arXiv 0912.07822009-12math.PR

零功率变化的高斯和非高斯过程

Gaussian and non-Gaussian processes of zero power variation

Francesco Russo, Frederi Viens

摘要：

本文考虑一类随机过程$X$，它是鞅$M$的Volterra卷积.当$M$是布朗运动时，$X$是高斯的，并且该类包括分数布朗运动和具有或不具有齐次增量的其他高斯过程。令$m$为奇数。在$M$的二次变差的某些技术条件下，证明了当$M$在$r$的小区间上的增量方差相关的量$δ^{2}(r) $满足$δ(r) = o(r^{1/(2m)}) $时，$m$幂变差存在且为零.在齐次增量高斯过程的情形下，证明了$δ$是$X$的正则度量，并证明了关于$δ$的条件是必要的，同时将零变差结果推广到非整数对称幂.在非齐次高斯情形下，当$m=3$时，定义了对称（广义Stratonovich）积分，证明了它的存在性，并证明了它的伊藤公式对$C^{6}$类的所有函数都成立.

英文摘要：

This paper considers the class of stochastic processes $X$ which are Volterra convolutions of a martingale $M$. When $M$ is Brownian motion, $X$ is Gaussian, and the class includes fractional Brownian motion and other Gaussian processes with or without homogeneous increments. Let $m$ be an odd integer. Under some technical conditions on the quadratic variation of $M$, it is shown that the $m$-power variation exists and is zero when a quantity $δ^{2}(r) $ related to the variance of an increment of $M$ over a small interval of length $r$ satisfies $δ(r) = o(r^{1/(2m)}) $. In the case of a Gaussian process with homogeneous increments, $δ$ is $X$'s canonical metric and the condition on $δ$ is proved to be necessary, and the zero variation result is extended to non-integer symmetric powers. In the non-homogeneous Gaussian case, when $m=3$, the symmetric (generalized Stratonovich) integral is defined, proved to exist, and its Itô's formula is proved to hold for all functions of class $C^{6}$.

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。

摘要：

英文摘要：

相关论文