(T-t_{M-1})\cdot b(t_{m},x),\quad Z^{u}_{T}:=\frac{B^{u}_{T}-B^{u}_{t_{M-1}}}{\sqrt{T-t_{M-1}}}\sim\mathcal{N}(0,\bm{1}_{q}),$		(3.7)
		$\displaystyle\sigma^{u}_{m}(x,\bm{\mu}):=\sqrt{T-t_{M-1}}\cdot\sigma(t_{m},x,[G\bm{\mu}]^{u}),\quad\theta^{u}_{m}(x,\bm{\mu}):=\sqrt{T-t_{M-1}}\cdot\theta(t_{m},x,[K\bm{\mu}]^{u}).$		(3.8)

For every $m\geq M$ or $m=\infty$ , it is obvious that $\Phi^{u}_{m}$ has a quadratic growth in the sense of (3.5). Suppose that $\Phi^{u}_{m+1}$ has a quadratic growth for some $m\leq M-1$ . Then, $\Phi^{u}_{m}$ also has a quadratic growth as $b_{m}$ and $\sigma^{u}_{m}$ have a linear growth by Assumption 2.1, and one concludes by a backward induction.

(ii) Firstly, notice that for any $\bm{\mu}_{0:\infty},\bm{\nu}_{0:\infty}\in(\mathcal{P}_{2}(\mathbb{R}^{d})^{I})^{\infty}$ with $\mu^{u}_{j}\preceq_{cv}\nu^{u}_{j},\;\forall j\in\mathbb{N},\;\forall u\in I$ , we have for $m\geq M$ or $m=\infty$ ,

\Phi^{u}_{m}\big(x_{0:M-1},\underbrace{x_{M},\cdots,x_{M}}_{\text{\#: $m-M+1$}};\bm{\mu}_{0:\infty}\big)=F\big(x_{0:M-1},\underbrace{x_{M},x_{M},\cdots}_{\text{\#: $\infty$}}\big)=\Phi^{u}_{m}\big(x_{0:M-1},\underbrace{x_{M},\cdots,x_{M}}_{\text{\#: $m-M+1$}};\bm{\nu}_{0:\infty}\big).

Next assume that $\Phi^{u}_{m+1}(x_{0:m+1};\cdot)$ is nondecreasing with respect to the convex order of $\bm{\mu}_{0:\infty}$ for some $m\leq M-1$ . Then, we obtain that

	$\displaystyle\Phi^{u}_{m}\big(x_{0:m};\bm{\mu}_{0:\infty}\big)$	$\displaystyle=\mathbb{E}\Big[\Phi^{u}_{m+1}\big(x_{0:m},b_{m}(x_{m})+\sigma^{u}_{m}(x_{m},\bm{\mu}_{m})Z^{u}_{m+1};\bm{\mu}_{0:\infty}\big)\Big]$
		$\displaystyle\leq\mathbb{E}\Big[\Phi^{u}_{m+1}\big(x_{0:m},b_{m}(x_{m})+\sigma^{u}_{m}(x_{m},\bm{\nu}_{m})Z^{u}_{m+1};\bm{\mu}_{0:\infty}\big)\Big]$
		(by Assumption 3.1 and Lemma 3.4, since $\Phi^{u}_{m+1}(x_{0:m},\cdot;\bm{\mu}_{0:\infty})$ is convex)
		$\displaystyle\leq\mathbb{E}\Big[\Phi^{u}_{m+1}\big(x_{0:m},b_{m}(x_{m})+\sigma^{u}_{m}(x_{m},\bm{\nu}_{m})Z^{u}_{m+1};\bm{\nu}_{0:\infty}\big)\Big]=\Phi^{u}_{m}\big(x_{0:m};\bm{\nu}_{0:\infty}\big).$

One completes proof by a backward induction. If $t_{M}>T$ , then for the case of $m=M-1$ , one should modify the above derivation as in (3.7)-(3.8). ∎

As $F$ has a quadratic growth in the sense of (3.5), the integrability of $F(\bar{X}^{u}_{t_{0}:t_{M-1}},\bar{X}^{u}_{T},\bar{X}^{u}_{T},\cdots)$ and $F(\bar{Y}^{u}_{t_{0}:t_{M-1}},\bar{Y}^{u}_{T},\bar{Y}^{u}_{T},\cdots)$ is guaranteed by Lemma 4.2 (i). To see this,

	$\displaystyle\mathbb{E}\Big[\big\|F\big(\bar{X}^{u}_{t_{0}:t_{M-1}},\bar{X}^{u}_{T},\bar{X}^{u}_{T},\cdots\big)\big\|\Big]$	$\displaystyle\leq C\left(1+\frac{1}{2}\sum_{i=0}^{\infty}\mathrm{e}^{-Khi}\Big(\mathbb{E}\Big[\big\|\bar{X}^{u}_{t_{i}\wedge T}\big\|^{2}\Big]+\mathbb{E}\Big[\big\|\bar{X}^{u}_{t_{i+1}\wedge T}\big\|^{2}\Big]\Big)h\right)$
		$\displaystyle\leq C\bigg(1+\frac{\widehat{C}h}{1-\mathrm{e}^{-Kh}}\bigg).$

Then, we define $\{\mathcal{X}^{u}_{m}\}$ as follows:

\mathcal{X}^{u}_{m}:=\mathbb{E}\Big[F\big(\bar{X}^{u}_{t_{0}:t_{M-1}},\bar{X}^{u}_{T},\bar{X}^{u}_{T},\cdots\big)\Big|\mathcal{F}_{m}\Big],\quad\text{for any $m\in\mathbb{N}\cup\{\infty\}$ and $u\in I$}.

Recall that $\bar{\mu}^{u}_{m}:=\mathcal{L}(\bar{X}^{u}_{m})$ and $\bar{\bm{\mu}}_{m}:=\{\bar{\mu}^{u}_{m}\}_{u\in I}$ .

Lemma 3.6.

For every $m\in\mathbb{N}\cup\{\infty\}$ and $u\in I$ , we have $\Phi^{u}_{m}\big(\bar{X}^{u}_{t_{0}\wedge T:t_{m}\wedge T};\bar{\bm{\mu}}_{0:\infty}\big)=\mathcal{X}^{u}_{m}.$

The proof is similar to that of Lemma 4.5 in 22, but we provide one here for completeness.

Proof of Lemma 3.6.

The proof follows from a backward induction. Firstly, it is obvious that for $\forall m\geq M\text{ or }m=\infty$ as

\Phi^{u}_{m}\big(\bar{X}^{u}_{t_{0}:t_{M-1}},\underbrace{\bar{X}^{u}_{T},\cdots,\bar{X}^{u}_{T}}_{\text{\#: $m-M+1$}};\bar{\bm{\mu}}_{0:\infty}\big)=F\big(\bar{X}^{u}_{t_{0}:t_{M-1}},\underbrace{\bar{X}^{u}_{T},\bar{X}^{u}_{T},\cdots}_{\text{\#: $\infty$}}\big)=\mathcal{X}^{u}_{m}.

Assume that $\Phi^{u}_{m+1}(\bar{X}^{u}_{0:m+1};\bar{\bm{\mu}}_{0:\infty})=\mathcal{X}^{u}_{m+1}$ holds for some $m\leq M-1$ . Then, we prove for the case of $m$ ,

	$\displaystyle\mathcal{X}^{u}_{m}$	$\displaystyle=\mathbb{E}\Big[\mathcal{X}^{u}_{m+1}\Big\|\mathcal{F}_{m}\Big]=\mathbb{E}\Big[\Phi^{u}_{m+1}\big(\bar{X}^{u}_{0:m+1};\bar{\bm{\mu}}_{0:\infty}\big)\Big\|\mathcal{F}_{m}\Big]$
		$\displaystyle=\mathbb{E}\Big[\Phi^{u}_{m+1}\big(\bar{X}^{u}_{0:m},b_{m}(\bar{X}^{u}_{m})+\sigma^{u}_{m}(\bar{X}^{u}_{m},\bar{\bm{\mu}}_{m})Z^{u}_{m+1};\bar{\bm{\mu}}_{0:\infty}\big)\Big\|\mathcal{F}_{m}\Big]$
		$\displaystyle=\Phi^{u}_{m}\big(\bar{X}^{u}_{0:m};\bar{\bm{\mu}}_{0:\infty}\big),$

where we used (3.6) for the last line. If $t_{M}>T$ , then for the case of $m=M-1$ , one should modify the above derivation as in (3.7)-(3.8). ∎

Similarly, for every $u\in I$ , we define

\Psi^{u}_{m}:(\mathbb{R}^{d})^{m+1}\times(\mathcal{P}_{2}(\mathbb{R}^{d})^{I})^{\infty}\longrightarrow\mathbb{R},\quad\forall m\in\mathbb{N}\cup\{\infty\},

by
$\bullet$ Set for $m\geq M$ or $m=\infty$ :

\Psi^{u}_{m}\big(x_{0:M-1},\underbrace{x_{M},\cdots,x_{M}}_{\text{\#: $m-M+1$}};\bm{\mu}_{0:\infty}\big)=F\big(x_{0},\cdots,x_{M-1},\underbrace{x_{M},x_{M},\cdots}_{\text{\#: $\infty$}}\big),

where $F:(\mathbb{R}^{d})^{\infty}\rightarrow\mathbb{R}$ is a convex function with quadratic growth in the sense of (3.5).
$\bullet$ Set for $m<M$ :

	$\displaystyle\Psi^{u}_{m}\big(x_{0:m};\bm{\mu}_{0:\infty}\big)$	$\displaystyle=\Big(Q^{u}_{m+1}\Psi^{u}_{m+1}\big(x_{0:m},\cdot;\bm{\mu}_{0:\infty}\big)\Big)\big(x_{m},\theta^{u}_{m}(x_{m},\bm{\mu}_{m})\big)$
		$\displaystyle=\mathbb{E}\Big[\Psi^{u}_{m+1}\big(x_{0:m},b_{m}(x_{m})+\theta^{u}_{m}(x_{m},\bm{\mu}_{m})Z^{u}_{m+1};\bm{\mu}_{0:\infty}\big)\Big].$

Recall that $\bar{\nu}^{u}_{m}:=\mathcal{L}(\bar{Y}^{u}_{m})$ , and $\bar{\bm{\nu}}_{m}:=(\bar{\nu}^{u}_{m})_{u\in I}$ . By a similar argument as the proof of Lemma 3.6, we obtain that for any $m\in\mathbb{N}\cup\{\infty\}$ and $u\in I$ ,

\Psi^{u}_{m}\big(\bar{Y}^{u}_{t_{0}\wedge T:t_{m}\wedge T};\bar{\bm{\nu}}_{0:\infty}\big)=\mathcal{Y}^{u}_{m}:=\mathbb{E}\Big[F\big(\bar{Y}^{u}_{t_{0}:t_{M-1}},\bar{Y}^{u}_{T},\bar{Y}^{u}_{T},\cdots\big)\Big|\mathcal{F}_{m}\Big].

(3.9)

The main result of this section is the following multi-marginal convex ordering result.

Proposition 3.7.

Under Assumptions 2.1 and 2.2, for every convex function $F:(\mathbb{R}^{d})^{\infty}\rightarrow\mathbb{R}$ with quadratic growth in the sense of (3.5), we have

\mathbb{E}\Big[F\big(\bar{X}^{u}_{t_{0}:t_{M-1}},\bar{X}^{u}_{T},\bar{X}^{u}_{T},\cdots\big)\Big]\leq\mathbb{E}\Big[F\big(\bar{Y}^{u}_{t_{0}:t_{M-1}},\bar{Y}^{u}_{T},\bar{Y}^{u}_{T},\cdots\big)\Big],\quad\forall u\in I.

Proof.

We start by proving by a backward induction that

\Phi^{u}_{m}\leq\Psi^{u}_{m},\quad\forall m\in\mathbb{N}\cup\{\infty\},\;\forall u\in I.

(3.10)

It follows from the definition of $\{\Phi^{u}_{m}\}$ and $\{\Psi^{u}_{m}\}$ that for $m\geq M$ or $m=\infty$ ,

\Phi^{u}_{m}\big(x_{0:M-1},\underbrace{x_{M},\cdots,x_{M}}_{\text{\#: $m-M+1$}};\bm{\mu}_{0:\infty}\big):=F\big(x_{0:M-1},\underbrace{x_{M},x_{M},\cdots}_{\text{\#: $\infty$}}\big)=:\Psi^{u}_{m}\big(x_{0:M-1},\underbrace{x_{M},\cdots,x_{M}}_{\text{\#: $m-M+1$}};\bm{\mu}_{0:\infty}\big).

Assume now $\Phi^{u}_{m+1}\leq\Psi^{u}_{m+1}$ for some $m\leq M-1$ , and consider for any $x_{0:m}\in(\mathbb{R}^{d})^{m+1}$ and $\bm{\mu}_{0:\infty}\in(\mathcal{P}_{2}(\mathbb{R}^{d})^{I})^{\infty}$ :

	$\displaystyle\Phi^{u}_{m}\big(x_{0:m};\bm{\mu}_{0:\infty}\big)$	$\displaystyle=\mathbb{E}\Big[\Phi^{u}_{m+1}\big(x_{0:m},b_{m}(x_{m})+\sigma^{u}_{m}(x_{m},\bm{\mu}_{m})Z^{u}_{m+1};\bm{\mu}_{0:\infty}\big)\Big]$
		$\displaystyle\leq\mathbb{E}\Big[\Phi^{u}_{m+1}\big(x_{0:m},b_{m}(x_{m})+\theta^{u}_{m}(x_{m},\bm{\mu}_{m})Z^{u}_{m+1};\bm{\mu}_{0:\infty}\big)\Big]$
		(by Assumption 3.1 (iv), Lemma 3.4 (iii), and Lemma 3.5 (i))
		$\displaystyle\leq\mathbb{E}\Big[\Psi^{u}_{m+1}\big(x_{0:m},b_{m}(x_{m})+\theta^{u}_{m}(x_{m},\bm{\mu}_{m})Z^{u}_{m+1};\bm{\mu}_{0:\infty}\big)\Big]$
		(since $\Phi^{u}_{m+1}\leq\Psi^{u}_{m+1}$ by assumption)
		$\displaystyle=\Psi^{u}_{m}\big(x_{0:m};\bm{\mu}_{0:\infty}\big),$

which completes the proof of $\Phi^{u}_{m}\leq\Psi^{u}_{m}$ , and one proceeds by a backward induction. If $t_{M}>T$ , then for the case of $m=M-1$ , one should modify the above derivation as in (3.7)-(3.8). Consequently, for any $u\in I$ , we have the following

	$\displaystyle\mathbb{E}\Big[F\big(\bar{X}^{u}_{t_{0}:t_{M-1}},\bar{X}^{u}_{T},\bar{X}^{u}_{T},\cdots\big)\Big]$	$\displaystyle=\mathbb{E}\Big[\Phi^{u}_{0}\big(\bar{X}^{u}_{0};\bar{\bm{\mu}}_{0:\infty}\big)\Big]$
		$\displaystyle\leq\mathbb{E}\Big[\Phi^{u}_{0}\big(\bar{Y}^{u}_{0};\bar{\bm{\mu}}_{0:\infty}\big)\Big]\quad\text{(by Lemma \ref{A:lemma:4}, since $X^{u}_{0}\preceq_{cv}Y^{u}_{0}$)}$
		$\displaystyle\leq\mathbb{E}\Big[\Phi^{u}_{0}\big(\bar{Y}^{u}_{0};\bar{\bm{\nu}}_{0:\infty}\big)\Big]\hskip 12.23447pt\text{(by Lemma \ref{A:lemma:4} and Proposition \ref{A:prop:2})}$
		$\displaystyle\leq\mathbb{E}\Big[\Psi^{u}_{0}\big(\bar{Y}^{u}_{0};\bar{\bm{\nu}}_{0:\infty}\big)\Big]\hskip 11.66573pt\text{(by \eqref{A:eqn:10})}$
		$\displaystyle=\mathbb{E}\Big[F\big(\bar{Y}^{u}_{t_{0}:t_{M-1}},\bar{Y}^{u}_{T},\bar{Y}^{u}_{T},\cdots\big)\Big].$

∎

3.2 Functional convex order

3.2.1 On the dependence of trajectory

Recall the notation $t_{m}:=m\cdot h$ . Firstly, we define two interpolators as follows, which extends Definition 5.1 in 22 from $T$ to $\infty$ .

