Functional convex order for the scaled McKean-Vlasov processes

Abstract

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 $(\Omega ,\mathcal{F},({\mathcal{F}}_t{)}_{t\ge 0},\mathbb{P})$ 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 $p\ge 2$, 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 $\sigma ⪯\theta$ with respect to the partial matrix order, the convex order for the initial random variable $X_0{⪯}_{cv}{Y}_0$ 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 $\mathbb{E}F(X)\le \mathbb{E}F(Y)$; for a convex functional $G$ defined on the product space involving the path space and its marginal distribution space, we have $\mathbb{E}G(X,({\mu }_t{)}_{t\in [0,T]})\le \mathbb{E}G(Y,({\nu }_t{)}_{t\in [0,T]})$ under appropriate conditions. The symmetric setting is also valid, that is, if $\theta ⪯\sigma$ and $Y_0\le X_0$ with respect to the convex order, then $\mathbb{E}F(Y)\le \mathbb{E}F(X)$ and $\mathbb{E}G(Y,({\nu }_t{)}_{t\in [0,T]})\le \mathbb{E}G(X,({\mu }_t{)}_{t\in [0,T]})$. The proof is based on several forward and backward dynamic programming principles and the convergence of the Euler scheme of the McKean-Vlasov equation.