Functional convex order for the scaled McKean-Vlasov processes

We establish the functional convex order results for two scaled 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 $$\begin{cases} dX_{t}= b(t, X_{t}, \mu_{t})dt+\sigma(t, X_{t}, \mu_{t})dB_{t}, \;\;X_{0}\in L^{p}(\mathbb{P}),\\ dY_{t}\,= b(t, \,Y_{t}\,,\, \nu_{t})dt+\theta(t, \,Y_{t}\,,\, \nu_{t})dB_{t}, \;\;Y_{0}\in L^{p}(\mathbb{P}), \end{cases}$$ where $p\geq2$, for every $t\in[0, T]$, $\mu_t$, $\nu_t$ denote the probability distribution of $X_t$, $Y_t$ respectively and the drift coefficient $b(t, x, \mu)$ is affine in $x$ (scaled). If we make the convexity and monotony assumption (only) on $\sigma$ and if $\sigma\preceq\theta$ with respect to the partial matrix order, the convex order for the initial random variable $X_0 \preceq_{\,cv} Y_0$ can be propagated to the whole path of process $X$ and $Y$. That is, if we consider a convex functional $F$ defined on the path space with polynomial growth, we have $\mathbb{E}F(X)\leq\mathbb{E}F(Y)$; for a convex functional $G$ defined on the product space involving the path space and its marginal distribution space, we have $\mathbb{E}\,G\big(X, (\mu_t)_{t\in[0, T]}\big)\leq \mathbb{E}\,G\big(Y, (\nu_t)_{t\in[0, T]}\big)$ under appropriate conditions. The symmetric setting is also valid, that is, if $\theta \preceq \sigma$ and $Y_0 \leq X_0$ with respect to the convex order, then $\mathbb{E}\,F(Y) \leq \mathbb{E}\,F(X)$ and $\mathbb{E}\,G\big(Y, (\nu_t)_{t\in[0, T]}\big)\leq \mathbb{E}\,G(X, (\mu_t)_{t\in[0, T]})$. The proof is based on several forward and backward dynamic programming principles and the convergence of the Euler scheme of the McKean-Vlasov equation.