Definition 3.1.

(i) We define the piecewise affine interpolator $i_{h}:(\mathbb{R}^{d})^{\infty}\rightarrow\mathcal{C}^{d}_{\infty}$ , i.e.

i_{h}(x_{0:\infty})(t)=\frac{1}{h}\big[(t_{m+1}-t)x_{m}+(t-t_{m})x_{m+1}\big],\quad\forall m\in\mathbb{N},\;\forall t\in[t_{m},t_{m+1}].

(ii) We define the functional interpolator $I_{h}$ by

\mathcal{C}^{d}_{\infty}\ni\alpha:=\{\alpha_{t}\}_{t\geq 0}\longmapsto I_{h}(\alpha):=i_{h}\big(\alpha_{t_{0}:t_{\infty}})\in\mathcal{C}^{d}_{\infty}.

Before stating the main result of this section, i.e. Theorem 3.9, we first introduce the following result, which can be seen as a variant of Lemma 5.1 in 22 or Lemma 2.2 in 26.

Lemma 3.8.

Under Assumption 2.1, $I_{h}(\bar{X}_{T\wedge\cdot}^{u,h})$ weakly converges to $X^{u}_{T\wedge\cdot}$ as $h\rightarrow 0$ for the $\|\cdot\|_{K}$ -norm topology.

Proof.

The stated result follows immediately from

		$\displaystyle\mathbb{E}\Big[\big\\|I_{h}(\bar{X}_{T\wedge\cdot}^{u,h})-X_{T\wedge\cdot}^{u}\big\\|^{2}_{K}\Big]=\mathbb{E}\bigg[\int_{0}^{\infty}\mathrm{e}^{-Kt}\Big\|I_{h}(\bar{X}^{u,h}_{T\wedge\cdot})(t)-X^{u}_{t\wedge T}\Big\|^{2}\mathrm{d}t\bigg]$
		$\displaystyle\quad\leq 2\mathbb{E}\bigg[\int_{0}^{\infty}\mathrm{e}^{-Kt}\Big\|I_{h}(\bar{X}^{u,h}_{T\wedge\cdot})(t)-\bar{X}^{u,h}_{t\wedge T}\Big\|^{2}\mathrm{d}t\bigg]+2\mathbb{E}\bigg[\int_{0}^{\infty}\mathrm{e}^{-Kt}\Big\|\bar{X}^{u,h}_{t\wedge T}-X^{u}_{t\wedge T}\Big\|^{2}\mathrm{d}t\bigg]$
		$\displaystyle\quad\leq 2\mathbb{E}\left[\int_{0}^{[T]^{h}}\mathrm{e}^{-Kt}\left(\big\|\bar{X}^{u,h}_{[t]^{h}}-\bar{X}^{u,h}_{t}\big\|^{2}+\big\|\bar{X}^{u,h}_{[t]^{h}+h}-\bar{X}^{u,h}_{t}\big\|^{2}\right)\mathrm{d}t\right]$
		$\displaystyle\qquad+2\mathbb{E}\left[\int_{[T]^{h}}^{T}\mathrm{e}^{-Kt}\left(\big\|\bar{X}^{u,h}_{[T]^{h}}-\bar{X}^{u,h}_{t}\big\|^{2}+\big\|\bar{X}^{u,h}_{T}-\bar{X}^{u,h}_{t}\big\|^{2}\right)\mathrm{d}t\right]$
		$\displaystyle\qquad+2\mathbb{E}\left[\int_{T}^{[T]^{h}+h}\mathrm{e}^{-Kt}\big\|\bar{X}^{u,h}_{[T]^{h}}-\bar{X}^{u,h}_{T}\big\|^{2}\mathrm{d}t\right]+\frac{2\widehat{\gamma}}{K}h^{1\wedge 2\rho}$
		$\displaystyle\quad\leq\frac{2\widehat{\kappa}h}{K}\Big(2+3\mathrm{e}^{-K[T]^{h}}-4\mathrm{e}^{-KT}-\mathrm{e}^{-K[T]^{h}-Kh}\Big)+\frac{2\widehat{\gamma}h^{1\wedge 2\rho}}{K}$
		$\displaystyle\quad\rightarrow 0,$

as $h\rightarrow 0$ , where we used Jensen’s inequality, the definition of the interpolator $I_{h}$ as well as Lemma 4.2. ∎

Theorem 3.9.

Assume that Assumptions 2.1 and 2.2 are in force. For every $u\in I$ , let $X^{u}:=\{X^{u}_{t}\}_{t\geq 0}$ , $Y^{u}:=\{Y^{u}_{t}\}_{t\geq 0}$ denote the unique solutions of systems (2.1) and (2.2), and for every $t\geq 0$ , let $\mu^{u}_{t}$ , $\nu^{u}_{t}$ denote the probability distributions of $X^{u}_{t}$ and $Y^{u}_{t}$ . Then, we have

(a) Marginal convex order: $\mu^{u}_{t}\preceq_{cv}\nu^{u}_{t}$ , $\forall t\geq 0,\;\forall u\in I$ .

(b) Functional convex order: for any convex function $F:(\mathcal{C}^{d}_{\infty},\|\cdot\|_{K})\rightarrow\mathbb{R}$ with quadratic growth, one has

\mathbb{E}\Big[F\big(X^{u}\big)\Big]\leq\mathbb{E}\Big[F\big(Y^{u}\big)\Big],\quad\forall u\in I.

Proof.

(a) Let $\varphi\in\mathbb{C}_{cv}(\mathbb{R}^{d},\mathbb{R})$ with linear growth. Proposition 3.3 implies that $\bar{\mu}^{u,h}_{[t]^{h}}\preceq_{cv}\bar{\nu}^{u,h}_{[t]^{h}}$ for every $t\geq 0$ , where we used the notation $[t]^{h}:=\lfloor\frac{t}{h}\rfloor\cdot h$ . Consequently, we have

\mathbb{E}\Big[\varphi\big(\bar{X}^{u,h}_{[t]^{h}}\big)\Big]\leq\mathbb{E}\Big[\varphi\big(\bar{Y}^{u,h}_{[t]^{h}}\big)\Big],\quad\forall\varphi\in\mathbb{C}_{cv}(\mathbb{R}^{d},\mathbb{R})\text{ with linear growth}.

(3.11)

It follows from Lemma 4.2 that $\bar{X}^{u,h}_{[t]^{h}}$ weakly converges to $X^{u}_{t}$ as $h\rightarrow 0$ , and $\big\{\bar{X}^{u,h}_{[t]^{h}}\big\}_{h}$ is uniformly integrable by the Vitali convergence theorem. Moreover, notice that

\Big|\varphi\big(\bar{X}^{u,h}_{[t]^{h}}\big)\Big|\leq C\Big(1+\big|\bar{X}^{u,h}_{[t]^{h}}\big|\Big),

which further implies the uniform integrability of $\big\{\varphi\big(\bar{X}^{u,h}_{[t]^{h}}\big)\big\}$ . Thus, we have $\mathbb{E}[\varphi(\bar{X}^{u,h}_{[t]^{h}})]\rightarrow\mathbb{E}[\varphi(X^{u}_{t})]$ , and similarly, $\mathbb{E}[\varphi(\bar{Y}^{u,h}_{[t]^{h}})]\rightarrow\mathbb{E}[\varphi(Y^{u}_{t})]$ . One completes the proof of the first part by letting $h\rightarrow 0$ on both sides of (3.11), and by using Lemma 3.2.

(b) Firstly, it follows from Proposition 2.4 (i) that $F(X_{T\wedge\cdot}^{u})$ and $F(Y_{T\wedge\cdot}^{u})$ are in $L^{1}(\mathbb{P})$ since $F$ has a quadratic growth. We define a function $F_{h}$ as follows:

(\mathbb{R}^{d})^{\infty}\ni x:=(x_{0},\cdots,x_{\infty})\longmapsto F_{h}(x):=F\big(i_{h}(x)\big)\in\mathbb{R}.

The function $F_{h}$ is obviously convex since $i_{h}$ is a linear interpolator, and has a quadratic growth on $(\mathbb{R}^{d})^{\infty}$ in the sense of (3.5). To see the latter,

	$\displaystyle\Big\|F_{h}(x)\Big\|$	$\displaystyle=\Big\|F\big(i_{h}(x)\big)\Big\|\leq C\bigg(1+\int_{0}^{\infty}\mathrm{e}^{-Kt}\big\|i_{h}(x)(t)\big\|^{2}\mathrm{d}t\bigg)$
		$\displaystyle\leq C\bigg[1+\sum_{m=0}^{\infty}\frac{1}{h}\int_{t_{m}}^{t_{m+1}}\mathrm{e}^{-Kt}\Big((t_{m+1}-t)\|x_{m}\|^{2}+(t-t_{m})\|x_{m+1}\|^{2}\Big)\mathrm{d}t\bigg]$
		$\displaystyle\leq C\bigg[1+\sum_{m=0}^{\infty}\mathrm{e}^{-Kt_{m}}\frac{\|x_{m}\|^{2}+\|x_{m+1}\|^{2}}{2}h\bigg]$
		$\displaystyle=:C\Big(1+\\|x\\|_{K}^{2}\Big).$

Furthermore, recall that $M:=\frac{T}{h}$ if $\frac{T}{h}\in\mathbb{N}$ , otherwise $M:=\lfloor\frac{T}{h}\rfloor+1$ . Then,

F_{h}\big(\bar{X}^{u}_{t_{0}:t_{M-1}},\bar{X}^{u}_{T},\bar{X}^{u}_{T},\cdots\big)=F\big(i_{h}\big(\bar{X}^{u}_{t_{0}:t_{M-1}},\bar{X}^{u}_{T},\bar{X}^{u}_{T},\cdots\big)\big)=F\big(I_{h}\big(\bar{X}_{T\wedge\cdot}^{u}\big)\big).

It then follows from Proposition 3.7 that

$\displaystyle\mathbb{E}\Big[F\big(I_{h}\big(\bar{X}_{T\wedge\cdot}^{u}\big)\big)\Big]$	$\displaystyle=\mathbb{E}\Big[F\big(i_{h}\big(\bar{X}^{u}_{t_{0}:t_{M-1}},\bar{X}^{u}_{T},\bar{X}^{u}_{T},\cdots\big)\big)\Big]$
	$\displaystyle=\mathbb{E}\Big[F_{h}\big(\bar{X}^{u}_{t_{0}:t_{M-1}},\bar{X}^{u}_{T},\bar{X}^{u}_{T},\cdots\big)\Big]$
	$\displaystyle\leq\mathbb{E}\Big[F_{h}\big(\bar{Y}^{u}_{t_{0}:t_{M-1}},\bar{Y}^{u}_{T},\bar{Y}^{u}_{T},\cdots\big)\Big]$
	$\displaystyle=\mathbb{E}\Big[F\big(i_{h}\big(\bar{Y}^{u}_{t_{0}:t_{M-1}},\bar{Y}^{u}_{T},\bar{Y}^{u}_{T},\cdots\big)\big)\Big]=\mathbb{E}\Big[F\big(I_{h}\big(\bar{Y}_{T\wedge\cdot}^{u}\big)\big)\Big].$	(3.12)

The function $F$ is $\|\cdot\|_{K}$ -continuous as it is convex with $\|\cdot\|_{K}$ -quadratic growth, see (24, Lemma 2.1.1). Thanks to Lemma 3.8, we have that $I_{h}(\bar{X}_{T\wedge\cdot}^{u})$ weakly converges to $X^{u}_{T\wedge\cdot}$ for the $\|\cdot\|_{K}$ -norm topology, and moreover, $F(I_{h}(\bar{X}_{T\wedge\cdot}^{u}))$ weakly converges to $F(X_{T\wedge\cdot}^{u})$ since $F$ is continuous under the $\|\cdot\|_{K}$ -norm topology. Similarly, $F(I_{h}(\bar{Y}_{T\wedge\cdot}^{u}))$ weakly converges to $F(Y_{T\wedge\cdot}^{u})$ . As $F$ has a quadratic growth, we have for some positive constant $C$ ,

\big|F\big(I_{h}\big(\bar{X}_{T\wedge\cdot}^{u}\big)\big)\big|\leq C\Big(1+\big\|I_{h}\big(\bar{X}_{T\wedge\cdot}^{u}\big)\big\|^{2}_{K}\Big).

Thanks to Lemma 3.8, together with the Vitali convergence theorem, we have the uniform integrability of $\big\|I_{h}\big(\bar{X}_{T\wedge\cdot}^{u}\big)\big\|^{2}_{K}$ . Therefore, $F(I_{h}(\bar{X}_{T\wedge\cdot}^{u}))$ is also uniformly integrable. Consequently, $\mathbb{E}[F(I_{h}(\bar{X}_{T\wedge\cdot}^{u}))]\rightarrow\mathbb{E}[F(X_{T\wedge\cdot}^{u})]$ . Similarly, $\mathbb{E}[F(I_{h}(\bar{Y}_{T\wedge\cdot}^{u}))]\rightarrow\mathbb{E}[F(Y_{T\wedge\cdot}^{u})]$ . Thus, by letting $h\rightarrow 0$ on both sides of the inequality (3.12), we have

\mathbb{E}\Big[F\big(X_{T\wedge\cdot}^{u}\big)\Big]\leq\mathbb{E}\Big[F\big(Y_{T\wedge\cdot}^{u}\big)\Big],\quad\forall T\geq 0.

(3.13)

Notice from Lemma 4.2 (ii) that

\displaystyle\mathbb{E}\Big[\big\|X^{u}_{T\wedge\cdot}-X^{u}\big\|^{2}_{K}\Big]=\mathbb{E}\bigg[\int_{T}^{\infty}\mathrm{e}^{-Kt}\big|X^{u}_{T}-X^{u}_{t}\big|^{2}\mathrm{d}t\bigg]\leq 2\widehat{C}\frac{\mathrm{e}^{-KT}}{K}\longrightarrow 0.

Thus, we conclude the proof by letting $T\rightarrow\infty$ on both sides of (3.13). ∎

3.2.2 Extended functional convex order

The main result of this section is the following corollary.

Corollary 3.10.

