Monotone convex order for the McKean-Vlasov processes

In this paper, we establish the monotone convex order between two $\mathbb{R}$-valued McKean-Vlasov processes $X=(X_t{)}_{t\in [0,T]}$ and $Y=(Y_t{)}_{t\in [0,T]}$ defined on a filtered probability space $(\Omega, $\mathcal{F}$, ($\mathcal{F}${t}){t\geq0}, $\mathbb{P}$)$ by \begin{align} &dX_{t}=b(t, X_{t}, \mu_{t})dt+\sigma(t, X_{t}, \mu_{t})dB_{t}, \quad X_{0}\in L^{p}($\mathbb{P}$); $\text{with}$; p\geq 2,\nonumber\ &dY_{t}=\beta(t, Y_{t}, \nu_{t})dt+\theta(t, ,Y_{t}, \nu_{t}),dB_{t}, ,\quad Y_{0}\in L^{p}($\mathbb{P}$), \nonumber \end{align} where $\forall, t\in [0, T],: \mu_{t}=$\mathbb{P}$\circ X_{t}^{-1}, :\nu_{t}=$\mathbb{P}$\circ Y_{t}^{-1}. $ If we make the convexity and monotony assumption (only) on $b$ and $\mathrm{\mid }\sigma \mathrm{\mid }$ and if $b\le \beta$ and $\mathrm{\mid }\sigma \mathrm{\mid }\le \mathrm{\mid }\theta \mathrm{\mid }$, then the monotone convex order for the initial random variable $X_0{⪯}_{\text{mcv}}{Y}_0$ can be propagated to the whole path of processes $X$ and $Y$. That is, if we consider a non-decreasing convex functional $F$ defined on the path space with polynomial growth, we have $\mathbb{E}F(X)\le \mathbb{E}F(Y)$; for a non-decreasing convex functional $G$ defined on the product space involving the path space and its marginal distribution space, we have $$\mathbb{E}$, G(X, (\mu_{t}){t\in [0, T]})\leq $\mathbb{E}$, G(Y, (\nu{t}){t\in [0, T]})$ under appropriate conditions. The symmetric setting is also valid, that is, if $Y_0{⪯}_{\text{mcv}}{X}_0$ and $\mathrm{\mid }\theta \mathrm{\mid }\le \mathrm{\mid }\sigma \mathrm{\mid }$, then $$\mathbb{E}$, F(Y)\leq $\mathbb{E}$, F(X)$and$$\mathbb{E}$, G(Y, (\nu{t}){t\in [0, T]})\leq $\mathbb{E}$, G(X, (\mu{t})_{t\in [0, T]})$. The proof is based on several forward and backward dynamic programming principle and the convergence of the truncated Euler scheme of the McKean-Vlasov equation.