对数正态连续级联：聚集性质和估计。应用于金融时间序列
Log-Normal continuous cascades: aggregation properties and estimation. Application to financial time-series
摘要：
对数正态连续随机级联构成了一类多重分形过程，已经成功地应用于各个领域。与此模型相关的几个统计问题进行了研究。我们首先快速但广泛的审查其主要属性，并表明，这些属性中的大多数可以分析研究。然后，我们在小不稳定性极限$λ^2\ll 1$中发展了这些过程的近似理论，即，当多重分形度较小时。这使我们能够证明，与这些过程相关的概率分布具有一些非常简单的跨时间尺度的聚集性质。这样的控制过程中的属性在不同的时间尺度，使我们能够解决的问题，参数估计。我们表明，一个必须区分两种不同的渐近制度：第一个，称为“低频制度”，对应于采取的样本，其整体大小增加，而第二个，称为“高频制度”，对应于采样过程中，在一个不断增加的采样率。我们发现，第一个政权导致收敛估计，而在高频率政权，情况要复杂得多：只有不确定性系数$λ^2$可以估计使用一致的估计。然而，我们表明，在实际情况下，可以检测的性质的渐近制度（低频率与高频率），从而决定是否其他参数的估计是可靠的或不。最后，我们说明了如何我们的结果参数估计和聚集属性，允许一个成功地使用这些模型的建模和预测的金融时间序列。
英文摘要：
Log-normal continuous random cascades form a class of multifractal processes that has already been successfully used in various fields. Several statistical issues related to this model are studied. We first make a quick but extensive review of their main properties and show that most of these properties can be analytically studied. We then develop an approximation theory of these processes in the limit of small intermittency $λ^2\ll 1$, i.e., when the degree of multifractality is small. This allows us to prove that the probability distributions associated with these processes possess some very simple aggregation properties accross time scales. Such a control of the process properties at different time scales, allows us to address the problem of parameter estimation. We show that one has to distinguish two different asymptotic regimes: the first one, referred to as the ''low frequency regime'', corresponds to taking a sample whose overall size increases whereas the second one, referred to as the ''high frequency regime'', corresponds to sampling the process at an increasing sampling rate. We show that, the first regime leads to convergent estimators whereas, in the high frequency regime, the situation is much more intricate : only the intermittency coefficient $λ^2$ can be estimated using a consistent estimator. However, we show that, in practical situations, one can detect the nature of the asymptotic regime (low frequency versus high frequency) and consequently decide whether the estimations of the other parameters are reliable or not. We finally illustrate how both our results on parameter estimation and on aggregation properties, allow one to successfully use these models for modelization and prediction of financial time series.
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