Assume that Assumptions 2.1 and 2.2 are in force. For every $u\in I$ , let $X^{u}:=\{X^{u}_{t}\}_{t\geq 0}$ , $Y^{u}:=\{Y^{u}_{t}\}_{t\geq 0}$ denote the unique solutions of systems (2.1) and (2.2), and for every $t\in[0,\infty)$ , let $\mu^{u}_{t}$ , $\nu^{u}_{t}$ denote the probability distributions of $X^{u}_{t}$ and $Y^{u}_{t}$ . In addition, let $\bm{\mu}_{t}:=\{\mu^{u}_{t}\}_{u\in I}$ and $\bm{\nu}_{t}:=\{\nu^{u}_{t}\}_{u\in I}$ . Then, for any function $G$ defined as

\mathcal{C}^{d}_{\infty}\times\mathcal{C}_{\infty}(\mathcal{H}(\mathbb{R}^{d}))\ni\big(\alpha,\{\bm{\eta}_{t}\}_{t\geq 0}\big)\longmapsto G\big(\alpha;\{\bm{\eta}_{t}\}_{t\geq 0}\big)\in\mathbb{R}

satisfying the following conditions:

(i) $G$ is convex in $\alpha$ with quadratic growth in the sense that

\big|G\big(\alpha;\{\bm{\eta}_{t}\}_{t\geq 0}\big)\big|\leq C\bigg[1+\int_{0}^{\infty}\mathrm{e}^{-Kt}|\alpha_{t}|^{2}\mathrm{d}t+\sup_{t\geq 0}\bm{d}_{\mathcal{H}}^{2}(\bm{\eta}_{t},\bm{\delta}_{0})\bigg],

(ii) G is continuous in $\{\bm{\eta}_{t}\}_{t\geq 0}$ with respect to the distance $\bm{d}_{\mathcal{C}}$ defined in (1.5), and is nondecreasing in $\{\bm{\eta}_{t}\}_{t\geq 0}$ with respect to the convex order in the sense that for any $\alpha\in\mathcal{C}_{\infty}^{d}$ and $\{\bm{\eta}_{t}\}_{t\geq 0},\;\{\tilde{\bm{\eta}}_{t}\}_{t\geq 0}\in\mathcal{C}_{\infty}(\mathcal{H}(\mathbb{R}^{d}))$ such that $\eta^{u}_{t}\preceq_{cv}\tilde{\eta}^{u}_{t}$ , $\forall u\in I,\;\forall t\geq 0$ ,

G\big(\alpha;\{\bm{\eta}_{t}\}_{t\geq 0}\big)\leq G\big(\alpha;\{\tilde{\bm{\eta}}_{t}\}_{t\geq 0}\big),

one has

\mathbb{E}\Big[G\big(X^{u};\{\bm{\mu}_{t}\}_{t\geq 0}\big)\Big]\leq\mathbb{E}\Big[G\big(Y^{u};\{\bm{\nu}_{t}\}_{t\geq 0}\big)\Big],\quad\forall u\in I.

The proof of Corollary 3.10 relies on the following proposition. Recall that $M:=\frac{T}{h}$ if $\frac{T}{h}\in\mathbb{N}$ , otherwise $M:=\lfloor\frac{T}{h}\rfloor+1$ , and $t_{m}:=m\cdot h$ .

Proposition 3.11.

Let $\{\bar{X}^{u}_{0:\infty}\}_{u\in I},\;\{\bar{Y}^{u}_{0:\infty}\}_{u\in I},\;\bar{\bm{\mu}}_{0:\infty},\;\bar{\bm{\nu}}_{0:\infty}$ be respectively random variables and probability distribution ensembles defined by (2.6) and (2.7). Under Assumptions 2.1 and 2.2, for any function $\tilde{G}$ defined as

(\mathbb{R}^{d})^{\infty}\times\mathcal{H}(\mathbb{R}^{d})^{\infty}\ni\big(x_{0:\infty},\bm{\eta}_{0:\infty}\big)\longmapsto\tilde{G}\big(x_{0:\infty};\bm{\eta}_{0:\infty}\big)\in\mathbb{R}

satisfying the following conditions:

(i) $\tilde{G}$ is convex in $x_{0:\infty}$ with quadratic growth in the sense that

\big|\tilde{G}\big(x_{0:\infty};\bm{\eta}_{0:\infty}\big)\big|\leq C\Bigg[1+\sum_{m=0}^{\infty}\mathrm{e}^{-Kt_{m}}\frac{|x_{m}|^{2}+|x_{m+1}|^{2}}{2}h+\sup_{m\geq 0}\bm{d}_{\mathcal{H}}^{2}\big(\bm{\eta}_{m},\bm{\delta}_{0}\big)\Bigg],

(ii) $\tilde{G}$ is nondecreasing in $\bm{\eta}_{0:\infty}$ with respect to the convex order in the sense that for any $x_{0:\infty}\in(\mathbb{R}^{d})^{\infty}$ and $\bm{\eta}_{0:\infty},\;\tilde{\bm{\eta}}_{0:\infty}\in\mathcal{H}(\mathbb{R}^{d})^{\infty}$ such that $\eta^{u}_{i}\preceq_{cv}\tilde{\eta}^{u}_{i}$ , $\forall i\in\mathbb{N},\;\forall u\in I$ ,

\tilde{G}\big(x_{0:\infty};\bm{\eta}_{0:\infty}\big)\leq\tilde{G}\big(x_{0:\infty};\tilde{\bm{\eta}}_{0:\infty}\big),

one has

\mathbb{E}\Big[\tilde{G}\big(\bar{X}^{u}_{t_{0}:t_{M-1}},\bar{X}^{u}_{T},\bar{X}^{u}_{T},\cdots;\bar{\bm{\mu}}_{0:\infty}\big)\Big]\leq\mathbb{E}\Big[\tilde{G}\big(\bar{Y}^{u}_{t_{0}:t_{M-1}},\bar{Y}^{u}_{T},\bar{Y}^{u}_{T},\cdots;\bar{\bm{\nu}}_{0:\infty}\big)\Big],\quad\forall u\in I.

Proof.

Similar to $\{\Phi^{u}_{m}\}$ and $\{\Psi^{u}_{m}\}$ , we define

\left\{\begin{aligned} &\widehat{\Phi}^{u}_{m}\big(x_{0:M-1},\underbrace{x_{M},\cdots,x_{M}}_{\text{\#: $m-M+1$}};\bm{\mu}_{0:\infty}\big):=\tilde{G}\big(x_{0:M-1},\underbrace{x_{M},x_{M},\cdots}_{\text{\#: $\infty$}};\bm{\mu}_{0:\infty}\big),\quad m\geq M\text{ or }m=\infty,\\ &\widehat{\Phi}^{u}_{m}\big(x_{0:m};\bm{\mu}_{0:\infty}\big):=\Big(Q^{u}_{m+1}\widehat{\Phi}^{u}_{m+1}\big(x_{0:m},\cdot;\bm{\mu}_{0:\infty}\big)\Big)\big(x_{m},\sigma^{u}_{m}(x_{m},\bm{\mu}_{m})\big),\quad m<M;\\ &\widehat{\Psi}^{u}_{m}\big(x_{0:M-1},\underbrace{x_{M},\cdots,x_{M}}_{\text{\#: $m-M+1$}};\bm{\mu}_{0:\infty}\big):=\tilde{G}\big(x_{0:M-1},\underbrace{x_{M},x_{M},\cdots}_{\text{\#: $\infty$}};\bm{\mu}_{0:\infty}\big),\quad m\geq M\text{ or }m=\infty,\\ &\widehat{\Psi}^{u}_{m}\big(x_{0:m};\bm{\mu}_{0:\infty}\big):=\Big(Q^{u}_{m+1}\widehat{\Psi}^{u}_{m+1}\big(x_{0:m},\cdot;\bm{\mu}_{0:\infty}\big)\Big)\big(x_{m},\theta^{u}_{m}(x_{m},\bm{\mu}_{m})\big),\quad m<M.\end{aligned}\right.

(3.14)

The rest of the proof proceeds in a similar manner to that of Proposition 3.7. ∎

Next, we extend the definition of $i_{h}$ to the space $\mathcal{H}(\mathbb{R}^{d})^{\infty}$ as follows: for any $t\in[t_{m},t_{m+1}]$ ,

\displaystyle\mathcal{H}(\mathbb{R}^{d})^{\infty}\ni\bm{\mu}_{0:\infty}\longmapsto i_{h}(\bm{\mu}_{0:\infty})(t):=\left\{\frac{1}{h}\big[(t_{m+1}-t)\mu^{u}_{m}+(t-t_{m})\mu^{u}_{m+1}\big]\right\}_{u\in I}\in\mathcal{H}(\mathbb{R}^{d}),

and similarly, for $I_{h}$ , we define

\mathcal{C}_{\infty}(\mathcal{H}(\mathbb{R}^{d}))\ni\{\bm{\mu}_{t}\}_{t\geq 0}\longmapsto I_{h}\big(\{\bm{\mu}_{t}\}_{t\geq 0}\big):=i_{h}(\bm{\mu}_{t_{0}:t_{\infty}})\in\mathcal{C}_{\infty}(\mathcal{H}(\mathbb{R}^{d})).

Recall that $\bar{\mu}^{u}_{t}:=\mathcal{L}(\bar{X}^{u}_{t})$ , $\bar{\nu}^{u}_{t}:=\mathcal{L}(\bar{Y}^{u}_{t})$ , for $u\in I$ , and $\bar{\bm{\mu}}_{t}:=\{\bar{\mu}^{u}_{t}\}_{u\in I}$ , $\bar{\bm{\nu}}_{t}:=\{\bar{\nu}^{u}_{t}\}_{u\in I}$ . We define for every $t\geq 0$ , $\tilde{\bar{\bm{\mu}}}_{t}:=\{\tilde{\bar{\mu}}^{u}_{t}\}_{u\in I}:=I_{h}\big(\{\bar{\bm{\mu}}_{t}\}_{t\geq 0}\big)(t)$ , and similarly, $\tilde{\bar{\bm{\nu}}}_{t}:=\{\tilde{\bar{\nu}}^{u}_{t}\}_{u\in I}:=I_{h}\big(\{\bar{\bm{\nu}}_{t}\}_{t\geq 0}\big)(t)$ .

Proof of Corollary 3.10.

We start by proving that

$\displaystyle\bm{d}_{\mathcal{C}}\big(\{\tilde{\bar{\bm{\mu}}}_{t}\}_{t\geq 0},\{\bm{\mu}_{t}\}_{t\geq 0}\big)$	$\displaystyle=\sup_{t\geq 0}\bm{d}_{\mathcal{H}}\big(\tilde{\bar{\bm{\mu}}}_{t},\bm{\mu}_{t}\big)=\sup_{t\geq 0}\sup_{u\in I}\mathcal{W}_{2}\big(\tilde{\bar{\mu}}^{u}_{t},\mu^{u}_{t}\big)$
	$\displaystyle\leq\sup_{t\geq 0}\sup_{u\in I}\mathcal{W}_{2}(\tilde{\bar{\mu}}^{u}_{t},\bar{\mu}^{u}_{t})+\sup_{t\geq 0}\sup_{u\in I}\mathcal{W}_{2}(\bar{\mu}^{u}_{t},\mu^{u}_{t})$
	$\displaystyle\rightarrow 0,$	(3.15)

where we used the definition of $\bm{d}_{\mathcal{C}}$ , $\bm{d}_{\mathcal{H}}$ for the first line, and the triangle inequality of $\mathcal{W}_{2}$ for the second line. To prove (3.15), on the one hand, for $t\in[t_{m},t_{m+1}]$ , we have

$\displaystyle\sup_{t\geq 0}\sup_{u\in I}\mathcal{W}^{2}_{2}(\tilde{\bar{\mu}}^{u}_{t},\bar{\mu}^{u}_{t})$	$\displaystyle\leq\sup_{t\geq 0}\sup_{u\in I}\mathbb{E}\bigg[\Big\|\mathds{1}_{\big\{U_{m}\leq\frac{t_{m+1}-t}{h}\big\}}\bar{X}^{u}_{m}+\mathds{1}_{\big\{U_{m}>\frac{t_{m+1}-t}{h}\big\}}\bar{X}^{u}_{m+1}-\bar{X}^{u}_{t}\Big\|^{2}\bigg]$
	$\displaystyle\leq\sup_{t\geq 0}\sup_{u\in I}\mathbb{E}\bigg[\Big\|\bar{X}^{u}_{m}-\bar{X}^{u}_{t}\Big\|^{2}\bigg]+\sup_{t\geq 0}\sup_{u\in I}\mathbb{E}\bigg[\Big\|\bar{X}^{u}_{m+1}-\bar{X}^{u}_{t}\Big\|^{2}\bigg]$
	$\displaystyle\leq 2\widehat{\kappa}h$
	$\displaystyle\rightarrow 0$	(3.16)

as $h\rightarrow 0$ , where $\{U_{m}\}$ is a sequence of random variables with probability distribution $\mathcal{U}([0,1])$ , independent of $\{X^{u}_{0},Y^{u}_{0},B^{u}\}_{u\in I}$ , and we used the fact that $[(t_{m+1}-t)\mu^{u}_{m}+(t-t_{m})\mu^{u}_{m+1}]/h$ is the distribution of the random variable

\mathds{1}_{\big\{U_{m}\leq\frac{t_{m+1}-t}{h}\big\}}\bar{X}^{u}_{m}+\mathds{1}_{\big\{U_{m}>\frac{t_{m+1}-t}{h}\big\}}\bar{X}^{u}_{m+1}

for the first line, and Lemma 4.2 (i) for the third line. On the other hand, also from Lemma 4.2, we have

\sup_{t\geq 0}\sup_{u\in I}\mathcal{W}_{2}^{2}(\bar{\mu}^{u}_{t},\mu^{u}_{t})\leq\sup_{t\geq 0}\sup_{u\in I}\mathbb{E}\Big[\big|X^{u}_{t}-\bar{X}^{u}_{t}\big|^{2}\Big]\longrightarrow 0,

which, combined with (3.16), completes the proof of (3.15).

We define a function $G_{h}$ as follows

(\mathbb{R}^{d})^{\infty}\times\mathcal{H}(\mathbb{R}^{d})^{\infty}\ni\big(x_{0:\infty},\bm{\eta}_{0:\infty}\big)\longmapsto G_{h}\big(x_{0:\infty};\bm{\eta}_{0:\infty}\big):=G\big(i_{h}(x_{0:\infty});i_{h}(\bm{\eta}_{0:\infty})\big)\in\mathbb{R}.

