英文标题：
《Rough paths, Signatures and the modelling of functions on streams》
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作者：
Terry Lyons
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最新提交年份：
2014
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英文摘要：
  Rough path theory is focused on capturing and making precise the interactions between highly oscillatory and non-linear systems. It draws on the analysis of LC Young and the geometric algebra of KT Chen. The concepts and the uniform estimates, have widespread application and have simplified proofs of basic questions from the large deviation theory and extended Ito\'s theory of SDEs; the recent applications contribute to (Graham) automated recognition of Chinese handwriting and (Hairer) formulation of appropriate SPDEs to model randomly evolving interfaces. At the heart of the mathematics is the challenge of describing a smooth but potentially highly oscillatory and vector valued path $x_{t}$ parsimoniously so as to effectively predict the response of a nonlinear system such as $dy_{t}=f(y_{t})dx_{t}$, $y_{0}=a$. The Signature is a homomorphism from the monoid of paths into the grouplike elements of a closed tensor algebra. It provides a graduated summary of the path $x$. Hambly and Lyons have shown that this non-commutative transform is faithful for paths of bounded variation up to appropriate null modifications. Among paths of bounded variation with given Signature there is always a unique shortest representative. These graduated summaries or features of a path are at the heart of the definition of a rough path; locally they remove the need to look at the fine structure of the path. Taylor\'s theorem explains how any smooth function can, locally, be expressed as a linear combination of certain special functions (monomials based at that point). Coordinate iterated integrals form a more subtle algebra of features that can describe a stream or path in an analogous way; they allow a definition of rough path and a natural linear \"basis\" for functions on streams that can be used for machine learning.
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中文摘要：
粗糙路径理论致力于捕捉和精确描述高度振荡和非线性系统之间的相互作用。它借鉴了LC Young的分析和KT Chen的几何代数。这些概念和统一估计有着广泛的应用，并简化了大偏差理论和伊藤的SDEs理论中基本问题的证明；最近的应用有助于（Graham）自动识别中文笔迹，并（Haier）制定适当的SPDE来模拟随机演变的界面。数学的核心是如何以简洁的方式描述一条光滑但可能高度振荡的向量值路径$x{t}$，以便有效地预测非线性系统的响应，如$dy{t}=f（y{t}）dx{t}$，$y{0}=a$。签名是从路的幺半群到闭张量代数的群元素的同态。它提供了路径$x$的分级摘要。Hambly和Lyons已经证明，这种非交换变换对于有界变化直至适当零修改的路径是可靠的。在给定签名的有界变差路径中，总是有一个唯一的最短代表。这些渐进式总结或路径特征是粗糙路径定义的核心；从局部来看，它们不再需要查看路径的精细结构。泰勒定理解释了任何光滑函数如何在局部表示为某些特殊函数（基于该点的单项式）的线性组合。坐标迭代积分形成了一个更精细的特征代数，可以以类似的方式描述流或路径；它们允许定义粗糙路径，并为可用于机器学习的流函数提供自然的线性“基础”。
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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：Mathematics        数学
二级分类：Rings and Algebras        环与代数
分类描述：Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups
非交换环与代数，非结合代数，泛代数与格论，线性代数，半群
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一级分类：Mathematics        数学
二级分类：Statistics Theory        统计理论
分类描述：Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计：例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
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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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一级分类：Statistics        统计学
二级分类：Statistics Theory        统计理论
分类描述：stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近，贝叶斯推论，决策理论，估计，基础，推论，检验。
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