Prediction of linear fractional stable motions using codifference

2025-07-21

AI Summary

This paper addresses the problem of predicting future increments of linear fractional stable motion (LFSM) under α-stable innovations—where classical covariance-based prediction fails due to infinite variance when α < 2. Methodologically, it proposes a novel codifference-based forecasting framework, distinguishing two regimes: for α > 1, it employs conditional expectation; for α ≤ 1, it introduces semimetric projection, thereby uniformly circumventing the infinite-variance obstacle. Theoretically, this work constitutes the first application of codifference to LFSM prediction, uncovering a fourth type of serial dependence and a “selective memory” phenomenon, while disentangling the distinct roles of increment kurtosis and serial dependence in fractal volatility dynamics. Empirically, the method significantly outperforms fractional Brownian motion benchmarks in high-frequency foreign exchange rate and volatility forecasting; its efficacy and practicality are validated on both simulated and real-world data.

Technology Category

Cognitive Modeling & Cognitive Systems: Neural Spike CodingReasoning under Uncertainty: Stochastic OptimizationMachine Learning: Other Foundations of Machine Learning

Application Category

Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networksSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingGraph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

The linear fractional stable motion (LFSM) extends the fractional Brownian motion (fBm) by considering $α$-stable increments. We propose a method to forecast future increments of the LFSM from past discrete-time observations, using the conditional expectation when $α>1$ 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 $α<2$. 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.

Problem

• Predicting linear fractional stable motion increments from past data
• Using codifference instead of covariance for serial dependence
• Improving forecasts for high-frequency FX rates and volatilities

Innovation

• Uses codifference for serial dependence analysis
• Employs conditional expectation for alpha>1 prediction
• Applies semimetric projection for alpha<=1 cases