一个非常类似于多重分数布朗运动的过程
A process very similar to multifractional Brownian motion
摘要：
在Ayache和Taqqu（2005）中，通过将分数布朗运动（fBm）的常数参数$H$替换为依赖于时间$t$的足够光滑的函数参数$H(.)$来获得多重分数布朗（mBm）运动。在这里，我们考虑的过程$Z$通过替换在小波展开的fBM的索引$H$由一个函数$H(.)$取决于二进点$k/2^j$。Benassi等人（2000年）引入了这一过程，用分段常数Hurst指数和连续路径对fBm进行建模。本文研究了函数参数满足$β>\sup_{t\in \rit} H(t)$阶一致Hölder条件的情形，结果表明，在这种情形下，过程$Z$与mBm在以下意义上非常相似：i）$Z$与mBm之间的差满足$d>\sup_{t\in \R} H(t)$阶一致Hölder条件; ii）作为副产品，可以推导出在每个点$t\in \R$处$Z$的逐点Hölder指数是$H(t)$，并且$Z$与具有Hurst参数$H(t)$的fBm相切。
英文摘要：
In Ayache and Taqqu (2005), the multifractional Brownian (mBm) motion is obtained by replacing the constant parameter $H$ of the fractional Brownian motion (fBm) by a smooth enough functional parameter $H(.)$ depending on the time $t$. Here, we consider the process $Z$ obtained by replacing in the wavelet expansion of the fBm the index $H$ by a function $H(.)$ depending on the dyadic point $k/2^j$. This process was introduced in Benassi et al (2000) to model fBm with piece-wise constant Hurst index and continuous paths. In this work, we investigate the case where the functional parameter satisfies an uniform Hölder condition of order $β>\sup_{t\in \rit} H(t)$ and ones shows that, in this case, the process $Z$ is very similar to the mBm in the following senses: i) the difference between $Z$ and a mBm satisfies an uniform Hölder condition of order $d>\sup_{t\in \R} H(t)$; ii) as a by product, one deduces that at each point $t\in \R$ the pointwise Hölder exponent of $Z$ is $H(t)$ and that $Z$ is tangent to a fBm with Hurst parameter $H(t)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