Then, it follows from Proposition 3.11 that

$\displaystyle\mathbb{E}\Big[G\big(I_{h}(\bar{X}_{T\wedge\cdot}^{u});\{\tilde{\bar{\bm{\mu}}}_{t}\}_{t\geq 0}\big)\Big]$	$\displaystyle=\mathbb{E}\Big[G\Big(i_{h}\big(\bar{X}^{u}_{t_{0}:t_{M-1}},\bar{X}^{u}_{T},\bar{X}^{u}_{T},\cdots\big);i_{h}\big(\bar{\bm{\mu}}_{t_{0}:t_{\infty}}\big)\Big)\Big]$
	$\displaystyle=\mathbb{E}\Big[G_{h}\big(\bar{X}^{u}_{t_{0}:t_{M-1}},\bar{X}^{u}_{T},\bar{X}^{u}_{T},\cdots;\bar{\bm{\mu}}_{t_{0}:t_{\infty}}\big)\Big]$
	$\displaystyle\leq\mathbb{E}\Big[G_{h}\big(\bar{Y}^{u}_{t_{0}:t_{M-1}},\bar{Y}^{u}_{T},\bar{Y}^{u}_{T},\cdots;\bar{\bm{\nu}}_{t_{0}:t_{\infty}}\big)\Big]$
	$\displaystyle=\mathbb{E}\Big[G\Big(i_{h}\big(\bar{Y}^{u}_{t_{0}:t_{M-1}},\bar{Y}^{u}_{T},\bar{Y}^{u}_{T},\cdots\big);i_{h}\big(\bar{\bm{\nu}}_{t_{0}:t_{\infty}}\big)\Big)\Big]$
	$\displaystyle=\mathbb{E}\Big[G\big(I_{h}(\bar{Y}^{u}_{T\wedge\cdot});\{\tilde{\bar{\bm{\nu}}}_{t}\}_{t\geq 0}\big)\Big].$	(3.17)

Under assumptions imposed on $G$ , together with Lemma 3.8 and the convergence result established in (3.15), we have

\mathbb{E}\Big[G\big(I_{h}(\bar{X}_{T\wedge\cdot}^{u});\{\tilde{\bar{\bm{\mu}}}_{t}\}_{t\geq 0}\big)\Big]\longrightarrow\mathbb{E}\Big[G\big(X^{u}_{T\wedge\cdot};\{\bm{\mu}_{t}\}_{t\geq 0}\big)\Big],\quad\text{as $h\rightarrow 0$},

(idem for $Y$ ). We conclude the proof by letting $h\rightarrow 0$ , and then $T\rightarrow\infty$ on both sides of (3.17). ∎

4 Proofs of the well-posedness and the Euler scheme

4.1 Proof of Lemma 2.3

First, let us introduce two complete metric spaces $\mathbb{D}_{K}$ and $\mathcal{D}_{K}$ as follows

\mathbb{D}_{K}:=\bigg\{\mu:[0,\infty)\rightarrow\mathcal{P}_{2}(\mathbb{R}^{d})\text{ s.t. }\int_{0}^{\infty}\mathrm{e}^{-Kt}\mathcal{W}_{2}^{2}(\mu_{t},\delta_{0})\mathrm{d}t<\infty\bigg\},\quad\text{and}

\displaystyle\mathcal{D}_{K}:=\bigg\{\bm{\mu}:[0,1]\rightarrow\mathbb{D}_{K}^{I}

\displaystyle\text{ s.t. $u\rightarrow\mu^{u}$ measurable, and $\sup_{u\in I}\int_{0}^{\infty}\mathrm{e}^{-Kt}\mathcal{W}_{2}^{2}(\mu^{u}_{t},\delta_{0})\mathrm{d}t<\infty$}\bigg\}

endowed with the metrics $\bm{d}_{\mathbb{D}}$ and $\bm{d}_{\mathcal{D}}$ , respectively, which are given by

\bm{d}^{2}_{\mathbb{D}}(\mu,\nu):=\int_{0}^{\infty}\mathrm{e}^{-Kt}\mathcal{W}_{2}^{2}(\mu_{t},\nu_{t})\mathrm{d}t,\quad\text{and }\bm{d}_{\mathcal{D}}^{2}(\bm{\mu},\bm{\nu}):=\sup_{u\in I}\int_{0}^{\infty}\mathrm{e}^{-Kt}\mathcal{W}_{2}^{2}(\mu^{u}_{t},\nu^{u}_{t})\mathrm{d}t.

Consider the map $\Gamma$ , defined as follows:

\mathcal{D}_{K}\ni\bm{\mu}:=\big\{\{\mu^{u}_{t}\}_{t\geq 0}\big\}_{u\in I}\longmapsto\Gamma(\bm{\mu}):=\big\{\mathcal{L}\{X^{u,\bm{\mu}}_{t}\}_{t\geq 0}\big\}_{u\in I}\in\mathcal{D}_{K},

where $\mathcal{L}(X^{u,\bm{\mu}}_{t})$ is the law of the solution to the system with the given $\bm{\mu}$ at time $t$ , i.e.

\mathrm{d}X^{u,\bm{\mu}}_{t}=b(t,X^{u,\bm{\mu}}_{t})\mathrm{d}t+\sigma(t,X^{u,\bm{\mu}}_{t},[G\bm{\mu}_{t}]^{u})\mathrm{d}B^{u}_{t},\quad X^{u,\bm{\mu}}_{0}=X^{u}_{0}\text{ and $t\geq 0$}.

(4.1)

By similar arguments as Bayraktar et al. in 3, there exists a unique pathwise solution $\{X^{u,\bm{\mu}}\}_{u\in I}$ to (4.1) due to the Lipschitz properties of $b,\sigma$ , and the map $\Gamma$ is well-defined, namely, $\Gamma(\bm{\mu})\in\mathcal{D}_{K}$ .

Next, we show that $\Gamma$ is a contraction with respect to the metric $\bm{d}_{\mathcal{D}}$ . We start by considering for any $\bm{\mu}:=\{\mu^{u}\},\bm{\nu}:=\{\nu^{u}\}\in\mathcal{D}_{K}$ ,

l(t):=\mathbb{E}\Big[\mathrm{e}^{-Kt}\big|X^{u,\bm{\mu}}_{t}-X^{u,\bm{\nu}}_{t}\big|^{2}\Big].

Using Itô’s formula, we have

l(t)=\mathbb{E}\bigg[\int_{0}^{t}\mathrm{e}^{-Ks}\Big(-K\big|X^{u,\bm{\mu}}_{s}-X^{u,\bm{\nu}}_{s}\big|^{2}+2\big(X^{u,\bm{\mu}}_{s}-X^{u,\bm{\nu}}_{s}\big)\cdot\widehat{b}^{u,\bm{\mu},\bm{\nu}}_{s}+\mathrm{tr}\Big(\widehat{\sigma}^{u,\bm{\mu},\bm{\nu}}_{s}\big(\widehat{\sigma}^{u,\bm{\mu},\bm{\nu}}_{s}\big)^{\top}\Big)\Big)\mathrm{d}s\bigg],

where $\widehat{b}^{u,\bm{\mu},\bm{\nu}}_{s}$ and $\widehat{\sigma}^{u,\bm{\mu},\bm{\nu}}_{s}$ are defined as:

\widehat{b}^{u,\bm{\mu},\bm{\nu}}_{s}:=b(s,X^{u,\bm{\mu}}_{s})-b(s,X^{u,\bm{\nu}}_{s}),\quad\text{and }\widehat{\sigma}^{u,\bm{\mu},\bm{\nu}}_{s}:=\sigma(s,X^{u,\bm{\mu}}_{s},[G\bm{\mu}_{s}]^{u})-\sigma(s,X^{u,\bm{\nu}}_{s},[G\bm{\nu}_{s}]^{u}),

respectively. By (2.3), we have

\mathbb{E}\Big[\big(X^{u,\bm{\mu}}_{s}-X^{u,\bm{\nu}}_{s}\big)\cdot\widehat{b}^{u,\bm{\mu},\bm{\nu}}_{s}\Big]\leq-c_{0}\mathbb{E}\Big[\big|X^{u,\bm{\mu}}_{s}-X^{u,\bm{\nu}}_{s}\big|^{2}\Big].

Then, using the fact that for any $A\in\mathbb{M}^{d\times q}(\mathbb{R})$ , $\|A\|_{\text{F}}\leq\sqrt{d\wedge q}\|A\|$ , where $\|A\|_{\text{F}}$ denotes the Frobenius norm of $A$ , defined as $\|A\|_{\text{F}}:=\sqrt{\sum_{i,j}|a_{ij}|^{2}}$ , we obtain that

	$\displaystyle\mathbb{E}\Big[\mathrm{tr}\Big(\widehat{\sigma}^{u,\bm{\mu},\bm{\nu}}_{s}\big(\widehat{\sigma}^{u,\bm{\mu},\bm{\nu}}_{s}\big)^{\top}\Big)\Big]$	$\displaystyle\leq(d\wedge q)\mathbb{E}\Big[\big\\|\widehat{\sigma}^{u,\bm{\mu},\bm{\nu}}_{s}\big\\|^{2}\Big]$		(4.2)
		$\displaystyle\leq 2(d\wedge q)L^{2}\mathbb{E}\bigg[\big\|X^{u,\bm{\mu}}_{s}-X^{u,\bm{\nu}}_{s}\big\|^{2}+\int_{I}\mathcal{W}_{2}^{2}(\mu^{u}_{s},\nu^{u}_{s})\mathrm{d}u\bigg],$

where we used the Lipschitz property as well as (1.3) for the second line. Combining these two estimates gives

	$\displaystyle l(t)$	$\displaystyle\leq-\Big(K+2\big(c_{0}-(d\wedge q)L^{2}\big)\Big)\int_{0}^{t}l(s)\mathrm{d}s+2L^{2}(d\wedge q)\int_{0}^{t}\int_{I}\mathrm{e}^{-Ks}\mathcal{W}_{2}^{2}(\mu^{u}_{s},\nu^{u}_{s})\mathrm{d}u\mathrm{d}s$
		$\displaystyle=:-C_{1}\int_{0}^{t}l(s)\mathrm{d}s+C_{2}\int_{0}^{t}\int_{I}\mathrm{e}^{-Ks}\mathcal{W}_{2}^{2}(\mu^{u}_{s},\nu^{u}_{s})\mathrm{d}u\mathrm{d}s,$

where $C_{1}:=K+2(c_{0}-(d\wedge q)L^{2})>0$ due to (2.4), and $C_{2}:=2(d\wedge q)L^{2}$ . It then follows from the Gronwall’s inequality that

l(t):=\mathbb{E}\Big[\mathrm{e}^{-Kt}\big|X^{u,\bm{\mu}}_{t}-X^{u,\bm{\nu}}_{t}\big|^{2}\Big]\leq\bigg(C_{2}\int_{0}^{\infty}\int_{I}\mathrm{e}^{-Ks}\mathcal{W}_{2}^{2}(\mu^{u}_{s},\nu^{u}_{s})\mathrm{d}u\mathrm{d}s\bigg)\mathrm{e}^{-C_{1}t}\longrightarrow 0,\quad\text{as $t\rightarrow\infty$}.

Letting $t\rightarrow\infty$ and using the fact that $\kappa_{1}:=c_{0}-8L^{2}q>0$ , we obtain that

\displaystyle\mathbb{E}\bigg[\int_{0}^{\infty}\mathrm{e}^{-Kt}\big|X^{u,\bm{\mu}}_{t}-X^{u,\bm{\nu}}_{t}\big|^{2}\mathrm{d}t\bigg]\leq\frac{C_{2}}{C_{1}}\sup_{u\in I}\int_{0}^{\infty}\mathrm{e}^{-Kt}\mathcal{W}_{2}^{2}(\mu^{u}_{t},\nu^{u}_{t})\mathrm{d}t\leq\frac{1}{7}\sup_{u\in I}\int_{0}^{\infty}\mathrm{e}^{-Kt}\mathcal{W}_{2}^{2}(\mu^{u}_{t},\nu^{u}_{t})\mathrm{d}t.

Then, we obtain

\bm{d}_{\mathcal{D}}^{2}\big(\Gamma(\bm{\mu}),\Gamma(\bm{\nu})\big)\leq\sup_{u\in I}\mathbb{E}\bigg[\int_{0}^{\infty}\mathrm{e}^{-Kt}\big|X^{u,\bm{\mu}}_{t}-X^{u,\bm{\nu}}_{t}\big|^{2}\mathrm{d}t\bigg]\leq\frac{1}{7}\bm{d}^{2}_{\mathcal{D}}(\bm{\mu},\bm{\nu}).

Thus, $\Gamma$ is a contraction from $\mathcal{D}_{K}$ to itself. Then, by the Banach fixed-point theorem, there exists a unique solution to

X^{u}_{t}=X^{u}_{0}+\int_{0}^{t}b(s,X^{u}_{s})\mathrm{d}s+\int_{0}^{t}\sigma(s,X^{u}_{s},[G\bm{\mu}_{s}]^{u})\mathrm{d}B^{u}_{s},\quad t\geq 0

with $\mu^{u}_{t}=\mathcal{L}(X^{u}_{t})$ and $\bm{\mu}_{t}:=\{\mu^{u}_{t}\}_{u\in I}$ . Notice that $\{X^{u}_{t}\}_{t\geq 0}$ is $t$ -continuous as the Lebesgue integral is obviously continuous combined with the fact that the Itô integral admits a continuous modification. Consequently, $\bm{\mu}\in\mathcal{H}(\mathcal{C}^{d}_{\infty})$ .

The proof of the moment estimate follows from the Lipschitz property of the coefficient as well as (2.3). Using Itô’s formula gives

$\displaystyle\mathbb{E}\Big[\mathrm{e}^{-Kt}\big\|X^{u}_{t}\big\|^{2}-\big\|X^{u}_{0}\big\|^{2}\Big]$	$\displaystyle=\mathbb{E}\bigg[\int_{0}^{t}\mathrm{e}^{-Ks}\Big(-K\big\|X^{u}_{s}\big\|^{2}+2X^{u}_{s}\cdot b(s,X^{u}_{s})+\mathrm{tr}\Big(\big(\sigma\sigma^{\top}\big)(s,X^{u}_{s},[G\bm{\mu}_{s}]^{u})\Big)\Big)\mathrm{d}s\bigg]$
	$\displaystyle\leq\mathbb{E}\bigg[\int_{0}^{t}\mathrm{e}^{-Ks}\Big(-K\big\|X^{u}_{s}\big\|^{2}+2\kappa_{2}\big\|X^{u}_{s}\big\|-2c_{0}\big\|X^{u}_{s}\big\|^{2}$
	$\displaystyle\qquad+3\kappa_{2}^{2}(d\wedge q)+3L^{2}(d\wedge q)\|X^{u}_{s}\|^{2}+3L^{2}(d\wedge q)\int_{I}\mathbb{E}\Big[\big\|X^{u}_{s}\big\|^{2}\Big]\mathrm{d}u\Big)\mathrm{d}s\bigg]$
	$\displaystyle\leq\mathbb{E}\bigg[\int_{0}^{t}\mathrm{e}^{-Ks}\Big(\big(-2c_{0}+13L^{2}(d\wedge q)\big)\big\|X^{u}_{s}\big\|^{2}$
	$\displaystyle\qquad+\Big(3(d\wedge q)+\frac{1}{C_{3}}\Big)\kappa_{2}^{2}+3L^{2}(d\wedge q)\int_{I}\mathbb{E}\Big[\big\|X^{u}_{s}\big\|^{2}\Big]\mathrm{d}u\Big)\mathrm{d}s\bigg],$	(4.3)

where we used Young’s inequality for the last inequality, i.e.

2\kappa_{2}\big|X^{u}_{s}\big|\leq C_{3}\big|X^{u}_{s}\big|^{2}+\frac{\kappa_{2}^{2}}{C_{3}},\quad\text{with $C_{3}:=K+10L^{2}(d\wedge q)$}.

Letting $t\rightarrow\infty$ on both sides of (4.3), we obtain that

	$\displaystyle\sup_{u\in I}\mathbb{E}\Big[\big\\|X^{u}\big\\|^{2}_{K}\Big]$	$\displaystyle=\sup_{u\in I}\mathbb{E}\bigg[\int_{0}^{\infty}\mathrm{e}^{-Kt}\|X^{u}_{t}\|^{2}\mathrm{d}t\bigg]$
		$\displaystyle\leq\frac{1}{2\big(c_{0}-8L^{2}(d\wedge q)\big)}\bigg(\sup_{u\in I}\mathbb{E}\Big[\big\|X^{u}_{0}\big\|^{2}\Big]+\Big(3(d\wedge q)+\frac{1}{C_{3}}\Big)\frac{\kappa_{2}^{2}}{K}\bigg)$
		$\displaystyle<\infty.$

4.2 Proof of Lemma 2.5

By Lemma A.1 in 6, it suffices to show that for any $t\geq 0$ ,

\text{$I\ni u\longmapsto\bar{\mu}^{u,h}_{t}:=\mathcal{L}(\bar{X}^{u,h}_{t})\in\mathcal{P}(\mathbb{R}^{d})$ is measurable}.

(4.4)

Firstly, we will prove by a forward induction that

\text{$I\ni u\longmapsto\mathcal{L}(\bar{X}^{u,h}_{t_{m}},B^{u})\in\mathcal{P}(\mathbb{R}^{d}\times\mathcal{C}^{q}_{\infty})$ is measurable for each $m=0,1,2,\cdots$}.

