标题： 线性分数稳定运动的预测使用协方差 显示英文标题

标题： Prediction of linear fractional stable motions using codifference

摘要： 线性分数稳定运动（LFSM）通过考虑$\alpha$稳定增量，扩展了分数布朗运动（fBm）。我们提出了一种方法，从过去的离散时间观测中预测LFSM的未来增量，当$\alpha>1$时使用条件期望，否则使用半度量投影。它依赖于协差分，而不是协方差，协差分描述了该过程的序列依赖性。实际上，协方差通常用于预测fBm，但当$\alpha<2$时它是无限的。该方法及其准确性的某些理论性质得到了研究，模拟研究和对实际数据的应用证实了该方法的相关性。在预测高频外汇汇率时，基于LFSM的方法优于fBm。它在波动率时间序列的预测中也表现出有前景的性能，能够正确分解粗糙波动率分形动态中增量峰度的贡献和其序列依赖性的贡献。此外，命中比率的分析表明，除了独立性、持久性和反持久性外，分数过程中还存在第四种序列依赖模式，其特征是受少数大增量控制的选择性记忆。 显示英文摘要

摘要： The linear fractional stable motion (LFSM) extends the fractional Brownian motion (fBm) by considering $\alpha$-stable increments. We propose a method to forecast future increments of the LFSM from past discrete-time observations, using the conditional expectation when $\alpha>1$ or a semimetric projection otherwise. It relies on the codifference, which describes the serial dependence of the process, instead of the covariance. Indeed, covariance is commonly used for predicting an fBm but it is infinite when $\alpha<2$. Some theoretical properties of the method and of its accuracy are studied and both a simulation study and an application to real data confirm the relevance of the approach. The LFSM-based method outperforms the fBm, when forecasting high-frequency FX rates. It also shows a promising performance in the forecast of time series of volatilities, decomposing properly, in the fractal dynamic of rough volatilities, the contribution of the kurtosis of the increments and the contribution of their serial dependence. Moreover, the analysis of hit ratios suggests that, beside independence, persistence, and antipersistence, a fourth regime of serial dependence exists for fractional processes, characterized by a selective memory controlled by a few large increments.