论文标题
由高斯噪声驱动的无限尺寸路径伏特拉工艺 - 概率和应用
Infinite Dimensional Pathwise Volterra Processes Driven by Gaussian Noise -- Probabilistic Properties and Applications
论文作者
论文摘要
我们研究了相对于Hilbert-varued高斯过程的Hölder连续轨迹，构成的伏特拉过程的概率和分析特性。为此，我们将Volterra缝纫引理从\ cite {harangtindel}扩展到两个维度的情况，以构建Young Type的二维操作员值的Volterra积分。我们证明，与无限维伏特拉过程相关的协方差运算符可以用这样的二维积分来表示，该积分扩展了此类协方差运算符的当前表示概念。然后，我们讨论了这些结果的一系列应用，包括建立与可能不规则协方差结构驱动的伏特拉过程相关的粗糙路径，以及对沿着不规则轨迹的高斯工艺产生的不规则协方差结构的描述。此外，我们考虑了由高斯噪声驱动的无限尺寸分数Ornstein-uhlenbeck过程，这可以看作是Rosenbaum等人提出的挥发性模型的扩展。在\ cite {eleuchrosenbaum}中。
We investigate the probabilistic and analytic properties of Volterra processes constructed as pathwise integrals of deterministic kernels with respect to the Hölder continuous trajectories of Hilbert-valued Gaussian processes. To this end, we extend the Volterra sewing lemma from \cite{HarangTindel} to the two dimensional case, in order to construct two dimensional operator-valued Volterra integrals of Young type. We prove that the covariance operator associated to infinite dimensional Volterra processes can be represented by such a two dimensional integral, which extends the current notion of representation for such covariance operators. We then discuss a series of applications of these results, including the construction of a rough path associated to a Volterra process driven by Gaussian noise with possibly irregular covariance structures, as well as a description of the irregular covariance structure arising from Gaussian processes time-shifted along irregular trajectories. Furthermore, we consider an infinite dimensional fractional Ornstein-Uhlenbeck process driven by Gaussian noise, which can be seen as an extension of the volatility model proposed by Rosenbaum et al. in \cite{ElEuchRosenbaum}.
