论文标题
长时间的Hurst SDES及其千古式手段的规律性
Long time Hurst regularity of fractional SDEs and their ergodic means
论文作者
论文摘要
分数布朗运动可以被视为由$（t，h）\ in {\ mathbb {r} _ {+} \ times（0,1）} $索引的高斯字段，其中$ h $是hurst参数。在紧凑的时间间隔内，众所周知，它几乎可以随着时间的流逝而持续不断，Lipschitz在$ h $中连续。首先，我们将此结果扩展到整个时间间隔$ \ mathbb {r} _ {+} $，并考虑简单和矩形增量。然后，我们考虑由分数布朗尼运动驱动的SDE，并具有收缩漂移。事实证明，解决方案及其千古的手段几乎肯定在$ h $中连续不断地逐渐及时。该结果用于统计应用程序的单独工作。我们还推断出$ h $中不变的度量的敏感性结果。 证明基于分数布朗运动和分数Ornstein-uhlenbeck过程的差异估计值，Garsia-Rodemich-Rumsey引理的多参数版本和组合论证，以估计高斯变量物的产物的预期。
The fractional Brownian motion can be considered as a Gaussian field indexed by $(t,H)\in {\mathbb{R}_{+}\times (0,1)}$, where $H$ is the Hurst parameter. On compact time intervals, it is known to be almost surely jointly Hölder continuous in time and Lipschitz continuous in $H$. First, we extend this result to the whole time interval $\mathbb{R}_{+}$ and consider both simple and rectangular increments. Then we consider SDEs driven by fractional Brownian motion with contractive drift. The solutions and their ergodic means are proven to be almost surely Hölder continuous in $H$, uniformly in time. This result is used in a separate work for statistical applications. We also deduce a sensibility result of the invariant measure in $H$. The proofs are based on variance estimates of the increments of the fractional Brownian motion and fractional Ornstein-Uhlenbeck processes, multiparameter versions of the Garsia-Rodemich-Rumsey lemma and a combinatorial argument to estimate the expectation of a product of Gaussian variables.