(4.5)

This holds for $m=0$ due to the measurability of $u\mapsto\mathcal{L}(X^{u}_{0})$ , the i.i.d. property of $\{B^{u}\}$ and the independence between $\{X^{u}_{0},B^{u}\}$ . Next, suppose that (4.5) holds up to $k-1$ for some $k\in\mathbb{N}^{*}$ . To complete the proof of (4.5), it suffices to show that

I\ni u\longmapsto\mathbb{E}\Big[f\big(\bar{X}^{u,h}_{t_{k}}\big)g(B^{u})\Big]\in\mathbb{R}

(4.6)

is measurable for all bounded continuous functions $f,g$ . From (2.6), we can write $\bar{X}^{u,h}_{t_{k}}$ in the form of $\bar{X}^{u,h}_{t_{k}}=H_{k}\big(u,\bar{X}^{u,h}_{t_{k-1}},B^{u}_{t_{k}},B^{u}_{t_{k-1}}\big),$ for some $H_{k}:I\times\mathbb{R}^{d}\times\mathbb{R}^{q}\times\mathbb{R}^{q}\rightarrow\mathbb{R}^{d}$ . Notice that $H_{k}(u,\cdot,\cdot,\cdot)$ is continuous on $\mathbb{R}^{d}\times\mathbb{R}^{q}\times\mathbb{R}^{q}$ for each $u\in I$ , and $H_{k}(\cdot,x,b_{1},b_{2})$ is measurable on $I$ for each $(x,b_{1},b_{2})\in\mathbb{R}^{d}\times\mathbb{R}^{q}\times\mathbb{R}^{q}$ . Then, $H_{k}$ is a Carathéodory function, and hence is jointly measurable ((10, Lemma 4.12)). This finishes the verification of (4.6).

To prove (4.4), it suffices to show that

I\ni u\longmapsto\mathbb{E}\Big[f\big(\bar{X}^{u,h}_{t}\big)g(B^{u})\Big]\in\mathbb{R}

is measurable for all bounded continuous functions $f,g$ . Similarly, we can write $\bar{X}^{u,h}_{t}$ in the form of $H_{t}(u,\bar{X}^{u,h}_{t_{m}},B^{u}_{t},B^{u}_{t_{m}})$ for some $m\in\mathbb{N}$ and jointly measurable $H_{t}$ . We then complete the proof of (4.4) by using (4.5).

4.3 Proof of Proposition 2.4

We first present an elementary result that will be essential for the proof of Proposition 2.4.

Lemma 4.1 (Lemma 7.1 in 7).

Let $y:[0,\infty)\rightarrow[0,\infty)$ be a non-negative differentiable function. Suppose

y(t)-y(r)\leq-a_{1}\int_{r}^{t}y(s)\mathrm{d}s+a_{2}\int_{r}^{t}\sqrt{y(s)}\mathrm{d}s+\int_{r}^{t}a_{3}(s)\mathrm{d}s,\quad\forall t>r\geq 0,

for some $a_{1}>0$ , $a_{2}\in\mathbb{R}$ and non-negative and continuous function $a_{3}$ . Then,

y(t)\leq\max\Bigg\{y(0),\Bigg(\frac{a_{2}}{2a_{1}}+\sqrt{\frac{\sup_{0\leq s\leq t}a_{3}(s)}{a_{1}}+\frac{a_{2}^{2}}{4a_{1}^{2}}}\Bigg)^{2}\Bigg\},\quad\forall t\geq 0.

In particular, $y(t)\leq\max\Big\{y(0),\frac{\sup_{0\leq s\leq t}a_{3}(s)}{a_{1}}\Big\}$ if $a_{2}=0$ .

The proof of Proposition 2.4 follows immediately from the following lemma, i.e. Lemma 4.2, combined with the definition of the exponentially decaying $L^{2}$ -norm $\|\cdot\|_{K}$ , defined in (1.4).

Lemma 4.2.

Assume that Assumption 2.1 is in force.

(i)

There exist $h_{0},\widehat{C}>0$ such that

\sup_{t\geq 0}\sup_{u\in I}\mathbb{E}\Big[\big|X^{u}_{t}\big|^{2}\Big]\vee\sup_{h\in(0,h_{0})}\sup_{t\geq 0}\sup_{u\in I}\mathbb{E}\Big[\big|\bar{X}^{u,h}_{t}\big|^{2}\Big]<\widehat{C}.

Moreover, there exists a constant $\widehat{\kappa}$ such that for any $h\in(0,h_{0})$ ,

\sup_{t\geq 0}\sup_{u\in I}\mathbb{E}\Big[\big|\bar{X}^{u,h}_{t}-\bar{X}^{u,h}_{t_{m}}\big|^{2}\Big]\leq\widehat{\kappa}(t-t_{m})\leq\widehat{\kappa}h,

where we used the notation $t_{m}:=\lfloor\frac{t}{h}\rfloor\cdot h$ .

(ii)

Recall $h_{0}>0$ defined in (i). Then, there exists $\widehat{\gamma}>0$ such that for any $h\in(0,h_{0})$ ,

$\sup_{t\geq 0}\sup_{u\in I}\mathbb{E}\Big[\big|X^{u}_{t}-\bar{X}^{u,h}_{t}\big|^{2}\Big]\leq\widehat{\gamma}h^{1\wedge 2\rho}.$

Proof.

(i) Using Itô’s formula, we have

\displaystyle\mathbb{E}\Big[\big|X^{u}_{t}\big|^{2}\Big]-\mathbb{E}\Big[\big|X^{u}_{0}\big|^{2}\Big]=\mathbb{E}\bigg[\int_{0}^{t}2X^{u}_{s}\cdot b(s,X^{u}_{s})+\mathrm{tr}\big(\sigma\sigma^{\top}(s,X^{u}_{s},[G\bm{\mu}_{s}]^{u})\big)\mathrm{d}s\bigg].

Therefore, the functions

\alpha^{u}(t):=\mathbb{E}\Big[\big|X^{u}_{t}\big|^{2}\Big],\qquad\alpha(t):=\int_{I}\mathbb{E}\Big[\big|X^{u}_{t}\big|^{2}\Big]\mathrm{d}u=\int_{I}\alpha^{u}(t)\mathrm{d}u

are differentiable, and

\mathbb{E}\Big[\big|X^{u}_{t}\big|^{2}\Big]-\mathbb{E}\Big[\big|X^{u}_{r}\big|^{2}\Big]=\mathbb{E}\bigg[\int_{r}^{t}2X^{u}_{s}\cdot b(s,X^{u}_{s})+\mathrm{tr}\big(\sigma\sigma^{\top}(s,X^{u}_{s},[G\bm{\mu}_{s}]^{u})\big)\mathrm{d}s\bigg]

(4.7)

for all $t>r\geq 0$ . From (2.3), we have

X^{u}_{s}\cdot b(s,X^{u}_{s})\leq-c_{0}\big|X^{u}_{s}\big|^{2}+|b(s,0)|\big|X^{u}_{s}\big|,

ans hence

\displaystyle\mathbb{E}\Big[X^{u}_{s}\cdot b(s,X^{u}_{s})\Big]\leq\mathbb{E}\Big[-c_{0}\big|X^{u}_{s}|^{2}+|b(s,0)|\big|X^{u}_{s}\big|\Big]\leq-c_{0}\alpha^{u}(s)+|b(s,0)|\sqrt{\alpha^{u}(s)},

(4.8)

where the last inequality uses Jensen’s inequality. For the rest of integrand in (4.7), we deduce by a similar argument as in (4.2) that

		$\displaystyle\mathrm{tr}\big(\sigma\sigma^{\top}(s,X^{u}_{s},[G\bm{\mu}_{s}]^{u})\big)=\big\\|\sigma(s,X^{u}_{s},[G\bm{\mu}_{s}]^{u})\big\\|^{2}_{\text{F}}\leq(d\wedge q)\big\\|\sigma(s,X^{u}_{s},[G\bm{\mu}_{s}]^{u})\big\\|^{2}$
		$\displaystyle\qquad\leq(d\wedge q)\Big(\big\\|\sigma(s,0,[G\bm{\delta}_{0}]^{u})\big\\|+L\big\|X^{u}_{s}\big\|+L\cdot\mathcal{WOP}_{2}\big([G\bm{\mu}_{s}]^{u},[G\bm{\delta}_{0}]^{u}\big)\Big)^{2}$
		$\displaystyle\qquad\leq 3(d\wedge q)\Big(\big\\|\sigma(s,0,[G\bm{\delta}_{0}]^{u})\big\\|^{2}+L^{2}\big\|X^{u}_{s}\big\|^{2}+L^{2}\int_{I}W^{2}_{2}\big(\mu^{u}_{s},\delta_{0}\big)\mathrm{d}u\Big)$

with $\bm{\delta}_{0}:=(\delta^{u}_{0})_{u\in I}$ and $\delta^{u}_{0}:=\delta_{0},\;\forall u\in I$ , where the second line uses the Lipschitz property of $\sigma$ , and the last line uses (1.3). Consequently, we obtain that

\mathbb{E}\Big[\mathrm{tr}\big(\sigma\sigma^{\top}(s,X^{u}_{s},[G\bm{\mu}_{s}]^{u})\big)\Big]\leq 3(d\wedge q)\Big(\big\|\sigma(s,0,[G\bm{\delta}_{0}]^{u})\big\|^{2}+L^{2}\alpha^{u}(s)+L^{2}\alpha(s)\Big).

Combining this with (4.8) gives

	$\displaystyle\alpha^{u}(t)-\alpha^{u}(r)\leq$	$\displaystyle-2\Big(c_{0}-\frac{3}{2}L^{2}(d\wedge q)\Big)\int_{r}^{t}\alpha^{u}(s)\mathrm{d}s+2\kappa_{2}\int_{r}^{t}\sqrt{\alpha^{u}(s)}\mathrm{d}s$
		$\displaystyle+3L^{2}(d\wedge q)\int_{r}^{t}\alpha(s)\mathrm{d}s+3(d\wedge q)\kappa_{2}^{2}(t-r),$		(4.9)

where $\kappa_{2}:=\sup_{s\geq 0}|b(s,0)|\vee\sup_{s\geq 0,u\in I}\|\sigma(s,0,[G\bm{\delta}_{0}]^{u})\|<\infty$ by assumption. Integrating over $u\in I$ gives

\alpha(t)-\alpha(r)\leq-2\big(c_{0}-3L^{2}(d\wedge q)\big)\int_{r}^{t}\alpha(s)\mathrm{d}s+2\kappa_{2}\int_{r}^{t}\sqrt{\alpha(s)}\mathrm{d}s+3(d\wedge q)\kappa_{2}^{2}(t-r).

Since $\alpha(t)$ is non-negative and differentiable, using Lemma 4.1 with $a_{1}:=2(c_{0}-3L^{2}(d\wedge q))>0$ , $a_{2}:=2\kappa_{2}$ , and $a_{3}:=3\kappa_{2}^{2}(d\wedge q)$ , we have $\alpha(t)\leq C_{1}<\infty$ , for any $t\geq 0$ , where

\displaystyle C_{1}:=\int_{I}\mathbb{E}\Big[\big|X^{u}_{0}\big|^{2}\Big]\mathrm{d}u\vee\left(\frac{\kappa_{2}}{2\big(c_{0}-3L^{2}(d\wedge q)\big)}+\sqrt{\frac{3\kappa_{2}^{2}(d\wedge q)}{2\big(c_{0}-3L^{2}(d\wedge q)\big)}+\frac{\kappa_{2}^{2}}{4\big(c_{0}-3L^{2}(d\wedge q)\big)^{2}}}\right)^{2}.

Substituting $\alpha(t)\leq C_{1}$ into (4.9) yields

	$\displaystyle\alpha^{u}(t)-\alpha^{u}(r)\leq$	$\displaystyle-2\Big(c_{0}-\frac{3}{2}L^{2}(d\wedge q)\Big)\int_{r}^{t}\alpha^{u}(s)\mathrm{d}s+2\kappa_{2}\int_{r}^{t}\sqrt{\alpha^{u}(s)}\mathrm{d}s$
		$\displaystyle+3(d\wedge q)(L^{2}C_{1}+\kappa_{2}^{2})(t-r).$		(4.10)

Since $\alpha^{u}(t)$ is non-negative and differentiable, using Lemma 4.1 again, we have $\alpha^{u}(t)\leq C_{2}$ , uniformly in $t\geq 0$ and $u\in I$ , with

\displaystyle C_{2}:=\sup_{u\in I}\mathbb{E}\Big[|X^{u}_{0}|^{2}\Big]\vee\left(\frac{\kappa_{2}}{2\Big(c_{0}-\frac{3}{2}L^{2}(d\wedge q)\Big)}+\sqrt{\frac{3(d\wedge q)(L^{2}C_{1}+\kappa_{2}^{2})}{2\Big(c_{0}-\frac{3}{2}L^{2}(d\wedge q)\Big)}+\frac{\kappa_{2}^{2}}{4\Big(c_{0}-\frac{3}{2}L^{2}(d\wedge q)\Big)^{2}}}\right)^{2},

which completes the proof of

\sup_{u\in I}\sup_{t\geq 0}\mathbb{E}\Big[\big|X^{u}_{t}\big|^{2}\Big]\leq C_{2}<\infty.

Next, we claim that for some $h_{0}\in(0,\infty)$

\sup_{h\in(0,h_{0})}\sup_{m\geq 1}\sup_{u\in I}\mathbb{E}\Big[\big|\bar{X}^{u,h}_{t_{m}}\big|^{2}\Big]<\infty.

