英文标题：
《Martingale Benamou--Brenier: a probabilistic perspective》
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作者：
Julio Backhoff-Veraguas, Mathias Beiglb\\\"ock, Martin Huesmann, Sigrid
K\\\"allblad
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最新提交年份：
2019
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英文摘要：
In classical optimal transport, the contributions of Benamou-Brenier and McCann regarding the time-dependent version of the problem are cornerstones of the field and form the basis for a variety of applications in other mathematical areas. We suggest a Benamou-Brenier type formulation of the martingale transport problem for given $d$-dimensional distributions $\\mu, \\nu $ in convex order. The unique solution $M^*=(M_t^*)_{t\\in [0,1]}$ of this problem turns out to be a Markov-martingale which has several notable properties: In a specific sense it mimics the movement of a Brownian particle as closely as possible subject to the conditions $M^*_0\\sim\\mu, M^*_1\\sim \\nu$. Similar to McCann\'s displacement-interpolation, $M^*$ provides a time-consistent interpolation between $\\mu$ and $\\nu$. For particular choices of the initial and terminal law, $M^*$ recovers archetypical martingales such as Brownian motion, geometric Brownian motion, and the Bass martingale. Furthermore, it yields a natural approximation to the local vol model and a new approach to Kellerer\'s theorem. This article is parallel to the work of Huesmann-Trevisan, who consider a related class of problems from a PDE-oriented perspective.
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中文摘要：
在经典的最优运输中，贝纳莫·布雷尼尔（BenamouBrenier）和麦肯（McCann）对该问题的时间相关版本的贡献是该领域的基石，并为其他数学领域的各种应用奠定了基矗对于给定的凸序$d$维分布，我们提出了鞅输运问题的Benamou-Brenier型公式。这个问题的唯一解$M ^*=（M\\u t ^*）{t\\in[0,1]}$是一个马尔可夫鞅，它有几个显著的性质：在特定意义上，它尽可能地模仿布朗粒子的运动，受条件$M ^*\\u 0\\sim\\mu，M ^*\\u 1\\sim\\nu$的约束。与McCann的位移插值类似，M ^*$提供了$\\ mu$和$\\ nu$之间的时间一致性插值。对于初始定律和终端定律的特殊选择，$M ^*$恢复了原型鞅，如布朗运动、几何布朗运动和Bass鞅。此外，它还为局部vol模型提供了一种自然近似，并为Kellerer定理提供了一种新的方法。本文与Huesmann Trevisan的工作平行，他从面向PDE的角度考虑了一类相关的问题。
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分类信息：

一级分类：Mathematics 数学
二级分类：Probability 概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Mathematics 数学
二级分类：Classical Analysis and ODEs 经典分析与颂歌
分类描述：Special functions, orthogonal polynomials, harmonic analysis, ODE\'s, differential relations, calculus of variations, approximations, expansions, asymptotics
特殊函数、正交多项式、调和分析、Ode、微分关系、变分法、逼近、展开、渐近
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一级分类：Quantitative Finance 数量金融学
二级分类：Mathematical Finance 数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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