标题： 建模平稳数据的一类广义 Ornstein-Uhlenbeck 过程 显示英文标题

标题： Modeling stationary data by a class of generalised Ornstein-Uhlenbeck processes

摘要： 一个Ornstein-Uhlenbeck（OU）过程可以被视为离散时间AR$(1)$过程的连续时间插值。 基于这一事实，我们在此工作中分析了将OU视为将Wiener过程映射到Ornstein-Uhlenbeck过程的线性算子时进行迭代的效果，以此构建了一组更高阶的Ornstein-Uhlenbeck过程，即OU$(p)$，类似于高阶自回归过程AR$(p)$的方式。 我们证明了对于$p \ge 2$，一般情况下会得到与AR$(p)$不同协方差的过程，并且对于从实际数据中以等间距时间点采样的各种连续时间过程，OU$(p)$模型优于相应的AR$(p)$模型。 技术上，我们的OU算子组合易于操作，并且其参数可以高效计算，因为正如我们所展示的，OU算子的迭代会产生可以用基本OU过程的线性组合表示的过程。 利用这个表达式，我们得到了迭代OU过程的协方差的闭合公式，从而可以通过最大似然估计OU$(p)$过程的参数，或者作为一种替代方法，通过匹配相关性来估计参数，后者类似于矩估计法。 显示英文摘要

摘要： An Ornstein-Uhlenbeck (OU) process can be considered as a continuous time interpolation of the discrete time AR$(1)$ process. Departing from this fact, we analyse in this work the effect of iterating OU treated as a linear operator that maps a Wiener process onto Ornstein-Uhlenbeck process, so as to build a family of higher order Ornstein-Uhlenbeck processes, OU$(p)$, in a similar spirit as the higher order autoregressive processes AR$(p)$. We show that for $p \ge 2$ we obtain in general a process with covariances different than those of an AR$(p)$, and that for various continuous time processes, sampled from real data at equally spaced time instants, the OU$(p)$ model outperforms the appropriate AR$(p)$ model. Technically our composition of the OU operator is easy to manipulate and its parameters can be computed efficiently because, as we show, the iteration of OU operators leads to a process that can be expressed as a linear combination of basic OU processes. Using this expression we obtain a closed formula for the covariance of the iterated OU process, and consequently estimate the parameters of an OU$(p)$ process by maximum likelihood or, as an alternative, by matching correlations, the latter being a procedure resembling the method of moments.