关于一类分形函数的$p^{\text{th}}$变分
On the $p^{\text{th}}$ variation of a class of fractal functions
摘要：
连续函数$f$沿细分序列的$p^{\text{th}}$变化的概念是具有积分器$f$的路径伊藤积分理论的关键。在这里，我们分析了一类分形函数的$p^{\text{th}}$变化，包含Takagi-van der Waerden和Weierstraffield函数。我们使用的概率参数表明，这些功能有线性$p^{\text{th}}$变化的参数$p\ge1$，这可以解释为倒数赫斯特参数的功能。此外，它示出，如果功能是从（一个歪斜的版本）的帐篷地图，那么斜率的$p^{\text{th}}$变化可以计算从$p^{\text{th}}$时刻（非对称）无限伯努利卷积。最后，我们提供了这些时刻的递归公式，并使用它来讨论的$p^{\text{th}}$的变化，这发生在路径伊藤演算时，$p\ge3$是一个奇数的一个有符号的版本的存在性和不存在性。
英文摘要：
The concept of the $p^{\text{th}}$ variation of a continuous function $f$ along a refining sequence of partitions is the key to a pathwise Itô integration theory with integrator $f$. Here, we analyze the $p^{\text{th}}$ variation of a class of fractal functions, containing both the Takagi--van der Waerden and Weierstraß functions. We use a probabilistic argument to show that these functions have linear $p^{\text{th}}$ variation for a parameter $p\ge1$, which can be interpreted as the reciprocal Hurst parameter of the function. It is shown moreover that if functions are constructed from (a skewed version of) the tent map, then the slope of the $p^{\text{th}}$ variation can be computed from the $p^{\text{th}}$ moment of a (non-symmetric) infinite Bernoulli convolution. Finally, we provide a recursive formula of these moments and use it to discuss the existence and non-existence of a signed version of the $p^{\text{th}}$ variation, which occurs in pathwise Itô calculus when $p\ge3$ is an odd integer.
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