(4.11)

To see this, write

		$\displaystyle\big\|\bar{X}^{u,h}_{t_{m+1}}\big\|^{2}-\big\|\bar{X}^{u,h}_{t_{m}}\big\|^{2}$
		$\displaystyle=2\bar{X}^{u,h}_{t_{m}}\cdot\big(\bar{X}^{u,h}_{t_{m+1}}-\bar{X}^{u,h}_{t_{m}}\big)+\big\|\bar{X}^{u,h}_{t_{m+1}}-\bar{X}^{u,h}_{t_{m}}\big\|^{2}$
		$\displaystyle=2\bar{X}^{u,h}_{t_{m}}\cdot b(t_{m},\bar{X}^{u,h}_{t_{m}})h+2\bar{X}^{u,h}_{t_{m}}\cdot\sigma(t_{m},\bar{X}^{u,h}_{t_{m}},[G\bar{\bm{\mu}}^{h}_{t_{m}}]^{u})\Delta B^{u}_{t_{m}}+\big\|b(t_{m},\bar{X}^{u,h}_{t_{m}})\big\|^{2}h^{2}$
		$\displaystyle\quad+\big\|\sigma(t_{m},\bar{X}^{u,h}_{t_{m}},[G\bar{\bm{\mu}}^{h}_{t_{m}}]^{u})\Delta B^{u}_{t_{m}}\big\|^{2}+2hb(t_{m},\bar{X}^{u,h}_{t_{m}})\cdot\sigma(t_{m},\bar{X}^{u,h}_{t_{m}},[G\bar{\bm{\mu}}^{h}_{t_{m}}]^{u})\Delta B^{u}_{t_{m}},$

where we denote $\Delta B^{u}_{t_{m}}:=B^{u}_{t_{m+1}}-B^{u}_{t_{m}}$ , and let $\beta^{u,h}_{m}:=\mathbb{E}[|\bar{X}^{u,h}_{t_{m}}|^{2}]$ , and $\beta^{h}_{m}:=\int_{I}\mathbb{E}[|\bar{X}^{u,h}_{t_{m}}|^{2}]\mathrm{d}u:=\int_{I}\beta^{u,h}_{m}\mathrm{d}u$ . Using (2.3), (1.3), (2.5) and the Lipschitz properties of $b,\sigma$ , we have

\mathbb{E}\Big[\bar{X}^{u,h}_{t_{m}}\cdot b(t_{m},\bar{X}^{u,h}_{t_{m}})\Big]\leq\mathbb{E}\Big[\kappa_{2}\big|\bar{X}^{u,h}_{t_{m}}\big|-c_{0}\big|\bar{X}^{u,h}_{t_{m}}\big|^{2}\Big]\leq\kappa_{2}\sqrt{\beta^{u,h}_{m}}-c_{0}\beta^{u,h}_{m},

\mathbb{E}\Big[\big|b(t_{m},\bar{X}^{u,h}_{t_{m}})\big|^{2}\Big]\leq 2\kappa_{2}^{2}+2L^{2}\mathbb{E}\Big[\big|\bar{X}^{u,h}_{t_{m}}\big|^{2}\Big]\leq 2\kappa_{2}^{2}+2L^{2}\beta^{u,h}_{m},

(4.12)

$\displaystyle\mathbb{E}\Big[\big\\|\sigma(t_{m},\bar{X}^{u,h}_{t_{m}},[G\bar{\bm{\mu}}^{h}_{t_{m}}]^{u})\big\\|^{2}\Big]$	$\displaystyle\leq 3\kappa_{2}^{2}+3L^{2}\mathbb{E}\Big[\big\|\bar{X}^{u,h}_{t_{m}}\big\|^{2}\Big]+3L^{2}\mathcal{WOP}_{2}^{2}\big([G\bar{\bm{\mu}}^{h}_{t_{m}}]^{u},[G\bm{\delta}_{0}]^{u}\big)$
	$\displaystyle\leq 3\kappa_{2}^{2}+3L^{2}\mathbb{E}\Big[\big\|\bar{X}^{u,h}_{t_{m}}\big\|^{2}\Big]+3L^{2}\int_{I}\mathcal{W}^{2}_{2}(\bar{\mu}^{u,h}_{t_{m}},\delta_{0})\mathrm{d}u$
	$\displaystyle\leq 3\kappa_{2}^{2}+3L^{2}\beta^{u,h}_{m}+3L^{2}\int_{I}\beta^{u,h}_{m}\mathrm{d}u$
	$\displaystyle=3\kappa_{2}^{2}+3L^{2}\beta^{u,h}_{m}+3L^{2}\beta^{h}_{m}.$	(4.13)

Combining these three estimates gives

\displaystyle\beta^{u,h}_{m+1}-\beta^{u,h}_{m}\leq\bigg[3q\big(\kappa_{2}^{2}+L^{2}\beta^{u,h}_{m}+L^{2}\beta^{h}_{m}\big)-2c_{0}\beta^{u,h}_{m}+2\kappa_{2}\sqrt{\beta^{u,h}_{m}}\bigg]h+2(\kappa_{2}^{2}+L^{2}\beta^{u,h}_{m})h^{2}.

(4.14)

Integrating over $u\in I$ yields

	$\displaystyle\beta_{m+1}^{h}-\beta_{m}^{h}$	$\displaystyle\leq\bigg[3q\kappa_{2}^{2}+\big(6qL^{2}-2c_{0}\big)\beta^{h}_{m}+2\kappa_{2}\sqrt{\beta^{h}_{m}}\bigg]h+2(\kappa_{2}^{2}+L^{2}\beta^{h}_{m})h^{2}$
		$\displaystyle\leq\bigg[3q\kappa_{2}^{2}+\big(8qL^{2}-c_{0}\big)\beta^{h}_{m}+\frac{\kappa_{2}^{2}}{C_{3}}\bigg]h+2(\kappa_{2}^{2}+L^{2}\beta^{h}_{m})h^{2}$		(4.15)

since

2\kappa_{2}\sqrt{\beta^{h}_{m}}\leq C_{3}\beta^{h}_{m}+\frac{\kappa_{2}^{2}}{C_{3}},\quad\text{with }C_{3}:=c_{0}+2qL^{2}>0.

Rewrite (4.15) as

\beta^{h}_{m+1}\leq(1-\kappa_{h}h)\beta^{h}_{m}+C_{4}h,

where $\kappa_{h}:=c_{0}-8qL^{2}-2L^{2}h$ and $C_{4}:=5q\kappa_{2}^{2}+\kappa_{2}^{2}/C_{3}$ . From (2.4), we can choose $h_{0}$ such that $\inf_{h\in(0,h_{0})}\kappa_{h}>0$ and $1-h\kappa_{h}\in(0,1)$ for all $h\in(0,h_{0})$ . Then, for all $h\in(0,h_{0})$ ,

	$\displaystyle\beta^{h}_{m+1}$	$\displaystyle\leq(1-\kappa_{h}h)\beta_{m}^{h}+C_{4}h\leq(1-\kappa_{h}h)^{2}\beta^{h}_{m-1}+(1-\kappa_{h}h)C_{4}h+C_{4}h\leq\cdots$
		$\displaystyle\leq(1-\kappa_{h}h)^{m+1}\beta^{h}_{0}+\sum_{j=0}^{m}(1-\kappa_{h}h)^{j}C_{4}h$
		$\displaystyle\leq\int_{I}\mathbb{E}\Big[\big\|X^{u}_{0}\big\|^{2}\Big]\mathrm{d}u+\frac{C_{4}}{\kappa_{h_{0}}}=:C_{5}.$

Substituting this back to (4.14) gives

	$\displaystyle\beta^{u,h}_{m+1}-\beta^{u,h}_{m}$	$\displaystyle\leq\bigg[3q\kappa_{2}^{2}+(3qL^{2}-2c_{0})\beta^{u,h}_{m}+3qL^{2}C_{5}+2\kappa_{2}\sqrt{\beta^{u,h}_{m}}\bigg]h+2(\kappa_{2}^{2}+L^{2}\beta^{u,h}_{m})h^{2}$
		$\displaystyle\leq\bigg[3q\kappa_{2}^{2}+\big(8qL^{2}-c_{0}\big)\beta^{u,h}_{m}+3qL^{2}C_{5}+\frac{\kappa_{2}^{2}}{C_{6}}\bigg]h+2(\kappa_{2}^{2}+L^{2}\beta^{u,h}_{m})h^{2}$

since

2\kappa_{2}\sqrt{\beta^{u,h}_{m}}\leq C_{6}\beta^{u,h}_{m}+\frac{\kappa_{2}^{2}}{C_{6}},\quad\text{with }C_{6}:=c_{0}+5qL^{2}>0.

Following the same derivation as before, we obtain that

\beta^{u,h}_{m+1}\leq(1-\kappa_{h}h)\beta^{u,h}_{m}+C_{7}h\leq\sup_{u\in I}\mathbb{E}\Big[\big|X^{u}_{0}\big|^{2}\Big]+\frac{C_{7}}{\kappa_{h_{0}}}=:C_{8},

where $C_{7}:=5q\kappa_{2}^{2}+3qL^{2}C_{5}+\kappa_{2}^{2}/C_{6}$ , and thus complete the proof of (4.11).

Then, using (4.11)-(4.13) and the Lipschitz properties of $b,\sigma$ , we have for any $t\in[t_{m},t_{m+1}]$ and $h\in(0,h_{0})$ ,

	$\displaystyle\mathbb{E}\Big[\big\|\bar{X}^{u,h}_{t}-\bar{X}^{u,h}_{t_{m}}\big\|^{2}\Big]$	$\displaystyle\leq 2\mathbb{E}\Big[\big\|b(t_{m},\bar{X}^{u,h}_{t_{m}})\big\|^{2}(t-t_{m})^{2}+q\big\\|\sigma(t_{m},\bar{X}^{u,h}_{t_{m}},[G\bar{\bm{\mu}}^{h}_{t_{m}}]^{u})\big\\|^{2}(t-t_{m})\Big]$
		$\displaystyle\leq C_{9}(t-t_{m})\leq C_{9}h,$		(4.16)

where $C_{9}:=2q(5\kappa_{2}^{2}+5L^{2}C_{8}+3L^{2}C_{5}).$ Combining (4.16) with (4.11), we can easily verify that

\sup_{h\in(0,h_{0})}\sup_{t\geq 0}\sup_{u\in I}\mathbb{E}\Big[\big|\bar{X}^{u,h}_{t}\big|^{2}\Big]=:C_{10}<\infty,

(4.17)

which completes the proof of part (b.i) by letting $\widehat{C}:=C_{2}\vee C_{10}$ and $\widehat{\kappa}:=C_{9}$ .

(ii) Recall $h_{0}$ defined in (i). Firstly, notice that the definition of continuous time Euler scheme (2.8) implies that

\mathrm{d}\bar{X}^{u,h}_{t}=b([t]^{h},\bar{X}^{u,h}_{[t]^{h}})\mathrm{d}t+\sigma\big([t]^{h},\bar{X}^{u,h}_{[t]^{h}},[G\bar{\bm{\mu}}^{h}_{[t]^{h}}]^{u}\big)\mathrm{d}B^{u}_{t},\quad\bar{X}^{u,h}_{0}=X^{u}_{0},

(4.18)

where $[t]^{h}:=\lfloor\frac{t}{h}\rfloor\cdot h$ . In addition, we define $\widehat{\sigma}^{u,h}_{s}$ as follows:

\widehat{\sigma}^{u,h}_{s}:=\sigma\big(s,X^{u}_{s},[G\bm{\mu}_{s}]^{u}\big)-\sigma\big([s]^{h},\bar{X}^{u,h}_{[s]^{h}},[G\bar{\bm{\mu}}^{h}_{[s]^{h}}]^{u}\big).

(4.19)

Then, using Itô’s formula, we have

\displaystyle\mathbb{E}\Big[\big|X^{u}_{t}-\bar{X}^{u,h}_{t}\big|^{2}\Big]=\mathbb{E}\bigg[\int_{0}^{t}2\big(X^{u}_{s}-\bar{X}^{u,h}_{s}\big)\cdot\Big(b\big(s,X^{u}_{s}\big)-b\big([s]^{h},\bar{X}^{u,h}_{[s]^{h}}\big)\Big)+\mathrm{tr}\Big(\widehat{\sigma}^{u,h}_{s}\big(\widehat{\sigma}^{u,h}_{s}\big)^{\top}\Big)\mathrm{d}s\bigg].

This implies that the functions

\gamma^{u,h}(t):=\mathbb{E}\Big[\big|X^{u}_{t}-\bar{X}^{u,h}_{t}\big|^{2}\Big],\quad\text{and }\gamma^{h}(t):=\int_{I}\mathbb{E}\Big[\big|X^{u}_{t}-\bar{X}^{u,h}_{t}\big|^{2}\Big]\mathrm{d}u=\int_{I}\gamma^{u,h}(t)\mathrm{d}u

are differentiable, and

	$\displaystyle\gamma^{u,h}(t)-\gamma^{u,h}(r)$	$\displaystyle=\mathbb{E}\bigg[\int_{r}^{t}2\big(X^{u}_{s}-\bar{X}^{u,h}_{s}\big)\cdot\Big(b\big(s,X^{u}_{s}\big)-b\big([s]^{h},\bar{X}^{u,h}_{[s]^{h}}\big)\Big)+\mathrm{tr}\Big(\widehat{\sigma}^{u,h}_{s}\big(\widehat{\sigma}^{u,h}_{s}\big)^{\top}\Big)\mathrm{d}s\bigg]$
		$\displaystyle\leq\mathbb{E}\bigg[\int_{r}^{t}2\big(X^{u}_{s}-\bar{X}^{u,h}_{s}\big)\cdot\Big(b\big(s,X^{u}_{s}\big)-b\big([s]^{h},\bar{X}^{u,h}_{[s]^{h}}\big)\Big)+(d\wedge q)\big\\|\widehat{\sigma}^{u,h}_{s}\big\\|^{2}\mathrm{d}s\bigg]$

for all $t>r\geq 0$ . By adding and subtracting terms, we have

\displaystyle\mathbb{E}\Big[\big(X^{u}_{s}-\bar{X}^{u,h}_{s}\big)\cdot\Big(b\big(s,X^{u}_{s}\big)-b\big([s]^{h},\bar{X}^{u,h}_{[s]^{h}}\big)\Big)\Big]\leq-c_{0}\gamma^{u,h}(s)+C_{11}\sqrt{\gamma^{u,h}(s)}h^{\rho\wedge\frac{1}{2}},

with $C_{11}:=\tilde{L}(1+\sqrt{C_{10}})+L\sqrt{C_{9}}$ , where the inequality uses (2.3), Assumption 2.1 (i), the Cauchy-Schwarz inequality, (4.17) and (4.16). Similarly, we have

\mathbb{E}\Big[\big\|\widehat{\sigma}^{u,h}_{s}\big\|^{2}\Big]\leq 6\tilde{L}^{2}(1+2C_{2})h^{2\rho}+8L^{2}\gamma^{u,h}(s)+8L^{2}\gamma^{h}(s)+16C_{9}L^{2}h.

Combining these two estimates, we have

	$\displaystyle\gamma^{u,h}(t)-\gamma^{u,h}(r)$	$\displaystyle\leq 2\int_{r}^{t}\Big(-c_{0}\gamma^{u,h}(s)+C_{11}\sqrt{\gamma^{u,h}(s)}h^{\rho\wedge\frac{1}{2}}\Big)\mathrm{d}s$
		$\displaystyle\quad+(d\wedge q)\int_{r}^{t}\Big[6\tilde{L}^{2}(1+2C_{2})h^{2\rho}+8L^{2}\gamma^{u,h}(s)+8L^{2}\gamma^{h}(s)+16C_{9}L^{2}h\Big]\mathrm{d}s$		(4.20)

for all $t>r\geq 0$ . Integrating over $u\in I$ gives

\gamma^{h}(t)-\gamma^{h}(r)\leq-2\big(c_{0}-8L^{2}(d\wedge q)\big)\int_{r}^{t}\gamma^{h}(s)\mathrm{d}s+2C_{11}h^{\rho\wedge\frac{1}{2}}\int_{r}^{t}\sqrt{\gamma^{h}(s)}\mathrm{d}s+C_{12}h^{2\rho\wedge 1}(t-r)

with $C_{12}:=(6\tilde{L}^{2}(1+2C_{2})+16C_{9}L^{2})(d\wedge q)$ . By Lemma 4.1 with $a_{1}=2(c_{0}-8L^{2}(d\wedge q))>2\kappa_{1}>0$ , $a_{2}=2C_{11}h^{\rho\wedge\frac{1}{2}}$ and $a_{3}=C_{12}h^{2\rho\wedge 1}$ , we have for any $t\geq 0$ and $h\in(0,h_{0})$ ,

\gamma^{h}(t)\leq\Bigg(\frac{C_{11}}{2\big(c_{0}-8L^{2}(d\wedge q)\big)}+\sqrt{\frac{C_{12}}{2\big(c_{0}-8L^{2}(d\wedge q)\big)}+\frac{C_{11}^{2}}{4\big(c_{0}-8L^{2}(d\wedge q)\big)^{2}}}\Bigg)^{2}h^{2\rho\wedge 1}=:C_{13}h^{2\rho\wedge 1}.

