英文标题：
《On a class of generalized Takagi functions with linear pathwise
quadratic variation》
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作者：
Alexander Schied
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最新提交年份：
2015
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英文摘要：
We consider a class $\\mathscr{X}$ of continuous functions on $[0,1]$ that is of interest from two different perspectives. First, it is closely related to sets of functions that have been studied as generalizations of the Takagi function. Second, each function in $\\mathscr{X}$ admits a linear pathwise quadratic variation and can thus serve as an integrator in F\\\"ollmer\'s pathwise It\\=o calculus. We derive several uniform properties of the class $\\mathscr{X}$. For instance, we compute the overall pointwise maximum, the uniform maximal oscillation, and the exact uniform modulus of continuity for all functions in $\\mathscr{X}$. Furthermore, we give an example of a pair $x,y\\in\\mathscr{X}$ such that the quadratic variation of the sum $x+y$ does not exist.
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中文摘要：
我们考虑$[0,1]$上的连续函数类$\\mathscr{X}$，它从两个不同的角度引起了人们的兴趣。首先，它与作为Takagi函数的推广而研究的函数集密切相关。第二$\\mathscr{X}$中的每个函数都允许线性路径二次变化，因此可以作为F\\\'ollmer的路径It\\=o演算中的积分器。我们推导了类$\\mathscr{X}$的几个一致性质。例如，我们计算$\\mathscr{X}中所有函数的整体点态最大值、一致最大振荡和精确一致连续模此外，我们给出了一对$x，y\\in\\mathscr{x}$的例子，使得$x+y$之和的二次变化不存在。
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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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