零幂变化的高斯和非高斯过程及相关的随机分析
Gaussian and non-Gaussian processes of zero power variation, and related stochastic calculus
摘要：
考虑一类由$X\left( t\right) =\int_{0}^{T}G\left( t,s\right) dM\left( s\right) $为$t\in\lbrack0,T]$定义的随机过程$X$，其中$M$是平方可积连续鞅，$G$是确定性核.令$m$为奇数。在$M$的二次变差$\left[ M\right] $可微且$\mathbf{E}\left[ \left\vert d\left[ M\right] (t)/dt\right\vert ^{m}\right] $有限的假设下，证明了当$M$在$r$的小区间上的增量方差相关的量$δ^{2}\left( r\right) $满足$δ\left( r\right) =o\left( r^{1/(2m)}\right) $时，$m$的次幂变差$$ \lim_{\varepsilon\rightarrow0}\varepsilon^{-1}\int_{0}^{T}ds\left( X\left( s+\varepsilon\right) -X\left( s\right) \right) ^{m} $$存在且为零.当$M$是维纳过程时，$X$是高斯过程;这类过程包括分数布朗运动和其他有或没有平稳增量的高斯过程。当$X$是高斯的且有平稳增量时，$δ$是$X$的单变量典范度量，并证明了关于$δ$的条件是必要的。在非平稳高斯情形下，当$m=3$时，定义了对称（广义Stratonovich）积分，证明了它的存在性，并建立了$C^{6}$类所有函数的伊藤公式.
英文摘要：
We consider a class of stochastic processes $X$ defined by $X\left( t\right) =\int_{0}^{T}G\left( t,s\right) dM\left( s\right) $ for $t\in\lbrack0,T]$, where $M$ is a square-integrable continuous martingale and $G$ is a deterministic kernel. Let $m$ be an odd integer. Under the assumption that the quadratic variation $\left[ M\right] $ of $M$ is differentiable with $\mathbf{E}\left[ \left\vert d\left[ M\right] (t)/dt\right\vert ^{m}\right] $ finite, it is shown that the $m$th power variation $$ \lim_{\varepsilon\rightarrow0}\varepsilon^{-1}\int_{0}^{T}ds\left( X\left( s+\varepsilon\right) -X\left( s\right) \right) ^{m} $$ exists and is zero when a quantity $δ^{2}\left( r\right) $ related to the variance of an increment of $M$ over a small interval of length $r$ satisfies $δ\left( r\right) =o\left( r^{1/(2m)}\right) $. When $M$ is the Wiener process, $X$ is Gaussian; the class then includes fractional Brownian motion and other Gaussian processes with or without stationary increments. When $X$ is Gaussian and has stationary increments, $δ$ is $X$'s univariate canonical metric, and the condition on $δ$ is proved to be necessary. In the non-stationary Gaussian case, when $m=3$, the symmetric (generalized Stratonovich) integral is defined, proved to exist, and its Itô formula is established for all functions of class $C^{6}$.
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