论文标题
凸阶的最佳基于运输的表征
An optimal transport based characterization of convex order
论文作者
论文摘要
为了概率，$μ，ν$和$ρ$定义成本函数\ begin {align*} c（μ，ρ）：= \ sup_ {π\inπ（μ，ρ）} \ int \ int \ langle x，y \ langle x，y \ rangle \ rangle \ rangle \ rangle \ range（dx，dx，dx，dy），\ quad C(ν,ρ):=\sup_{π\in Π(ν,ρ)} \int \langle x,y\rangle\, π(dx,dy), \end{align*} where $\langle\cdot, \cdot\rangle$ denotes the scalar product and $Π(\cdot,\cdot)$ is the一组耦合。我们表明，$ \ mathbb {r}^d $上有两个概率的$μ$和$ν$具有有限的第一瞬间的订单（即$μ\preceq_ccν$）iff $ c（μ，μ，ρ）这概括了Carlier的结果。我们的证明依赖于$ \ int f \，dν-\ int f \，dμ$的定量限制，所有$ 1 $ -LIPSCHITZ函数$ f $，这是通过最佳运输双重性和Brenier定理获得的。在此结果的基础上，我们获得了凸秩序众所周知的一维表征的新证明。我们还描述了用于调查凸顺序的新计算方法和对数学金融中独立于模型的套利策略的应用。
For probability measures $μ,ν$ and $ρ$ define the cost functionals \begin{align*} C(μ,ρ):=\sup_{π\in Π(μ,ρ)} \int \langle x,y\rangle\, π(dx,dy),\quad C(ν,ρ):=\sup_{π\in Π(ν,ρ)} \int \langle x,y\rangle\, π(dx,dy), \end{align*} where $\langle\cdot, \cdot\rangle$ denotes the scalar product and $Π(\cdot,\cdot)$ is the set of couplings. We show that two probability measures $μ$ and $ν$ on $\mathbb{R}^d$ with finite first moments are in convex order (i.e. $μ\preceq_cν$) iff $C(μ,ρ)\le C(ν,ρ)$ holds for all probability measures $ρ$ on $\mathbb{R}^d$ with bounded support. This generalizes a result by Carlier. Our proof relies on a quantitative bound for the infimum of $\int f\,dν-\int f\,dμ$ over all $1$-Lipschitz functions $f$, which is obtained through optimal transport duality and Brenier's theorem. Building on this result, we derive new proofs of well-known one-dimensional characterizations of convex order. We also describe new computational methods for investigating convex order and applications to model-independent arbitrage strategies in mathematical finance.