Substituting this back to (4.20), we obtain that

\gamma^{u,h}(t)-\gamma^{u,h}(r)\leq-2\big(c_{0}-4L^{2}(d\wedge q)\big)\int_{r}^{t}\gamma^{u,h}(s)\mathrm{d}s+2C_{11}h^{\rho\wedge\frac{1}{2}}\int_{r}^{t}\sqrt{\gamma^{u,h}(s)}\mathrm{d}s+C_{14}h^{2\rho\wedge 1}(t-r)

with $C_{14}:=(6\tilde{L}^{2}(1+2C_{2})+16C_{9}L^{2}+8L^{2}C_{13})(d\wedge q)$ . Using Lemma 4.1 again, we obtain that $\gamma^{u,h}(t)\leq C_{15}h^{2\rho\wedge 1}$ , uniformly in $u\in I$ and $t\geq 0$ , for any $h\in(0,h_{0})$ , where $C_{15}$ is defined as

C_{15}:=\Bigg(\frac{C_{11}}{2\big(c_{0}-4L^{2}(d\wedge q)\big)}+\sqrt{\frac{C_{14}}{2\big(c_{0}-4L^{2}(d\wedge q)\big)}+\frac{C_{11}^{2}}{4\big(c_{0}-4L^{2}(d\wedge q)\big)^{2}}}\Bigg)^{2}.

Thus, we complete the proof by letting $\widehat{\gamma}:=C_{15}$ . ∎

5 Applications to graphon mean-field games

In this section, we provide an example to illustrate how the (extended) functional convex order results can be used to compare value functions in linear-quadratic(LQ) graphon mean-field games (MFGs). Given two graphons $M,G$ , we first define the corresponding graphon-weighted means $Z^{u},Z^{u,\sigma}:\mathcal{H}(\mathbb{R})\rightarrow\mathbb{R}$ by

Z^{u}(\bm{\mu}):=\int_{I}\int_{\mathbb{R}}M(u,v)z\mu^{v}(\mathrm{d}z)\mathrm{d}v,\quad\text{and }Z^{u,\sigma}(\bm{\mu}):=\int_{I}\int_{\mathbb{R}}G(u,v)\sigma(z)\mu^{v}(\mathrm{d}z)\mathrm{d}v.

Then, let us consider the following one-dimensional controlled dynamics:

\mathrm{d}X^{u,\alpha}_{t}=\alpha^{u}_{t}\mathrm{d}t+Z^{u,\sigma}(\bm{\mu}_{t})\mathrm{d}B^{u}_{t},\quad X^{u,\alpha}_{0}=x^{u}\in\mathbb{R},

where $\alpha:=\{\alpha^{u}\}_{u\in I}$ , and $\alpha^{u}:=\{\alpha^{u}_{t}\}_{t\in[0,T]}$ represents the control of agent $u$ . The function $\sigma:\mathbb{R}\rightarrow\mathbb{R}_{+}$ is Lipschitz and convex; $\{B^{u}\}_{u\in I}$ is a sequence of independent one-dimensional Brownian motions; $\bm{\mu}:=\{\bm{\mu}_{t}\}_{t\in[0,T]}$ is a $\mathcal{H}(\mathbb{R})$ -valued flow of probability measure ensembles; and the map $I\ni u\mapsto x^{u}$ is measurable satisfying $\sup_{u\in I}|x^{u}|^{2}<\infty$ . Agent $u$ aims to maximize her cost function $\mathcal{J}^{u,\bm{\mu}}$ defined as follows:

\displaystyle\mathcal{J}^{u,\bm{\mu}}(t,x^{u},\alpha^{u}):=\frac{1}{2}\mathbb{E}\bigg[\int_{t}^{T}\Big(\big(X^{u,\alpha}_{s}-Z^{u}(\bm{\mu}_{s})\big)^{2}+\big(\alpha^{u}_{s}\big)^{2}\Big)\mathrm{d}s+\big(X^{u,\alpha}_{T}-Z^{u}(\bm{\mu}_{T})\big)^{2}\bigg].

To solve this problem, we first fix $\bm{\mu}$ . The value function for agent $u$ , defined as $\mathcal{V}^{u,\bm{\mu}}(t,x^{u}):=\inf_{\alpha^{u}}\mathcal{J}^{u,\bm{\mu}}(t,x^{u},\alpha^{u})$ , solves the following Hamilton-Jacobi-Bellman (HJB) equation:

\left\{\begin{aligned} &-\partial_{t}\mathcal{V}^{u,\bm{\mu}}(t,x^{u})-\mathcal{H}^{u,\bm{\mu}}\big(t,x^{u},\partial_{x}\mathcal{V}^{u,\bm{\mu}}(t,x^{u}),\partial_{xx}\mathcal{V}^{u,\bm{\mu}}(t,x^{u})\big)=0,\quad\text{in $(0,T)\times\mathbb{R}$},\\ &\mathcal{V}^{u,\bm{\mu}}(T,x^{u})=\frac{1}{2}\big(x^{u}-Z^{u}(\bm{\mu}_{T})\big)^{2},\hskip 156.49014pt\text{in $\mathbb{R}$},\end{aligned}\right.

(5.1)

with the Hamiltonian $\mathcal{H}^{u,\bm{\mu}}$ defined by

\displaystyle\mathcal{H}^{u,\bm{\mu}}\big(t,x,p,M\big):=\inf_{\alpha^{u}}\bigg\{\frac{1}{2}\big(\alpha^{u}_{t}\big)^{2}+p\cdot\alpha^{u}_{t}\bigg\}+\frac{1}{2}\big(x-Z^{u}(\bm{\mu}_{t})\big)^{2}+\frac{1}{2}\big(Z^{u,\sigma}(\bm{\mu}_{t})\big)^{2}\cdot M.

When there is no ambiguity, we write $\mathcal{J}^{u},\mathcal{V}^{u},\mathcal{H}^{u}$ instead of $\mathcal{J}^{u,\bm{\mu}},\mathcal{V}^{u,\bm{\mu}},\mathcal{H}^{u,\bm{\mu}}$ to simplify notation. In the LQ setting, a natural candidate for the value function is given by

\mathcal{V}^{u}(t,x):=\frac{\eta^{u}_{t}}{2}x^{2}+h^{u}_{t}x+\beta^{u}_{t}.

Then, the optimal control for agent $u$ takes the form

\alpha^{u,*}_{t}=-\partial_{x}\mathcal{V}^{u}(t,x^{u})=-\eta^{u}_{t}x^{u}-h^{u}_{t},

where the first equality comes from the definition of $\mathcal{H}^{u}$ , and we denote $\alpha^{*}:=\{\alpha^{u,*}\}_{u\in I}$ . Substituting the above $\mathcal{V}^{u}$ and $\alpha^{u,*}$ into (5.1), we obtain the following ordinary differential equation (ODE) system:

\left\{\begin{aligned} &(\eta^{u}_{t})^{\prime}=(\eta^{u}_{t})^{2}-1,\hskip 180.10599pt\eta^{u}_{T}=1,\\ &(h^{u}_{t})^{\prime}=\eta^{u}_{t}h^{u}_{t}+Z^{u}(\bm{\mu}_{t}),\hskip 151.9376pth^{u}_{T}=-Z^{u}(\bm{\mu}_{T}),\\ &(\beta^{u}_{t})^{\prime}=-\frac{1}{2}\big(Z^{u}(\bm{\mu}_{t})\big)^{2}+\frac{1}{2}(h^{u}_{t})^{2}-\frac{1}{2}\eta^{u}_{t}\big(Z^{u,\sigma}(\bm{\mu}_{t})\big)^{2},\qquad\beta^{u}_{T}=\frac{1}{2}\big(Z^{u}(\bm{\mu}_{T})\big)^{2}.\end{aligned}\right.

(5.2)

Notice that the first equation is a Riccati equhation, which has a unique closed-form solution independent of $u\in I$ . Moreover, $\eta^{u}_{t}\equiv 1$ , for any $t\in[0,T]$ (see (2.50) in 12). Hence, in the following discussion, we denote it by $\eta$ and consider it as known.

The previous analysis is valid for any plug-in $\bm{\mu}$ . Now we consider a special $\bm{\mu}^{*}$ defined as:

\bm{\mu}^{*}_{t}=\{\mu^{u,*}_{t}\}_{u\in I}:=\big\{\mathcal{L}(X^{u,\alpha^{*}}_{t})\big\}_{u\in I},\qquad\forall t\in[0,T],

where $\{X^{u,\alpha^{*}}\}_{u\in I}$ is the unique strong solution, see (3, Proposition 2.1), of the graphon mean-field system

\mathrm{d}X^{u,\alpha^{*}}_{t}=\big(-\eta_{t}X^{u,\alpha^{*}}_{t}-h^{u}_{t}\big)\mathrm{d}t+Z^{u,\sigma}(\bm{\mu}^{*}_{t})\mathrm{d}B^{u}_{t},\quad X^{u,\alpha^{*}}_{0}=x^{u}\in\mathbb{R}.

(5.3)

We denote $m^{u}_{t}:=\mathbb{E}[X^{u,\alpha^{*}}_{t}]$ , and notice that $m^{u}_{t}=\int_{\mathbb{R}}x\mu^{u,*}_{t}(\mathrm{d}x)$ . Then, we are left to verify the unique solvability of the system

\left\{\begin{aligned} &(h^{u}_{t})^{\prime}=\eta_{t}h^{u}_{t}+\int_{I}M(u,v)m^{v}_{t}\mathrm{d}v,\qquad h^{u}_{T}=-\int_{I}M(u,v)m^{v}_{T}\mathrm{d}v,\\ &(m^{u}_{t})^{\prime}=-\eta_{t}m^{u}_{t}-h^{u}_{t},\hskip 69.13998ptm^{u}_{0}=x^{u}.\end{aligned}\right.

(5.4)

To solve the system (5.4), we use similar techniques as (11, Section 5) by converting the existence analysis into a fixed-point problem. More precisely, we view $m^{u}_{t}=m(u,t)$ as a function of $(u,t)$ . Below we derive an equation for $m^{u}_{t}$ by eliminating $h^{u}_{t}$ . Denote the function space $D_{\Lambda}$ consisting of continuous $\mathbb{R}$ -valued functions on $I\times[0,T]$ , equipped with the norm $\|\check{m}\|:=\sup_{u,t}|\check{m}(u,t)|$ . Define the operator $\Lambda$ as follows: for $\check{m}\in D_{\Lambda}$ ,

\displaystyle\big(\Lambda\check{m}\big)(u,t):=\int_{0}^{t}\mathrm{e}^{r-t}\bigg\{\mathrm{e}^{r-T}\int_{I}M(u,v)\check{m}(v,T)\mathrm{d}v+\int_{r}^{T}\mathrm{e}^{r-l}\bigg(\int_{I}M(u,v)\check{m}(v,l)\mathrm{d}v\bigg)\mathrm{d}l\bigg\}\mathrm{d}r.

If we assume in addition that

\text{$\int_{I}M(u,v)h(v)\mathrm{d}v$ is continuous in $u\in I$, \quad for any bounded, measurable $h:I\rightarrow\mathbb{R}$},

(5.5)

then $\Lambda$ is from $D_{\Lambda}$ to itself.

The solution of (5.4) further reduces to finding a fixed point to the equation:

\check{m}(u,t)=x^{u}\mathrm{e}^{-t}+\big(\Lambda\check{m}\big)(u,t).

Denote $c_{M}:=\sup_{u\in I}\int_{I}M(u,v)\mathrm{d}v\leq 1$ . We have the bound for the operator norm:

	$\displaystyle\\|\Lambda\\|$	$\displaystyle\leq c_{M}\cdot\sup_{t\in[0,T]}\bigg\{\int_{0}^{t}\bigg[\mathrm{e}^{-(T+t-2r)}+\int_{r}^{T}\mathrm{e}^{-(t+l-2r)}\mathrm{d}l\bigg]\mathrm{d}r\bigg\}$
		$\displaystyle=c_{M}\cdot\big(1-\mathrm{e}^{-T}\big)$
		$\displaystyle\leq 1.$

Hence, $\Lambda$ is a contraction and (5.4) has a unique solution. With this solution $\{h^{u}\}_{u\in I}$ , the dynamics (5.3) is then determined, and so is $\{\mathcal{L}(X^{u,\alpha^{*}})\}$ . Finally, $\{\beta^{u}\}$ is determined by the third equation in (5.2) as it only depends on $\{h^{u}\}$ , $\{m^{u}\}$ and $\{\mathcal{L}(X^{u,\alpha^{*}})\}$ . We remark that $\bm{\mu}^{*}$ is indeed the unique graphon MFG equilibrium. This follows from the facts that $\alpha^{*}:=\{\alpha^{u,*}\}_{u\in I}$ is the unique optimal control as the the cost function is strictly convex in $\alpha^{u}$ , and the uniquely solvable system (5.4) does not change for different plug-in $\bm{\mu}$ ’s.

Moreover, as the solution $\{m^{u},h^{u}\}_{u}$ of the system (5.4) is continuous on $I\times[0,T]$ due to the definition of $D_{\Lambda}$ and (5.5). We have from the first equation in (5.4) that $t\mapsto h^{u}_{t}$ is Lipschitz continuous since the boundedness of the derivative $(h^{u}_{t})^{\prime}$ . Thus, the optimal dynamics (5.3) satisfies Assumption 2.1 (i)-(ii) and Assumption 2.2 (i)-(iii) thanks to Example 2.1. Besides, the unique solvability of (5.4) implies that $h^{u}_{t}$ depends on $t$ and $m_{t}:=\{m^{u}_{t}\}_{u\in I}$ , i.e. $h^{u}_{t}=h^{u}(t,m_{t})$ .

Now let us introduce another controlled dynamics:

\mathrm{d}Y^{u,\alpha}_{t}=\alpha^{u}_{t}\mathrm{d}t+\widehat{Z}^{u,\theta}(\bm{\nu}_{t})\mathrm{d}B^{u}_{t},\quad Y^{u,\alpha}_{0}=y^{u}\in\mathbb{R},

(5.6)

and its associated cost function

\mathcal{J}^{u,\bm{\nu}}(t,y^{u},\alpha^{u}):=\frac{1}{2}\mathbb{E}\bigg[\int_{t}^{T}\Big(\big(Y^{u,\alpha}_{s}-Z^{u}(\bm{\nu}_{s})\big)^{2}+\big(\alpha^{u}_{s}\big)^{2}\Big)\mathrm{d}s+\big(Y^{u,\alpha}_{T}-Z^{u}(\bm{\nu}_{T})\big)^{2}\bigg].

(5.7)

where $\widehat{Z}^{u,\theta}(\bm{\nu}):\mathcal{H}(\mathbb{R})\rightarrow\mathbb{R}$ is defined by

\widehat{Z}^{u,\theta}(\bm{\nu}):=\int_{I}\int_{\mathbb{R}}K(u,v)\theta(z)\nu^{v}(\mathrm{d}z)\mathrm{d}v;

the function $\theta:\mathbb{R}\rightarrow\mathbb{R}$ is Lipschitz, and the map $I\ni u\mapsto y^{u}\in\mathbb{R}$ is measurable satisfying $\sup_{u\in I}|y^{u}|^{p}<\infty$ for some $p>2$ . By a similar reasoning as before, there exists a unique graphon MFG equilibrium to (5.6)-(5.7), and denote the optimal control and the corresponding flow of measure ensembles by $\widehat{\alpha}^{*}:=\{\widehat{\alpha}^{u,*}\}_{u\in I}$ and $\bm{\nu}^{*}:=\{\nu^{u,*}\}_{u\in I}$ , respectively.

We assume in addition the following two conditions:
$\bullet$ $x^{u}=y^{u}$ , for any $u\in I$ ;
$\bullet$ for any fixed $u\in I$ , we have for any $(v_{1},v_{2},z_{1},z_{2})\in I^{2}\times\mathbb{R}^{2}$ that

G(u,v_{1})G(u,v_{2})\sigma(z_{1})\sigma(z_{2})\leq K(u,v_{1})K(u,v_{2})\theta(z_{1})\theta(z_{2}).

