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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 $(Ω, \mathcal{F}, (\mathcal{F}_{t})_{t\geq0}, \mathbb{P})$ by \[\begin{cases} dX_{t}= b(t, X_{t}, μ_{t})dt+σ(t, X_{t}, μ_{t})dB_{t}, \;\;X_{0}\in L^{p}(\mathbb{P}),\\ dY_{t}\,= b(t, \,Y_{t}\,,\, ν_{t})dt+θ(t, \,Y_{t}\,,\, ν_{t})dB_{t}, \;\;Y_{0}\in L^{p}(\mathbb{P}), \end{cases}\] where $p\geq2$, for every $ t\in[0, T]$, $μ_t$, $ν_t$ denote the probability distribution of $X_t$, $Y_t$ respectively and the drift coefficient $b(t, x, μ)$ is affine in $x$ (scaled). If we make the convexity and monotony assumption (only) on $σ$ and if $σ\preceqθ$ 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 $\mathbb{E}\,G\big(X, (μ_t)_{t\in[0, T]}\big)\leq \mathbb{E}\,G\big(Y, (ν_t)_{t\in[0, T]}\big)$ under appropriate conditions. The symmetric setting is also valid, that is, if $θ\preceq σ$ and $Y_0 \leq X_0$ with respect to the convex order, then $\mathbb{E}\,F(Y) \leq \mathbb{E}\,F(X)$ and $\mathbb{E}\,G\big(Y, (ν_t)_{t\in[0, T]}\big)\leq \mathbb{E}\,G(X, (μ_t)_{t\in[0, T]})$. The proof is based on several forward and backward dynamic programming principles and the convergence of the Euler scheme of the McKean-Vlasov equation.