论文标题
暗影烈士 - 孔雀问题的随机大规模运输方法
Shadow martingales -- a stochastic mass transport approach to the peacock problem
论文作者
论文摘要
给定一个真实概率的家族测量$（μ_t）_ {t \ geq 0} $增加了凸顺序（孔雀），我们描述了一种系统的方法，可以随时创建一个完全适合边缘的martingale。我们方法的关键对象是孔雀中的一项度量的阻塞阴影，这是\ cite {beju16，nustta17}中引入的（受阻）阴影的概括。作为输入数据，我们采用了越来越多的度量家族$（ν^α）_ {α\ in [0,1]} $，带有$ν^α（\ Mathbb {r}）=α$，是$μ_0$的章节，称为$μ_0$。然后，对于任何$α$，我们定义了度量$（η^α_t）_ {t \ geq 0} $，通过将$η^α_t$等于$ n^α_t$等于$ n^α_t$的$ n^α_t$等于$ n^α_t$的$ n^α_t$。我们确定参数化$（ν^α）_ {α\在[0,1]} $中的条件，使得这种构造会导致唯一的martingale量度$π$，即阴影martingale，而没有任何假设在孔雀上。对于左列参数化$（ν_ {\ text {lc}}^α）_ {0,1]} $ in [0,1]} $，我们将阴影martingale识别为Martingale最佳运输问题的连续时间版本的独特解决方案。 此外，我们的方法丰富了有关可预测表示属性（PRP）的知识，因为任何影子Martingale都带有极端马尔可夫Martingales中的规范choquet代表。
Given a family of real probability measures $(μ_t)_{t\geq 0}$ increasing in convex order (a peacock) we describe a systematic method to create a martingale exactly fitting the marginals at any time. The key object for our approach is the obstructed shadow of a measure in a peacock, a generalization of the (obstructed) shadow introduced in \cite{BeJu16,NuStTa17}. As input data we take an increasing family of measures $(ν^α)_{α\in [0,1]}$ with $ν^α(\mathbb{R})=α$ that are submeasures of $μ_0$, called a parametrization of $μ_0$. Then, for any $α$ we define an evolution $(η^α_t)_{t\geq 0}$ of the measure $ν^α=η^α_0$ across our peacock by setting $η^α_t$ equal to the obstructed shadow of $ν^α$ in $(μ_s)_{s \in [0,t]}$. We identify conditions on the parametrization $(ν^α)_{α\in [0,1]}$ such that this construction leads to a unique martingale measure $π$, the shadow martingale, without any assumptions on the peacock. In the case of the left-curtain parametrization $(ν_{\text{lc}}^α)_{α\in [0,1]}$ we identify the shadow martingale as the unique solution to a continuous-time version of the martingale optimal transport problem. Furthermore, our method enriches the knowledge on the Predictable Representation Property (PRP) since any shadow martingale comes with a canonical Choquet representation in extremal Markov martingales.