Notice that $\widehat{\alpha}^{u,*}_{t}:=-\eta_{t}Y^{u,\widehat{\alpha}^{*}}_{t}-h_{t}^{u}=-\eta_{t}Y^{u,\widehat{\alpha}^{*}}_{t}-h^{u}(t,\{\mathbb{E}[Y_{t}^{u,\widehat{\alpha}^{*}}]\}_{u\in I}),$ where $\{h^{u}_{t}\}$ also solves (5.4) since the equation of $\{m^{u}_{t}\}$ , i.e. the expectation of the optimal controlled dynamics, does not change if we switch the diffusion coefficient from $Z^{u,\sigma}$ to $\widehat{Z}^{u,\theta}$ . Thus, Assumption 2.2 (iv)-(v) are satisfied by Example 2.1. Consequently, Theorem 3.9 in this paper implies the comparison of the following two value functions $\mathcal{V}^{u,\bm{\mu}^{*}}(x^{u}),\mathcal{V}^{u,\bm{\nu}^{*}}(y^{u})$ , which are defined as:

		$\displaystyle\mathcal{V}^{u,\bm{\mu}^{}}(x^{u}):=\inf_{\alpha^{u}}\mathcal{J}^{u,\bm{\mu}^{}}(0,x^{u},\alpha^{u})=\mathcal{J}^{u,\bm{\mu}^{}}(0,x^{u},\alpha^{u,}),$
		$\displaystyle\mathcal{V}^{u,\bm{\nu}^{}}(y^{u}):=\inf_{\alpha^{u}}\mathcal{J}^{u,\bm{\nu}^{}}(0,y^{u},\alpha^{u})=\mathcal{J}^{u,\bm{\nu}^{}}(0,y^{u},\widehat{\alpha}^{u,}).$

More precisely, $\mathcal{V}^{u,\bm{\mu}^{*}}(x^{u})\leq\mathcal{V}^{u,\bm{\nu}^{*}}(y^{u})$ , $\forall u\in I$ .

References

[1] A. Alfonsi, J. Corbetta, and B. Jourdain (2019) Sampling of probability measures in the convex order by wasserstein projection. arXiv preprint arXiv:1709.05287. Cited by: §3.1.
[2] A. Aurell, R. Carmona, and M. Lauriere (2022) Stochastic graphon games: II. the linear-quadratic case. Applied Mathematics & Optimization 85 (3), pp. 39. Cited by: §1.
[3] E. Bayraktar, S. Chakraborty, and R. Wu (2023) Graphon mean field systems. The Annals of Applied Probability 33 (5), pp. 3587–3619. Cited by: §1, §1, §2.1, Example 2.1, §4.1, §5.
[4] E. Bayraktar, X. He, and D. Kim (2025) Graphon particle systems with common noise. arXiv preprint arXiv:2507.03265. Cited by: §1.
[5] E. Bayraktar and D. Kim (2024) Concentration of measure for graphon particle system. Advances in Applied Probability 56 (4), pp. 1279–1306. Cited by: §1.
[6] E. Bayraktar, R. Wu, and X. Zhang (2023) Propagation of chaos of forward–backward stochastic differential equations with graphon interactions. Applied Mathematics & Optimization 88 (1), pp. 25. Cited by: §1, Example 2.1, §4.2.
[7] E. Bayraktar and R. Wu (2022) Stationarity and uniform in time convergence for the graphon particle system. Stochastic Processes and their Applications 150, pp. 532–568. Cited by: §1, §1, Example 2.1, Lemma 4.1, Abstract.
[8] E. Bayraktar and X. Zhang (2023) Solvability of infinite horizon mckean–vlasov fbsdes in mean field control problems and games. Applied Mathematics & Optimization 87 (1), pp. 13. Cited by: §1.
[9] J. Bergenthum and L. Rüschendorf (2007) Convex ordering criteria for Lévy processes. Advances in Data Analysis and Classification 1 (2), pp. 143–173. Cited by: §1.
[10] B. Bouchard and M. Nutz (2015) Arbitrage and duality in nondominated discrete-time models. The Annals of Applied Probability 25 (2), pp. 823–859. Cited by: §4.2.
[11] P.E. Caines and M. Huang (2021) Graphon mean field games and their equations. SIAM Journal on Control and Optimization 59 (6), pp. 4373–4399. Cited by: §1, §5.
[12] R. Carmona and F. Delarue (2018) Probabilistic theory of mean field games with applications i. Springer. Cited by: §5.
[13] F. Coppini, A. D. Crescenzo, and H. Pham (2025) Nonlinear graphon mean-field systems. arXiv preprint arXiv:2402.08628. Cited by: §1.1, §1, §2.1.
[14] A. D. Crescenzo, F. D. Feo, and H. Pham (2026) Linear-quadratic optimal control for non-exchangeable mean-field sdes and applications to systemic risk. ESAIM: Control, Optimisation and Calculus of Variations 32, pp. 34. Cited by: §1.
[15] M.F. Djete (2025) A non-exchangeable mean field control problem with controlled interactions. arXiv preprint. Cited by: §1.
[16] B. Hajek (1985) Mean stochastic comparison of diffusion. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 68 (3), pp. 315–329. Cited by: §1.
[17] B. Jourdain and G. Pagès (2023) Convex ordering of solutions to one-dimensional SDEs. arXiv preprint arXiv:2312.09779. Cited by: §1.
[18] B. Jourdain and G. Pagès (2025) Convex ordering for stochastic volterra equations and their euler schemes. Finance and Stochastics 29 (1), pp. 1–62. Cited by: §1.
[19] N.E. Karoui, M. Jeanblanc‐Picquè, and S.E. Shreve (1998) Robustness of the black and scholes formula. Mathematical Finance 8 (2), pp. 93–126. Cited by: §1.
[20] D. Lacker and A. Soret (2023) A label-state formulation of stochastic graphon games and approximate equilibria on large networks. Mathematics of Operations Research 48 (4), pp. 1987–2018. Cited by: §1.
[21] H. Leblanc, T.L. Gouic, J. Liandrat, and M. Tournus (2023) Extending the wasserstein metric to positive measures. arXiv preprint arXiv:2303.02183. Cited by: §1.1.
[22] Y. Liu and G. Pagès (2023) Functional convex order for the scaled mckean–vlasov processes. The Annals of Applied Probability 33 (6A), pp. 4491–4527. Cited by: §1.1, §1, §1, §3.1.1, §3.1.2, §3.1, §3.2.1, §3.2.1, Lemma 3.5, Abstract.
[23] L. Lovász (2012) Large networks and graph limits. American Mathematical Society. Cited by: §1.
[24] R. Lucchetti (2006) Convexity and well-posed problems. Springer Science & Business Media. Cited by: §3.2.1.
[25] C. Martini (1999) Propagation of convexity by markovian and martingalian semigroups. Potential Analysis 10 (2), pp. 133–175. Cited by: §1.
[26] G. Pagès (2014) Convex order for path-dependent derivatives: a dynamic programming approach. arXiv preprint arXiv:1407.6348. Cited by: §1, §3.2.1.
[27] A. Schied and M. Stadje (2007) Robustness of delta hedging for path-dependent options in local volatility models. Journal of Applied Probability 44 (4), pp. 865–879. Cited by: §1.
[28] L. Tangpi and X. Zhou (2024) Optimal investment in a large population of competitive and heterogeneous agents. Finance and Stochastics 28 (2), pp. 497–551. Cited by: §1.

		$\displaystyle\mathbb{E}\Big[\big\\|I_{h}(\bar{X}_{T\wedge\cdot}^{u,h})-X_{T\wedge\cdot}^{u}\big\\|^{2}_{K}\Big]=\mathbb{E}\bigg[\int_{0}^{\infty}\mathrm{e}^{-Kt}\Big\|I_{h}(\bar{X}^{u,h}_{T\wedge\cdot})(t)-X^{u}_{t\wedge T}\Big\|^{2}\mathrm{d}t\bigg]$
		$\displaystyle\quad\leq 2\mathbb{E}\bigg[\int_{0}^{\infty}\mathrm{e}^{-Kt}\Big\|I_{h}(\bar{X}^{u,h}_{T\wedge\cdot})(t)-\bar{X}^{u,h}_{t\wedge T}\Big\|^{2}\mathrm{d}t\bigg]+2\mathbb{E}\bigg[\int_{0}^{\infty}\mathrm{e}^{-Kt}\Big\|\bar{X}^{u,h}_{t\wedge T}-X^{u}_{t\wedge T}\Big\|^{2}\mathrm{d}t\bigg]$
		$\displaystyle\quad\leq 2\mathbb{E}\left[\int_{0}^{[T]^{h}}\mathrm{e}^{-Kt}\left(\big\|\bar{X}^{u,h}_{[t]^{h}}-\bar{X}^{u,h}_{t}\big\|^{2}+\big\|\bar{X}^{u,h}_{[t]^{h}+h}-\bar{X}^{u,h}_{t}\big\|^{2}\right)\mathrm{d}t\right]$
		$\displaystyle\qquad+2\mathbb{E}\left[\int_{[T]^{h}}^{T}\mathrm{e}^{-Kt}\left(\big\|\bar{X}^{u,h}_{[T]^{h}}-\bar{X}^{u,h}_{t}\big\|^{2}+\big\|\bar{X}^{u,h}_{T}-\bar{X}^{u,h}_{t}\big\|^{2}\right)\mathrm{d}t\right]$
		$\displaystyle\qquad+2\mathbb{E}\left[\int_{T}^{[T]^{h}+h}\mathrm{e}^{-Kt}\big\|\bar{X}^{u,h}_{[T]^{h}}-\bar{X}^{u,h}_{T}\big\|^{2}\mathrm{d}t\right]+\frac{2\widehat{\gamma}}{K}h^{1\wedge 2\rho}$
		$\displaystyle\quad\leq\frac{2\widehat{\kappa}h}{K}\Big(2+3\mathrm{e}^{-K[T]^{h}}-4\mathrm{e}^{-KT}-\mathrm{e}^{-K[T]^{h}-Kh}\Big)+\frac{2\widehat{\gamma}h^{1\wedge 2\rho}}{K}$
		$\displaystyle\quad\rightarrow 0,$

	$\displaystyle\Big\|F_{h}(x)\Big\|$	$\displaystyle=\Big\|F\big(i_{h}(x)\big)\Big\|\leq C\bigg(1+\int_{0}^{\infty}\mathrm{e}^{-Kt}\big\|i_{h}(x)(t)\big\|^{2}\mathrm{d}t\bigg)$
		$\displaystyle\leq C\bigg[1+\sum_{m=0}^{\infty}\frac{1}{h}\int_{t_{m}}^{t_{m+1}}\mathrm{e}^{-Kt}\Big((t_{m+1}-t)\|x_{m}\|^{2}+(t-t_{m})\|x_{m+1}\|^{2}\Big)\mathrm{d}t\bigg]$
		$\displaystyle\leq C\bigg[1+\sum_{m=0}^{\infty}\mathrm{e}^{-Kt_{m}}\frac{\|x_{m}\|^{2}+\|x_{m+1}\|^{2}}{2}h\bigg]$
		$\displaystyle=:C\Big(1+\\|x\\|_{K}^{2}\Big).$

$\displaystyle\mathbb{E}\Big[\mathrm{e}^{-Kt}\big\|X^{u}_{t}\big\|^{2}-\big\|X^{u}_{0}\big\|^{2}\Big]$	$\displaystyle=\mathbb{E}\bigg[\int_{0}^{t}\mathrm{e}^{-Ks}\Big(-K\big\|X^{u}_{s}\big\|^{2}+2X^{u}_{s}\cdot b(s,X^{u}_{s})+\mathrm{tr}\Big(\big(\sigma\sigma^{\top}\big)(s,X^{u}_{s},[G\bm{\mu}_{s}]^{u})\Big)\Big)\mathrm{d}s\bigg]$
	$\displaystyle\leq\mathbb{E}\bigg[\int_{0}^{t}\mathrm{e}^{-Ks}\Big(-K\big\|X^{u}_{s}\big\|^{2}+2\kappa_{2}\big\|X^{u}_{s}\big\|-2c_{0}\big\|X^{u}_{s}\big\|^{2}$
	$\displaystyle\qquad+3\kappa_{2}^{2}(d\wedge q)+3L^{2}(d\wedge q)\|X^{u}_{s}\|^{2}+3L^{2}(d\wedge q)\int_{I}\mathbb{E}\Big[\big\|X^{u}_{s}\big\|^{2}\Big]\mathrm{d}u\Big)\mathrm{d}s\bigg]$
	$\displaystyle\leq\mathbb{E}\bigg[\int_{0}^{t}\mathrm{e}^{-Ks}\Big(\big(-2c_{0}+13L^{2}(d\wedge q)\big)\big\|X^{u}_{s}\big\|^{2}$
	$\displaystyle\qquad+\Big(3(d\wedge q)+\frac{1}{C_{3}}\Big)\kappa_{2}^{2}+3L^{2}(d\wedge q)\int_{I}\mathbb{E}\Big[\big\|X^{u}_{s}\big\|^{2}\Big]\mathrm{d}u\Big)\mathrm{d}s\bigg],$	(4.3)

	$\displaystyle\sup_{u\in I}\mathbb{E}\Big[\big\\|X^{u}\big\\|^{2}_{K}\Big]$	$\displaystyle=\sup_{u\in I}\mathbb{E}\bigg[\int_{0}^{\infty}\mathrm{e}^{-Kt}\|X^{u}_{t}\|^{2}\mathrm{d}t\bigg]$
		$\displaystyle\leq\frac{1}{2\big(c_{0}-8L^{2}(d\wedge q)\big)}\bigg(\sup_{u\in I}\mathbb{E}\Big[\big\|X^{u}_{0}\big\|^{2}\Big]+\Big(3(d\wedge q)+\frac{1}{C_{3}}\Big)\frac{\kappa_{2}^{2}}{K}\bigg)$
		$\displaystyle<\infty.$

		$\displaystyle\big\|\bar{X}^{u,h}_{t_{m+1}}\big\|^{2}-\big\|\bar{X}^{u,h}_{t_{m}}\big\|^{2}$
		$\displaystyle=2\bar{X}^{u,h}_{t_{m}}\cdot\big(\bar{X}^{u,h}_{t_{m+1}}-\bar{X}^{u,h}_{t_{m}}\big)+\big\|\bar{X}^{u,h}_{t_{m+1}}-\bar{X}^{u,h}_{t_{m}}\big\|^{2}$
		$\displaystyle=2\bar{X}^{u,h}_{t_{m}}\cdot b(t_{m},\bar{X}^{u,h}_{t_{m}})h+2\bar{X}^{u,h}_{t_{m}}\cdot\sigma(t_{m},\bar{X}^{u,h}_{t_{m}},[G\bar{\bm{\mu}}^{h}_{t_{m}}]^{u})\Delta B^{u}_{t_{m}}+\big\|b(t_{m},\bar{X}^{u,h}_{t_{m}})\big\|^{2}h^{2}$
		$\displaystyle\quad+\big\|\sigma(t_{m},\bar{X}^{u,h}_{t_{m}},[G\bar{\bm{\mu}}^{h}_{t_{m}}]^{u})\Delta B^{u}_{t_{m}}\big\|^{2}+2hb(t_{m},\bar{X}^{u,h}_{t_{m}})\cdot\sigma(t_{m},\bar{X}^{u,h}_{t_{m}},[G\bar{\bm{\mu}}^{h}_{t_{m}}]^{u})\Delta B^{u}_{t_{m}},$