论文标题
缩放的McKean-Vlasov过程的功能凸顺序
Functional convex order for the scaled McKean-Vlasov processes
论文作者
论文摘要
我们为两个缩放的McKean-Vlasov处理$ x =（x_ {t}）_ {t \ in [0，t]} $和$ y =（y_ _ {t}）_ {t \ in [0，t]} $定义了在[0，t]} $（the [0，T] （\ Mathcal {f} _ {t}）_ {t \ geq0}，\ Mathbb {p}）$ by \ [\ [\ begin {cases} dx_ {t} = b（t，x__ {t，x_ {t，x_ {t}，μ__{t} t}） μ_{t}）db_ {t}，\; \; x_ {0} \ in L^{p}（\ Mathbb {p}），\\ dy_ {t} \，= b（t，= b（t，\，\ \，y__ _ _ { \，y_ {t} \ ,, \，ν_{t}）db_ {t}，\; \; y_ {0} \ in l^{p}（\ Mathbb {p}） 其中$ p \ geq2 $，对于[0，t] $，$μ_t$，$ν_t$的每个$ t \表示分别为$ x_t $，$ y_t $的概率分布，而漂移系数$ b（t，x，x，μ）in In $ x $（缩放）。如果我们对$σ$进行凸度和单调假设（仅），如果相对于部分矩阵订单，则在$σ$上进行凸面和单调假设，则可以在整个过程中传播到process $ x $ y $ x $ y $ $ y $ $ y_0 $的初始随机变量$ x_0 \ preceq _ {\ preceq _ {\，cv} y_0 $。也就是说，如果我们考虑在路径空间上定义的凸功能$ f $，则具有多项式增长，我们具有$ \ m athbb {e} f（x）\ leq \ leq \ mathbb {e} f（y）$;对于涉及路径空间及其边际分配空间的产品空间上定义的凸功能$ g $，我们有$ \ mathbb {e} \，g \ big（x，（μ_t）_ {t \ in [0，t]} \ big） t]} \ big）$在适当的条件下。对称设置也是有效的，也就是说，如果$θ\ preceqσ$和$ y_0 \ leq x_0 $相对于convex顺序，则$ \ mathbb {e} \，f（y）\ leq \ leq \ leq \ leq \ mathbb {e} （ν_t）_ {t \ in [0，t]} \ big）\ leq \ mathbb {e} \，g（x，（μ_t）_ {t \ in [0，t]}）$。该证明基于几个前向和向后的动态编程原理以及McKean-Vlasov方程的Euler方案的收敛性。
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.
