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On the 𝑝th variation of a class of fractal functions
Proceedings of the American Mathematical Society ( IF 0.8 ) Pub Date : 2020-09-11 , DOI: 10.1090/proc/15171 Alexander Schied , Zhenyuan Zhang
Proceedings of the American Mathematical Society ( IF 0.8 ) Pub Date : 2020-09-11 , DOI: 10.1090/proc/15171 Alexander Schied , Zhenyuan Zhang
Abstract:The concept of the 
th variation of a continuous function 
along a refining sequence of partitions is the key to a pathwise Itô integration theory with integrator 
. Here, we analyze the 
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 
th variation for a parameter 
, 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 
th variation can be computed from the 
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 
th variation, which occurs in pathwise Itô calculus when 
is an odd integer.
中文翻译：
一类分形函数的第五变分
摘要：连续函数沿分区细化序列的第三次变化的概念是带积分器的循序渐进Itô积分理论的关键。在这里，我们分析一类分形函数的th次变化，其中既包含Takagi-van der Waerden函数又包含Weierstraß函数。我们使用概率论证法来证明这些函数的参数具有线性变化，这可以解释为函数的互惠赫斯特参数。此外，还表明，如果从帐篷地图（的倾斜版本）构建函数，则可以根据（非对称）无限伯努利卷积的第一个矩。最后，我们提供了这些矩的递归公式，并用它来讨论th变体的有符号版本的存在与否，当有奇数整数时，该变体在有路径的Itô演算中发生。
更新日期：2020-11-09
th variation of a continuous function along a refining sequence of partitions is the key to a pathwise Itô integration theory with integrator . Here, we analyze the 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 th variation for a parameter , 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 th variation can be computed from the 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 th variation, which occurs in pathwise Itô calculus when is an odd integer. 中文翻译：
一类分形函数的第五变分
摘要：
连续函数沿分区细化序列的第三次变化的概念是带积分器的循序渐进Itô积分理论的关键。在这里，我们分析一类分形函数的th次变化，其中既包含Takagi-van der Waerden函数又包含Weierstraß函数。我们使用概率论证法来证明这些函数的参数具有线性变化，这可以解释为函数的互惠赫斯特参数。此外，还表明，如果从帐篷地图（的倾斜版本）构建函数，则可以根据（非对称）无限伯努利卷积的第一个矩。最后，我们提供了这些矩的递归公式，并用它来讨论th变体的有符号版本的存在与否，当有奇数整数时，该变体在有路径的Itô演算中发生。
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